l0ti<^t:t ^mx^ '^km^tm 1903 Cornell University Library 3 1924 031 218 724 olin.anx Cornell University Library The original of tliis book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31 924031 21 8724 HEAT. H EAT Pf G. TAIT, M.A., Sec. R.S.E. FORMERLY FELLOW OF ST. PETER's COLLEGE, CAMDRIUCE PROFESSOR OF NATURAL PHILOSOPHY IN THE UNIVERSITY OP EDINBURGH. ^onbon : MACMILLAN AND CO. 1884. r& Rightt of Translation and Rep-oduction are Reserved. LONDON : R. Clav, Sons, and Taylor, BREAD STKHET HILU PREFACE. This work originated in an article which I contributed in 1876 to the Handbook to the Loan Collection of Scientific Apparatus (at South Kensington). As that article was based upon the system of teaching the subject of Heat which, after many years' experience, I have found best adapted to the wants of intelligent students taking up the subject for the first time, Mr. Macmillan asked me to develop it in the form of an elementary treatise. In 1876 — 7 a large portion of the work was written ; the greater part of which was put in type, and had the advantage of Clerk- Maxwell's valuable criticism. Some of the earlier chapters were utilised when I had to give at short notice an evening discourse at the British Association Meeting in 1876. Pressing work of a very different character, such as the examination of the Challenger Thermometers, interfered from time to time with my writing by occupying most of my leisure. I have at last managed to finish the book, as nearly as possible on the lines laid down at starting ; but it cannot, under the circumstances, be expected to have the unity which it might have secured by being continuously ■written. vi PREFACE. It may be asked — Why publish another text-book on a subject which is already thoroughly treated in the excellent (and strictly scientific) works of Clerk-Maxwell and Balfour Stewart ? The only answer, and it may be a sufficient orie, is : — Clerk-Maxwell's work is on the Theory of Heat, and is specially fitted for the Study ; that of Stewart is rather for the Physical Laboratory ; so that there still remains an opening for a work suited to the Lecture Room. By that expression I mean a work suitable for students who, without any intention of entering on a scientific career, whether theoretical or experimental, are yet desirous of knowing accurately the more prominent facts and theories of modern science to such an extent as to give them an intelligent interest in physical phenomena. In addition to what is stated in the text (§ ii) as to the special arrangement and division of the subject in the present work, it may be well to say a few words here as to the way in which the important subjects of tem- perature, and especially of absolute temperature, have been treated. Temperature is first introduced as a mere condition de- termining which of two bodies in contact shall part with heat to the other (§ 6). In this sense it is compared to the pressure of the air in a receiver, the air itself being the analogue of heat (§ 52). When two such receivers are made to communicate with one another, air passes from that in which the pressure is greater to that in which it is less. And this is altogether independent of the relative quantities of air in the two receivers. In § 57 it is pointed out that there is an absolute method PREFACE. vii of defining temperature, which must therefore be the sole scientific method. But it is also stated that, in experi- mental work, few quantities are directly measured in terms of the strictly scientific units in which they are ultimately to be expressed. In § 60 the reader is told that what is wanted, to prepare him for the absolute measurement of temperature, is "something which shall be at once easy to comprehend and easy to reproduce, and which shall afterwards require only very slight modifications to reduce it to the rigorous scale." Thus (§61) a temporary centigrade scale is defined by means of mixtures of ice-cold water and boiling water. And this is stated to " accord so closely with the absolute scale, that careful experiment is required even to show that it does not exactly coincide with it." In terms of temperatures thus defined, the questions of Expansibility, Latent and Specific Heats, &c., are explained from the experimental point of view. But, before these are taken up, the first rapid resumi of the whole subject introduces Carnot's Cycles, and his principle of Reversibility, with Thomson's application of them to the absolute measurement of temperature. It is pointed out (§ 95) that Carnot's results still leave a certain option in the formal definition of absolute temperature, and that Thomson found it possible so to frame his definition (which is given at this stage) as to make a very close agreement between the new scale and that of an ordinary air-thermometer. Thus, the use of air-thermo- meter temperatures is justified by their close approximation to absolute temperatures, as well as by their comparative viii PREFACE. simplicity and convenience, for the chapters which follow ; and they are thenceforth employed until (by means of the Indicator Diagram) it has been fully pointed out what is the scientific importance of absolute temperature. It is then shown, in § 405, how to compare absolute tempera- ture with that given by the air-thermometer. All a priori notions as to strict logical order, however valuable, and in fact necessary, in a formal treatise, must be set aside in an elementary work, if they be found in practice to impede rather than assist the progress of the average student. And it is solely on this ground that the above system of exposition has been adopted in the present work. He who expects to find this work, elementary as it is, everywhere easy reading, will be deservedly disappointed. No branch of science is free from real and great difficulties, even in its elements. Any one who thinks otherwise has either not read at all, or has confined his reading to pseudo-science. The reader, who wishes to know more of the general science of Energy than was admissible into the present work, will find a connected historical sketch of it in my little book on Thermodynamics. There he will also find more of the purely analytical development of the subject, specially from W. Thomson's point of view. P. G. TAIT. College, Edinburgh, December 1st, 1883, CONTENTS. CHAPTER I. PAGE FUNDAMENTAL PRINCIPLES 1 CHAPTER II. INTRODUCTORY 8 CHAPTER III. DIGRESSION ON FORCE AND ENERGY I3 CHAPTER IV. PRELIMINARY SKETCH OF THE SUBJECT 21 CHAPTER V. DILATATION OF SOLIDS Jl CHAPTER VI. DILATATION OF LIQUIDS. AND GASES 88 CHAPTER VII. THERMOMETERS 103 d X CONTENTS. CHAPTER VIII. PACE CIIAK&E OF MOLECULAR ST.^TE. MELTING AND SOLIDIFICATION Il8 CHAPTER IX. CHANGE OF MOLECULAR STATE. VAPr^RISATION AND CON- DENSATION . .... 131 CHAPTER X. CHANGE OF TEMPERATURE. SPECIFIC HEAT. ... . I49 CHAPTER XI. THERMO-ELECTRICITY .... . ... 163 CHAPTER XII, OTHER EFFECTS OF HEAT . . . .... 182 CHAPTER XIII. CO.MBINATION AND DISSOCIATION . . . I93 CHAPTER XIV. CONDUCTION OF HEAT . . 205 CHAPTER XV. CONVECTION 230 .CHAPTER XVI. RADIATION 239 CHAPTER XVII. RADIATION AND ABSORPTION . 250 CONTENTS. xi CHAPTER XVII r. I'AGE RADIATION 276 CHAPTER XIX. UNITS AND DIMENSIONS 2(;C CHAPTER XX. \V.\TT'S INDICATOR DIAGRAM . ... 298 CHAPTER XXI. ELEMENTS OF THERMODYNAMICS 324 CHAPTER XXII. NATURE OF HEAT . ... 350 HEAT. CHAPTER I. FUNDAMENTAL PillNCIPLES. 1. In dealing with any branch of physical science it is absolutely necessary to keep well in view the fundamental and all-important principle that Nothing can be learned as to the physical world save by observation and experiment, or by mathematical deductions from, data so obtained. On such a text volumes might be written ; but they are unnecessary, for the student of physical science will feel at each successive stage of his progress more and more pro- found conviction of its truth. He must receive it, at starting, as the unanimous conclusion of all who have in a legitimate manner made true physical science the subject of their study ; and, as he gradually gains knowledge by this — the only — method, he will see more and more clearly the abso- lute impotence of all so-called metaphysics, or a priori reasoning, to help him to a single step in advance. 2. Man has been left entirely to himself as regards the acquirement of physical knowledge. But he has been gifted with various senses (without which he could .not even know 3S B 2 HEAT. [CHAP. that the physical world exists) and with reason to enable him to control and -understand their indications. Reason, unaided by the senses, is totally helpless in such matters. The indications given by the senses, unless inter- preted by reason, are in general utterly unmeaning. But when reason and the senses work harmoniously together they open to us an absolutely illimitable prospect of mysteries to be explored. This is the test of true science — there is no resting-place — each real advance discloses so much that is new and easily accessible, that the investigator has but scant time to co-ordinate and consolidate his know- ledge before he has additional materials poured into his store. 3. To sight without reason, the universe appears to be filled with light — except, of course, in places surrounded by opaque bodies. Reason, controlling the indications of sense, shows us that the sensation of sight is our own property ; and that what we understand by brightness, &c., does not exist out- side our minds. It shows us also that the sensation of colour is purely subjective, the only differences possible between different so-called rays of light outside the eye being merely in the extent, form, and rapidity of the vibra- tions of the luminiferous medium. To hearing, without reason, the air of a busy town seems to be filled with sounds. Reason, interpreting the indications of sense, tells us that if we could see the particles of air we should observe among them (superposed upon their rapid motions among one another) simply a comparatively slow agitation of the nature of alternate compressions and dilatations. And our classification of sounds as to loudness, pitch, and quality, is merely the subjective correlative of what in the air-particles is objectively the' amount of com- I.] FUNDAMENTAL PRINCIPLES. 3 pression, the rapidity of its alternations, and the greater or less complexity of the alternating motion. A blow from a stick or a stone produces pain and a bruise ; but the motion of the stick or stone before it reached the body is as different from the sensation produced by the blow as is the alternate compression and dilatation of the air from the sensation of sound, or the ethereal wave- motion from the sensation of light. Hence to speak of what we ordinarily mean by light, or sound, as existing outside ourselves, is as absurd as to speak of a swiftly-moving stick or stone as pain. But no incon- venience is occasioned if we announce the intention to use the terms light and sound for the objective phenomena, and to speak of their subjective effects as "luminous impressions" or " noise," as the case may be. In this sense there is outside us energy of motion of every kind, but in the mind mere corresponding impressions of brightness and colour, noise or harmony, pain, &c. &c.. 4. It would seem therefore that we must be extremely cautious in our own interpretation of the immediate evidence of our senses as to heat> And the very first instance that occurs to us fully justifies this caution. Touch, in succession, various objects on the table. A paper-weight, especially if it be metallic, is usually cold to the touch ; books, paper, and especially a woollen table- cover, comparatively warm. Test themj however, by means of a thermometer, not by the sense of touch, and in all proba- bility you will find little or no difference in what we call their temperatures. In fact, as we shall presently see, any number of bodies of any kind shut up in an enclosure (within which there is no fire or other source of heat) all tend to acquire ultimately the same temperature. Why then do some feel cold, others warm, to the touch ? 4 HEAT. [chap. The reason is simply this — the sense of touch does not inform us directly of temperature, but of the rate at which our finger gains or loses heat. As a rule, bodies in a room are colder than the hand, and heat always tends to pass from a warmer to a colder body. Of a number of bodies, all equally colder than the hand, that one will seem coldest to the touch which, is able most rapidly to convey away heat from the hand. The question therefore is one of conduction of heat. And to assure ourselves that it is so, reverse the process : let us, in fact, try an. experiment, though an exceed- ingly simple one; for the essence of experiment is to modify the circumstances of a physical phenomenon so as to increase its value as a test. Put the paper-weight, the books, and the woollen table-cloth into an oven, and raise them all to one and the same temperature considerably above that of the hand. The woollen cloth will still be comparatively cool to the touch, while the metal paperr weight may be much too hot to hold. The order of these bodies, as to warm and cold in the popular sense, is in fact reversed ; and this is so, because the hand is now receiviiig heat from all the various bodies experimented on, and it receives most rapidly from those bodies which in their previous condition were capable of abstracting heat most rapidly. However it may be in the moral world, in the physical universe the giving and taking powers of one and the same body are strictly correlative and equal. 5. Thus -the direct indications of sense are in general utterly misleading as to the relative temperatures of different bodies. In a baker's or a sculptor's oven, at temperatures far above the boiling-point of water (on one occasion even 320° F.), so high indeed that a beef-steak was cooked in thirteen minutes, Tillet in France, and Blagden and Chantrey in I.] FUNDAMENTAL PRINCIPLES. 5 England, remained for nearly an hour in comparative com- fort. But, though their clothes gave them no great incon- venience, they could not hold a metallic pencil-case without being severely burned. On the other hand, great care has to be taken to cover with hemp, or wool, or other badly conducting substance, every piece of metal which has to be handled in the intense cold to which an Arctic expedition is subjected ; for contact with very cold metal produces sores almost uadistinguish- able from burns,, though due to a directly opposite cause. Both of these phenomena,, however, ultimately depend on the comparative facility with which heat is conducted by metals. 6. Even from the instance just given, the reader cannot fail to see that there is. a profound distinction between heat and temperature. Heat, whatever it may be, is something which can be transferred from one portion of matter to another;, the consideration of temperatures is virtually that of the mere conditions which determine whether or not ■ there shall be a transfer of heat, and in which direction the transfer is to take place- In fact, we may without risk of misleading the student tell him, even at the beginning of his work, that from one point of view the quantity of heat in a body bears a very close analogy to the water-po-wer stored up in a cistern or reservoir, while the temperature of the body is as closely analogous to the elevation of the cistern. He must notice, however, that , water-power does not depend upon mere quantity of water 1 to be capable of driving a mill the water must have ■" head," or elevation. But, in another and quite different sense, heat in a body is analogous to the water, not to the water-power.. Heat tends to pass from a hotter to a colder body, just as water 6 HEAT. [chap. tends to flow from a cistern at a higher to another at a lower level. Thus heat in a hot body is in this property analo- gous to (or at least behaves lik*) water at a high level, and vice vers&. Two such connected and yet .diSerent analogies can be safely presented to the student at an early stage ; for they will certainly help his conceptions, and their diiference will prevent his being in any way misled by either. 7. For all that, as -will presently be seen, heat^thotigh not material — has .objective existence in as complete a sense as ■matter has. This may appear, at first sight, paradoxical ■; but we must remember that so-called paradoxes are merely facts as yet unexplained, and therefore still apparently inconsistent with others, already understood dn their full -significance. When we say that matter has objective existence, we mean that it is something which exists altogether indepen- dently of the senses and brain-processes by which alone we are informed of its presence. An exact, or adequate con- ception of it, if it could be formed, would probably be something very different frpm any conception which our senses will ever enable us to form ; but the object of all pure physical science is to endeavour to grasp more and more perfectly the nature and laws of the external world, by using the imperfect means which are at our command — ' reason acting as interpreter as well as judge ; while the senses are merely the witnesses,-^-who may be more or less untrustworthy and incompetent, but are nevertheless of inconceivable value to us, because they are our only available ones. 8. Without further discussion we may state once for all that our conviction of the objective reality of inatter is based niainly upon the fact, discovered solely by experiment, that we I.] FUNDAMENTAL PRINCIPLES. 7 cannot in the slightest degree alter its quantity. We cannot destroy, nor can we produce, even the smallest portion of matter. But reason requires us to be consistent in our logic ; and thus, if we find anything else in the physical world whose quantity we cannot alter, we are bound to admit it to have objective reality as truly as matter has, however strongly our senses may predispose us against the concession. Heat, therefore, as well as Light, Sound, Electric Currents, &c., though not forms of matter, must be looked upon as being as real as matter, simply because they have been found to be forms of Energy (Chap III. below), which in all its constant mutations satisfies the test which we adopt as conclusive of the reality of matter. But the student must here be again most carefully warned to distinguish between heat and the mere sensation of warmth; just as he distinguishes between the energy of motion of a cudgel and the pain produced by the blow. The one is the thing to be measured, the other is only the more or less imperfect reading or indication given by the instrument with which we attempt to measure it in terms of some one of its effects. 9. There is one other point which must be insisted on as a necessary preliminary to all physical inquiries, to wit, the condition under which alone it is possible that physical science can exist. We may enunciate it as follows : — Under the same physical conditions the same physical results will cilways be produced, irrespective altogether of time or place. It requires no comment whatever, if the terms employed be fully understood and be interpreted in the strictest sense. It is, in fact, merely the assertion (based entirely on observation and experiment) of the existence of definite and unchanging laws to which all physical processes are found to be subject. CHAPTER II. INTRODUCTORY. 10. Until all physical science is reduced to the deduction of the innumerable mathematical consequences of a few known and simple laws, it will be impossible completely to avoid some confusion and repetition, whatever be the arrangement of its various parts which we adopt in bringing them before a beginner. When we confine ourselves to one definite branch of the subject, all of whose fundamental laws can be distinctly formulated, there need be no such confusion. Here in fact the mathematician has it all in his own hands. He is the skilled artificer with his plan and his trowel, and the hodmen have handed up' to him all the requisite bricks and mortar. This has long been known and recognised as a fact, but it has not often been put so neatly as in the following extract : — "That which is properly called Physical Science is the knowledge of relarions between natural phenomena and their physical antecedents, as necessary sequences of cause and effect; these relations being investigated by the aid of mathematics— that is, by a method in which processes of reasoning, on all questions that can be brought under the CHAP. 11.] INTRODUCTORY. 9 categories of quantity and of space-cOftditions, are rendered perfectly exact, and simplified and made capable of general application to a degree almost inconceivable by the un- initiated, through the use of conventional symbols. There is no admission for any but a mathematician into^ this school of philosophy. But there is a lower department of natural science, most valuable as a precursor and auxiliar}', which we may call Scientific Phenomenology; the office of which is to observe and classify phenomena,, and by induction to infer the laws that govern them. As, however, it is unable to determine these laws to be necessary results of the action of physical forces, they remain merely empirical until the higher science interprets them.. But the inferior and auxiliary science has of late assumed a position to which it is by no means entitled.. It gives itself airs, as if it were the mistress instead of the handmaid, and often conceals its own incapacity and want of scientific purity by high-sound- ing language as to the mysteries of nature. It may even complain of true science, the knowledge of causes, as merely mechanical. It will endue matter with mysterious qualities and occult powers, and imagines that it discerns in the physical atom 'the promise and potency of all terrestrial life.' " * Thus all who have even a Slight acquaintance with the subject know that the laws of motion, and the law of gravita- tion, contain absolutely all of Physical Astronomy, in the sense in which that term is commonly employed : — viz., the investigation of the motions and mutual perturbations of a number of masses (usually treated as mere points, or at least as rigid bodies) forming any system whatever of sun, planets, and satellites. But, as soon as physical science points out that we must * Church Quarterly Review, April 1876, p. 149. lo HEAT. [chap. take account of the plasticity and elasticity of each mass of such a system, the amount and distribution of liquid on its surface, possibly of magnetic and other actions between them, and the resistance due to the medium in which they move ; the simplicity of the data of the mathematical problem is gone; and physical astronomy, except in its grainder outlines, becomes as much confused as any other branch of science. So it is with the Dynamics of Solid and Fluid bodies : — so long as we are content to view solids as perfectly smooth and rigid, and fluids as incompressible and frictionless, the difficulties of Dynamics, though often enormously great, are entirely mathematical, it falls naturally into quite distinct and separate heads, and the classification of its various problems is comparatively simple. Introduce ideas of strain, and fluid friction, with consequent development of heat, and the confusion due to imperfect or impossible classification comes in at once. Each problem, instead of being treated by itself, has to borrow, sometimes over and over again, from others ; and the only fully satisfactory and uncom- plicated mode of attacking such a subject (were it con- ceivable) would be to work it all out at once. II. Hence, in dealing with the general subject of heat we shall find it quite impossible to lay down definite lines of demarcation. Divide it as we choose, each part will be found to req»jire for its development something borrowed from another. All that we can do under these conditions (the existence of which simply means that we do not yet know all about heat in the same sense as we may be said to know the laws of motion and of gravitation) is to make our classification confessedly somewhat "indefinite, and freely to assume throughout, when needed, results of other parts of our II.] INTRODUCTORY. n subject which we have not yet discussed. The advantages of this method, at least in my own experience, have been found much to outweigh its obvious but as yet inevitable disadvantages. But, to reduce the latter as far as possible, I shall first go over the whole subject briefly, so as to point out its main features and their mutual relations ; explaining some of the more important things, though not giving their experimental proof ; and thus the student will be from the outset fairly prepared to take up in turn each of our divi- sions of the subject with as much detail as is consistent with the dimensions of an elementary treatise. And another advantage will be gained, inasmuch as such a rapid and general glance at the whole subject will admit of a iiumber of useful and even important digressions which would seriously impair the consistency of the more detailed and definite part of the work. 12. The classification I liave found convenient is as follows : — 1. Nature of Heat. 2. Effects of Heat. 3. Measurement of Heat and of Temperature. 4. Sources of Heat. 5. Transference of Heat. 6. Transformations of Heat. As already pointed out, there is no hard and fast line drawn between any two of these heads, in fact the explanations of many even among ordinary phenomena belong in part to more than one of them. It is well, however, to remark that there is a very intimate connection between the three heads (i), (4), and (6) above, which contain among them the chief recent advances of the Dynamical Theory of Heat, or Thermodynamics, as it is commonly called. Hence (i) will be more fully developed- 12 HEAT. [chap. ii. than the other heads in our first rapid rtsuvik of the whole subject, while its farther develoi^men twill be wholly merged in that of (4) and (6), which may then profitably be studied together. Again (3) depends, at least in all its ordinary practical forms, on some application or other of one of the group (2). (5) stands to a great extent by itself, but na- turally divides itself into three perfectly distinct processes, all of which are of great scientific as well as practical importance. CHAPTER III. DIGRESSION ON FORCE AND ENERGY. 13. We must now take up briefly and in order these divisions of our subject ; but before we can do so intelligibly, a digression into the elements of Dynamics is absolutely essen- tial. This involves, in fact, the citation and explanation of a passage in Newton's Principia which, till very lately, seems to have altogether escaped' the notice of scientific men. The reader will find that the consideration of this passage, especially when he sees it rendered into the terms used in modem science, will greatly facilitate his farther progress with reg3.rd to the nature of heat. Newton's Third Law of Motion is to the effect that — '•' To every action there is always an equal and contrary re- action; or, the mutual actions of any two bodies are always equal and oppositely directed." This law Newton first shows to hold for ordinary pres- sures, tensions, attractions, &c., that is, for what we com-, monly call forces exerted on one another by two bodies ; also for impacts, or impulses, which are merely the time- integrals of forces. But he proceeds to point out that the same law is true in another and much higher sense. He says ; — " If the action of an agent be measured by- the product of its 14 HEAT. [chap. force into its velocity; and if, similarly, the reaction of the resistaiice be measured by the velocities of its several parts into their several forces, whether these arise from friction, cohesion, ■weight, or acceleration; — action and reaction, in all combina- tions of machines, will be equal and ofposite." The actions and reactions which are here stated to be equal and opposite are no longer simple forces, but the products of forces into- their velocities ; i.e. they are what are now called Rates of doing Work; the time-rate of increase, or the increase per second, of a very tangible and real something : for the measurement of which Watt intro- duced the practical unit of a horse-power, tlie rate at which an agent works, when it lifts 33,odo' pounds one foot high per minute, against the earth's attraction. 14. Now let the reader think of the difference between raising a hundredweight a,nd endeavouring to raise a ton. With a moderate exenionhe can raise the hundredweight a few feet, and in its descent it might be employed to drive machinery, or to do some other species of work. Let. him tug as he pleases at the ton, he will not be- able to lift it; and therefore after all his exertion, it will not be capable of doing any work by descending again. In both cases the first interpretation- of Newton's Third Laiv has been verified^ With whatever force he pulled either of the masses, that mass reacted with an equal force. But the second interpretation cannot be applied to the ton ; for it did not acquire any velocity, it was not moved. Hence. as no work was spent upon it, it has. not acquired the power of doing work. On the other hand, the hundred- weight was moved, work was done upon it, and that work was stored up in it in its mised position, ready for use at any future time, Newton's statement implies that in this case the work spent in raising the hundredweight is III.] DIGRESSION ON FORCE AND ENERGY. i; stored up (without change of amount) in the mass when raised. 15. Thus it appears t\\sX force is a mere name; but that the product of a force into the displacerhent of its point of application has an objective existence. [Even those who are so metaphysical as not to see that the product of a mere name into a displacement can have objective existence, may perhaps see that the quotient of a horse-power by a velocity is not likely to be more than a mere name.] In facti modern science shows us that farce is merely a convenient term employed for the present (very usefully) to shorten what would otherwise be cumbrous expressions; but it is not to be regarded as a thing ; amy more than the bank rate of interest is to be looked upon as a sum of money, or than the birth-rate of a country is to be looked upon as the actual group of children born in a year. And a very simple mathematical operation shows us that it is precisely the same thing to say — The horse-power of an agent, or the amount of work done by an agent in each second, is the product of the force into the average velocity of the agent i and to say — Force is the rate at ivhich an agent does work per unit of length.* 16. In the special illustration of Newton's words which we have just given, the resistance was a weight, that of a hundredweight or of a ton. When the resistance was * In symbols — this is merely ^ = fv = f — , the first statement, dt ■' ■' at' whence ^=/, the second. dx ^' i6 HEAT. [chap. overcome, work was done, and it was stored up for use in the raised mass — in a form which could be made use of at any future time. Following a hint given by Young, we now employ the term energy to signify the power of doing work, in whatever that power may consist. The raised mass, then, we say, possesses in virtue of its elevation an amount of energy precisely equal to the work spent in raising it. This dormant, or passive, form, is called Potential Energy. Excellent instances of potential energy are supplied by water at a high level, or with a "head," as it is technically called, in virtue of which it can in its descent drive machinery ; — by the wound-up " weights " of a clock, which in their descent keep it going for a week ; — by gunpowder, the chemical affinities of whose constituents are called into play by a spark; &c. &c. Another example of it is suggested by the word " Cohe- sion" employed in Newton's statement (§ 13), and which must be taken to include what are called molecular forces in general, such as for instance those upon which the elasticity of a solid depends. When we draw a bow we do work, because the force exerted has a velocity ; but the drawn bow (like the raised weight) has in potential energy the equivalent of the work so spent. That can in turn be expended upon the arrow ; and what then 1 17. Turn again to Newton's words (§ 13) and we see that he speaks of one of the forms of resistance as arising from " acceleration." In fact, the arrow, by its inertia, resists being set in motion ; work has to be spent in propelling it : — but the moving arrow has that work in store in virtue of its motion. It appears from Newton's previous statements that the measure of the rate at which work is spent in producing III.] DIGRESSION ON FORCE AND ENERGY. 17 acceleration* is the product of the momentum into the accele- ration in the direction of motion, and the energy produced is measured by half the product of the mass into the square of the velocity produced in it. This active form is called Kinetic Enei^y, and it is the double of this to which the term Vis Viva (erroneously translated Living Force) has been applied. As instances of ordinary' kinetic energy, or of mixed kinetic and potential energies, take the following : — A current of water capable of driving an undershot wheel ; winds, which also are used for driving machinery; the energy of water-waves or of sound-waves ; the radiant energy which comes to us from the sun, whether it affect our nerves of touch or. of sight (and therefore be called radiant heat or light) or produce chemical decomposition, as of carbonic acid and water in the leaves of plants, or of silver-salts in photography (and be therefore Called actinism) ; the energy of motion of the particles of a gas, upon which its pressure depends, &c. [When the motion is vibratory the energy is generally half potential, half kinetic] 18. These explanations and definitions being premised, we can now translate Newton's words (without alteration of their meaning) into the language of modern science, as follows : — Work done on any system of bodies (in Newton's statement the parts of any machine) has its equivalent in work done against friction, molecular forces, or gravity, if there be no acceleration ; but if there be acceleration, part of the work is expended in overcoming the resistance to acceleration, and the * Let i) be the acceleration, in the direction of motion, of a mass M whose velocity is v. Then Newton's expression for the rate of spending work against the re-istance to acceleration is Mv . i/, and the whole work spent in giving the velocity v to the mass M, originally at rest, is 4 MvK C i8 HEAT. [chap. additional kinetic energy develoJ>ed is equivalent to the work so spent. But we have just seen that when work is spent against molecular forces, as in drawing a bow or winding up a spring, it is stored up as potential energy. Also it is stored up in a similar form when done against gravity, as in raising a weight. Hence it appears that, according to Newton, whenever work is spent, it is stored up either as potential or as kinetic energy :— except possibly in the case of work done against friction, about whose fate he gives us no information. Thus Newton expressly tells us that (except possibly when there is friction) ejiergy is indestructible — it is changed from one form to another, and so on, but never altered in 'quantity. To make this beautiful statement complete, all that is requisite is to know what becomes of work done against friction. 19. Here, of course, experiment is requisite. Newton, unfortunately, seems to have forgotten that savage men had long since been in the habit of making it whenever they wished to procure fire. The patient rubbing of two dry sticks together, or (still better) the drilling of a soft piece of wood with the slightly blunted point of a hard piece, is known to all tribes of savages as a means of setting both pieces of wood on fire. Here, then, heat is undoubtedly produced, bid it is produced by the expenditure of work. In fact, work done against friction has its equivalent in the heat produced. This Newton failed to see, and thus his. grand generalization was left, though on one point only, incom- plete. The converse transformation, that of heat into work, dales back to the time of Hero at least. But the knowledge that a certain process will produce a certain result does not necessarily imply even a notion of the " why " ; and Hero III.] DIGRESSION ON FORCE AND ENERGY. 19 as little imagined that in his CEolipile heat was converted into work, as do savages that work can be converted into heat. Rumford and Davy, at the very end of last century, by totally different experimental processes, showed conclusively that the materiality of heat could not be maintained ; and thus gave the means of completing Newton's statement. 20. One particular case of the Conservation of Energy had been formulated long before Newton's time, under the title of the Impossibility of the " Perpetual Motion." This was employed by Stevinus as the basis of Statics. In 1775 the French Academy of Sciences refused to consider any scheme pretending to give work without corresponding and equivalent expenditure. The multiplied experiments of some of the most ingenious men who have ever lived have been directed to obtaining the Perpetual Motion, and their absolute failure in every case may be taken as proof of the impossibility. One mode of reasoning from this acknowledged impossibility may be here explained by a particular instance, as it will be found very useful in some of the more theoreti- cal parts of our subject. We employ it to show that in all cases of natural laws, such as the laws of gravitation and of magnetic attraction, the work spent in moving a body through a certain course in one direction is exactly restored by letting it return to its first position, not merely by the original path but by any other; always on the supposition that friction is avoided. Suppose there could be two courses, from A to B, by the one of which more work would be spent on the mass than by the other. Let these amounts be W and w. If such were the case the Perpetual Motion could be produced. Apply frictionless constraint to guide the mass, so that in its ascent it shall travel along the course A w B, and in its c 2 20 HEAT. [CHAP. III. descent, along B "W A. From A to B the amount w of work is spent against the forces of the system— from B to A these forces refund the amount W. On the whole, after a complete cycle, the mass is restored to A with an amount W — w oi energy additional to what it possessed at starting. Every time the mass goes round the double course in the same direction it gains the difference between the larger quantity and the smaller one,. and therefore at the end of each complete cycle, that amount may be drained off to turn some machine ; — to do useful work. We here assume that the work spent in one course would be exacdy recovered by letting the mass retrace its steps ; in other words, that the operation is reversible. This term will be fully discussed later. In general, if there were one way of doing a thing at less cost than another, and if the more costly operation were reversible, it would be possible to get unlimited amounts of useful work from nothing. We are now prepared to undertake the short general glance over our subject promised in § ii, according to the plan there laid down. CHAPTER tV. PRELIMINARY SKETCH OF THE SUBJECT. 21. Nature of Heat.— Heat, under the name of fire, rtrhich seems to have included everything either really or apparently of the nature of flame, such as sun and moon, stars, planets, and comets, lightning and Aurora, &c., as well as ordinary fire, was in old times regarded as one of the so- called Jvur Elements, of which, or of some of which, it was imagined that everything in the physical world was neces- sarily composed. The tendency of early experimental science, which took its first great impulse, if not its absolute origin, from Gilbert of Colchester {circa 1570), was to regard heat, light, elec- tricity, &c., as forms of matter — excessively subtle and refined — capable of freely pervading and combining with all ordinary gross matter. They were, in fact, classed as Imponderables, because a heated or electrified body, for instance, was not found to be increased in weight by the heat or electricity which it was supposed to have imbibed. 22. In this sense heat was supposed to be an excessively light species of matter, and was called Caloric, or, in certain of its manifestations (especially in some chemical processes). Phlogiston : — though, as has been recently shown,* the latter * Cium down, rroc. R.S.E., 1870. 23 HEAT. [CHAP term was also used to describe certain forms of what we now call chemical potential energy. (The meaning of this term will be obvious from the statement above (§ i6) regarding the energy of gunpowder,} The notion of the materiality of heat or caloric, in spite of experiments proving the contrary, and of several shrewd guesses* as to their true nature, was all but universally accepted and taught till about 1840. Then commenced that rapid revolution which has entirely altered the generally accepted views of heat, just as a few years previously the Material or Corpuscular Theory of Light may be said to have received its death-blow; or, as in somewhat more recent times, the so-called Electric Fluids have been ban- ished from all British science except some portions of that spurious and worse than useless kind which is commonly called popular. 33. One by one the Imponderables have been displaced from their old and proud position in science ; but they have all died hard, and among many of the continental schools of physicists, and especially of the German, the electric * " And tryall hath taught me that there are liquors, in which the bare admixture of milk, oyle, or other liquors — nay, or of cold water, will presently occasion a notable heat : and I sometimes imploy a menstruum, in which nothing but a little flesh being put, though no visible Ebullition ensue, there will in a few minuts be excited a Heat, intense enough to be troublesome to him that holds the Glasse. And yet it seems not necessary that this should be ascrib'd to a true fermen- tation, which may rather proceed from the perturb'd motion of the Corpuscles of the menstruum, which being by the adventitious liquor or other body put out of their wonted motion, and into an inordinate one, there is produced in the menstruum a brisk confus'd Agitation of the small parts that compose it ; and in such an %i'tation (from what cause soever it proceeds), the nature pf heat seems mainly to consist. " — Some Considerations touching the Usefulness of Experimental Naturall Philosophy. By Robert Boyle. Part II., Section i, p. 45. 1663. IV.] PRELIMINARY SKETCH OF THE SUBJECT. 23 fluids still retain (thanks to an exceedingly ingenious idea of W, Weber, quite as good in its way as Newton's Fits of Easy Reflection and Transmission) their position as, in a certain sense, forms of imponderable matter, acting on one another according to a very peculiar law, quite different from anything met with in other branches of physics. But even there the end must soon come, when this last trace of the imponderables shall be handed over for preservation in the museum of the scientific antiquary. 24. The experiments, which, as stated in § 22, first proved that heat is not matter, are due to Ruraford and Davy, and date from 1798 and 1799 respectively. The explanation of the heat produced by friction which was given by those who believed heat or caloric to be matter was simply this : The body in its solid state, or rather in its massive state, before you began to abrade filings from it, possessed, at any particular temperature, a certain quantity of heat. It had a certain capacity for heat, as it was called ; in other words, it required so much heat to be mixed up with its particles in order to make the temperature of the whole that which was observed. But if you could make it more capacious — if you could give it greater capacity for heat — then it would hold more heat without becoming of a higher temperature. On the other hand, if by any process whatever you could diminish its capacity for heat, then, of course, it would become hotter in itself, and even give out heat to surrounding bodies ; so that, according to the notion of the supporters of this theory, the production of heat by friction or abrasion is due to the fact that you make the capacity of a body for heat smaller by reducing it to powder. For of course, when its capacity for heat is thus made smaller, it must part with some of the heat it had at first ; or if it retains it, it must necessarily show the effect of the 24 HEAT. [CHAP. heat more than it did before, and must therefore rise in temperature. Now this reasoning is, so far, perfectly philosophical, and we can say nothing against it as a mode of reasoning. But it involves the fallacious assumption that heat is matter, and therefore indestructible. 25. Next, see how well Rumford laid hold of that point, and how he proceeds by experiment to try if possible to satisfy his doubts about it. He says : — " If this were the case, then, according to the modern doctrines of latent heat, and of caloric, the capacity for heat of the parts of the metal so reduced to chips, ought not only to be changed, but the change undergone by them should be sufficiently great to account for all the heat produced." Rumford found no difference, so far as his form of experiment enabled him to test them, between the ca- pacities for heat of the abraded metal and of the metal before the abrasion had taken place : so that if this additional experiment had only been a satisfactory one : — and Rumford did not see how to make it thoroughly satis- factory : — the fact that heat is not matter would have been conclusively established in 1798. What Rumford really did want was a test to bring the abraded metal and the non- abraded metal, if possible, precisely to the same final state. He tried to do that by throwing them into water — equal masses of the metal in lumps and filings, each raised to the same high temperature, into equal quantities of water at the same lower temperature, to see whether they would produce different changes of temperature, each in its own Vessel of water. 26. But then they were not in the same final state. The filings were not in the same state as the solid metal ; they might have been very considerably compressed, or they might have been distorted in shape, and in virtue of these IV.] PRELIMINARY SKETCH OF THE SUBJECT. 25 they might have had a certain quantity of latent heat which Rumford could not discover by this process. The simplest legitimate process which we know of for completely answer- ing this question, which was Rumford's sole difficulty, is a chemical process. Dissolve the lumps and an equal weight of the filings in equal quantities of an acid. At the end of the operation, of course, there can be no doubt that the chemical substances produced will be precisely the same, whether you begin with lumps or with fiUngs. If there be any mysterious difference as to the capacity for heat in them, that will be shown during the process of solution. In general, in dissolving a metal in an acid, there is a develop- ment of heat ; but if there were any difference in the quantity of heat which the lumps and an equal weight of filings contained when at the same temperature — that is to say, if heat could by any possibility be matter — there would necessarily have been a greater development of heat in one vessel than in the other. Had Rumford tried that one additional experiment, he would have had the sole credit of having established the non-materiality of heat. 27. Rumford found that, in spite of inevitable loss of heat in his operations (which consisted in boring a cannon with a blunt borer), the work of a single horse for two hours and twenty minutes was sufficient to raise to the boiling-point about nineteen pounds of water, besides heating the cannon and a,ll the machinery engaged in the process. Here is his final reasoning : — "In reasoning on this subject, we must not forget to consider that most remarkable circumstance, that the source of heat generated by friction in these experiments appeared evidently to be inexhaustible. "It is hardly necessary to add, that anything which any insulated body or system of bodies can continue to furnish without limitation, cannot possibly be a material substance. It appears to me to be ex- tremely difficult, if not quite impossible, to form any distinct idea of 26 HEAT. [chap. anything capable of being excited and communicated in the manner in which heat was excited and communicated in these experiments, except it be motion." 28. When we make a calculation frona the data furnished by Rumford's paper, we find that, supposing heat to be a form of energy, and taking 30,000 foot-pounds per minute as the work of a horse, the mechanical equivalent of heat is 940 foot-pounds. The meaning of this statement is, that if you were to expend the amount of work designated as 94a foot-pounds in stirring a single pound of water, that pound of water at the end of the operation would be 1° Fahrenheit hotter than before you commenced. We can put it in another form, which is perhaps still more striking. In the fall down a cascade or waterfall 940 feet high there would be 940 foot-pounds of work done by gravity upon each pound of water ; and therefore if all the energy which the moving water has as it reaches the bottom of the fall were spent simply in heating the water, the result would be that the water in the pool at the bottom of the fall would be 1° Fahrenheit hotter than the water at the top. 29. Davy first showed that by rubbing two pieces of ice together — by simply expending work in the friction of two pieces of ice — ice could be melted. Now a believer in the caloric theory would have argued thus : two pieces of ice when rubbed together cannot possibly melt one another, because in order to melt them heat must be furnished to them. But if the heat can only come from themselves when they are rubbed together, if it cannot come from surround- ing bodies, they cannot possibly melt; because to melt one another they would have first to part with some of their heat. To Davy's experiment, which seems to decide the ques- tion against the calorists, there was therefore this possible IV.] PRELIMINARY SKETCH OF THE SUBJECT. 27 objection, that the heat might have come from some external source, so that he tried a second form of experiment. He rubbed two pieces of metal together, keeping them sur- roaiided by ice, and in the exhausted receiver of an air- pump, so as to remove every possible disturbing cause, or even source of suspicion, from his experiment ; and still he found that these two pieces of metal when rubbed together produced heat and melted the ice, every precaution having been taken to prevent heat from getting at them from every side. 30. It is curious that his reasoning upon the subject is extremely inconclusive, although his experiments themselves completely settle the question. He says : — " From this experiment it is evident that ice by friction is converted into water, and according to the supposition its capacity is diminished ; but h is a well known fact that the capacity of water for heat is much greater than that of ice ; and ice must have an absolute quantity of lieat added to it before it can be converted into water. Friction, conse- quently, does not diminish the capacities of bodies for heat. " And there he stops. But some years afterwards he came to this conclusion from these experiments : — " Heat, then, or that power which prevents the actual contact of the corpuscles of bodies, and which is the cause of our own sensations of heat and cold, nuty be defined as a peculiar motion, probably a vibration, of the corpuscles of bodies tending to separate them. It may with propriety be called the repulsive motion. Bodies exist in different states, and these states depend upon the action of attraction and of the repul- sive power on their corpuscles, or, in other words, on their different quantities of repulsion and attraction. " 31. Davy explains by these experiments the difference between a solid and a liquid, and that between a liquid and a gas. In general the melting of a solid is produced by communicating heat to it. In other words, according to Davy's explanation, the particles of the solid are set in 28 HEAT. [chap. vibration, and thus, in consequence of repeated impacts upon one another, push one another aside. And as he also says, you may consider this repulsive motion to have a complete analogy to the so-called centrifugal force in a planetary orbit, for the faster one particle is moving about another, the larger necessarily is the orbit into which it will be forced. The particles of a solid, then, are forced from one another by this repulsive action of heat, and it assumes what we call the liquid state. Increase still farther the amount of heat communicated to the body, the cohesive forces are at length wholly overcome, and you have free particles, as in a gas, flying about and impinging upon one another, but only for very brief periods coming near enough in the course of their gyrations to bring into play the molecular forces again. When, however, the molecular forces do come into play for a moment, you may have two particles adhering together, but they are soon knocked asunder by a blow from a third particle. 32. There is one other sentence, however, which must be quoted from Davy, for it shows when and how he finally got over his difficulties and confusion of reasoning. In 1812 he enunciated this proposition : — - " The immediate cause Ojf the phenomenon of heat, then, is motion, and the laws of its communication are precisely the same as the laws of the communication of motion. '' When Davy was in a position to make that statement he had only to take it in addition to the second interpretation of Newton's third law {ante § 18),' and the whole dynamical theory, of heat was in his possession. Still, that publication of Davy's in 18 1.2, like the earlier ones of Rumford and of Davy himself, remained almost unnoticed — looked upon, perhaps, as an ingenious guess, or something of that sort, but IV.] PRELIMINARY SKETCH OF THE SUBJECT. 29 as something whidi it was not worth the trouble of philoso- phers to consider ; and it was not until Joule's time, some- where about 1840, that the subject was fairly taken up, and that justice was rendered to their real value. 33. Notice, however, how distinctly these two great leaders were men who based their work directly upon experiment. There is no d priori guessing, or anything of that kind, about either Rumford's or Davy's work. They simply set to work to find out what heat is. They did not speculate on what it might be. But both before and after their time there have been numbers of philosophers who have, without trying a single experiment, or at best trying only the roughest forms of experiment, endeavoured to discover by ct, priori reasoning what heat is. 34. The investigations of Colding and Joule, dating from about 1840, cannot here be treated in full; because, though it was to them that the final dethronement of caloric from its old position as an imponderable is unquestionably due, these exquisite experiments (especially those of Joule) extended to all forms of energy, and therefore included an immense range of subjects entirely beyond the scope of this work. In so far, however, as they bear upon our present ques- tion, these researches re-established the conclusions of Rumford and Davy ; and they supplied a much more exact determination of the dynamical (or, as it is commonly but inaccurately called, mechanical) equivalent of heat than that above deduced (§ 28) from some of Rumford's experimental data. 35. In fact the extensive, and exceedingly, accurate, ex- periments of Joule led, in 1843 and subsequent years, by processes depending directly on friction, to numbers varying from 770 to 774 foot-pounds of energy as the equivalent 30 HEAT. [CHAP. of one unit of heat (defined below, § 37) on the Fahrenheit scale. The number finally assigned by Joule (for the lati- tude of Manchester) is 772, and it is almost certainly not in error by anything approaching to i per cent In 1853 Joule verified this result by means of a very accurate determination of the specific heats of air, and a direct experimental proof (given in 1845) that the heat developed by the sudden compression of air is very nearly the equivalent of the work expended. 36. Direct measurements of the heat produced by the expenditure of mechanical energy were made in various ways by Colding, in 1843, and have been repeated in many forms by Hirn, Regnault, &c., since the publication (in 1849) of Joule's final result. The direct verification of the fact that heat disappears when work is done by a heat-engine, unsuccessfully at- tempted by Sdguin in 1839, was first effected by Hirn in 1857- A great variety of indirect methods of approximating to the mechanical equivalent of heat have been successfully applied within the last thirty years by different experi- menters. The earliest determinations of this kind are, of course, those of Joule, eff"ected in 1843 and subsequent years, by means of magneto-electricity. 37. The results of all such experiments are briefly summed up in the exceedingly important statement known as the First Law of Thermodynamics. — When equal quan- tities of mechanical effect are produced by any means from purely thermal sources, or lost in purely thermal effects, equal quantities of heat are put out of existence or are generated. And, in the latitude of Manchester, 772 foot-pounds of work are capable of raising the temperature of i lb. of water from 50° F. to 51° E. IV.j PRELIMINARY SKETCH OF THE SUBJECT. 31 38. Though by experiments such as those we have mentioned (but not, as yet, described or explained) Heat has been proved to be a form of Energy — whether kinetic or potential — we have made no progress towards the dis- covery of the mechanism upon which it depends. And, referring again to §§ 30, 31, it may be stated generally that we have as yet very little information about the nature of the internal heat-motions, &c., of the particles of solids and liquids. 39. But we have obtained from several kinds of experi- ments, entirely different from these, what must be called a probable, rather than a plausible, explanation of the effects of heat on gases — along with at least a general idea of the nature of the motions upon which heat depends in such bodies. This idea, hinted at in § 3r, is due originally to D. Bernoulli ; * but it has been resuscitated and immensely developed of late years by Herapath, Joule, Claiisius, Clerk-Maxwell, Boltzmann, and others. It has been found capable of explaining a very great number of the known physical properties of gases: and it seems destined, in a comparatively short time, to acquire all the claims to acceptance which can be demanded from a physical theory of the motions, collisions, &c., of particles of matter whose exceedingly minute dimensions put them altogether and for ever beyond the range of the most perfect possible micro- scope. 40. From another absolutely distinct point of view we have obtained very remarkable information as to the nature of the motion upon which Radiant Heat depends. But here we are especially favoured, because we have complete experimental evidence that the nature and the mechanism of propagation of Radiant Heat are precisely the same as • Hydrodynamica, sectio dtcima ; Argentorali, 1 738. 32 HEAT. [chap. those of Light (§ 3). Thus we have two quite independent sense-organs adapted for the study of these allied pheno- mena. 41. Resume of %% 20^40, — Heat is novf proved 'to be a form of energy. Hence the First Law of Thermodynamics, which merely states the equivalence of heat to work, with the requisite numerical datum called Joule's Equivalent, The mechanism upon which heat-energy depends is (pro- bably at least) approximately known so far as regards heat in a gas, and as regards radiant heat. Beyond these we have, as yet, little information on the subject. 42. Effects of Heat. — These are exceedingly varied and numerous, and in our present rapid sketch we cannot allude to any but the more common or more prominent of them. These we may classify as follows : — a. Change of Dimensions, or of Stresses, in Solids ; and of Volume, or of Pressure, in Fluids. b. Change of Molecular State. c. Change of Temperature. d. Electric Effects. e. Effects in starting Chemical Changes. 43 {a). The change of dimensions and stresses of solid bodies by heat is known by experience, even of the com- monest kind, to every one. We mention a few instances taken at random, but it will be good exercise for the student to try to recollect others, and the same exercise will be found profitable in the other departments of this subject. The " shrinking on," as it is called, of a wheel-tire, and of coil after coil on the core of a wrought-iron gun, is accomplished by heating the tire or coil, slipping it on while expanded by heat, and then cooling it suddenly or gradually as may be necessary. IV.] PRELIMINARY SKETCH OF THE SUBJECT. 33 Rails are not laid down end to end, but with a small interval ; else on a summer day they might expand so much more than the ground supporting their bearings as to pucker, and displace one another. The huge metal tubes of the Menai bridge are not rigidly fixed at each end, else they would tear themselves or their supports ; they are free to expand and contract as their temperature changes, one end of each being supported on rollers. Uncompensated clocks and watches (if their average rate be exactly adjusted) go too slow in summer and too fast in winter ; the former simply, the latter mainly, on account of the change, by heat, of the dimensions of their moving parts. The touch of a finger on the graduated limb of a delicate meridian circle produces a perceptible change in the measured zenith-distance of a star.~- A harp tuned in a warm room rises notably in pitch when taken out into frosty air. Telegraph wires are seen to " sag " more and more as the temperature of the air rises. Massive walls pressed outwards (by overloading of the roofs or floors of buildings) have been forcibly restored to their vertical position by the contraction of iron rods passed through them ; nuts being screwed tight upon the rods up to the exterior surface, while the rods were in a state of expansion by heat. 44 (a continuea). The grand circulations constantly going on in the ocean and in the atmosphere are mainly due to the •expansion of water and of air by heat. Bulk for bulk, the heated portions are lighter than the colder, and rise above them in virtue of the ordinary hydrostatic laws, which explain the floating of oil on water or the rise of a ba;lloon in the air. On a smaller scale the rise of what we call smoke D 34 HEAT. [chap. from a chimney, ventilation of mines by a fire at the bottom of an " upcast " shaft, &c., are examples of the same effect. 45 {b and c). Take a piece of very cold ice. Though the assertion may appear a little startling at first, it is really a stone — ^just as much as is a lump of rock-salt or galena — only that its molecular or crystalline structure is somewhat more complex. It becomes warmer, Just as other stones, by every fresh application of heat — up to a certain point, which we call its meltingpoint — but you cannot make it any hotter. Heat now does not change its temperature, but changes its molecular state. Precisely the same is true of the rock-salt only that the temperature of its melting-point is considerably higher than that of the ice. Suppose sufficient heat to have been applied just to melt all the ice. It is still the same substance from the chemical point of view, its temperature is still that of the melting- point, but it is a liquid instead of a soUd. 46 (b and c continued). Apply more heat to the water. Its effect is now to make the water warmer : in scientific language, the temperature of the water rises. Every fresh application of heat raises the temperature more and more till it reaches what is called the boiling-point, but here the rise of temperature again stops. Earth er application of heat produces a new alteration in the molecular state, and the liquid changes into steam or water vapour. 47 (b and c continued) . Suppose heat to have been applied till the whole of the liquid has, without farther rise of tem- perature, been converted into vapour — saturated steam, as it is technically called — we can now, by applying more heat,- raise the tem-perature of the steam, so that it becomes what is called superheated steam, and is practically a gas. But this gas cannot be heated, indefinitely farther without the production of another molecular change — this time what iv.J PRELIMINARY SKETCH OF THE SUBJECT. 35 is commonly called a chemical change — dissociation : the analysis or separation of the water-gas into its constituents, oxygen and hydrogen. Experiment has not yet told us whether or not still farther applications of heat may be capable of altering the physical or chemical nature of either of these now merely fnixeti gases. 48. Thus the successive effects, produced by continuous application of heat to a piece of very cold ice, are — 1. Heating. 2. Melting. 3. Heating. 4. Evaporation. 5. Heating. 6. Dissociation. 7. Heating. Water-substance is familiar to all, and, as we have seen, gives an excellent and instructive example of the various successive changes of state and of temperature. Few other substances oifer, at least with our present very limited ex- perimental facilities, so complete a series. On the other hand there are substances, such as carbo/i, which we cannot even melt; others, such as uncombined hydrogen, which, till the very end of 1877, we knew only in the gaseous form. 49 (d). Certain crystals, such as tourmalines when heated, attract other bodies in the same way as do sealing-wax or glass which have been electrified by friction. This is, how- ever, still a very obscure branch of our subject, and has as yet yielded no results of great importance, though it may possibly become the source of enormous additions to our knowledge. 36 HEAT. [CHAP. 49 {d continued). But what is commonly called Thermo- electricity is already of immense practical as well as theo- retical importance, as the reader will see when we come to the question oi measurement of heat and temperature. The fundamental phenomenon of Thermo-electricity is the following, discovered by Seebeck about 1820 : — When one of the junctions of a closed circuit of two metals is raised to a higher temperature than the other, a current of electricity passes round the circuit and (in general) increases in intensity with increasing difference of temperature of the junctions. The direction of the current is, of course, reversed if the cold junction be now made the hotter. If, for instance, the ends of an iron and of a copper wire be joined by twisting, soldering, or otherwise, we form what is called a circuit or re-entrant path, round which an electric current can travel. Let A and B in the sketch be the (necessarily) two junc- tions. To a person going completely round the circuit in the direction indicated by the arrows, these junctions would be distinguished from one another by the fact that, while at B he is passing from iron to copper, and at A from copper to iron, if he reversed the direction of his motion round the circuit, these characteristics of the two junctions would IV.] PRELIMINARY SKETCH OF THE SUBJECT. 37 be interchanged. Now suppose the whole to be at any ordinary temperature, and a lamp flame to be applied (for a moment only) at A. A current of electricity will pass round the circuit in the direction of the arrows, and will continue, though becoming gradually weaker, till A at length cools down again, so as to have the same temperature as B. This is Seebeck's discovery. What we have just said as to the characteristic difference between the two junctions shows us that, if the lamp be now applied for a moment to B, the current produced will run round the circuit in the opposite, direction to that indicated by the arrows. In fact, under the conditions specified, the current passes from copper to iron through the warmer jurution. (This statement applies to all cases in which neither junction is made very hot. It would only confuse the student at present to give him the general law, of which the above is merely a particular case. The whole subject will be thoroughly discussed later.) 50 ( d continued). Suppose now a closed circuit consisting of any number of pieces of iron and copper wire arranged alternately as below. It need not be longer (and therefore need not resist electricity more) than the circuit of two just described. Thsse may be in fact cut up into shorter pieces and re-arranged. Call the successive junctions in order Aj, Bj, A2, B2, &c. Then it is obvious that a person going 38 HEAT. [chap. round the cifcuit in the direction indicated by the arrows will at each A junction pass from copper to iron, and at each B junction from iron to copper. Hence all the A junctions, when heated, produce thermo-electric currents in the direction of the arrows. When the B junctions are all heated the current is the other way round. But in the simple circuit we had only one hot junction, while we may have here as many as we please ; it is obvious that by this arrangement we can increase the strength of the electric current (for the same difference of temperatures of the hot and cold junctions) in the same proportion as we increase the total number of junctions. For convenience of application, the wires or rods Aj Bp Bj Aj, Aj Bg, &c., are made of equal length, and packed IV.] PRELIMINARY SKETCH OF THE SUBJECT. 39 together as closely as possible into what is called a Thermo- electric Pile, so that all the A junctions are at one end and all the B junctions at the other. Thin paper, gutta percha, or other insulating material, is interposed between each two contiguous rods to prevent their touching one another, except where they are soldered together at the junctions. It has only to be added that as Seebeck (led by very curious reasoning) discovered that the currents produced by alter- nate bars of bismuth and antimony are far more powerful than those produced under the same circumstances of tem- perature and resistance by iron and copper — ^the pile is generally built up of alternate bars of the former pair of metals. The invention of this valuable instrument is due to Nobili (1834). 51 {«). There is little doubt that it is the heat developed by friction which inflames the particles of iron when the old " flint and steel " is used, just as it sets fire to amorphous phosphorus in contact with chlorate of potash in our modern " safety matches " ; or as the heat of a red-hot poker causes gunpowder to explode. In all such cases heat appears to induce ox promote chemical combination ; to perform, in fact, the reverse of the operation which we have already attributed to it as dissociation (§ 47). 52. Measurement of Heat and of Temperature. — The absolute distinction between the ideas of heat and of temperature cannot be too soon learned by the student. Heat, we have seen, is a real something, a form of energy. Temperature he may be content at first to look on as a mere condition which determines which of two bodies put in contact shall part with heat to the other. That such a state- ment is consistent with observed facts is shown by this, that if A is at the same temperature as B and also at the same temperature as C —no transfer of heat takes place between 40 HEAT. [CHAP. B and C, whatever be these bodies : i.e. bodies which have the same temperature as another body have themselves the same temperature. To refer to one of the analogies formerly employed, Heat may be compared to the quantity of air in a receiver, tem- perature to the pressure of that air. When two receivers, each containing air, are connected by a pipe, air is forced to pass from the receiver in which the pressure is the greater to that in which it is less. And this is altogether independent of the quantities of air in the two receivers : — that which parts with some of its air may be very small in comparison with the other, and. contain far less air ; it is the difference of pressure alone which determines the direction of the transfer. 53. But, just as one receiver may be more capacious than another, so as to contain more air at a given pressure — so one body may have more capacity for heat than another, and therefore contain more heat even when their temperatures are the same. And this difference of capacity may be due in part either to mere quantity of matter, (as when we compare an ocean with a pond,) or to a specific property of the substance itself — for, as we shall see later, mass for mass, water has thirtyfold the capacity for heat that mercury has. This specific property is known by the name of Specific Heat, a term derived from the old erroneous notions as to the nature of heatj but now fairly rooted in the language so as to be almost permanent, though as much calculated to mislead the student as is the celebrated " Centrifugal Force." 54. Thus the quantity of heat employed in the entire system of fires, furnaces, &c., of Great Britain during a whole year- — producing vivid incandescence of millions of tons of coal and of liquid iron — may be a mere trifle compared v;ith the heat required to raise the average IV.] PRELIMINARY SKETCH OF THE SUBJECT 41 temperature of the Atlantic Ocean by a single degree of the thermometer. 55. Referring again to the general principle enunciated in § 9, it is clear that we may measure heat in terms of a unit of heat, which may be defined as the heat required to melt a pound of ice at the freezing-point, to raise to the boiling-point a pound of ice-cold water, or in general to produce any definite physical change in a given mass of a given substance. Or, knowing as we do, that heat is a form of energy, we may dynamically measure a quantity of it by the number of foot pounds of work to which it is equivalent (§ 37). But 'the mode usually adopted in Britain is to define a U7iit of heat as the amount of heat required to raise a pound of water from the temperature called 50° F. to that called 5i»r. 56. Of course we may equally well adopt what are called metrical units, and the scale of the Centigrade Thermometer instead of that of Fahrenheit, but the change from one of these systems to another is one not of principle but of convenience, and, at the worst, involves a mere arithmetical operation of multipUcation or division by a definite nuniber. Such questions rarely rise to scientific importance, though they may raise (often justly) discussions as to comparative convenience. [Thus, there can be no question about the fact that the metre is inconveniently long, and the kilogramme incon- veniently massive, for the ordinary affairs of life. The average length of the arms of shop-girls, and the average quantity of tea or sugar wanted at a time by a small purchaser, have no conceivable necessary relation to the ten-millionth part of the quadrant of the earth's meridian passing through Paris or the maximum density of water. 42 HEAT. [chap. But the standard yard and pound were, no doubt, originally devised to suit these very requirements as regards the average dimensions of the shop-girl or the paying powers of the ordinary customer. Yet this invaluable superiority of our units over those of the metrical system is, with an almost over-refinement of barbarism, thrown away at once when we come to multiples or submultiples. The very lowest attempt at consistency should have rendered it impossible for any one who employs the decimal notation to use any but a decimal system of multiplication and sub- division of units. All the monstrosities of the old Logic with its Barbara celarent, &c., or of the Latin Grammar, with its As in presenti, &c., seem almost natural and'proper when compared with a statement like this : — 12 inches = i foot, 3 feet = .1 yard, 220 yards = i furlong, 8 furlongs = i mile. And even this is nothing to the awful complex of poles or roods, grains Troy and Avoirdupois, drachms and fluid ounces.] 57. The measurement of Temperature, upon which we have seen (§ 52) that the measure of Heat ultimately depends, presents absolutely unlimited choice. Any of the effects of heat (which we have already briefly discussed) may be employed, and we may use any material substance whatever for the purpose. It will be seen, however, when we have passed these preliminaries, that the Second Law of Thermo-dynamics enables us to lay down an absolute definition of tem- perature : — absolute in the sense that it is entirely inde- pendent of the physical properties of any particular kind of matter, and depends solely upon the laws of trans- formation of energy. 58. It will at once be obvious to the reader that, since IV.] PRELIMINARY SKETCH OF THE SUBJECT. 43 there is such an absolute scale of temperature, it must bs adopted in all really scientific reasoning from the results of experiments ; though it may happen to be very difficult to apply it directly, during the practical work of experiment and observation. 59. Few experimental measurements, certainly very few in which great precision is sought, are made in terms of the scientific or other units to which they are ultimately reduced. Thus, in constructing a screw micrometer for delicate astronomical or other measurements, it is not necessary, even if it were practicable, to give the screw exactly 10 or 100 threads to the inch. What is necessary is a good, i.e. a uniform, screw : — and the exact length of its step is of no consequence. When the observer has, once for all, measured with it a number of objects or angles of known magnitude, he knows the value of one whole turn, and thence the value of any reading whatever, by a simple arithmetical process. Very fine thermometers, for research, often have their tubes not only calibrated, but graduated before the bulbs are blown on them. The freezing and boiling-points are then determined with the utmost care in terms of the arbitrary scale of the instrument, which thus (by an arithmetical operation) becomes perfectly definite throughout. Far from being a drawback to the use of such an instru- ment, the arbitrary scale is often of positive advantage to the experimenter, for it prevents his being (as the most honest and careful experimenter is very liable to be) uncon- sciously influenced by a knowledge of the reading that is to be expected. And the, mere arithmetical difiiculty of passing from one scale to another is so trifling in com- parison with the difficulties of experiment that it would be wholly disregarded, even were there not the distinct 44 HEAT. [CHAP. gain thus secured, of absolute freedom from the bias of preconceived notions. 60. What we want at present, in our preliminary sketch of the subject, is not the rigorous scientific measure of temperature, but something which shall be at once easy to comprehend and easy to reproduce, and which shall after- wards require only very shght modifications to reduce it to the rigorous scale. Approximately pure water is (practically) to be obtained with ease everywhere. Now our fundamental principle (§ 9) assures us that the changes called melting and boiling will always take place, with the same substance, each under its own precise set of conditions. It will be found later that the only conditions which require to be insisted on here are those of temperature and pressure. Hence we merely assert for the present, that For a definite pressure there is a definite temptrature at which pure ice melts, and another defi.nite temperature at which pure water boils. 61. The idea of using two such fixed temperatures for determining a scale is due to Newton. For our present purpose we may suppose we have one vessel in which ice is melting, and another in which water is boiling, the barometer standing at 30 inches.* If we call the temperature of the cold water zero, or 0°, that of the boiling water 100°, we adopt what is called the Centigrade scale. Now it is obvious that, we may obtain every possible intermediate temperature by mixing portions of water from the two vessels in different proportions. And we may * The student may here take for granted that it would only confuse him were we now to give the scientifically rigorous dcliiiition of the temperature called the " boiling-point." IV.] PRELIMINARY SKETCH OF THE SUBJECT. 45 define the intermediate degrees by saying that the per- centage (by bulk or preferably) by weight of the hot water in each such mixture is its temperature in degrees. Thus 10° C. would be the temperature of a mixture of 10 lbs. of water at 100° C. with 90 lbs. of water at 0° C, and so on. This is obviously but one of an infinite number of ways in which the intermediate degrees might have been defined • but it has the advantage of directness and simplicity, while it is near enough to the absolute scale for our present purposes. And it has the farther advantage that it is the only one upon which the temperature of a mixture of equal weights of water at any two temperatures has a value exactly half- way between these. Until we can point out clearly the necessity for, and the possibility of, making improvements on this scale, we shall employ it for our work ; for it accords so closely with the absolute scale that careful experiment is required even to show that it does not exactly coincide with it. 62. The eflfect of heat which is most commonly em- ployed for measurements of heat and temperature is the change of volume of liquids (§ 42). We may take for granted that every reader has some little familiarity with an ordinary Thermometer. The liquid employed is usually mercury or alcohol. The precautions necessary in making and using Thermometers will be discussed later. The other effect of Heat most commonly employed for these purposes is the Thermo-electric one already described. (§ 49)- In two respects it is vastly superior to any of the others — iox, first, it is far more delicate; and, second, its indications can quite easily be made visible to the largest audience. This method is based on the fact that, at least for a small difference of temperatures of the junctions, the strength 46 HEAT. [chap. of the electric current is directly proportional to that difference. Hence a galvanometer, an instrument which measures the strength of the electric current, measures at the same time the difference of temperatures of the junctions. 63. Here a digression is necessary. The fundamental fact of Thermo-electricity (§ 49) belongs directly to our subject ; but the fundamental fact of Electro-magnetism, on which the action of the galvanometer depends, is entirely foreign to it. The fact, discovered by Oersted in 1820, is simply that the position of a freely suspended magnet is, in general, altered by the passage of an electric cttrrent in its neighbourhood. The figure represents all the essential features of the only form of this experiment which we require for illustration. A long copper wire, covered with silk or gutta-percha, to prevent contact of the wire with itself or with other metal, is wound into a circular coil ot a considerable number of turns, and its free ends are connected — one with a zinc, the other with a copper plate. The coil is placed vertically in the magnetic meridian, and a magnetized steel bar is suspended horizontally inside it by a fine silk thread. 'While the zinc and copper plates are dry, and not in metallic contact, the IV.] PRELIMINARY SKETCH OF THE SUBJECT. 47 magnetized bar of course takes the line of magnetic north, and therefore settles in the plane of the coil. But the moment the zinc and copper plates are dipped into a vessel containing water slightly acidulated, so as to cause an electric current to pass round the coil, the magnet is deflected, and tends, so far as it can, to set itself at right angles to the plane of the coiL As the figure is drawn above, the bar tends to turn one or other end towards the spectator. The actual position which it will finally take up depends upon the relative amounts of the forces exerted upon it by the earth's magnetism and by the electric current. If the current be very weak the magnet is very little deflected ; if the current be very strong the magnet is placed almost at right angles to the plane of the coil. Thus, from the amount of the deflection, the strength of the current may be calculated. The direction in which the current passes round the coil determines to which side of the coil the north pole of the magnet will be deflected. This can easily be shown by interchanging the copper and zinc plates, when we find the direction of the deflection reversed. The rule which is found to determine the direction of the deflection may be stated thus. Suppose that, in the figure, we are looking at the coil from the west side, then the left-hand end of the magnet is what is usually called its north pole. If, then, the current be so applied that positive electricity passes round the coils of wire in the direction in which the hands of a watch move, the north end of the magnet will move towards the east side of the coil, i.e. will move away from the spectator. 64. The galvanometer is usually constructed on a small scale, because the action of the current on the magnetic poles within the coils is greater as their radius is smaller (provided 48 HEAT. [chap. there be the same number of coils and the same strength of current). Also a small magnet has greater mobility ; and in general harder temper, so that it preserves its magnetiza- tion longer than a large one. Hence a double advantage is gained by diminishing, as far as possible, the dimensions of the apparatus. Another gain is thus insured : the number of coils, and with them the electro-magnetic action, may be greatly increased while the letigth of the wire, which by its resistance weakens the current, may actually be reduced. But all the more essential features of galvanometers are exhibited in the rude apparatus depicted in § (^t,. 65. Suppose, now, a small mirror to be fixed vertically to one side of the magnetized bar, and a ray of sunlight to be made to fall on it, and after reflection received on a white screen. Every motion of the magnet. would be at once indicated by the corresponding motion of the illumi- nated spot on the screen. And the farther off the screen is placed, the greater will be the motion of the spot for the same deflection of the mirror — i.e. of the magnet. Thus deflections, however small, may be magnified so as to become visible even to a very large audience. This makes the thermo-electric method invaluable for lecture-illustration. And here a fourth important advantage is gained, for the smaller the deflections of the magnet, the more nearly are they proportional to the strength of the current. When accurate measurement is our object, the little mirror is made concave, and a wire is adjusted vertically between the mirror and the source, of light at such a distance that a sharp image of it is formed on the screen. This, and not the ill-defined patch of light which it crosses, serves as an index, by which to read the deflections by the help of the scale of equal parts drawn on the screen. (i(>. The various successful devices for rendering galvano- IV.] PRELIMINARY SKETCH OF THE SUBJECT. 49 meters more and more sensitive, belong properly to the subject of electricity ; but it may be well to indicate here a few of the more important. The mirror, with a number of very small pieces of hard steel (magnetized to saturation), attached to it, need not weigh more than a fraction of a grain. It is supported by a si'ng/e fibre of unspun silk. The coils of wire should be as numerous, and of as small diameter as possible, to secure the greatest effect (§ 64) ; but the whole resistance of the wire should be as small as possible — since in the greater number of thermo-electric arrangements which we require for radiant heat, &c., the tnain resistance is in the galvano- meter coils. Thus the best copper should be employed, and the diameter of the wire should be so small as to allow of a great number of coils of small diameter, while not so small as to destroy this gain by the greater consequent resistance, and weakening of the current. The astatic '^xma.^Xe. may also be introduced with great advantage for very delicate instruments. Here two light needles, or sets of needles, are placed parallel to each other with their north poles oppositely di- rected, and their middle points connected by a thin aluminium wire, which is supported in a vertical position by a fibre of unspun silk. Each magnet has its own coil, and the ends of the wires are so connected, that (as in the figure) a current, passed through them, runs round them in opposite directions. Thus, as the direction of the current, as well as the direction E 53 HEAT. [chap. of the north pole, are each reversed in passing from one coil to the other, the electro-magnetic effects tend in the same direction, while those of the earth's magnetism tend in opposite directions. If we could make the magnetic moments of the needles exactly equal, and place them with their magnetic axes exactly parallel, terrestrial magnetism would not affect the apparatus at all, and the electro-magnetic effect would bo counteracted and balanced solely by the torsion of the silk fibre. Practically this cannot be done; but the effective directive force of the earth's magnetism can easily be reduced to xJirth of its whole amount, and even less ; thus securing at least a hundredfold greater delicacy by this arrangement. As an illustration of the delicacy thus attainable, it may be mentioned that it is easy to construct a galvanometer which will show the jo^m th or even the jooou th of a degree- Fahrenheit of difference of temperatures beween the two junctions of a copper-iron circuit (§ 49) at ordinary tem- peratures. 67. When a galvanometer is to be used sometimes for powerful, sometimes for weak currents, various devices may be employed ; such as making the coil in two or more nearly equal parts, one or more of which may be employed at a time. If two be employed, with the current running opposite ways in them, a much more powerful current will obviously be required to produce a given deflection than if it run the same Way in both. Another device is to use external, adjustable, magnetized bars so as to increase or diminish at pleasure the intensity of the field of force due to the earth's magnetism. This method is satisfactory for powerful currents, but not for weak ones, because the earth's horizontal magnetic force IV.] PRELIMINARY SKETCH OF THE SUBJECT. 51 is continually and rapidly changing, and when the greater part of it is neutralized, even a slight change may produce great percentage variation in the uncompensated part, on which alone the delicacy of the galvanometer depends. The great majority of those valuable improvements just de- scribed, were recently introduced by Sir W.Thomson (originally for practical telegraphic applications). Another, and by no means the least, consists in inclosing the magnets and mirror in a small, narrow, glass cell, with parallel faces. The vis- cosity of the air, and the large surface and small moment of inertia of the mirror, cause this instrument to pounce almost instantly upon its exact reading, without the long-continued and tantalizing oscillations which were unavoidable in the older instruments. This dead beat principle, as it is called, enables the experimenter to make accurate readings faster almost than he can record them ; and this not only effects an immense saving, both of time and trouble, but enables the observer to study a phenomenon in various ways before it has sensibly changed its conditions, a most desirable thing in itself, but only now made possible. 68. Sources of Heat. — As heat is only one of the many forms of energy, and as any one form of energy can in general be transformed, in whole or in part, into another, the sources of heat are as numerous as the forms of energy at our disposal. Our available sources of potential energy are mainly fuel. Under the head of fuel are included not merely coal, wood, and so on, but also all that may properly be called fuel — the zinc used in a galvanic battery, for instance, and various other things of that kind, including the food of animals. We have also : — Ordinary water-power. Tidal water-power. E 2 52 HEAT. [CHAP. In the kinetic form we have : — Winds. Currents of water, especially ocean currents. Hot springs and volcanoes. There are other very small sources known to us, some exceedingly small, such as diamonds for instance; but those named include our principal resources. And every one of these, by proper processes, can be made a Source of Heat. 69. Now comes the question. What are the sources of these supplies themselves ? They also can be classified under four heads. The first is primitive chemical affinity, which we may sup- pose to have existed between particles of matter from the earliest times, and still to exist between them, because these portions of matter have not combined with one another nor with other matter. If, for instance, when the materials of which the earth is composed were widely separated from one another, there were particles of meteoric iron and native sulphur which, when the materials fell together to form the earth, did not combine, but still remain separate from one another, the mutual chemical potential energy of the iron and sulphur remains to us as a portion of energy priipordially connected with the universe. But of tJiat, so far as we know, at least near the surface of the earth, there is very little. There may be towards the interior enormous masses of as yet uncombined iron and uncombined sulphur, or various other materials > but towards the surface, where they could be of any direct use to us, the quantities of these are excessively small. The second source is solar radiation, by far the most abundant source we have. Then we have two very IV.] PRELIMINARY SKETCH OF THE SUBJECT. 53 instructive forms, the energy of the earth's rotation about its axis, and the internal heat of the earth. 70. Now compare our available stock with the sources from which we derive it, and we easily see how the two are connected. Our supplies of fuel are almost entirely due to the sun, In times long gone by, the sun's rays by their energy, as absorbed in the green leaves of plants, decomposed carbonic acid and stored up the carbon. That carbon, and various other things stored up ages ago along with it, we have still as an immense reserve fund of coal. For food we are mainly indebted to the sun again, because the food of animals must ultimately be vegetable food, even of the animals which live upon animal food. For ordinary water-power we are also indebted to the sun, because it is mainly the energy of the radiation from the sun which eva- porates water from the plains or seas, so as to be con- densed again at such an height that it has potential energy in virtue of its elevation. Ordinary currents of water are a mere transformation of this potential energy, because water on a height converts part of its potential energy into kinetic energy of visible motion as it flows down. But when we come to tidal water-power, we must look to another source. If we employ tidal power for the purpose of driving an engine, we take it in the rise of the water as the tide wave passes us. We secure a portion of water at a certain elevation, wait till the tide has gone back, and then take advantage of the descent of that portion of water. Now, if we were to go on doing this for any considerable period of time, and over large tracts of sea-coast, we should find that the effect would be to gradually slacken the rate of rotation of the earth. Winds and ocean currents are almost entirely due to 54 HEAT. [CHAP. solar radiant heat. And hot springs and volcanoes, which have never been employed for any direct production of work, depend mainly, at least, upon the internal heat of the earth ; partly, perhaps, on potential energy of chemical affinity. 71. It is obvious, then, that the sun is the great source of almost all our available energy ; and we can carry the investigation still one step farther back, so as to inquire into the source of the sun's enormous store of energy. It will be seen later, that no known chemical source can ac- count for more than a very small fraction of it ; and that the only adequate known source is the potential energy which its parts must have possessed, in virtue of their mutual gravitation, while they were widely dispersed throughout a space of dimensions at least equal to those of the known solar system. 72. Transference of Heat. — Under this head it is usual to classify as the three ways in which heat can be transferred from one place to another, the processes of Conduction. Radiation. Convection. As will be seen presently, this list involves some confu- sion, for as commonly understood it includes either too much Of too little. 73. By Conduction of Heat we usually mean that com- paratively slow transfer of heat from part to part of a body, or from one body to another with which it is in direct contact. It is mainly by conduction that any part of the earth's internal heat reaches the surface, and that the sun's heat penetrates into the crust. The study of the laws of conduction, and of the relative conducting powers of 17.] PRELIMINARY SKETCH OF THE SUBJECT. 55 various substances, as well as of the conducting power of one and the same substance at different temperatures, is of very high practical value, as well as of intense scientific interest. By Radiation of Heat we usually mean the process by which heat is transferred from one body to another with the velocity of light — even through space altogether devoid of what is called tangible matter. It is entirely by radiation that we obtain heat from the sun, as also (in very small quantity) from other stars. The subject of radiation and the connected subject of absorption have, of late, received unexpectedly an enormous extension. The practical pro- cess of Spectrum Analysis, to which they have recently led, is one of the most important means at our command for the study alike of the most distant stellar and nebular systems, and of the almost inconceivably minute grained structure of matter. By Convection of Heat is usually meant the carriage of heat-energy from one locality to another along with the particular portion of matter with which it is associated. The Gulf-stream is a vast convection-current, whereby the solar heat of the tropics is carried into the North Atlantic. Every form of ventilation, whether of coal-mines or of private houses, at least if heat be directly the effective agent, is a mere case of convection. 74. To avoid *hat the student might feel to be perplexing novelties, it may be well to adhere in this elementary book to the classification just given. But it may not be amiss to make here a remark or two upon it. In the first place we have absolutely no proof that radia- tion from the sun is in any of the forms of energy which we call heat, while it is passing through interplanetary space. That it is a form of energy, and that it depends upon some species of vibration of a medium, we have absolute proof. S6 HEAT. [chap. But it seems probable that we are no more entitled to call it heat than to call an electric current heat ; for, though an electric current is a possible transformation of heat- energy and can again be frittered down into heat, it is not usually looked upon as being itself heat. Just so the energy of vibrational radiations is a transformation of the heat of a hot body, and can again be frittered down into heat — but in the interval of its passage through space devoid of tangible matter, or even while passing (unabsorbed) through tangible matter, it is not necessarily Heat. Again, in dealing with Thermo-electricity, we shall find that a current of electricity has a convective power for heat: — i.e. that when passed through a metal bar whose temperature is not the same at all points it may carry heat Irom the hot to the colder parts, or vice versa; and that when passed through a circuit, made up partly of one metal, partly of another, and at the same temperature throughout, it carries heat from one of the two junctions to deposit it in the other. These facts render the usual hard and fast classification of § 7 2 somewhat inadequate ; but without farther advert- ing to this we merely mention that the apparent exceptions can easily be treated under the head of Transformations of Heat, as will be seen when we go over the various divisions of the subject more fully. This is clearly a case of some- what illogical arrangement, but, as already explained (§ ii), we have decided to adopt it, lest we should perplex rather than assist the beginner. The student who has got suc- cessfully over the earlier stages will easily surmount this difficulty also. 75. Transformations of Heat. — These are, of course, as numerous as the other forms of Energy, since we have already seen that any form of energy can be transformed (at IV.] PRELIMINARY SKETCH OF THE SUBJECT. 57 least partially) into any other. On this point nothing farther need be said at present. But there is another point of the case which is of the utmost importance. We may state it thus : — Given a certain quantity of energy in one form and under given conditions, how much of it can by means of a given kind of apparatus, be converted into some btlier definitely assigned form, the rest being either untrans- formed, or transformed, in vi^hole or in part, into some third form ? 76. Now part at least of the enormous amount of waste which takes place in an ordinary steam-engine is familiar to all. Not to speak of the unburned fuel which is allowed to escape as smoke, the very ascent of the column of smoke is due to wasted heat, and there is constant and large leak- age from the furnaces, boilers,, and cylinders. The very fact that the stoke-hole is so intolerably hot is due to the same waste. But there is unavoidable as well as unnecessary waste. It has" been satisfactorily demonstrated that in the very best engine, even if it were theoretically perfect, and working at ordinary ranges of temperature, only som.ewhere about one-fourth — very rarely so much, but at the best about one-fourth— of the heat which is actually employed is con- verted into work ; that is to say, three-fourths of the coals, or three-fourths of the heat employed, are absolutely wasted under the most favourable circumstances. What is it that determines this ? Why is it that the whole of a quantity of work or of potential energy can be converted into heat, Tvhile the heat cannot be converted again, except in part, into the higher form of work or potential energy ? 77. The answer is included entirely in the word "higher," just used. When energy is to be converted from a higher form into a lower, the process can in general be carried out 58 HEAT. [chap. ia its entirety ; but when it comes to be a question of reversal — going up hill, as it were — then it is only a fraction, in general (even under the most favourable circumstances) only a small fraction, of the lower kind of energy which can be raised up again into the higher form. All the rest usually sinks down still lower in the process. When it is low already, and part of it has to be elevated and transformed into a higher order, a large part of it must inevitably be still farther degraded; in general the larger part of it. This is one of the most important advances ever made in science, and has most stupendous bearing on the future of the whole visible universe. 78. In all transformations of energy we find experiment- ally that there is a tendency for the useful energy to run down in the scale, — so that, the quantity being unaltered, the quality becomes deteriorated, or the availability becomes less ; and thus we are entitled to enunciate, as Sir William Thomson did very early after the new ideas were brought into full development, the principle of Dissipation of Energy in Nature. 79. The principle of dissipation, or degradation, as it may perhaps preferably be called, is simply this, that as every operation going on in nature involves a transformation of energy, and every transformation involves a certain amount of degradation (degraded energy meaning energy less capable of being transformed than before), energy is as a whole con- tinually becoming less and less transformable. 80. Thus, as long as there are changes going on in nature, the energy of the universe is falling lower and lower in the scale, and we can at once see what its ultimate form must be, so far at all events as our knowledge yet extends. Its ultimate form must be that of heat so diffused as to give all bodies the same temperature. Whether this be a high IV.] PRELIMINARY SKETCH OF THE SUBJECT. 59 temperature or a low temperature does not matter, because whenever heat is- so diffused as to produce uniformity of temperature, it is in a condition from which it cannot raise itself again. In order to procure any work from heat, it is absolutely necessary to have a hotter body and a colder one. When therefore all the energy in the universe is transformed into heat, and so distributed as to raise all bodies to the same temperature, it is impossible — at all events by any process yet known, or even conceivable — to raise any part of that energy into a more available form. Why it is so, and what (slight) exceptions there are to this general state- ment, will form an important subject for discussion farther on. For the present purpose we may hold it as a perfectly general law of nature. 81. The grand question, therefore, with regard to all transformations of energy is To what extent can they be carried out? This, of course, is a question to which nothing but experiment, or reasoning ultimately based on experi- ment, can possibly give an answer. And, so far at least as the transformation of heat into work, and various allied transformations, are concerned, it has been answered in the most brilliant and satisfactory manner. 82. Perhaps no purely physical idea has done so much to simplify science, or led to so many singular and novel pre- dictions (subsequently verified by experiment) as has Carnot's idea of a Cycle, or his farther idea of a Reversible Cycle, of operations. It has supplied not merely the legitimate mode of finding the relation between the heat taken in and the work done by an engine, but also the test of perfection for a heat- engine, an absolute definition of temperature, the effect of pressure on the melting-points of solids, and innumerable important groups of associated properties of matter and 6o HEAT. ' [chap. energy under various conditions. To a great extent these are included in the statement of the Second Law of Thermodynamics. — If an e?igine be such that, when it is worked backwards, the physical and mechanical agencies in every part of its motions are all reversed (see § 89), it produces as much mechanical effect as can be pro- duced by any thermody?iamic engine, with the same tempera- tures of source and refrigerator, from a given quantity of heat. 83. It is to be particularly observed here that Reversibility (see §§ 88, 89) is the sole test of perfection of an engine. Also in the working of a reversible heat-engine nothing is said about the nature of the working substance ; the tempe- ratures of source and refrigerator, and the quantity of heat supplied, are the sole determining factors of the work which can be done. The importance of this proposition, as regards actual and proposed engines, cannot be over-estimated. 84. Carnot's work is upon the Motive Power of Heat, and was published in 1824. It forms no inconsiderable portion of Sir W. Thomson's many scientific claims that he recog- nised at the right moment the full merits of this all but forgotten volume, and recalled the attention of scientific men to it in 1848 ; pointing out, among other things, that it enabled us to give, for the first time, an absolute definition of Temperature. Although Carnot (seemingly against his own convictions) reasons on the assumption that heat is matter, and therefore indestructible ; and although, in con- sequence, some of his investigations are net quite exact, his work is of inestimable value, because it has furnished us, not only with a correct basis on which to reason but, with a physical method ©f extraordinary novelty and power, which enables us at once to apply mathematical reasoning to all questions of this kind. These, then, are his two great claims— first, the setting thermo- dynamics upon a proper IV] PRELIMINARY SKETCH OF THE SUBJECT. 6i physical and experimental basis j and, second, in the fur- nishing us with a means of reasoning upon it which was absolutely new in physical mathematics, and which has been, not merely in Carnot's hands, but in the hands of a great many of his successors, as fruitful in new discoveries as the idea of the conservation of energy itself. 85. In order to reason upon the working of a heat- engine (suppose it for simplicity a steam-engine), we must imagine a set of operations, such that at the end of the series the steam or water is brought back to the exact state in which it was at starting. That is what Carnot calls a cycle of operations, and of it Carnot says, that only for such a cycle are you entitled to reason upon the rela- tion, between the work done and the heat spent. If a quantity of steam were allowed merely to expand, losing •heat in the process and doing work, we should have no right whatever to say that the quantity of heat which has disappeared is the equivalent of the work which is given out, because at the end of the operation the steam is in a different state as to pressure and temperature from that in which it was at the beginning. It was saturated steam at a certain temperature, let us say, to start with, and at the end of the operation it may still, if proper adjustments be made, be saturated steam, but it is neces- sarily at a different temperature, and therefore we have no right to assume that it possesses intrinsically the same amount of energy as it did in its former state. We have no right whatever to reason upon the quantity of heat which appears to have gone, as compared with the work which has been done, when the working substance begins in one state ■ and ends in another. But if we can by any process bring the working substance back to its initial state, then we are entitled to assert that it must contain neither more nor less 62 HEAT. [chap than it did at first, and therefore of course we are also entitled to reason upon all the external things that have taken place during the operation, and to determine the condition of equivalence among them. 86. The hypothetical operation which Carnot introduced for the purpose of reasoning on this subject is, like most great ideas, excessively simple — when found. Let us imagine two bodies, each maintained constantly at a definite temperature. These will be called the hot body and the cold body respectively. In addition to these suppose a body whichj as regards other bodies, is neither cold nor hot, being incapable of absorbing heat or of giving it out — a body which is a non-conductor of heat. Suppose the walls of the working cylinder and the piston to be non- conductors of heat, but the bottom of the cylinder a perfect conductor. Suppose a quantity of water and steam in the cylinder, both at the same temperature, that of the cold body. Place the cylinder on the non-conductor and expend work in pressing down the piston, the contents will become warmer, and some steam will be liquefied. Continue this process till the temperature rises to that of the hot body — ^ then transfer the cylinder to it. Now allow the piston to rise, the contents remain at the temperature of the hot body, fresh steam is generated, and work is done. Arrest this process at any stage and transfer the cylinder to the non- conducting body. If we now allow the contents farther to expand, more work is done, but the temperature gradually sinks. Continue this process till the temperature falls to that of the cold body, to which, therefore, without loss or gain of heat, it may now be transferred. Next apply work to compress it at the constant temperature of the cold body till (by condensation) the contents have become exactly as .they were at starting. The cylinder must now be transferred IV.] PRELIMINARY SKETCH OF THE SUBJECT 63 to the non-conducting stand, and everything is as it was at first — save that some heat -was taken from the hot body in the second operation, and heat was given to the cold body during the fourth. Also it is evident that more work has been done during the second and third operations than was spent in the first and fourth, for the temperature, and there- fore the pressure, of the contents was greater during the expansion than during the compression. Of course this operation may be repeated any number of times. 87. Notice particularly what the peculiarity of the opera- tion is. The steam or expanding substance, whatever it is — for air or anything else would do equally well — must always be in contact with bodies at its own temperature, or else with non-conducting bodies. If it were in contact with a body of a lower temperature, there would be a waste of heat. Heat would pass by conduction from the cylinder to external bodies, and would of course be wasted as regards work. The same would happen if the cylinder were to be removed from the non-conducting body and placed upon the cold body, before its contents had been allowed to expand sufficiently to cool down to the temperature of the cold body: some heat would then be conducted away at once, and be lost to the engine. So, throughout the whole of Carnot's operations, it is essential that there should be no direct transfer of heat at all except while heat is being taken in from the hot body or given out to the cold body : the temperature of the contents of the cylinder being in each of these cases the same as that of the body with which they are for the moment in contact. 88. We now come to another point, also perfectly novel, and of great importance. Suppose the operations in Carnot's cycle to be performed in exactly the reverse order. Begin, for instance, with the hot body, but do not allow the piston 64 HEAT. [chap. to rise there. Take the cylinder from the hot body when the water and the steam below the piston have acquired the higher temperature. Lift it to the non-conducting body, and then allow the piston to rise. Let it continue to rise, doing work all the time, till the temperature sinks to that of the cold body ; place it on the cold body ; allow the steam to expand still farther,— it will be in that case giving out work but taking in heat. When it has risen to its former highest point, place it back again on the non-conducting body, force the piston back to the same extent as that to which it rose when (in Carnot's direct set of operations) it was first placed on that body. Everything has taken place in precisely the reverse order to that in which it took place before. Finish then upon the hot body, and press home. There is sent back in that final operation precisely the quantity of heat taken from the cold body ; but during the two first operations the piston was in contact with steam at a lower temperature, and therefore at a lower pressure than during the two last. And, therefore, in the reversed method of working the engine, work has on the whole to be spent in taking heat from the cold body and depositing it in the hot body. 89. These are the grand ideas which Carnot introduced. Their two distinctive features are, Jirsi, the idea of a com- plete cycle of operations, at the end of which the working substance, whatever it is, is brought back to precisely its primary condition; — this cycle can be repeated over and over again indefinitely. Secondly, the notion of making the cycle a reversible one, so that all the operations can be performed in the reverse order ; i.e. instead of taking in heat at any stage, heat is given out ; — instead of work being done by the engine at any stage, work is spent upon it. With these reversals of each part of the IV.] PRELIMINARY SKETCH OF THE SUBJECT. 65 operation, the whole cycle can be gone over the reverse way. 90. Now Camot, considering heat as a material sub- stance, says that obviously it has done work in the direct series of operations by being let down from the higher temperature to the lower, just as water might do work by being let down through a turbine or other water-engine, doing work in proportion to the quantity that comes down and the height through which it is allowed to descend. In the reversed operations work is spent in pumping up the heat again from the cold body to the hot one. Here it is of course assumed that the quantity of heat which reaches the condenser is the same as that which leaves the boiler ; — i.e. virtually that heat is matter, and therefore unchangeable in quantity. We now know that this notion of the nature of heat is erroneous, but Carnot's reasoning does not there- fore lose its value, because the change of a word or two only is required to render it perfectly applicable with our modern knowledge of the subject. 91. One point which appears to show conclusively that Carnot's analogy (not his result) was incorrect, is that nothing is easier than to let heat down at once from the hot body to the cold, without the performance of any work. If the hot body be put into direct communication with the cold body, the same quantity of heat might be allowed to go down from one to the other, and yet give no work at all. Before we give the correct statement of the question here involved it may be well to complete this brief account of Carnot's contributions to the problem. For even with his false assumption he obtained much more than has yet been stated. We have yet to show why he introduced the notion of reversibility. And this is virtually what he said : — If an engine be reversible (as this cycle of operations has been 66 HEAT. [CHAP. shown to be), it does as tnuch work as can be got from a given quantity of heat under the same given circumstances. So that, — no matter what be the substance which is expand- ing and contracting, if a certain quantity of heat be let down from a source at a certain temperature through a reversible- engine to a sink at another definite temperature, then the quantity of useful work which can be got from that heat will be absolutely'ihe same. Reversibility is thus the sole necessary condition of equi- valence between two heat-engines. This is an enormous step in physical science. The reasoning is independent altogether of the properties of any particular substance. Whether steam, or air, or ether, &c., be the working substance, there is the same crucial test of the perfection of an engine. That test is, if a heai-aigiiie be reversible it is perfect, — perfect not in the popular sense, but in a scientific sense ; that is to say, it is as good as it is possible physically to make it. 92. Carnot demonstrates this property by a simple reductio ad absurdum exactly analogous to that of § 20. He says that a reversible heat-engine is a perfect one ; for, if not, suppose there could be one more perfect, and let these two engines be employed in conjunction. Let the more perfect engine be employed in taking a quantity of heat, conveying it to the condenser from the boiler, and giving from it a larger quantity of work than the reversible engine could do. Part only of the work thus done will be required to enable the reversible engine to pump the same amount of heat back again. Every time the heat goes down, it is through the more perfect engine ; every time it comes up, it is through the less perfect engine, and therefore the double system necessarily gains more work than it spends. Thus we have an engine which will not merely go for ever. IV.] PRELIMINARY SKETCH OF THE SUBJECT. 67 but which will for ever, without any drain on its sources, steadily do work on external bodies. Such a consequence, as we have seen (§ 20), is incon- sistent with all experimental results, and therefore the supposition which led to it, viz. — that there can be a more perfect engine than a reversible one, — is necessarily false. Such is Carnot's proof that (on the assumption that heat is matter) a reversible heat-engine is a perfect engine. It requires very little indeed, as a moment's reflection shows, to make this reasoning consistent with modern knowledge of heat. 93. We have now to consider the cycle in the light of the conservation of energy ; so that, if work be produced from heat at all, some of that heat must have disappeared in its production. Therefore, under no circumstances — if the engine be doing external work at all — can the quantity of heat which reaches the condenser ever be equal to that which leaves the boiler. If no heat has been wasted by conduction or in other unprofitable ways, — the difference between the quantity which leaves the source and the quantity which reaches the condenser during a complete cycle must be precisely the equivalent of the external work which has been done. Taking that into account, suppose we could make an engine more perfect than a reversible one. Work the two together, as before. Make the re- versible engine continually restore to the source as much as the other takes from it. Then, as it is less perfect, it will require less work to be employed on it, when reversed (to restore to the source or boiler that quantity of heat), than is furnished by the other engine; and therefore, on the whole, while there is a pumping up of heat and letting it down which exactly compensate one another, at least so far as the source is concerned, there is a gain of work. But a heat- F 2 68 HEAT. [chap. engine, however complex, can only work by expenditure of heat— so that, as the arrangement in question takes no heat from the boiler, it must obtain it from the Condenser. The result amounts to this, that by taking, as the condenser for a compound engine such as that supposed, any limited portion of the available universe, the engine would con- tinue to give out work till it had removed all heat from that portion. It may safely be assumed as axiomatic that this cannot be done ; all experimental facts are against it. Thus we have it, ex absurdo, that there can be no engine more perfect than a reversible one. This question will be more carefully considered in a later chapter. 94, Since all reversible engines are perfect, they are all of equal efficiency : that is, they all give precisely the same amount of work from the same quantity of heat, under the same conditions. It follows that these conditions alone determine how much work can be produced, by a perfect engine, from a given quantity of heat. Now, the tempera- tures of the boiler and condenser are the only particulars in which this set of perfect engines agree. Suppose each to be worked till it has employed a given quantity of heat, then each would do the same amount of work. That is to say. All perfect heat-engines under the same conditions convert into work the same fraction of the heat used, and the value of this fraction depends only upon the temperatures em- ployed. Hence follows immediately Thomson's absolute method of measuring temperature. (Refer again to § 82.) For it is obvious that the relation between the temperatures of the boiler and the condenser can now be defined in terms of the fraction just spoken of. 95. The terms of the definition are to a certain extent at our option. That which was finally fixed on by Thomson was chosen so as to make as near an agreement as possible IV.] PRELIMINARY SKETCH OF THE SUBJECT. 69 between the new scale and that of the ordinary air- thermometer, and therefore to make the introduction of this method, the only scientific one, produce as small a dislocation of previous conventions as possible. The full reasons for this particular choice will be afterwards explained. Meanwhile we simply state the definition in Thomson's words : — The temperatures of two bodies are proportional to the quantities of heat respectively taken in and given out in localities at one temperature and at the other, respectively, by a material system subjected to a complete cycle of perfectly reversible thermo-dynamic operations, and not allowed to part with or take in heat at any other temperature : or, the absolute values of two temperatures are to one another in the proportion of the heat taken in to the heat rejected in a perfect thermo- dynamic engine, working with a source and refrigerator at the higher and lower of the temperatures respectively." * 96. Suppose we keep a body at the temperature of boiling water, under the condition that the barometer shall be at a height of 30 inches. (See foot-note to § 61.) Suppose we keep another body at the temperature of melting ice, with the barometer at the same height. Suppose we could measure what amount of heat is taken in, and what amount given out, by a perfect engine working between these two temperatures ; these amounts of heat, as will be shown later, would be found nearly in the proportion of 374 to 274. These particular numbers, have been chosen for the terms of the ratio because their diSerence is 160. In the ordinary- centigrade scale we call the freezing temperature zero, and we call the temperature of boiUng water, under the 30 inches of pressure of the atmosphere, 100°. Thus, as we see by the experiment, in the case above mentioned, that » Trans. R. S. E., May 1854. 70 HEAT. [chap. IV. for 374 taken in, 274 are given out, the temperature of boiling water will on this scale be represented by 374°, and that, of freezing water or melting ice by 274°, the range between these being the ordinary 100° of the centigrade thermometer. Hence the curious and very important result, that a body cooled down 274 centigrade degrees below zero is absolutely deprived of heat. This limit, which is commonly called the absolute zero of temperature, is perhaps more correctly called the zero of absolute tem- perature. And it is obvious that the temperature of a body could not be reduced so as to be lower than this. 97. The student is recommended fully to master the brief statement we have now given : for he w 11 then be much better able to understand the relations of the various parts of our great subject than he would have been without such knowledge ; and he will also understand why, in the more detailed study of them which follows, it is constantly necessary to anticipate, by borrowing materials from a later chapter. 98. Rksum'e c/§§ 41 — 97. — Temperature is a mere con- dition of a body as regards Heat. The utmost fraction of a given quantity of Heat which can be converted into useful work depends solely on the temperature of the body with which it is associated, and upon the lowest temperature available. Carnot's inestimable services to science consist in his suggestion of , Cycles ; and especially of Reversible Cycles, for the criterion of a perfect engine. Second Law of Thermo- dynamics, and Absolute measurement of Temperature. CHAPTER V. DILATATION OF SOLIDS. 99. Taking the process explained in § 61 as a mode of defining temperatures between o°and 100° of the Centigrade scale, we proceed to determine the dilatation of various bodies when they are raised to various temperatures within that range. One most important result of this inquiry will be found to be the (approximate) measurement of temperatures by means of these dilatations, so that we shall be enabled to get rid of the theoretically very simple but practically very cumbrous process which we first adopted for defining temperature. The order of simplicity of result would lead us to com- mence with gases such as air and its constituents, but as we cannot experiment on them except when they are enclosed in solid vessels, simplicity of process requires that we commence with solids. The importance of this inquiry from the merely popular point of view consists mainly in the common applications of the results to such matters as the regulation of watches and clocks, and the correction of measuring-rods for changes of temperature. Its pure scientific interest is of a far higher order. 100. A Solid which has the same properties at all points is called homogeneotts, otherwise it is heterogeneous. But even a homogeneous solid has not necessarily the same properties 72 ' HEAT. [chap. in all directions. The grain of wood, of fibrous iron, the crystalline forms and cleavage planes of minerals, the planes of deposition and of cleavage in slates and sandstone rocks, and many analogous phenomena, familiarize us with the idea of solids which are homogeneous but not isotropic. An isotropic body, then, is one from which, if a small sphere were cut, it would be impossible to tell by any operation on it how it originally lay in the solid — it has, in fact, precisely the same properties in all directions. Probably there does not exist any solid absolutely isotropic as regards all its physical properties, but such substances as non-crystalline metals (lead, gold, silver, copper,) and well-annealed glass may be taken as approximately isotropic as well as homo- geneous. We commence our work with a rod or bar of such a material, and we define thus : — The co-efficient of linear dilatation at any temperature is the ratio -frhich the increment of length of the bar when its temperature is raised one degree bears to the original length. loi. As the bar is supposed to be homogeneous, and to be raised to the same temperature throughout, each inch of its length increases by the same amount; so that the whole increase of length is directly proportional to the original length. To obtain a result independent of the dimensions of the specimen chosen, we must therefore take the ratio of the increase to the whole original length : — which is the state- ment made in the definition. The linear dilatation might be put in the more formal terms of a specific property of the substance, by operating on a bar of unit length, and then defining its increase of length for one degree of temperature as the dilatation required. But, practically, no one operates on an exact unit, and therefore we must make our definition such as to be independent of the length of the specimen operated on ; and of course independent of the standard of length. v.] DILATATION OF SOLIDS. 7i 102.. As tliis. work is not designed to teach the details of experiiiQental methodSy which are in all cases much more readily and also more appropriately taught in the laboratory than in the lecture-room or by books, it is sufficient to say that the co-efficient of dilatation of a bar is usually obtained by actual measurement ©f its length,, or of the distance between two marks upon it,, first when it is surrounded by melting ice,, and again when it has been for some time immersed in a bath of water or other suitable liquid at a definite temperature. The temperature of the bath is then al'tered, and another measurement made. The measurements may be made directly with a microscope and screw,, or com- parisons may be made between the lengths of the heated bar and of another kept permanently in melting ice in a second bath placed alongside of the first.. Methods in which one end of ttie bar is clamped, while the other in expanding moves a bent lever or a mirror, are not so trust- worthy as those in which the distance between two. known points of the bar is directly measured. Their principal use, and it is. often a very important one, is in lecture illustra- tions. The principle- on which they depend is very simple. One end of the bar or rod is clamped in. a pillar A ;. the other end,, passing, fireely through a hole in the pillar B,, presses against the back of a plane mirror supported on a horizontal axis at C, perpendicular to. the rod.. (The reader may easily supply the diagram for himself.)) Any advance of the free end of the rod must of course be due to. expan- sion, and its amount can be at once determined by the change of direction of a ray of light reflected from the mirror, for that change is double the angle through which the mirror rotates. And the tangent of the angle through which the mirror rotates is the ratio of the increase of length of the bar to the height of the axis C above it Thus by 74 HEAT. [chap. lowering C the sensibility of the arrangement admits of almost unlimited increase. The annexed cut represents the old form of this arrangement, where a bent lever is used instead of the mirror. A method of extreme delicacy has recently been intro- duced by Fizeau. It depends upon what are called Newton's rings, an optical phenomenon caused by the retardation of light in passing twice through the air-space between a lens and a flat plate of glass. The colour pro- duced varies with the distance of the plate from the lens : — so that if one be fixed, and the other carried on the end of an expanding rod, the changes of colour enable us to measure with great accuracy the expansion of the rod. 103. The general result of such measurements is found to be the very simplest possible : — The ratio of the increase of length to the length at zero is proportional to the rise of temperature. v.] DILATATION OF SOLIDS. 75 Let /, represent the length of an assigned portion of the bar at temperature /"C, then the above experimental result is at once stated in the form ^' 7 '° ^kt (a) where k is the coefficient of proportionality. Hence, if we put /= I, we see by the definition (§ 100) that k is the coefficient of linear dilatation at zero. The value of k is usually so small that to determine it with any accuracy we must use a considerable change of temperature. Taking then the extreme temperatures of our temporary scale — i.e. measuring the length of the same bar, first in melting ice and then in boiling water — we have —J r = 100 k. The coefficient of linear dilatation is therefore the one- hundredth part of the ratio which the increase of length of the -bar between the temperatures o"C and ioo°C bears to the length at o'C. 104. It is rarely found that two specimens even of the same material possess any one specific property to exactly the same amount. This remark applies even to natural crystals, much more therefore to artificial compounds such as glass, &c., so that it is impossible to give any perfectly definite general statements of Dilatation-Coefficients. The following numbers must therefore be taken merely as an illustration of the sort of results arrived at ; quite sufficient as a basis for some general remarks upon the subject, but altogether unfit for any application in delicate calculations or constructions. When an optician (worthy of the name) has discs of flint and crown glass supplied him for the 76. HEAT. [chap. pnrpose of making an achromatic lens, he does not look up tables of results of former experimeiiters to find the re- fractive index, he has to measure with great care for himself this particular constant for the material of each of the discs, and for more than one definite wave-length of light. The constructor of really high-class, electrical apparatus, be it the smallest galvanometer or the longest submarine cable, most carefully chooses his copper by a determination of its specific resistance. And similar statements, are, or ought to be, true in every practical branch of physical science.. Approximate Coefficients- of Linear Dilatation, of Isotropic Substances.. 120,000 120,000 Glass ' -0000085 Platinum -0000085 Steel -000012 1 = 8^ Copper -ooooigr = ^ Zinc -000029 = liSS. 105. The foregoing table, meagre as it purposely has been made, shows two marked results. First, the Coefficient of Linear Dilatation is in general very small; secondly, its values are very considerably different in different solids. From the first result it follows that, unless very delicate operations or very high temperatures are involved, tempe- rature-change of dimensions in solids may be neglected except when we are dealing with great lengths such as the iron girders of a bridge, or the rails on a railway. From the second result, coupled with the experimental law of § 103, it appears that we may by means of combina- tions of two different materials produce compensation, if we v.] DILATATION OF SOLIDS. 77 arrange so that the distance between two assigned points of- a complex system shall be increased bj' the expansion of some of its parts, and diminished to an equal amount by the expansion of others. Thus, to take the simplest con- ceivable arrangement, let us have two similar bars, of zinc and of copper, placed parallel to one another, and fastened at one end to a transverse piece. The condition that the dis- tance between two uprights, fixed to the free ends of these bars, shall remain constant at all ordinary tem- peratures is simply that the actual ex- pansion shall be the same for each bar. But, as the Coefficients for zinc and copper are to one another as 29 to 19, this will be at once secured by making the lengths of the bars as 19 to 29. Thus, by taking advantage of the different expansibilities of different substances, we can construct a measur- ing-rod whose length is independent of temperature. This is, practically, the same arrangement as that of Graham's mercurial Compensation Pen- dulum. A slightly more complex arrangement of the same kind gives the Gridiron Pendtdum of Harrison. But the reader who has mastered the simple case given above can have littie trouble with the more complex cases. HEAT. CHAP, io6. The balance-wheel and its spring perform for a watch or chronometer precisely the same function as the pendulum, assisted by gravity, performs for a clock. But here the application of the compensation process is con- siderably more complicated, because gravity is constant in the clock, while the coefficient of elasticity of the chron- ometer balance-spring depends on its temperature. How- ever, we may for the present consider only the question of compensating for the increased moment of inertia of the balance-wheel due to expansion. [The effect of heat on the balance-spring usually renders (^Z'^r-correction necessary, but the principle is the same.] When a rod, or preferably a flat strip, of wood or metal is bent it is known that the layers on the convex side are v.] DILATATION OF SOLIDS. 79 extended, those on the concave side compressed, relatively to the intermediate ones. Hence it is obvious that such a strip would bend itself if the layers on one side of its mesial plane were to be extended or contracted relatively to those on the other. Thus a strip formed by soldering or brazing together strips of two metals of different expansibilities vfill become curved by change of temperature, the more expan- sible metal being on the convex side when the whole is heated, and on the concave when it is cooled. Now suppose the rim of the balance-wheel to be con- structed of two concentric layers, the outer the more expan- sible. Also let it be cut into separate arcs sufficiently removed from one another to prevent interference. Then it is easy to see that the expansion of the radial bar tends to increase the moment of inertia of the whole, while the in- creased curvature of the arc tends to diminish it. A ribbon of two differently expansible metals similar to that just described was employed by Bre'guet for the purpose of con- structing an exceedingly delicate thermometer. The ribbon was usually coiled into a helix which was fixed vertically 8o HEAT- [chap. in a support .at its upper end, while the lower end carried a light index travelling on a horizontal dia;l. The riibbon is so thin that it almost instantly acquires the same tem- perature as the surrounding medium, and its capacity for heat is so small that it does so without sensibly .affecting the temperature of the medium. 107. One fact of exceeding great use in practice appears in the short table of § 104. The coefficient is nearly the same for platinum as for glass, and farther experiment shows that this is at least approximately true for a very wide range of temperature. In consequence 'Of this a platinum wire may be ^^ fused in " to a glass vessel :: — the glass, melted under the blow-pipe, adhering to the hot wire, and the two materials contracting equally as they are allowed gradually to cool, so that the junction is perfectly air-tight. If the attempt be made with a wire of gold or of any other metal whose melting-point is high, it almost certainly fails however thin be the wire : — either the glass cracks on cooling ; or the wire contracts more than the glass, so that the junction is not air-tight. 108. The facts that platinum is practically infusible in any ordinary furnace, and that its coefficient of linear dilata- tion changes very little through great ranges of temperature, have led to its employment for the rough measurement of high temperatures. Instruments of this rude kind are usually called Pyrometers. The principle on which they work is to keep a record of the length which a bar of platinum had while plunged in the furnace, so that it may be compared accurately with the length of the bar when cold. The bar is put into a hole bored in a piece of graphite or plumbago, whose dilatation is exceedingly small. The bar rests against the bottom of the hole and is kept there by a v.] DILATATION OF SOLIDS. 8i tightly-slipping plug of graphite or baked clay. As the platinum expands by heat it pushes this plug forward, but when it cools it does not draw it back. The principle is exactly that with which every one is familiar in the ordinary maximum thermometer. 109. When a body is aeolotropic, i.e., non-isotropic, its properties are not the same in all directions. But, though the circumstances are now less simple than in the case of isotropic bodies, they are usually by no means very complex. For in general, even in the most complex cases, these properties can be referred to three definite directions at right angles to one another. When these directions are ascertained, and measurements of a particular property (as of coefficients of linear dilatation, with which we are at present engaged) are made parallel to them, the value of the property for any other assigned direction may be calculated by a simple process. [It would require a little more of mathematical reasoning than can well be introduced into an elementary work to show that, whereas the properties of an isotropic body are analogous to those of a sphere, those of an aeolotropic body have the same relation to an ellipsoid. This and what follows we take for granted. An ellipsoid may be regarded as formed from a sphere, by selecting three diameters at right angles to one another, and extending or compressing the whole sphere to different amounts in directions parallel to these three hues. An eUipsoid is known if the direc- tions and lengths of its three principal axes (which are the three lines at right angles to one another) are known. When the lengths of any two of its axes are equal it becomes an ellipsoid of revolution {prolatexi \ht third axis is the longer, oblate if it is the shorter, of the three) ;. and when all three are equal it becomes a sphere. Thus when the amounts of G §2 HEAT. [chap. compression or extension parallel to two of the three diameters are equal, the sphere becomes an ellipsoid of revolution ; when all three amounts are equal, it remains a sphere. Every plane section of an ellipsoid is an ellipse. Every ellipse which has equal conjugate diameters in direc- tions perpendicular to one another is a circle. An ellipsoid which has two plane circular sections such that the line joining their centres is perpendicular to their planes is an ellipsoid of revolution, of which the line just mentioned is the axis. All lines through the centre of an ellipsoid of revolution, and equally inclined to the axis, are equal. If it have been formed from a sphere by one elongation and two equal contractions, or by one contraction and two equal elongations, there will be a set of diameters (all equally in- clined to the axis) which are each unaltered in length.] no. By far the most homogeneous substances we possess are natural crystals. None are absolutely isotropic, for all have planes of more or less perfect cleavage. But one great class of crystals, those of the first or regular system (with a few curious exceptions, probably only apparent), are isotropic as regards light, and are also isotropic as regards heat. Substances so very different as diamond, galena, rock-salt, &c. , belong to this system, and their coefficients of linear dilatation are the same in all directions. Crystals of the prismatic and rhombohedral systems, and in general bodies which have one optic axis as well as one axis of crystalline symmetry, have their greater or lesser coefficient of linear dilatation in the direction of that axis, and are equally expansible in all directions at right angles to that. All other crystalline bodies have three principal dilatation- coefficients, different from one another, and in directions at right angles to one another. v.] DILATATION OF SOLIDS. 83 III. Optical instruments, such as the reflecting goniometer, enable us to measure with very great accuracy the angles between the plane faces of crystals ; and measurements of angle are in general much more exact and useful for such inquiries as those in which we are now engaged than direct measures of length. It was, in fact, almost entirely by measurements of angles that Mitscherlich and others arrived at the results briefly stated in § no. As a simple illustration, suppose a square prism to be cut from an aeolotropic body, so that the diagonals of one of its ends are parallel to the two of the chief axes, and have therefore different coefficients of linear dila- tation. An application of heat will neces- sarily change the square into a rhombus as in the cut, and the ratio of the lengths of the diagonals is very accurately found by measurement of one of the angles. The defect of this method is that it gives the difference of the linear dilatations in the directions of the two diagonals, but not the actual amount of either. For if the diagonals were of unit length at 0° C, their lengths will be i + hi and I -I- kj at fC, and the tangent of half the measured angle of the rhombus is ° , which is practically 1+ {k,- k^t, because /e, and k^ are both exceedingly small. Suppose two such prisms cemented together as below. the edges brought into contact being those whose angle becomes more obtuse by heating. The upper surface will G 2 84 HEAT. [chap. form a continuous plane mirror when the whole is cool, but will consist of two inclined mirrors after heating. An almost infinitesimal difference of expansibility in different directions can be detected by observing with a small tele- scope the image of a distant object formed by reflection at the surfaces of the prisms. What are called twin-crystals are made up of two parts put together in a manner somewhat similar to that just described : and the phenomena are beautifully seen when a polished slab of " arrow-headed " selenite is slightly warmed. 112. By the optical method the differences between the principal dilatation coefficients are very accurately deter- mined. All that remains to be done is to measure one of them directly, by the methods already referred to (§ ro2). Here Fizeau's method, already described, may be used with advantage. When this is done it- is found that in somebodies, notably in Iceland spar, the difference of two dilatation coefficients is greater than either. In other words this substance, which has om axis of symmetry, expands parallel to its axis when heated, but simultaneously contracts equally in all directions perpendicular to the axis. It follows from the explanation in § 109 that there are an infinite number of directions, equally inclined to the axis, in which Iceland spar neither expands nor contracts. As a similar property is found in certain masses of marble, Brewster long ago suggested the use of a cylinder of this substance, cut in a proper direction, as an invariable pendulum. But the apparent anomaly of contraction by heating is not confined to crystalline bodies. We shall presently have to discuss it as shown by water, but a very instructive instance is furnished by india-iubber. v.] DILATATION OF SOLIDS. 85 If the spiral wire be extracted from an ordinary vulcanised india-rubber gas-pipe, and the pipe be then suspended vertically, with a weight attached to its lower end, it con- tracts (in some specimens by five or even ten per cent, of its length) and raises the weight, when steam is blown through it from a little boiler. Thus we have a contrac- tion, easily visible to a large audience, without any of the artificial processes alluded to in § 102 for the exhibition of expansion. 113. The experimental result of § 103 involves, as a consequence, the farther statement : — The coefficient of cubical dilatation is in ail cases the sum of the three chief coefficients of linear dilatation. This is easily seen by supposing a body to be divided into a number of brick-shaped portions, with their edges parallel to the three chief axes. The lengths of the several edges of each brick are increased by heating from 0° to r C in the ratios 1:14--^,/, X : 1 -\- kf, 1:1-1- k^t, and thus we nave for the final, in terms of the original, volume V\=V„^x^kf)(x+kJ){x^kf), or, what comes to the same thing, since the quantities k are always exceedingly small This is precisely the same formula as (a) of § 103, and thus the above statement is verified. When the body is isotropic the values of k are equal, and the coefficient of cubical dilatation is three times that of linear dilatation. Thus, for such bodies, only one 86 HEAT. [CHAP. determination (whichever may be the easier; or the more exact) need be made. In aeolotropic bodies there may be expansion, con- traction, or no change of volume, as a result of rise of ■ temperature. Thus, just as there may be change of volume without change of form, we may have change of form without change of volume. 114. Most of the modes of directly measuring change of volume depend on the use of expansible liquids. Hence, in describing their principle, wc shall suppose first a non- expansible vessel and a non-expansible liquid, and then show how to correct for the expansion of the vessel ; leaving to a later section to show how the expansion of any actual liquid may be measured and allowed for. Let a vessel containing the solid whose cubical dilatation is to be measured be filled up to the end of its (narrow) neck with an inexpansible liquid, and let the whole be weighed. Then let it be raised to a known temperature, the expansion (if any) of the solid will drive out some of - the liquid. Let it cool, and weigh it again. The difference of weight is the weight of the liquid driven out by the expansion of the solid. The correction for the expansion of the vessel, the material of which is usually glass or some other isotropic substance, is, by § 113, where k is its coefficient of linear dilatation. 115. The greater part of what precedes is only approxi- mately true, even for the small range of temperature (0° to 100° C.) to which our statements have been, in the main, confined. v.] DILATATION OF SOLIDS. 87 Measurements carried out at much higher temperatures have shown that the coefficients both of linear and of cubical dilatation are not constant, but increase slowly with rise of temperature. We are not aware of any experiments made with the view of deciding"whether, as is probable, these co- efficients become gradually less as the temperature is lowered below zero. It has been tacitly assumed, in what precedes, that heat- ing has not permanently altered the molecular state of the solid operated on. So long as this is the case the dimen- sions of the body are always practically the same at the same temperature, whether the body is being heated or is cooling. But it is familiar to every one that sudden cooling has often a marked effect on the properties of a body. The process of " annealing,'' as it is called, is devised essentially for the purpose of preventing such physical changes. The whole of this subject, with the exception of a few rules discovered tentatively, is still very obscure. Think, for instance, of the very different conditions of stress, &c., under which the different parts of a Rupert's drop are successively solidified. A recent great practical improve- ment in the manufacture of steel depends essentially upon causing it to solidify under considerable pressure. 116. Rhume of §§ 99 — nS- — The linear dilatation of a solid rod is approximately proportional to the rise of tem- perature. Hence, in isotropic solids, the cubical dilatation (being threefold the linear dilatation) is also proportional to the rise of temperature. Special exceptions, in the case of aeolotropic bodies. Applications to measuring-rods, compensation balances and pendulums, pyrometers, Bre'guet's metallic thermometers. Irregularities due to rapid cooling, &c. CHAPTER VI. DILATATION OF LIQUIDS AND GASES. 117. In fluids this question is much more simple than in solids, for we have cubical dilatajtion only. Taking liquids first, we find that in general their cubical dilatation is, like that of solids, proportional directly to the rise of temperature, but greater. Exceptionally great labour and experimental skill have been devoted to this point, in the case especially of two common liquids : — mercury, because of its importance in thermometers ; and water, because its expansion at ordinary ranges of temperature is anomalous, and because this anomaly is closely concerned with the laws of its circula- tion. We propose to devote several sections to each of these liquids, as they may be considered typical. Meanwhile we may merely remark that what we seek is the real dilatation ; not what is called apparent dilatation, which is affected by the dilatation of the vessel. 118. Let, as in § 114, a glass vessel with a very narrow neck be weighed when full of mercury at 0° C ; then let it be heated to t° C, and weighed again after cooling. Let these weights be W„ JV„ respectively, and let w be the weight of the empty vessel. CH. vi.l DILATATION OF LIQUIDS AND GASES. 89 Then, as the vessel at 0° holds an amount of mercury whose weight is it would hold at temperature i° an amount whose weight is (i + ski) ( W—W), provided mercury were not expansible. It actually holds an amount whose weight is only W, — w. The ratio of these two numbers is therefore that of the densities of mercury at 0° and at /°; and, if .^ be the mean coefficient of cubical dilatation of mercury between 0° and f°, we have (i + Z-/) ( W,-7^) = (I + 3^/) ( W—w), from which A' may at once be calculated. To verify this equation, suppose the cubical dilatation of the liquid to be the same as that of the vessel, z.e. then we have at once showing that the vessel remains just filled at all tem- peratures. 119. This process is a simple and effective one, although it involves the determination of the expansion of the vessel as well as of the liquid ; and it is so because the expansion of mercury is much greater than that of glass. But a very ingenious method of determining the expansion of a liquid, independently of that of the solid in which it is enclosed, was devised by Dulong and Petit, and applied with great skill by Regnault. The principle of the method is merely that of the simple hydrostatic equilibrium of two liquids, one in each of the branches of a U tube : viz., that the heights of the separate columns above the common surface 9° HEAT. [chap. (where the liquids meet) are inversely as their specific gravities. In practice, to avoid direct contact between the hot and cold liquids, an air-space is made to intervene, so that the essential parts of the arrangement are as in the sketch. The tube is now a double U, one half of it being surrounded by a vessel containing ice-cold water, the other by a vessel whose liquid contents may be raised to any desired tem- perature. The columns a /', a' l>' , of mercury are each exposed to the pressure of the atmosphere at their upper ends, and at their lower ends to the pressure of the air in /< b'. Hence, if the differences of level of the surfaces at a and /', and at a' and l>', be measured with a cathetometer, the mercury in each tube has a density inversely as the corresponding measured number. 1 20. Here are a few of Regnault's numbers for mer- cury : — The second column contains numbers derived directly from the experiment described ; the third is cal- culated froni the second : — VI.] DILATATION OF LIQUIDS AND GASES. gi Dilatation of Mercury. "empera- ture C. coefficient of dilataticn from 0°. Coefficient referred to vol. ato". True Coefficient. o° . — 0-0001791 . . 0-000179 5° • o'oooi8o3 0-0001815 . 0-000 1 80 lOO 00001815 o-oooi84i 0-000181 ISO o-ooor828 0-0001866 0-000182 200 0-0001840 0-OOOI89I . 0-000183 250 . 0-0001853 0-0001916 . 0-000184 300 0-0001866 0-0001941 . 0-000185 350 0-0001878 0-0001967 . 0-000186 We have extended the list to temperatures far higher than those already temporarily defined, in fact up to the boiling point of mercury. As given in the table they are measured by the air-thermometer presently to be described. Meanwhile the numbers above give, as the reader may easily verify, the following very simple formula for the usually employed coefficient of cubical dilatation of mercury at any temperature t° C : — A'l = 0-0001791 + 0-00000005;'. Thus if (he 250: — ■ ^j5o = 0-0001791+0 0000125 = 0-0001916, as given in the table. As this formula agrees with all the given numbers, it may safely be employed to calculate iT for any intermediate temperature. It will be observed that the coefficient of dilatation increases steadily with rise of temperature. This is found to be the case with the great majority of liquids, through the whole range of temperature in which we can experiment upon them. But the rate of increase of the coefficient is generally greater as the temperature of the liquid rises towards its boiling point. 92 HEAT. [chap The last column in the table gives the true coefficient of dilatation — ^being (according to the analogy of the defini- tion in § loo) the ratio which the increment of volume for one degree of rise of temperature bears to the origmal volume : — not (as is tabulated in the third column in accord- ance with common usage) to the volume at zero C. With solids, in general, the dilatation is so small that this dis- tinction is of little importance — but a comparison of the two last columns just given, shows that it cannot be neglected for a liquid like mercury. The formula for this true coefficient is very nearly A"", = 0"OOOl79I -|-O'O0OO0OO2/. In this case, and in that of water which follows, we abandon for a time the rule laid down in § 104, and enter into details. The reason is that we are now dealing with a perfectly definite substance, whose properties (under the same conditions, § 9) are the same at all times and in all places ; — not a substance like glass or brass, &c., of which different specimens may differ seriously from one another. 121. The behaviour of water between 0° and 100° C is, very different from that of mercury. From 0° upwards it contracts, more and more slowly, till it reaches its maximinn density almost exactly at 4°C. From 4° upwards it ex- pands, at first very slowly, then faster and faster to 100°. The 'usual class-illustration of this phenomenon is known as Hope's experiment. It depends on the simple hydro- static law that, when a heterogeneous fluid is in stable equilibrium under the action of gravity, the density increases from above downwards. Hope merely applied what is called a freezing mixture (a mixture of common salt with snow or pounded ice is quite VI.] DILATATION OF LIQUIDS AND GASES. 93 sufficient) to the middle of a cylindrical jar full of water. Thermometers were fixed by corks in holes in the side of the jar, so as to indicate the temperatures of the upper and lower stratum of water. Before the freezing mixture is applied, provided the tem- perature of the water is above 4° C, there is usually a slight excess of temperature indicated by the upper thermometer — showing that the warmer water is less dense than the colder. The first effect of the freezing mixture is to reduce the tem- perature shown by the lower thermometer, without perceptibly affecting the higher. This goes on till the lower thermometer reaches 4°C, for then it ceases to descend; very soon after- wards the temperature indicated by the higher thermometer begins to sink in its turn. But it does not stop at 4° C. It shows lower and lower temperatures till the water towards the top of the vessel begins to freeze, and is therefore at 0°. Thus water at 4° C is proved to be denser than at any other tem- perature from 0° to 100°, because it remains persistently at the bottom of the vessel whatever be the temperature of the water above it. Such an experiment, however, is not adapted for the measurement of coefficients of dilatation. Some experi- 9+ HEAT. [chap. menters have for this purpose weighed a lump of glass in water at different temperatures, others have operated by the process indicated in § 119. The following are from Kopp's paper {Foggendorff, 1847) : — the volume of water at 4° being taken as unit, the excess, of any of the other numbers over unit is the whole expansion from 4° to the corresponding temperature : — Dilatation of Water at Ordinary Pressure. 'emp. C. Vol. of Water. Temp. C. Vol. of Waler. 0° 1 '00012 20° . . . . I"ooi69 2 . I '00003 30 . I '00420 3 . . . i-ooooi 40 . . I '00766 4 I '00000 5'3 • . . I '01 189 5 I '0000 1 60 . , i'oi672 6 I 'OOOO3 70 ' . . . I "02237 8 I'OOOII 80 . . 1-02871 10 , I '00025 90. • 1-03553 15 • ■ . 1 -00082 100 . . , 1-04312 The numbers in this table, from 0° to 20°, are represented with a fair degree of approximation by the formula V, = 1 + 144,000 Hence the coefficient of dilatation of water (between these limits) is approximately 72,000. Matthiessen, Pierre, and Hagen have given experimental results, the mean of which tends to show that the denomi- nator of this fraction should be more nearly 68,000. According to Despretz, the volume of water when cooled more than four degrees below its temperature of maximum density increases somewhat faster than if the water be heated to the same number of degrees above that point. Water can be cooled several degrees below 0° C without freezing, provided it be not agitated. VI.] DILATATION OF LIQUIDS AND GASES. 95 It was found by Grassi that the compressibility of water increases, more and more rapidly, as its temperature is lowered towards, and even below, 4° C. It follows from this that, as will be sh6wn later, the maxmium density point is lowered by increase of pressure. The experimental data do not yet enable us to obtain very definite information, but we may say (roughly) that a pressure of 50 atmospheres lowers the maximum density point by i" C. 122. The most expansible of aU known liquids are those which require considerable pressure to keep them in the liquid state. The coefficients of dilatation of sulphurous acid, or carbonic acid, in hermetically sealed tubes, are considerably greater than that of air. The following are some of Thilorier's results for the density of liquid carbonic acid in presence of its vapour at different temperatures : — Temperature. Density. O'go ■ 20C. oC. . . . 0-83 ,+ 30 C. . . . o-6o. Thus the mean e3q)ansion for i'^ between 0° C and 30° C is o 'or 3 nearly. As a rough general rule it may be stated that (at ordinary temperatures) the more volatile a liquid is the greater is its co-efficient of dilatation. Alcohol, however, (whose co- efficient of dilatation is o'ooio5 at o°C.) is considerably more expansible than ether. 123. When we come to consider the expansion of gases, we find a problem quite different from those we have already treated. For a given quantity of a gas cannot be said to have any particular volume, unless we specify the vessel in which it is confined, or the pressure to which it is subjected^ The effects of moderate pressures upon the volumes of 96 HEAT. [CHAP. solids and of the majority of liquids are so small that we have not found it necessary to take them into account while discussing the expansion of such substances. In fact a solid or a liquid will expand very much as we have already described, even if it be confined in an envelop of consider- able strength — simply bursting the envelop if it does not otherwise yield. But, practically, no superior limit has yet been assigned to the volume which a given quantity of air will occupy at ordinary temperatures ; and on the . other hand a strong vessel containing a gas at atmospheric pres- sure would be melted or at least softened before the contained gas had been raised to a sufficient temperature to burst it. It follows, then, that we must measure either the dilata- tion (z>. ratio of increase of volume to original volume) of a gas kept at a constant pressure ; or the ratio of increase of pressure to original pressure, in a gas kept at constant volume ; each for a given rise of temperature. The first rough experiments, due to Charles about 1787, and the subsequent more perfect measures of Gay Lussac, indicated that the quantities just defined are not only the same in any one gas, but have the same common numerical value for all gases, at least for the range from. 0° to 100° C. Though these results are only approximately true, they are sufiiciently exact for many very important applications, both theoretical and practical, and therefore we will devote a section or two to the study of their consequences before Stating the more accurate results of modern experiments. 124. In 1662 was published by the Hon. Robert Boyle the experimental fact now called bdyle's law. The volume of a given mass of gas, kept at a given temperature, is inversely as the pressure. VI.] DILATATION OF LIQUIDS AND GASES. 97 Along with this we may take (in a somewhat extended form) the result stated in last section, which is called C(IARLES' LAW. The volume of a given mass of gas, kept at a constant pressure, increases by a definite fraction of its amount for a given rise of temperature. The two laws together may be stated in the simple form pv = C(i. + at), where C is a constant quantity for any one gas (depending ' in fact upon its quantity and quality), and a is the coefficient of its cubical dilatation. The first really good determination of the value of a was made by Rudberg. For our present purpose we may take a = o"oo3665. 125. Three remarks must, at this stage, be made on the statement of last section. First.- — Note the remarkable fact that the coefficient of dilatation is (practically) the same for all gases. This points to a common simplicity of behaviour, and gives a hint that a gas-thermometer is probably to be preferred to a mercurial one. Second. — Note that the two forms of statement (§§ 123, 124), which we have used in giving Charles' result, are really deducible from one another by the help of Boyle's law. This is obvious from the equation written in last section ; — because / and v are similarly involved in it. We have stated Charles' law as holding for all tempera- tures intermediate to 0° and 100° C. Practically this is true, though it seems to have been tacitly assumed. H gS HEAT. [chap. Third. — If we assume the formula of § 124 to hold for «// temperatures, we arrive at the very important result that p or V must vanish at a particular temperature, which is the same for all gases : i.e. if any gas be cooled to the particular temperature I I / = - - = - -T-r- = — 27^° nearly, a 0-003665 '^ -" which makes the expression i + ai vanish, then either: — it will cease to have any volume : — ox it will cease to press -upon the walls of the containing vessel. Thus we obtain a first rough hint of the position of the absolute zero of temperature, i.e. the temperature of a body altogether deprived of heat. It may be well to recall the student's attention, in passing, to the hint (given in § 96 above) that such an absolute zero does exist, and to state that the above rough approximation gives a fairly satisfactory notion of its position on the ordi- nary centigrade scale. The only lawful methods, however, of treating such questions are those based upon the general laws of Thermodynamics, which apply to all bodies : and not on the properties, simple in their statement though they be, of any particular class of substances. 126. If we would shift our zero-point, which is (§ 61) wholly at our option, to the absolute zero thus determined, we must put / — - for t, and the expression for the laws of a Boyle and Charles (§ 124) becomes pv = Jit, where ^ is a definite constant for each gas. As already stated (§ 123) this is only approximately true, and the divergences from it will be afterwards carefully con- sidered; but the degree of approximation is found to be closest in those gases which are most difficult to condense ; VI.] DILATATION OF. LIQUIDS AND GASES. 99 arid closer for any one gas the higher its temperature is raised, and (with limitations) the more it is rarefied. The Kinetic theory of gases, presently to be described, shows us that the smaller the mutual action of the particles of a gas, when not in contact, in comparison with the action during collision, the more nearly should the above equation be satisfied. Hence we speak of the ideal perfect gas as a substance between whose particles there is no action except at the instants of collision — and in which therefore (as it is proved in the Kinetic theory of gases) the product of the pressure and the volume is strictly propor- tional to the absolute temperature. The experiments of Joule and Thomson show that, under such circumstances, the absolute temperature, as indicated by the ideal perfect gas, is strictly identical with absolute temperature as defined (§ 96) by the help of Carnot's reversible cycle. Hence the a,ir-thermometer, when properly constructed, furnishes a measure of temperature much more in accord- ance with . thermodynamic theory than does the mercurial thermometer. But it is necessary again to observe (as in § 59) that it does not at all matter what instrument is used, provided it be a good one of its kind, and provided its indications can be translated into the corresponding absolute tet/tperatures as defined in § 95. 127. Recurring to the statements of § 123, we may now give an idea of the methods employed by Regnault in his classic investigations. The nature and use of the apparatus .he employed are, in all the more essential features, suffi- ciently indicated by the rudimentary diagram on the .next page, which represents a somewhat simplified form devised by Balfour Stewart. H 2 HEAT. [chap. Two vertical glass tubes, of equal bore (to get rid as far as possible of the effects of capillary forces), are fastened vertically into the lid of a closed vessel full of mercury. The capacity of this vessel can be altered at will by screwing in, or unscrewing, a plug or piston. One of the tubes is open at the top, the other terminates in a glass balloon A, containing the gas to be experi- mented on. This balloon, and as much of the stem as is necessary, are surrounded alternately by melting ice, and by steam from water boiling at i atmosphere (§ 134). When the change of volume at con- stant pressure is to be measured, the plug is screwed in or withdrawn till the mercury stands at the same level in each of the tubes at each of the tem- peratures selected. The pressure of the gas in a is then exactly that of the atmosphere for the time -being, and the volume of the gas in A and its stem must be accurately measured at each temperature. A- correction (small in comparison with the change of volume of the gas) must of course be made for the dilatation of the balloon by heat. When the change of pressure at constant volume is to be determined, the plug is adjusted at each temperature so as to bring the surface of the mercury to a definite point, a, on the tube carrying the balloon : and the difference of levels of the mercury in the two tubes is to be added to the column of mercury in the barometer for the estimation of the pressure of the gas. In addition to' the correction for VI.] DILATATION OF LIQUIDS AND GASES. loi dilatation of the balloon by heat, there is also to be applied a correction (determined by experiment) for its dilatation by increase of pressure. Other methods were employed by Regnault, such for instance as to fill with mercury, and weigh, a glass. vessel with a narrow neck. Thus the capacity of the vessel is ascertained. Then, filhng the vessel with dry air, to heat it to iob° C, to seal it hermetically while the contained air was thus rarefied ; open it when cold under mercury, and weigh the quantity of mercury which entered. But it is suflScient to say that the results of these experiments agreed satisfactorily with those obtained by the first- described apparatus. 128. The following are a few of Regnault's results: — The first column is the ratio of the volume of the gas at 100° to that at 0° C. under atmospheric pressure ; the second the pressure at ioo°C. when the gas is taken at a pressure of I atmosphere at 0° C. and heated without change of volume : — Hydrogen 1-3661 1-3667 Air . 1-3670 I -3665 Nitrogen 1-3670 1-3668 Carbonic Oxide . I -3669 1-3667 Carbonic Acid I -3710 1-3688 Sulphurous Acid . I -3903 1-3845 It is to be observed that, except in the case of Hydrogen, the numbers in the first column are larger than those in the second. Also that for the less easily condensible gases, such as the four first in the table, the numbers are all nearly equal — while for the easily condensible gases they differ widely, and the more so the more easily condensible is the gas. 129. The following numbers, also determined by Regnault, show the ratio in which the pressure is increased at constant I02 HEAT [chap. VI. volume, for the same range of temperature, at initial pressures greater and less than an atmosphere : — Pressure in; Ratio of increase of AtmDspheres at o°. pressure from 0° to 100° C- Air . 0-I444 . . 1-36482 I 'OOOO I -36650 4-8i 1-37091 Carbonic Acid 0-998 1-36856 4722, 1-38598 Thus, for a range of pressures from i to 4-8 atmospheres, the ratio for air changes by about -j^th, while in a range, of but I to 4-7 atmospheres tlie ratio for carbonic acid changes by more than -^^^ For dilatation under constant pressure,, in the same range of temperature, we have — Air . Pressure in Atmosplieres. I 3 '447 Ratio of volumes at 0° and loo" C. 1-36706 I -36964 Hydrogen I 1-36613 3 '349 1-36616 Carbonic Acid I I -37099 3'3i6 1-38455 We will return to this subject when we come to consider Thermodynamic principles as applied to the different states of matter. 130. J?hume o{ §§ 117 — 129. Dilatation of Liquids; especially Mercury and Water. Carbonic Acid. Dilatation of Gases. Laws of Boyle and Charles. Another hint of an Absolute Zero of Temperature. Regnault's results as to change of volume under constant pressure, and change of pressure at constant volume — for different gases, CHAPTER VI I. THERMOMETERS. 131. It seems now certain that the first inventor of the thermometer was Galileo, before 1597 (see Memoire sur la Determination de V Eelielle eiu Therniometre de FAcadeinie del Citiiento, par G. Libri, Ann. de C/iimie, xlv. 1830). His thermometer was an air-thermometer, consisting cf a bulb with a tube dipping into a vessel of liquid. The first use to which it was applied was to ascertain the temperature of the human body. The patient took the bulb in his mouth, and the air, expanding, forced the liquid down the tube, the liquid descending as the temperature of the bulb rose. From the height at which the liquid finally stood in the tube, the physician could judge whether or not the disease was of the nature of a fever. A similar instrument was afterwards used, for a similar purpose, by the physician Sagredo, who, till recently, was regarded as the inventor of the thermometer. Air-thermometers, however, are affected by changes in the pressure of the atmosphere, as well as by changes in the temperature of the enclosed air, and I04 HEAT. [chap. therefore, unless this disturbing cause is removed or ac- counted for, the reading of the thermometer is of no value. Thermometers, containing a liquid hermetically sealed up in glass, were first made under the direction of Rinieri (died 1647), by Giuseppe Moriani, who, for his skill in glass- blowing, was sumamed II Gonfia. Many of the readings recorded by Rinieri are to be found in the Memoirs of the Academy del Cimento, but these were long supposed to have lost their value, as the instruments themselves could not be compared with our present ther- mometric scale. In 1829, however, a numbet of these very thermometers were found by Antinori, and their graduations were com- pared with those of Rdaumur's scale, so that the readings of Rinieri can now be interpreted. ' One of the physical researches for which the Florentine Academy employed these thermometers, was to determine whether the melting of ice always takes place at the same temperature. This question they finally answered affirmatively. The next great step in thermometry was made by Newton, in his Scala graduum Caloris,m the Philosophical Transactions for 1701, where he proposes the melting of ice and the boiling of water as standard temperatures. 132. Fahrenheit of Dantzic, about 17 14, first constructed thermometers of which the graduation was uniform. These thermometers were much used in England, and Fahrenheit's graduation is still the most common in English-speaking countries. In Fahrenheit's scale the temperature of melting ice is marked 32°, and that of boiling water 212°. The Centigrade scale was introduced by Celsius, of Upsala. In it the freezing-point is marked 0°, and the boil- ing point 100°. The obvious simplicity of this mode of VII.] THERMOMETERS. 105 dividing the space between the points of reference has caused it to be very generally adopted, along with the French decimal system of measurement, by scientific men, especially on the Continent of Europe. The scale of Edaumur, in which the freezing-point is marked 0°, and the boiling point 80°, is still used for some medical and domestic purposes on the Continent of Europe. The existence of these three different thermometric scales furnishes an example of the inconvenience of the want of uniformity in systems of measurement. The division of the range from freezing to boiling points into 180°, was probably in imitation of the division of a semicircle into 180 degrees of arc. This arose from the fact that 360 has a great number of divisors. The selection of Fahrenheit's zero probably arose from its being supposed that the ordinary freezing mixture of ice and salt gives the lowest attainable temperature. If we sup- pose the same thermometer to have these three separate scales adjusted to it, or (still better) engraved side by side upon the tube, we easily see how to reduce from one scale to the other. , 3 3 f s IS r 1 00 (1 v no For, if /, c, r be the various readings of one temperature, it is obvious that / — 32 bears the same ratio to (212 — 32 or) 180, that c bears to 100, and r bears to 80. io6 HEAT. [chap. Hence 180 100 80 [So far, all is legitimate and also necessary. But, in this latter half of the 19th century, the progress of science (!) seems to demand that, in Britain at least (for quasi-Chinese examination purposes), " problems " should be set, wherever a tempting opportunity like this presents itself. To prepare the student for such an ordeal, we append a couple of possible specimens, with their solutions. These specimens are not of more than average grotesqueness, and scarcely of average absurdity ; but we cannot devote more space to such things, useful as they may be. Thus, to find the temperature which has the same numerical expression in the Fahrenheit and Centigrade scales, put / for c ot c for f, and we have /- 32 _ / 180 100' so that f — c = — 40°. Again, to find the temperature whose Centigrade reading is the sum of its Fahrenheit and Reaumur readings, take c = f + r, and we have , 180 80 c ^ 32 + c ■\- c, 100 100 whence c = — 20. Also, / = — 4, r -^ — 16. • We do not exaggerate, when we say that practice at rubbish like this may turn the scale in some competitions for desirable appointments.] 133. The glass tubes for liquid thermometers are drawn, not bored, and it is therefore necessary to go through the VII.] THERMOMETERS. 107 process of calibrating them : — first, rouglily, in order to reject all that are not nearly uniform in bore throughout; then more carefully, so as to discover and tabulate the defects of the selected tube. This is a very simple process. It consists in introducing a little column of mercury an inch or two in length ; and, by slightly inclining the tube, causing it to slide from point to point, and measuring its length in each successive posi- tion. The narrower the bore at the place occupied by the mercury, the longer will be the column. These records are taken into account in graduating the tube by the help of a dividing engine. » ^=» The size of the bulb (which, in standard instruments, is usually cylindrical) is then to be estimated by three quantities : — i, the section of the tube ; 2, the expansibility of the liquid to be used ; 3, the desired length of one degree of the scale. It varies of course directly as the first and third of these, and inversely as the second. The bulb, being blown, is gently heated in a spirit-lamp to expel a little air, and the open end of the tube is plunged into the liquid to be used. As the bulb cools, some of the liquid is forced into the stem by atmospheric pressure : and, on erecting the instrument, part of it passes into the bulb. This portion of liquid is now made to boil freely so that its vapour expels the air, and the end of the tube is again suddenly immersed in the liquid. When the whole has cooled, both bulb and stem are filled with the HEAT. [chap. liquid, and tlie excess is expelled by again warming the bulb. The liquid is once more boiled so as to expel the air from the stem, and the end of the tube is then hermetically sealed. Thus, in a properly filled thermometer, when it lies horizontally, the contained liquid is subject to the pressure of its own vapour only. 134. It is always desirable to have the range of a mer- curial thermometer made long enough to include both 0° and 100° C. When this cannot be done, or when instead of mercury there is a liquid which boils at a lower tempera- ture than water, definite points of the scale must be determined bv careful comparison with a standard instru- ment. ^^~^ When the range is long enough, the cardinal points of the scale are determined as follows : — Freezing-point. The bulb and part of the stem of the instrument are immersed, as in the sketch, in pounded ice, from which water is dripping away, and left for some time, VII.] THERMOMETERS. 109 till the upper surface of the mercury, which is just visible above the ice, is observed to take a definite position. This is carefully marked on the glass tube as the zero-point ; or, if the graduation has been previously eflfected according to an arbitrary scale (taking account of the calibration § 133), the zero-point is carefully observed and registered. Boiling point. The instrument is now to be immersed, as carefully as possible, in steam freely escaping from water boiling under a pressure of i atmosphere. It is found that the temperature of boiUng water is not steady, but that the temperature of the escaping steam is so. The annexed cut shows fully the arrangement usually adopted, and re- quires no explanation ; but we must show how to define accurately what is meant by a pressure of i atmosphere. (See § 61, footnote.) The pressure at a given depth under the surface in the column of mercury of a well-filled barometer (whose tube is wide enough to prevent any sensible effect due to capillarity) depends solely upon two things — the density of the mercury, and the intensity of the force of gravity. For a short column of mercury we may neglect the com- pression due to its own weight, so that the density (if the mercury be pure) depends practically upon the temperature alone. Hence, in defining barometric pressure, we must reduce the column to its length at 0° C. (See § 120.) The intensity of the force of gravity depends upon the latitude, and upon the height above the mean sea-level. Hence, both of these must be taken into account in defining an *' atmosphere." The definition, agreed on in this country, is the pressure when the barometer, reduced to 0° C, stands at 29'905 inches at the sea-level, in the latitude of London. The change of pressure, per degree of temperature of the HEAT. [chap. VII.] THERMOMETERS. Ill boiling-point of water, in the neighbourhood of ioo° C, is about 1-071 inch. It is rather less for temperatures lower than 100° C, and rather more for temperatures above that point. But the single number just given suffices for the practical determination of boiling-points near the sea-level. The further discussion of this question must be deferred till we are dealing with Changes of Moleadar State due to heat. (See below, § 165.) 135. The mercurial thermometer is useful through a range of temperature from a little above the melting point ( — 40° C.) of mercury, to a few degrees below its boiling point (350° C). When we desire to estimate temperatures about or below — 40° C, alcohol is the liquid generally em- ployed, as its freezing-point is about — 130^0. For tempera- tures much over 300° C. recourse must be had to an air-ther- mometer ; or, for rough purposes, to Pyrometers (§ 108), or Thermo-electric processes. When a thermometer of great sensitiveness is required for use at ordinary temperatures, carbonic acid is obviously (§ 122) a suitable liquid. The contrivances for what are called maximum or mini- mum thermometers — i.e. instruments to record the highest or lowest temperature during a given time — are practically innumerable. The simplest, and most common, depend upon capillary forces. A column of mercury, pushing before it a little iron index (which it does not wet), is usually em- ployed in a maximum thermometer ; — a column of alcohol 112 HEAT. [chap. pulling back with it a little index of enamel (which it thoroughly wets), is usually employed for a minimum instru- ment. The force in either case is due to the tendency of the surface-film of the liquid to preserve the smallest area possible under the conditions to which it is subject Other principles are employed, such as for instance a maximum thermometer in which the mercury has to pass through a very narrow part of the tube near the bulb. While ex- panding, it passes freely : but as soon as it begins to con- tract the column breaks at the narrow neck, and remains in the tube as a record of the highest temperature reached. 136. But all these devices are excessively imperfect, at least in comparison with continuous registration. This must, of course, be automatic. Many ingenious devices have been introduced for the purpose, but they are entirely super- seded by the introduction of photography. The photo- graphic process is so simple, and valuable, and at the same time so instructive, that we will devote a section or two to an explanation of it, and of some of its elementary consequences. It is appUcable of course, not to temperature alone, but to every variable quantity which can be suitably indicated by an instrument — -such as (for instance) the barometer, the electrometer, or the variation compass. Suppose the stem of a mercurial thermometer to be placed close in front of a narrow slit, so that light can pass only where the slit is not blocked by the column of mercury. Then, if a gas flame be placed in front of the thermometer, and a sheet of photographic paper be drawn uniformly along behind the slit, a permanent record of the successive positions occupied by the mercury will be formed by the •boundary between the blackened and the unaltered parts of the paper. The clockwork by which the paper is drawn VII.] THERMOMETERS. ,IJ3 along is easily made to mark it at every hour or minute, and the graduations of the thermometer, or some known marks made on its stem, may easily be recorded photographically. When such a sheet has been taken off, developed, and fixed, it presents an appearance somewhat like the sketch below : — ■ To find then what was the temperature at any instant, say 4''.4i'", all we have to do is to draw (as in the figure) the line D E which at that time must have coincided with the slit. The point E, where this line meets the boundary of the blackened part of the paper, indicates where the end of the mercury column was at the given instant. In the figure, as drawn, the temperature is 5°'2. 137. If we look at the diagram above, we see that the portion of the curve drawn obviously contains two maxima and three minima ; and we see that the character- istic of either kind of point is simply that the tangent to the curve shall at that point be horizontal, and shall not pass through the cume. At a maximum, such as A, the tangent is horizontal, and (near ^) wholly in the blackened space above the curve. Near a minimum, as B,. it is wholly in the white space below the curve. I 114 HEAT. [chap. But a point such as C in the sketch is obyiously neither a maximum nor a minimum, although the tangent is hori- zontal. At one side of C it is above, at the other below, the curve. Another way in which this may be put, and which is in fact quite obvious from the behaviour of the thermometer itself, is the statement that between two equal values of any varying quantity there always necessarily lies a maximum or a minimum. Therefore the nearer together (in point of time) are these equal values, the more nearly does either coincide (in time, and also in magnitude) with the maximum or minimum sought. [This simple consideration enables us to solve many problems which are usually thought to require the Differ- ential Calculus, or even the Calculus of Variations, for their proper treatment. Thus it suffices not merely for the in- vestigation of the law of refraction from the assumption that light takes the least time possible to pass from one point to another, but also for the investigation of brachisto- chrones, or lines of swiftest descent.] 138. The general principle on which air-th;rmometers are constructed, is easily apprehended from the sketch of Balfour Stewart's apparatus in § 127. If a metal ball be em- ployed, as in fact must be done if very high temperatures are to be measured, the gas must be preserved at atmospheric pressure, else there is danger of diffusion through the metal. Even in the case of a glass ball the constant pressure is desirable, not however on account of the permeability of the glass, but because of the alteration of its volume by in- creased pressure from within. There is not, as yet, any one particular form of air-thermometer which is in .general use; all the greater- experimenters having devised special forms for the special inquiries they had on hand. VII.] THERMOMETERS. ii; The Differential Thermometer of Leslie and Rumford is really a couple of air-therttiometers working against one" another. In its simplest form it consists merely of a tube . with a ball blown at each end. These balls contain air," and the contents of the two are kept separate by a column of sulphuric acid, whose motions indicate differences of pressure, and therefore of temperature, in the ttvo bulbs. This instrument is obviously unaffected by changes of baro-' metric pressure ; it can be made very sensitive, and it is therefore of great use. Leslie made all his experiments on radiation of heat with this instrument ; and, by varying the materials of which the balls (or one of them) were' made, he converted it into a hygrometer, a photometer, an sethrioscope, &c. ii6 HEAT. [chap. 139. The Mean or Average temperature during any assigned period can be obtained very accurately from the photographic record above described. Another method is to use a clock whose pendulum is not compensated (§105) ; and, from the gain or loss of time which it shows as com- pared with the normal mean time clock, to calculate the average length of the pendulum, and thence the average temperature. Another method, originally suggested by Moseley's obser- vations on the way in which, in consequence of alterations of temperature, sheet lead gradually tears itself off from a sloping roof, is that of T. Stevenson's Creeper. It is simply a flat bar of an expansible metal, provided with equidistant rows of teeth along its upper and lower ends, and made to rest on an inclined slab of slightly expansible material, in which horizontal grooves are cut as close to one another as possible. The teeth and grooves are so shaped as to offer very slight resistance to motion upwards, while preventing all sliding down. When the instrument is raised in teinpera- ture it expand-s ; and, as the lower end cannot move down- wards, the upper end must rise, so that its teeth reach a VII.] THERMOMETERS. 1 17 higher groove. On the other hand, when it contracts by cooling, the lower teeth must be drawn up into a higher groove, because the upper end cannot slide down. It is obvious from this description that the " creeper " gives a record depending mainly upon the number and extent of the fluctuations of temperature. 'How it acquires the potential energy, which it has in its elevated position, is a question to be treated later. 140. Rtsum& of §§ 131 — 139. Invention of the Ther- mometer. Newton's Fixed points. Scales of Fahrenheit, Celsius, and Rdaumur. Calibration, Graduation, and Filling of Liquid Thermometers, determination of Fixed points. Air Thermometer. Definition of " an Atmosphere." Ranges of different Thermometers. Maximum and Mini- mum Thermometers. Photographic Registering Thermo- meters. Properties of a Maximum, of a Minimum, and of a Maximum-Minimum. Differential Thermometer, Aver- aging Thermometers. The Creeper. CHAPTER VIII. CHANGE OF MOLECULAR STATE. MELTING AND SOLIDIFICATION. 141. In sections 45 and 48 above we have given a general sketch of the part of the subject to which tliis cliaptef is devoted. The student is recommended to re-read these sections before proceeding further. Wlien a similar recom- rnendation has to be made, it will be put simply' in the form "Refer again to §§— , — ." 142. Melting. In accordance with the principle of § 9, and the fundamental principle of § i, it is found that pressure and temperature alone require to be taken into account in the discussion of the melting of any definite solid. Hence the experimental law : — The pressure remaining the same, there is a definite melting- point for every solid ; and (provided the mass be stirred) however much heat be slowly applied, the temperature of the whole remains at the melting-point till the last particle is melted. This is one of the bases of Black's doctrine of Latent Heat. Our modern knowledge that heat is not matter leads us to regard the energy which escapes detection by the thermometer as being employed in tearing asunder the particles of the solid. This will not appear very startling CH. VIII.] CHANGE OF MOLECULAR STATE. 119 if we think Of the work required to reduce a mass to fine powder, every particle of which is still a portion of the solid. But the first clause of the statement leads to the important question of the influence of pressure upon the melting-point. This was first discovered by James Thomson, in 1849, ^nd ^'s calculations with regard to the lowering of the melting-point of ice by pressure were exactly verified experimentally by Sir W. Thomson in the same year. Hopkins, Bunsen, and others, have experimentally de- terminated the elevation by pressure of the melting-points of substances which expand on becoming liquid. 143. As an instance showing the nature of the experi- mental bases on which this statement rests, we refer again to the process of determining the zero-point of the thermo- metfer (§ 134). There nothing is s.iid about the rate at which the ice is melting : — i.e. about the quantity of heat supplied in a given time.' See also § 131. Water and mercury can be procured in very great purity, but the same statement cannot be made about the majority of other substances. Hence we give (as in § 104) only a short table of examples of approximate Melting-Points. Temperature C. Ice . . ; 0° Mercury — 40 Sulphur Ill Lead 335 Wraujht Iron 15°° P) Some metallic alloys, especially those containing bismuth (which have received the general name of " fusible metal "), melt at temperatures considerably under 100° C. The recently discovered metal, gallium, melts at about 30° C. On the other hand, platinum cannot be fused in 120 HEAT. [chap. any ordinary furnace ; and gas-coke (a form of carbon) has been softened only, not melted, by the most intense heat yet produced artificially — that of the electric arc. 144. As already stated, the effect of pressure upon the melting-point of a body was deduced from theory, and subsequently verified by experiment. The reason of its having previously escaped experimenters is probably to be found in the extremely small amount of the effect even when great pressures are applied. This is not the place to enter upon the theory, which will be discussed later ; but we may mention the theoretical result in the form that Bodies which contract in the act of melting have their melting-points lowered by increase of pressure, and vice versi. 145. The theoretical result for ice, exactly verified by experiment, is a lowering of the melting-point by o°-oo75 C. for each additional atmosphere (§ 134) of pressure. This may be roughly stated in the form that, under a pressure of one ton weight per square inch, ice melts at one degree Centigrade under its ordinary melting-point. [Along with this we have to remark that the density of ice is only about 0-92 of that of water, so that water-sub- stance contracts by eight per cent, in the act of melting.] Many of the consequences of this important fact were familiar to all before the fact itself was pointed out. One form in which it must have been well known for hundreds of years is the form in which we try the same experiment every time we make a snowball. Schoolboys know well that after a very frosty night the snow will not " make " : — their hands cannot apply sufficient pressure. But, if the snow be held long enough in the hands to be warmed nearly to its melting-point, it recovers the power of VIII.] CHANGE OF MOLECULAR STATE. 121 "making," or rather of "being made." Eyery time we see a wheel-track in snow we see the snow is crushed, and even after one loaded cart has passed over it, — certainly after two or three have passed, — the snow has been crushed into clear transparent ice. The same thing takes place by degrees after people enough have walked over a snow-covered pavement; and in all these cases this minute lowering of the freezing-point has led to the result. And now we see how it is that the enormous mass of a glacier moves slowly on like a viscous body, because in consequence of this most extraordinary property it behaves under great pressure precisely as if it were a viscous body. The pressure down the mass of a glacier must of course be very great, and as the mass is — especially in summer — freely percolated throughout by water, its, temperature can never (except on special occasions, and then near the free surface) fall notably below the freezing-point. Now, in the motion of the mass on its journey, there will be every instant places at which the pressure is greatest, — where in fact a viscous body, if it were placed in the position of the glacier ice, would give way. The ice, however, has no such power of yielding ; but it has what produces quite a similar result — wherever there is concentration of pressure at one particular place it melts, and as water occupies less bulk than the ice from which it is formed, there is immediate relief, and the pressure is handed on to some other place or part of the mass. The water is thus relieved from the pressure by the yielding caused by its own diminution of bulk on melting. The pressure is handed on; but the water remains still colder than the freezing-point, and therefore instantly becomes ice again. The only effect is that the glacier is melted for an instant at the place where there is the greatest pressure, and gives way there precisely as a 122 HEAT, [chap, viscous body would have done. But the instant it has given way and shifted off the pressure from itself it becomes ice again, and that process goes on continually throughout the whole mass ; and thus it behaves, though for special reasons of its own, precisely as a viscous fluid would do under the same external circumstances. The first who seems to have realised this on a small scale by experiment was Dollfuss-Ausset, who showed that by compressing a number of fragments of ice in a Bramali press, it was possible to melt them ; and when pressure was taken off them, to allow them to revert again into a solid block. But he found that withi very cold ice the experi- ment did not succeed.. Iji fact, as. we now see, even with his Bramah press, he couldi not apply pressure enough. By opening a holein the end. of a cylinder in which snow is compressed by a Bramah press, we obtain a cylinder or wire of solid ice gradually squeezed; through the hole, just as wires are made of soft or crystalline' metals. The mechanism of the process is different,, but. the results are exactly the same. Another simple but effective experiment may be made by passing a loop of wire round a bar of ice supported hori- zontally. A weight attached to the wire pulls it gradually through the ice, which melts before the wire and is imme- diately re-formed behind it : so that the wire passes entirely through (as in cutting cheese or soap) and yet leaves the bar as strong as ever. 146. Every one knows that when melted bees-wax partially solidifies, the crust if broken sinks in the liquid. The same phenomenon is observed in lava-lakes such as that of Hawaii, where the crust '' cracks in different directions, and first one half of the lake and then the other is covered with yiii.] CHANGE OF MOLECULAR STATE. 123 a fresh coating of red-hot lava, the crust tumbhng out of sight as it shrunk and cracked in coohng." * Just as the effect of pressure on the melting-point of ice enables us to account for the plasticity of glacier ice, so the effects now described enable us to explain with great pro- bability why it is that the materials forming the interior of the earth are practically rigid, and therefore solid, though at temperatures far above their ordinary melting-points. For, as will be seen when we consider (under the head of Conduttiori) underground temperatures, there can be little doubt that the temperature at a few hundred miles below the earth's surface must be very high— certainly far above the usual melting-point of lava. Yet, as Sir W. Thomson ■has shown from the amount of the tides, there can be no doubt that, as a whole, the earth is nearly as rigid as a globe of steel of the same size. These two apparently inconsistent conditions are at once reconciled, if we suppose the average materials of the earth to be such as, like lava, to expand on melting : — for the immense pressure to which they are subjected from super- incumbent strata is probably sufficient to raise their melting- point above their present temperature. Thus they may remain solid, even at a white- heat. But if by the cooling and shrinking of the lower strata within the solid crust this pressure should be anywhere considerably relieved, the mass affected would (almost explosively) melt with considerable expansion. This seems to have important bearings on earthquakes and upheavals. [It may be mentioned, in passing, that if the earth were liquid throughout, and of uniform density equal to its pre- sent mean density, the pressure within a few hundred miles * J. W. Nichol, Prcc. R, S. E., 1875-6; p. 117. 124 HEAT. [chap. of the surface would increase at the rate of somewhere about 800 atmospheres (or nearly s| tons weight per square inch) per mile of depth.] Hopkins has found that the melting-point of wax is raised about 10° C. by 500 atmospheres' pressure ; and Bunsen gives a rise of 3°"S C. as the effect of 100 atmospheres on the melting-point of paraffin. 147, We must now consider the second clause of the experimental statement of § 142. That the temperature of the mixture of solid and liquid should remain at the melting-point, however much heat be supplied, till the last particle is melted, was explained by Black (on the hypothesis of the materiality of heat) by the supposition that the liquid differs from the solid simply by having taken into combination a certain proportion of caloric. Thus the liquid was regarded as a species of chemical combination of the solid with an equivalent of caloric. As no effect was produced on the temperature of the body by this admixture, Black introduced for this supposed equivalent of caloric the name of Latent Heat. Relatively to the knowledge of his time, the name was a felicitous one, because it was found that when a liquid solidified it gave out exactly as much, heat as it had taken in on melting. Unfortunately, from our modern point of view, the term is not a felicitous one — and yet it is so ingrained in our language (like vacuum and centrifugal force) that we are not at all likely soon to get rid of it. But no difficulty will be found if we keep in mind that when we speak of latent heat we mean no more than this — that a certain amount of heat is in every case required to change the molecular state of a substance even when there is no alteration of its temperature. VIII.] CHANGE OF MOLECULAR STATE. 135 148. We are, as yet, almost wholly ignorant of the form in which energy exists in bodies generally ; and a great deal of mischief has been done (almost ever since the time of Black) by the assumption that' bodies must possess a certain amount of what has been called " sensible,'' or " thermo- metric " heat. Until we know, at least partially, the nature of the internal mechanism connecting the particles of matter, it is altogether vain to discuss questions concerning the heat present in a body, farther than is involved in estimating the amounts of energy supplied to it, and given out by it, in the various stages of an operation. Refer again to § 74. 149. But if we consider for a moment the amount of work necessary to grind to fine powder even the most friable of solids, and, going farther, think what almost infinitely more perfect breaking up befalls such sohds when they are melted, it is easy to see how a great ijart of the energy, supplied to a solid in the form of heat, must be applied to the mere mechanical work of pulling its particles into the position of comparatively slight constraint which they occupy in the liquid, against the molecular forces which originally main- tained them in the solid form. 150. With these explanations, we retain the term Latent Heat as a general one — applicable in all cases of change of molecular state — where energy in the heat form can be supplied to a body without producing alteration of its temperature. [The student should carefully notice the words in italics, for the same amount of energy can be given to the ice in many other ways, without either melting it or changing its temperature — e.g. by lifting it to the proper height against gravity, or by projecting it with the proper velocity.] Thus the latent heat of water is to be numerically measured as the number of units of heat i^ ^^'^ which must 126 HEAT. [CHAP. be communicated to a pound Of ice at cP C. to convert it into a pound of water ato' C. This is (§ 9) evidently a definite quantity. For the temperature mentioned in the definition requires that the pressure shall be i atmosphere. 151. The experimental determination of this quantity can be made in many ways : but all the usual ones depend upon the direct comparison of the effects of equal amounts of heat upon ice and upon water. The ordinary lecture-room form of the experiment con- sists in pouring over a pound of ice at 0° C. a pound of water at about 80° C., and carefully stirring the mixture. When proper precautions against loss of every kind are taken, it is found that under these circumstances the whole of the ice is just melted — the resulting temperature being still 0° C. Assuming, what is very nearly the case, that the amount of heat required to raise the temperature" of a pound of water one degree is the same throughout the whole range from o" to 80° C, it is obvious that the pound of hot water has lost 80 units of heat, which must have gone to the pound of ice with the result of melting it without raising its tempera- ture. Hence it follows that (in accordance with the defini- tion of latent heat, § 150), 80 is the latent heat of water. It appears from the experiments of Person and others that the correct value is more nearly 79"2S. The experimental part of the work presents no very great difficulty, but the mode of reasoning from the results depends to some extent upon the opinion we may form as to whether the transition from ice to water is an abrupt or a gradual process. 152, There can be no doubt that, in many substances at least, the transition from solid to liquid is gradual and not abrupt* Every one is familiar witli the softening of wax and VIII.] CHANGE OF MOLECULAR STATE. 127 paraffin, as they are gradually raised-in temperature. The welding of iron and of platinum, at high temperatures, is another case in point. And several excellent authorities state their conviction that something analogous occurs with ice. What farther we have to say on this question will be profitably deferred till we discuss the whole subject on thermodynamic principles. It is clear that a rapid increase of specific heat, just below the melting-point, might, if un- detected, lead to an over-estimate of the latent heat as we have defined it. But it is also clear that this explanation can only be valid if ice, from which water is trickling, is (all but its superficial layer) essentially a little colder than 0° C. Forbes and Balfour Stewart slate that the temperature of a mass of ice, which has been rapidly pounded, is invariably found to be a little below the freezing-point. 153. Water is almost exceptionally high among liquids as regards the amount of its latent heat. The following short table gives some general notions on the subject : — • Latent Heat of Fusion. Water 79'2S Phosphate of Soda (crystjllised) .... 67*0 Zinc 28'! Sulphur 9 '4 Lead 5'4 Mercury 2 '8 It is to be observed that the numbers here given denote the units of heat required just to melt one pound of each of these substances without change of temperature. 154. We have already alluded to the abrupt changes of volume which usually take place on melting. In genera], the change is of the nature of expansion ; but water, cast- ircn, type-metal, and some other bodies, contract.- As the' 128 HEAT. [chap. process of solidification is exactly the converse of that of melting, it is accompanied by disengagement of latent heat and return to the former volume. Hence the nicety with which iron and type-metal adjust themselves to every little crevice in a mould. Hence also the bursting of water-pipes during (not, as the majority even of "educated" people still think, after) a frost. The pressure requisite to prevent water from freezing in a closed vessel, nearly full, is enor- mous, provided the temperature be a few degrees under the usual freezing-point : sufficient, as has been found by trial, to burst the strongest bomb-shells. The amount of work which can be done by a pound of water, in freezing under given circumstances, can be at least approximately calcu- lated, as we will show when treating of the dynamical theory. In fact, while discussing the lowering of the freezing-point by pressure, we have pointed out (§ 145) that a pressure of a ton weight per square inch is re- quisite to prevent ice from being formed at a temperature even one degree under zero. vui.] CHANGE OF MOLECULAR STATE. 129 155. The process of solution of a solid in a liquid is, in so far at least as it is independent of chemical action, very closely analogous to melting; and, as in § 149, must obvi- ously require a supply of energy. This is usually taken from the body itself and from the menstruum, as well as from surrounding bodies, in the form of heat. Thus solution of a solid in a liquid is usually accom- panied by cooling. This is, in some cases, partially or wholly masked, sometimes even overcome, by the heat developed by chemical action. One of the commonest of these arrangements for pro- ducing cold {i.e. rendering heat latent), is to dissolve nitrate of ammonia in an equal weight of water. If both be taken at ordinary temperatures (say 10° C, or 50° F.), the tempera- ture of the solution falls to about — ^r5° C. A somewhat greater cooling is produced by pouring commercial hydro- chloric acid upon snov/. But by far the lowest temperatures yet attained by such means are procured by pouring ether upon solid carbonic acid and other bodies which are gaseous at ordinary tem- peratures and pressures. These temperatures are further considerably lowered by placing the mixture under a receiver, and exhausting the air. Natterer estimates the lowest tem- perature he has thus obtained at about — 140° C, more than half-way from freezing-point to absolute zero (§§ 96, 125). Perhaps even more striking results may yet be obtained from solid hydrogen. 156. When two solids, on being mixed as crystals or powder, melt one another, we have of course to supply from without the latent heat for each, except in so far as heat is chemically developed by the combination. Thus we explain the action of the well-known freezing mixture of snow and common salt, or of salt and pounded ice (§ 132). K J 3° HEAT. [chap. viti. If the salt has been previously cooled to o° C, the tempera- ture of the mixture falls to about — 20° C. 157. Conversely, when a substance in solution crystallises out, we have in general a development of heat. This is very well shown by supersaturated solutions of sulphate or acetate of soda, which suddenly crystallise when the smallest fragment of the solid is dropped in. In the acetate of soda the water is almost entirely taken up by the solid as water of crystallisation. 158. Hhum'eoi^^ 141-157. — Law of Melting. Effect of Pressure. Melting-points of some Solids. Behaviour of Ice and Lava. Consequences as regards Glaciers and the Strata under the Earth's Crust. Latent Heat of Fusion. Latent Heat in different Liquids. Solution. Freezing Mixtures. Heat developed in Solidification. CHAPTER IX. CHANGE OF MOLECULAR STATE. VAPORISATION AND CONDENSATION. 159. Refer again to §§ 46, 47. We have seen (§ 152) that it is not yet settled whether the change from the solid to the liquid state takes place precisely at the definite tem- perature called the melting-point or not. There is no doubt, however, that the transition from the liquid (and, in some cases at least, from the solid) state to the state of vapour takes place at all temperatures, but more freely the higher the temperature, up to the so-called boiling-point. Every one must have noticed that even in the coldest winter day — especially if it be windy — the ice on the pave- ments gradually dwindles away, though constantly appear- ing solid and hard. It is, in fact, always evaporating though very slowly. Whether it passes or not through the liquid state in a thin film on the surface is not yet known. And distillation of liquids is often practised on a large scale, in manufacturing operations, at temperatures kept, purposely below their boiling-points. Salt-pans form an excellent example. All but a very small fraction of the vapour con- stantly in the air, has been raised from the surface of oceans, lakes, or moist ground, at temperatures far beloTV the boiling-point, K 2 132 HEAT. [chap. 1 60. Boiling. Experiment enables us to state for this phenomenon a law precisely similar to that of § 142, viz. — The pressure remaining the same, there is a definite boiling- point for the free surface of every liquid ; a7id {provided the mass be stirred) however much heat be applied, the tempera- ture of the whole remains at the boiling-point till the last particle is evaporated. The effect of pressure upon the boiling-point can be cal- culated, as was that of pressure upon the melting-point ; but as we do not know of a substance which (at the same temperature and pressure) occupies less bulk in the form of vapour than in that of liquid, we may assert that the effect of pressure is in all cases to raise the boiling-point. So far as water is concerned, we have already (§ 60) given -some explanations and experimental data on this subject. We will now treat the whole experimental law more fully, and in a manner similar to that which we adopted for the a:nalogous law of melting. 161. We take water vapour as a type^ — as it is the most important, and has been therefore the most carefully studied. The first even approximately accurate statement of its ■behaviour is due to Dalton. Numerical data of great pre- cision, and extending through a wide range of temperatures and pressures, have recently been furnished by Regnault. And an experimental result of startling novelty, and of the greatest theoretical importance, due to Andrews, revealed, so lately as 1869, what is the true distinction between a vapour and a gas. 162. If a vessel of water be placed in an exhausted receiver, evaporation will immediately commence, and the process will go on with great rapidity until the pressure of the vapour in the receiver rises to a certain definite amount, which is found to depend solely upon the temperature. IX.] VAPORISATION AND CONDENSATION. 133 If the receiver is fitted with a piston, which is pushed in or pulled out so as greatly to increase or to diminish its volume, still the pressure of the vapour remains unchanged j 134 HEAT. [chap. for fresh vapour is formed when the volume is increased, and part is condensed into Hquid water when the volume is diminished. This is easily effected in practice by intro- ducing a drop of water into a tube filled with mercury, as for the Torricellian experiment. When the tube is inverted in a vessel of mercury, there is a definite amount of de- pression of the mercury column (as compared with the barometer), which is not altered by plunging the tube deeper into the mercury. [This is exhibited in the first of the above cuts, where a shows the level of the mercury in the barometer tube (with the Torricellian vacuum above it), while b is that in the tube with a drop of water inserted. The second of the above cuts represents the simultaneous behaviour in three barometer tubes of different lengths, into each of which a drop of water has been introduced.] The same result is ultimately arrived at, but not so rapidly, if the receiver contain air. The only effect of the air is to retard the production of vapour, but the process goes on as before until the pressure of vapour is that due to the particular temperature of the water— and the denser the air the greater the retardation. 163. Vapour, which is in equilibrium in contact with excess of liquid, is called saturated vapour, and Dalton's statements may be put in the form — The ultimate pressure of the saturated vapour of any liquid depends only upon the temperature. [N.B. — The student should remark that in many old books, and in too many modern ones, the word tension is improperly used in this connection instead of pressure. Chemists, especially, have been led to sanction this blunder, and almost invariably speak of vapour-tension. J 164. While a liquid is thus in equilibrium with its own vapour, suppose the vapour to be removed by means of an IX.] VAPORISATION AND CONDENSATION. 135 air-pump almost as fast as it is formed. Then, obviously, fresh vapour is furnished very rapidly from the liquid : and it is observed to come not merely from its surface, but in bubbles from the interior of the mass and specially from sharp points or edges of solids immersed in it. This free and rapid discharge of vapour is what is called boiling. And we thus see that — The boiling-point of a liquid is the temperature at which its saturated vapour has a pressure equal to that io which the Jree surface of the liquid is subjected. Pressure of Saturated Water Vapour — Regnault. erature C. Pressare in Atmo- Temperature C. Pressure in Atmo spheres. spheres. 0° . 0006 120° . 1-962 10 0-0I2 130 2-671 20 . . 0-023 140 . 3-576. 30 0042 15a 4-712 40 . . 0-072 160 . 6-120 5° • o-i2r 170 7-844 60 o'lgft 180 . 9-929 70 . . 0-306 190 12-425 80 0-466 200 15-380 90 . . . 0-691 210 18-848 TOO . I -000 220 . . . 22-882 no . 1-415 230 ^7 '535 165. On this very instructive table (given by Regnault at p. 728 of his magnificent Relation des Experiences, etc., Mem. de VAc, des Sciences, 1847,) a few remarks may be made. ' {a.) We see how very rapidly the pressure of saturated water-vapour rises as the temperature is raised. Thus the rise of temperature from 100° to 180" increases the pressure from one to nearly ten atmospheres, while an additional rise of only 40° C. raises the pressure to nearly twenty-three atmospheres. Compare this with the behaviour of a gas such as air 136 HEAT. [chap. (§ 124), where the pressure (at constant volume) rises ap- proxinnately in proportion to the absolute temperature ; and we see how very much- less dangerous, so far as the chances of an explosion are concerned, is an air-engine than a steam-engine working (with saturated steam) at the same high temperature. {b.) Water at ordinary temperatures may be made to boil by placing it in the receiver of an air-pump and producing a sufficient vacuum. Thus if it be taken at 10° C. (50° F.) it will boil when the pressure is reduced to o'oi2 of an atmosphere (or, more accurately, to 9'i6 millimetres of mercury). And a notable feature of this experiment is that the water is found to become gradually colder as the boiling proceeds, thus requiring farther exhaustion by the air-pump IX.] VAPORISATION AND CONDENSATION. 137 to maintain the process. This will be explained in a subsequent section, when we are dealing with the Latent Heat of steam. A striking form of this experiment consists in making water boil in an open flask so furiously that the greater part of the air is expelled by the steam, then corking the flask and inverting it. When cold water is poured on the bottom of the inverted flask, it condenses the steam and thus diminishes the pressure on the water, so that it im- mediately begins to boil. Boiling water, poured on, at once stops the boiling. If the air has been very completely expelled, the boiling by the application of suflSciently cold water can be produced even when the contents of the flask have cooled to the temperature of the air of the room. (c.) The temperature at which water boils (in a small apparatus like that sketched in § 134) may be used as a means of measuring the pressure to which it is subjected. Thus WoUaston made the thermometer take the place of the barometer in the measurement of the heights of mountains. It has, in fact, several advantages as regards portability, being far more compact and much less hable to breakage. The following roughly approximate formula, calculated from a table of Regnault's (p. 633 of the Relation, etc.) shows how the boiling-point varies with the ordinary fluctuations of the barometer near the sea-level. Pressure. Temperature C. Atmospheres. Inches. 99° ± T. 0*9647 ± 0*0348 T. 28'87 ± I '04 T. [This formula must not be employed for values of r much exceeding 2. For greater ranges, or for more accurate values within this range, Regnault's full table must be consulted.] 138 HEAT. [chap. Roughly speaking, the boiling-point of water is lowered by 1° C. for 960 feet of vertical ascent above the sea-level. At the top of Mont Blanc the Hypsometric Thentionieter was found by Bravais and Martins, in 1844, to stand at 84°-4 C. (d^ We see why a diminution of pressure reduces the solvent and cooking powers of boiling water, while an increase of pressure exalts them. Tea prepared at the top of Mont Blanc is poor stuff; and Papin's digester, which, is used for extracting everything soluble from bones, etc., is simply a very strong boiler, in which water, under the pressure of its saturated vapour, can be safely raised to temperatures far above its ordinary boiling-point. IX.] VAPORISATION AND CONDENSATION. 139 166. Approximate expressions for the amount of heat rendered latent in the evaporation of water at different temperatures were given by Watt and others. But the subject remained doubtful until Regnault cleared it up. He gives for what he calls the total heat of steam, at any temperature f C. the very simple formula — 605-5 + 0-305 t. Thus a pound of saturated water vapour at 0° C. gives out 606 '5 units of heat in condensing to water at 0° ; while if its temperature had been originally 100° C. it would have given out 606-5 + 30-5 (or 637 units) in passing to. water at 0°. In the next chapter the (very small) variation of specific heat of water with temperature will be treated. When it was taken into account along with the above formula, it was found by Regnault that the latent heat of steam at different temperatures falls from 606-5 at 0° C, to 536-5 at 100°, and to 464-3 at 200°. The latent heait at any temperature is here the number of units of heat required simply to convert a pound of water into steam of the same temperature. For most practical purposes this may be taken as indi- cating a decrease simply in proportion to the rise of tempe- rature, and amounting to 11-5 per cent, for each hundred degrees above the freezing-point. Watt's hypothetical state- ment was to the efFect that the heat required to change a pound of water at o°-C. into steam at any pressure whatever — i.e., the total heat of steam — is constant. If this were true the sums of the last three pairs of numbers should be 140 HEAT. [chap. approximately equal. They are not so, and they show that Watt's statement involves an error in defect amounting to about five per cent, of the whole for every hundred degrees above o° C. When we are dealing with the dynamical theory, we shall have occasion to discuss this question more fully. 167. According to Andrews, the latent heat of a pound of vapour, produced from certain common liquids by boiling at the ordinary atmospheric pressure, is as follows : — Latent Heat of Evaporation at i Atmosphere Pressure. Water ... . 536*0 Alcohol . 202 "4 Ether ... . . . 90*5 Bromine . . .... 45 '6 Here, again, water stands relatively very high. And, as we have already seen that the latent heat of steam increases as the temperature of evaporation is ; lower, we see what a very large amount of heat is required for the evaporation going on over the oceans i and how great , is the amount of heat set free_ when that vapour is condensed into fogs or clouds. This leads us 'to consider the phenomenon of condensation with some of its consequences. 168. A striking illustration of the great latent heat of steam is given by a very simple arrangement. We have merely to lead steam from a boiler into a vessel containing a measured quantity of ice-cold water. At first, the steam is condensed as soon as it reaches the water ; but as the water becomes warmer the' steam gradually advances from the end of the conducting pipe, forming an increasing bubble at whose surface condensation is steadily going on. As soon as the water is raised to the boiling-point, the IX.] ■ VAPORISATION, AND CONDENSATION. 141 steam passes freely through it. When this stage is reacherl, remove the steam-pipe and measure the volume of the water. It is found to have increased by less than a fifth of its original amount. This increase is, of course, in a condensed form, the steam, whose latent heat has raised by 100° C. the temperature of the mass of water. i6g. The latent heat of vaporisation is utilised in many ways, especially for cooling and for freezing. Thus water, put into a vessel of unglazed clay, is kept permanently cool in warm dry air, by the evaporation from the surface of the vessel. A similar result is produced when a glass vessel is employed, if it be wrapped in a wet clotli and placed in a current of air. In some parts of India ice is procured by exposing water at night in shallow unglazed saucers, laid upon rice-straw. More rapid effects may, of "course; be obtained by using instead of water highly- evaporable liquids such as sulphuric ether. A few drbps of ether, sprinkled on the bulb of a thermometer, produce an immediate contraction of the contents, which is greater as •the temperature of the air is higher. The cooling of. water, when it is made to boil at low temperatures (by reducing the pressure, as in § 165 (i) ) is due to the sam.e cause. This process, with a quantity of dry oatmeal or a large sur- face of sulphuric acid (to absorb the vajjour as it is formed^ was employed by Sir John Leslie for the purpose of making ice ; and is still, with various modifications, the basis of some of the most convenient domestic ice machines. 'The Cryophorus (whose principle will be explained wheii we are dealing with Daniell's Hygrometer, § 172) is another very curious illustration of the same fact. When a jet of carbonic acid, liquefied by pressure, is allowed to escape irito the air, the outer layers of the jet vaporise at once ; taking the requisite latent.' heat from the 142 HEAT. [chap. core of the jet, which is thus frozen into a solid and can be collected in a proper receiver as a snow-like mass. It appears that hydrogen has in this way been obtained as a steel-grey powder. A very striking experiment of the same class is the freezing of water in a white-hot platinum dish. This is easily effected by the help of liquid sulphurous acid, which evaporates very freely from the water while it remains suspended above the dish in what is called the " spheroidal state" (§ 213). If the dish be not sufficiently heated the experiment fails. Faraday succeeded in freezing even mercury in a white- hot vessel by a process of this kind. The mercury was in a little capsule resting on a mixture of solid carbonic acid and ether in the spheroidal state in a highly heated platinum crucible. 170. The reader will now easily understand why it is possible (as stated in § 155) to procure very low tempera- tures by exposing solidified carbonic acid, mixed with ether, in an exhausted receiver. 171. We must next consider the converse process, the condensation of vapour into liquid. And for the present, we confine our remarks to the behaviour of- water-substance. In § 162 we have already shown that condensation com- mences, in a vessel containing water-vapour alone, as soon as the pressure of the vapour exceeds that corresponding to saturation. We may state this, of course, in another form, viz. that condensation commences as soon as the tempera- ture falls below that correspionding, in Regnault!s table, § 164, to the pressure of the water-vapour present. And it is found by experiment that the pressure of air or other gas does not modify this result except in more or less retarding it. Hence, if we present to air containing water-Vapour a solid cooled below, the temperature of saturation corres- IX.] VAPORISATION AND CONDENSATION. 143 ponding to the vapour-pressure, condensation will take place in the layer of air immediately in contact with the solid, which will thus be covered witli a film of Dew. This film will become thicker, until, by the latent heat given up by the vapour in condensing, the solid has been raised to the temperature of saturation corresponding to the vapour- pressure. If the solid be at a temperature below 0° C. the film freezes as it is deposited, and becomes what is called Hoar Frost. In Hope's experiment (§121) the metal vessel containing the freezing mixture is rapidly covered with a layer of hoar frost, however warm the surrounding air may be, provided it contains a moderate amount of aqueous vapour. 172. The statements in last section show at once what is meant by the meteorological term, the Dew-point. It is the temperature at which saturated water-vapour would have the same pressure as that of the vapour present in the air at the time; and it could therefore be found directly from Regnault's table (§ 164) if we knew the quantity of vapour per cubic foot of air. But the converse process is that most commonly used, the dew-point being directly measured, and then employed for the purpose of estimating by Regnault's table the quantity of vapour present in the air at any time. Daniell's Hygrometer is an ingenious instrument by which a body is gradually cooled, so that we can measure its temperature at the instant when dew just begins to be deposited on it. It consists of a glass tube with a bulb at each end. One of these bulbs is made of black glass, to show at once the slightest trace of dew on its surface. In this bulb is a thermometer' (whose indications can be read through the clear glass tube joining the two bulbs) and a quantity of sulphuric ether The instrument-maker boils 144 HEAT. [chap. this ether ; and hermetically seals the second bulb when the ether-vapour has expelled the air from the whole apparatus (as in § 165 {b) ). To use the instrument, pour a few drops of ether on a piece of cambric wrapped round the second bulb. The evaporation of the ether cools this bulb and condenses the ether-vapour in it. More vapour is formed from the ether in the black bulb, and again condensed in the second bulb. Distillation in fact goes on between, the two bulbs ; and the black bulb, with its contents, becomes gradually colder on account of the latent heat required for the persistent formation of vapour. The temperature of the black bulb is read by the inclosed thermometer at the instant of the first appearance of dew. The thermometer is again read at the instant that the dew disappears, when the apparatus begins to be heated again to the temperature of the air. It is usual to assume the mean of these readings as the dew-point. Suppose water to be put in the instrument instead of ether, and snow and salt instead of ether to be applied to the empty bulb. . The rapid evaporation of part of the water from the full bulb freezes what is left. This is the Cryophorus (§ i6g). Another method of finding the dew-point is by the Wet and Dry Bulb Thermometers. Two similar thermometers are placed side by side ; one having its bulb covered with cambric, connected by a few threads with a small vessel of water, so as to be kept constantly moist. The thermometer with the naked bulb shows the temperature of the air, the other shows a lower temperature, which differs from the first in consequence of the evaporation constantly going on from its moist covering. When the air is nearly saturated with m'oisture, the evapora,tion'is very slow ; and the lowering of temperature smalL But if the air be dry, and especially if IX.] VAPORISATION AND CONDENSATION. 145 it be also warm, there is rapid evaporation and considerable lowering of temperature. Dr. Apjohn, who has studied this subject with particular care, gives for the determination of the dew-point by this arrangement the formula 40 30 In this formula 8 is the difference between the readings of the thermometers, / is the sought pressure of vapour in the air, /o that in Regnault's table corresponding to the temperature of the wet-bulb, and b is the height of the barometer in inches. To a certain extent this formula is, as yet, empirical. The full treatment of the question by theory remains to be discovered. 173. The theory of the formation of dew was first correctly given in Dr. Wells's £ssay of 18 14. It cannot be fully stated till we have dealt with Radiation. But it may suffice for the present to say that the cooling of stones, blades of grass, &c., by radiation, takes place most rapidly in clear, calm nights. Clouds, in general, radiate back a great deal of the heat they receive, and so prevent objects on the ground from cooling sufficiently. Winds also, unless the air is itself very cold, constantly warm again bodies cooled by radiation. Hence it is after clear, calm nights that the dew is usually most copious ; and when the radiation has been sufficiently rapid we have it in the form of hoar-frost. The temperature of the air itself cannot at any time fall much below the dew-point ; for, as soon as it does so, condensation takes place, and latent heat of vapour is set free. 174. What has been said in the preceding sections of the condensation of. aqueous vapour is true, generally, of L 146 HEAT.. [chap. the vapours of all substances which are liquid at ordinary temperatures and pressures. Some liquids, such as com- mercial sulphuric acid, have a scarcely measurable vapour- pressure under ordinary conditions; others, as sulphuric ether, evaporate with great rapidity. But it is only within comparatively recent times that it has been shown experimentally that all gases are really vapours. This has been done by reducing them to the liquid, and sometimes even to the solid, state. Mere cold effects this with many substances, mere pressure with others ; but some (especially those bodies which used to be called permanent gases) require the combined application of cold and pressufe. Faraday eifected the liquefaction of a great number of gases, mainly by pressure. On the other hand Cagniard de la Tour rendered liquids, with little change of density, gaseous (or, at all events, not liquid), by heating them under great pressure. It was Andrews, however, who first cleared up the whole subject by showing the nature of the distinction between what may now be called a true gas, and what may be called a true vapour. His experiments, on carbonic acid and other bodies, led to the experimental law that There is a Critical Temperature for every vaporous or gaseous, substance; such that, only when its temperature is below this, can the substance be reduced to the liquid form by any pressure, however great. The critical temperature for sulphurous acid is high, and this substance is easily liquefied at ordinary temperatures by an atmosphere or two of pressure. That for carbonic acid is 3P°.9C.,.and it requires, at ordinary temperatures, forty or fifty atmosphere.s for its condensation. Water vapour has to be raised to a temperature of about 41 2° C. before it ceases to be condensable by pressure. On the other hand, IX.] VAPORISATION AND CONDENSATION. 147 the critical temperature for Hydrogen, Oxygen, Nitrogen, and "Carbonic Oxide is so low that even considerable pressure requires to be assisted by the most powerful freezing mix- tures before the liquefaction of these substances is accom- plished. Cailletet and Pictet, separately but simultaneously, eflfected in the end of 1877 this complete verification of Andrews's law. 175. When a substance is at a temperature higher than its critical temperature, we may call it a true gas ; when at a lower temperature, a true vapour. The distinction is well shown by the following brief account trf Andrews's experi- ments on carbonic acid. At 13°.! C. carbonic acid gas was reduced by a pressure of forty-nine atmospheres, as measured by an air gauge, to ■gj of its original volume under one atmosphere, without undergoing any change of state. With a slight increase of pressure, liquefaction occurred, and the volume of the carbonic acid in the liquid state was -j^ of the original volume of the gas. During the process of liquefaction both liquid and gas were visible in the tube in which the experi- ment was made. After complete liquefaction, the carbonic acid continued sensibly to contract as the pressure was augmented. At 2i°-5 C, similar results were obtained, but liquefaction did not occur till a pressure of sixty atmospheres was reached. At 3i°-iC., which is o°'2 above the critical temperature of carbonic acid, the gas behaved in the same way as at lower temperatures, till a pressure of seventy-four atmospheres was attained, when a further increase of pressure produced a very rapid, but nowhere abrupt, dimi- nution of volume, unaccompanied ny by a evidence of liquefaction or of the presence (at any stage of the process) of two different states of matter in the experimental tube. After this the carbonic acid, now reduced to -gg^ of its L 2 148 HEAT. [chap, IX. original volume, continued slowly to diminish in volume as the pressure was increased. At higher temperatures this rapid fall became less manifest, and at 48° it could no longer be observed. [This question will be more fully dis- cussed in Chapter XX. below.] 176. For the formation of liquid water from ordinary vapour a nucleus of some kind must be present, else the vapour may remain in a state of saturation, sometimes even of supersaturation, without condensing. This has quite recently been shown by Aitken, who has traced the formation of clouds and fogs to the presence of excessively fine particles of dust in the air. These dust particles are found to be completely arrested by a plug of cotton-wool, through which air is made to pass. Each dust particle secures a share of the vapour, so that, when they are very numerous, the share of each is small, and a fog or mist is formed ; when they are few each gets more, and clouds or even rain-drops are produced. The smaller the number of nuclei the larger are the individual rain-drops. Aitken concludes from his experiments that, if the atmosphere were entirely free from such particles no mists or clouds could be formed, no rain would fall, and the air would get rid of its superfluous moisture only by direct deposition upon bodies exposed to it. 177. Rksuvi'e of §§ 159-176. — Vaporisation. Dalton's Law. Regnault's Determination of the Pressure of Satu- rated Water-vapour at different temperatures. Hypsometric Thermometer. Total Heat of Steam. Freezing by Evapo- ration. Condensation. Dew-point. Hygrometers. Dis- tinction between Gas and Vapour. Andrews's Critical Point. Cause of Clouds and Fogs. CHAPTER X. CHANGE OF TEMPERATURE. SPECIFIC HEAT. 178. Refer again to §§ 45-48. In these sections we have spoken of the gradual rise of temperature of water substance, as heat is steadily applied to it, in the solid, liquid, and gaseous forms, and the question immediately suggests itself : What amount of heat is required for each degree of rise of temperature by the substance in each of these three states ? In other words, does water substance become easier to heat or harder to heat by being changed from one of the molecular states to another ? Again, if a pound of water and a pound of mercury, at the same tem- perature, have equal quantities of heat communicated to them, will the heating effect be the same for both; or, if not, in what proportion will be their rise of temperature ? 179. Experiment has answered these and other analogous questions, in a form which shows that bodies differ in a very marked manner as to this effect of heat ; and we therefore define, as a property of each particular substance (under assigned conditions) what is called its Specific Heat, as follows : — The specific heat of a substance, under any specified con- ditions, is the number of units of heat required, under these I50 HEAT. [chap. conditions, to raise the temperature of one pound of the substance by 1° C. We have already defined a unit of heat as the amount of heat required to raise the temperature of a pound of water by i" C. \ it is therefore clear that, by our definition, the specific heat of water is i. It will be seen later (§ 184) that the specific heat of water varies so slowly with temperature that it is practically the same at 0° C. and at 10° C, i.e. at 32° F. and at 50° F. (see § 55). The term Specific Heat was originally devised by the Calorists, but it is still used in science, like Latent Heat, Centrifugal Force, etc., etc., as it is convenient and quite harmless. 180. There are many experimental processes for deter-, mining Specific Heat, but we cannot spare space for more than two or three so far as solids and liquids, under ordinary pressures, are concerned. We commence with the most readily intelligible of them; a process, in fact, long ago devised by Black. It will be found, when we are dealing with Radiation, that the quantity of heat radiated from the surface of a hot body, in a given time, depends (so far as the body itself is concerned) solely upon the temperature and the nature of the surface. Hence, if we take a thin shell of metal and fill it successively with different liquids, each at a temperature higher (or lower) than that of the surrounding bodies, we know that its rate of loss (or gain) of heat by radiation will be the same at any one temperature whatever be the sub- stance which fills it. Suppose, for simplicity, that the shell holds just one pound of hot water. Then, if the bulb of a thermometer be plunged in the water we know that, for every degree the thermometer falls, one unit of heat has been removed from the water by radiation and convection. Let us note the number of seconds which elapse for each' xj CHANGE OF TEMPERATURE. 151 degree of temperature as the thermometer falls, and we may then construct a table of the number of seconds required for the loss of one unit of heat by radiation from the shell at each successive degree of temperature. Now fill the shell with another hot liquid, mercury sup- pose. (To be thus employed the shell must be made of iron, for chemical reasons.) Note again the time required for each degree the thermometer sinks. It will be found to fall faster than when the contents were water ; although, in consequence of the great specific gravity of mercury, there are 13 "6 pounds of that liquid in the shell. In fact, the time of cooling through any given range of temperature is now less than half of what it was when the shell was filled with water. Hence we conclude that the specific heat of water is more than 27-2 times that of mercury, because the same change of temperature is produced in one pound of water and in i3"6 pounds of mercury, while the water loses more than twice as much heat as the mercurj'. The correct specific heat of mercury is found in this way to be about 317 only. 181. The process we have just described is liable to several objections. The most serious of these are (i) that the contents of the shell cannot, unless constantly stirred, have the same temperature throughout ; (2) that, unless the neck of the shell be closed, there is serious loss of heat by evaporation — a loss which is of greatly different amounts with different liquids at the same temperature. But for the beginner in experimental science, the process is very instructive ; especially if he employs a graphic method in deducing the results. We may take this opportunity of giving a general notion of this most important process. Although from its very nature it cannot pretend to any great accuracy, it is always free from large error ; and is 152 HEAT. [chap.. therefore an almost indispensable auxiliary to numerical calculation, in which the most expert calculators may make, serious mistakes. [Thus, suppose the results of the observations of tempe- rature in terms of the time, as furnished by the readings of the thermometer after the lapse of successive intervals, equal or not, be plotted (on mechanically ruled paper) as above ; intervals of time being measured horizontally, corresponding temperatures vertically. The figure represents, on a much. x] CHANGE OF TEMPERATURE. 153 reduced scale, the original plotting of the data of part of an actual experiment ', the cooling of the short bar in Forbes's method for measuring thermal conductivity (Chapter XIV. below). The horizontal line is divided into minutes of time, the vertical line (on its left side) into degrees centi- grade. Suppose a smooth curve to be drawn lilierd manu (the full curve in the figure), so as to agree as nearly as possible with the observed points. This curve will repre- sent an approximation (the more close the shorter are the intervals between the observations) to that which would have been obtained by the continuous photographic process of § 136. Then if rate of cooling at any temperature, say 186°, be required, all that has to be done is to draw a hori- zontal line through 186° on the scale of temperatures, and where this line meets the curve draw a tangent, producing it both ways to meet the two divided lines. As drawn,- it cuts oSt^° on the temperature scale, and i7™'6 on the time scale. Hence the rate of cooling at 186° is 73° in i7'"'6, or 4°'i5 per minute. Even the best mercury thermometers, when the column is rising or falling otherwise than very slowly,, are found to move occasionally by sudden starts, the surface film sticking for a moment in the tube and thus subjecting the column to pressure or tension. It is possible also that the bulb recovers by starts from its state of dilatation. Hence for the purpose of finding rates of cooling, as in the case above, it is sometimes better to make the observa- tions of temperature at successive equal intervals taken as the unit of time, and to plot the differences of the successive readings midway between the beginning and end of the interval in question. In the figure the corresponding points are put in and connected by dotted lines. The time scale is the same as before, but the degrees on the temperature 1 54 HEAT. [chap. scale are ten times as long as before. Its numbers are inserted on the- right side of the temperature scale. We now find it impossible to draw a smooth curve through the various points, but if we draw a smooth curve (the dotted one in the figure) which (perhaps not passing through any of them) shall show on the whole as much divergence of observed points from itself on one side as on the other, the ordinates of this curve will be themselves the rates of cooling required, and no drawing of tangents will be necessary. In the figure, the ordinate of this dotted curve, corresponding to 1 86°, is 4°'i2, thus agreeing within one per cent, with the result of the former method. Experience alone can guide us, in any case, as to which of these methods is to be preferred.] 182. Another method of determining specific heat is supplied by the ice^calorimeter, in which the amount of heat lost by a hot body in cooling to 0° C. is measured by the amount of ice which it melts. The general principle on which the calorimeter is constructed will be obvious from the figure. There are virtually three vessels, one within another, the two outermost containing ice and water at ordinary atmospheric pressure, and therefore always at 0° C. The middle vessel can part with no heat to the outer one unless its temperature changes; but this cannot be until all its contained ice is melted (§ 142). We have thus a very simple and fairly effective method of preventing loss of heat. Into the interior vessel the substance whose specific heat is to be measured is dropped, at a known temperature, and the whole is closed up. After a short time the whole contents of the vessel are again at 0° C, but the heat disengaged frbm the substance has melted some of the ice in the middle vessel. In the older and rougher forms of the instrument, as designed by Lavoisier and Laplace (from x.] CHANGE OF TEMPERATURE. iSS one of which the sketch is taken), the amount of ice melted was given by the amount of water which drained away from the middle vessel. Of course there was considerable error, as some of the water was necessarily retained by capillary forces. In the improved form, as designed by Bunsen and others, the middle vessel is filled with solid ice to begin with, and the amount melted is calculated from the dimi- nution of the volume (§ 145) of the contents, as shown by an external gauge containing mercury in- contact with the ice. 156 HEAT. [chap. The result of each experiment shows (§151) the number of units of heat lost by a known mass of a substance in passing from a known temperature to 0° C. By dividing the number of units of heat lost by the number of pounds of the substance and by the number of degrees of change of temperature, we obtain the Average Specific Heat of the substance through the range of temperature employed. This method is very troublesome, and requires great care and skill; but it is much more trustworthy than any other, when properly conducted. 183. The only other method we need describe is the common one called the Method of Mixture. Here we determine the specific heat of a substance by dropping it, when hot, into a liquid (usually water) at a lower tem- perature, and observing the temperature which the mixture acquires. For rough determinations this process is sufficient, and it has the additional advantage of being very easy. But the corrections, which are absolutely necessary if we wish to obtain exact results, are exceedingly troublesome and require great experimental skill. These corrections are required chiefly on account of the loss of heat by radiation and evaporation, which do not affect the method last described. Suppose M pounds of one substance, at temperature ly C, to be mixed with m pounds of another substance at a lower temperature, /° C. ; and suppose the mixture (all corrections made) to have the temperature t° C, which must obviously be intermediate between T and t. Then the heat lost by the one substance has gone to the other. If the specific heats of the two substances be .Sand s respectively, the loss of the first (in units of heat) is • MS{T—t). X.] CHANGE OF TEMPERATURE. 157 The gain of the other is (in the same units) m s (t — /) Equating these quantities, we find at once S m JT—t) ~s- m\t—t)' which gives the ratio of the specific heats ; or, if the second substance be water, the specific heat of the first substance. [It is to be observed that i'' is the average specific heat of Mitom. T\.o T, s that of m from r to A] It is well to observe that the temperature of the mixture, if no heat be lost by radiation or evaporation, is _ MS T + mst MS + ms' The quantities MS, ms, which appear in this expression, are called Water-equivalents of the substances. The water, equivalent of any mass is thus the number of pounds of water which would be altered in temperature to the same extent, by the same number of units of heat, as the given mass. [And we see that the temperature of a mixture is given in terms of the water-equivalents of the several substances and their several temperatures, by a formula exactly the same as that which gives the position of the centre of inertia of a number of particles in one line, in terms of their separate masses and positions.] Another term which is sometimes of use, is the thennal capacity of a. sybstance. This may be regarded as the water-equivalent of unit volume of the substance, and its numerical value is obviously the prodijct of the density by the specific heat of the substance. iS8 HEAT. [chap. 184. We will now give some general notions as to relative specific heats of solids and liquids. But before doing so, it is well to remark that, in general, specific heat rises with temperature ; and that (as above stated) Dur experimental determinations give directly the average specific heat of a substance through a certain range of temperature. From the results of such determinations, however, if they be sufficiently numerous and varied in their circumstances, it is not difficult to deduce the true specific heat as a function of the temperature. According to Regnault, a pound of water at 0° C. requires ioo'5 units of heat to raise it to 100° C, and 203'2 units to raise it to 200° C. These are represented by the formula H = i* + O'OO0,O2 t^ + o'ooo,ooo,3 t^, which gives, for the specific heat of water at any temperature, i° C, the expression I + o'ooo,04 t + o'ooo,ooo,9 t^. Thus the change for any ordinary ranges of temperature is extremely small. Refer again to § 61. The specific heat of ice at o°C. is almost exactly /lalf that of water. Regnault has shown that it diminishes as the temperature is lowered. That of glass may be roughly assumed as about 0-2. That of platinum is about 0-0355, ^"d varies very slightly even for large ranges of temperature. With other metals, such as copper, the increase of specific heat is (roughly) about 10/. c. for 100° C. It appears to be rather more in iron. But we have no very exact knowledge on this subject, as different specimens seem to give very different results. X.] CHANGE OF TEMPERATURE. IS9 Specific Heat of Elementary Solids. Lead 0-031 . , . 207 . . . 6-S Tin 0-056 ... 118 . . . 6-6 Copper .... 0-095 ■ • • 63-5 . . . 60 Iron . . . . 0-114 • • • 56 . . . 6-4 Sodium .... 0-293 ■ • • 23 . . . 6-7 Lithium. . . . 0-941 ... 7 . . . 6-6 The first column of the table gives the specific heat at ordinary temperatures, the second the atomic weight of the substance. The numbers in the third column are the products of those in the first and second. It will be seen that the numbers in this third column are within about 10 per cent, of one another. Hence it is concluded that the specific heat of an elementary solid is inversely as its atomic, weight. The small differences in the last column are attributed (in part at least) to the known fact that the specific heat of each substance increases as the temperature rises, and therefore that, as these specific heats are measured all at the same temperature, the different bodies compared are taken in different physical states, some being much more nearly at their melting-point than others. It is probable that very important information as to the nature of matter will be obtained from this experimental fact. Berhaps its value may be more easily seen if (by § 183) we put it in the form : — The atomic water-equivalent is nearly the same for all elementary solids. A similar law has been found to hold for groups of compound bodies of similar atomic composition. But the product of the specific heat and the atomic weight varies in general from group to group of such compounds. 185. In general, as we have seen, the specific heat of a i6o HEAT. [chap. substance in the liquid state is greater than in the solid state, Specific Heats of Elementary Liquids. Lead o"040 400° C. Tin 0-064 300° C. Mercury 0-033 30° C. The second column gives the mean temperature of the range through which the specific heat is measured. The specific heat of Alcohol, which it is important to keep in mind, is (at 30° C.) a little above o-6. 186. In the case of gases the problem of determination of specific heat, is not only more difficult, from the experi- mental point of view, than in the case of a solid or of a liquid, but it is also more complex from a theoretical point of view. For, in consequence of the great expansibility of gases, we have to specify the conditions under which the measurement is to be made. Thus the gas may be main- tained at the same volume, or at the same pressure, through- out the range of temperature employed. And we therefore speak of the Specific Heat at constant volume, or of the Specific Heat at constant pressure, Thermo-dynamics gives us a simple relation between these quantities for the ideal perfect gas (§ 126); and therefore for gases such as Air, Hydrogen, &c., it is only necessary to determine directly one of the two specific heats. This is a very happy cir- cumstance, because the experimental difficulties of deter- mining the specific heat at constant volume are extremely great. The mass of a gas at ordinary pressures is small compared with that of the containing vessel, unless the vessel be of unwieldy dimensions ; and we cannot get over this difficulty by compressing the gas, for we must then make the vessel strong (and therefore massive) in proportion. X.] CHANGE OF TEMPERATURE. l6i The measurement of the specific heat at constant pressure is effected by passing the gas, at a uniform rate, through two spiral tubes in succession. In the first of these it is heated to a known temperature, and in the second it gives up its heat to a mass of water in a calorimeter in which that spiral is immersed. From the volume of the gas which has passed through the spirals, we can calculate its mass, and the observed rise of temperature in the water of the calorimeter supplies the requisite additional datum. This process was devised by De la Roche and B^rard, but its details were greatly improved by Regnault. Further remarks on this subject, especially from the theoretical point of view, must be deferred for the present. 187. Regnault's experiments showed that, for gases like air, tne specific heat of a given mass of the gas is inde- pendent of the temperature and pressure, and therefore (§ 124) of the volume. Hence the specific heat of a given volume of such a gas varies directly as the density. Equal volumes of different gases of this class, at the same pressure, have approximately equal specific heats. Specific Heats at Constant Pressure. Air o'237 Oxygen .... 0"2I7 Nitrogen . . . 0-244 Hydrogen .... 3 '409 It will be seen that these numbers are very nearly in the inverse ratio of the densities of the various gases (see § 184). Experiments on sound have shown that the ratio of the two specific heats of such gases is about i '408, whence we see that the specific heat of air at constant volume is o-i68. l62 HEAT. [chap. X. The specific heat of water-vapour under the same con- dition is 0-48, a little less than that of ice (§ 184). Hence water-substance has twice as great a specific heat in the liquid state as in the solid or in- the vaporous state. 188. Resume of §§ 178-187. — Change of temperature of unit mass by a given quantity of heat. Specific Heat. Method of Cooling. [Digression on Graphic Methods.] Ice Calorimeter. Method of Mixtures. Water Equiva- lent. Thermal Capacity. Relation between Specific Heat and Atomic Weight. Specific Heat of Gases : {a) at constant volume, (B) at constant pressure. Specific Heat of Water-vapour. CHAPTER XL THERMO-ELECTRICITY. 189. We have already stated, in § 49, the fundamental fact of Thermo-electricity as discovered by Seebeck ; and we now propose to examine the subject more closely from the experimental point of view. For this purpose we must use the term Electro-motive Force. So far as we require its properties they may be enunciated as follows : — The strength of a current in a given circuit is directly proportional to the Electro-motive Force, and inversely proportional to the resistance. The energy of the current is the product of the Electro-motive Force (contracted as E.M.F.) and the strength of the current. It will be advantageous to commence with the following statement of an experimental result :— If e be the E.M.F. in a thermo-electric circuit when t„ and 4 are the temperatures of the junctions ; and e' the E.M.F. when the temperatures are t^ and t, ; then when the temperatures are 4 and 4 the E.M.F. is e-\-e'. [Consider this from the point of view given by the fol- lowing figure ; where the circuit is supposed to consist of /our wires, alternately of two metals (say iron and copper). See, again, § 49, where it is stated that the wires may be M 2 164 HEAT. [chap. soldered together at the junctions. No third substance, such as metal or solder, introduced into the solid circuit, is found • to affect* the result if it have the same temperature at the points where it meets each of the two metals. Let the temperatures of the four junctions be as in the figure. Then D and A together (if B and C had the same temperature with one another) would give E.M.F. e, and B and C together (if D and A had one common temperature) would give E.M.F. e. Hence as the figure is drawn, there would be total E.M.F. e + ^, provided the assumptions made (in brackets) above were superfluous. Iron Iron But the E.M.F. may be looked upon as due entirely to the difference of temperatures of C and D, for the tem- peratures of A and £ are equal, and therefore the copper wire A B produces no effect. Hence the contribution of each metal to the whole E.M.F. of such a circuit depends only on the temperatures of its ends, i.e. is independent of the temperature of the rest of the circuit.] 190. It follows from the experimental fact above, that we may break up any interval of temperature, /„ to 4> into ranges: — viz. t„ to /„ i^ to t„ Sac, 4-i to i„ ; and the E.M.F. in a simple circuit of two metals, when the junctions. XI.] THERMO-ELECTRICITY. are at t„ and t„ will be the sum of its separate values when the junctions are successively at 4 and /■„ t^ and t^, &c, and 4-, and /„• We shall now look upon these as successive equal ranges of temperature, and we shall define the thermo-electric power of a circuit of two metals, at mean temperature t, as the E.M.F. which is produced when one junction is kept half a degree above, the other half a degree below, /. We thus get the means of representing in a diagram the relative thermo-electric ,positions of any two metals at different temperatures. Thus we may erect little rectangles one degree in breadth, as below, upon a line representing one of the metals. The area of each rectangle is the thermo-electric power corresponding to the temperature at the middle of its base ; and, by taking the intervals of temperature small enough, it is obvious that the final upper boundary of the group of rectangles will become a continuous line. This will repre- sent at every point the relative thermo-electric situation x)f the second metal with regard to the first, as the first is represented by the line on which temperatures are measured, And the sum of any number of the rectangles (giving the E.M.F. for any two assigned temperatures of the junc- 1 66 HEAT. [chap. tions) is now represented by the area of the corresponding portion of the curve. 191. But there is a much more general experimental result than that in § 189. ' At every temperature the algebraic sum. of the thermo-electric powers of metals a and /3, and IB and y, is the thermo-electric power of u, and y. [Suppose one of the iron wires in the diagram of § 189 to be replaced by a third metal, say gold; and let the temperatures of the ends of one of the copper wires be t„ and of the other /j. Then A and D (if £ and C had the same temperature) Iron Gold would give E. M. F. depending on iron-copper alone at the temperatures t^, 4 ; B and C (if A and D had the same temperature), would give E. M. F. due to a gold-copper circuit with its junctions at the same two temperatures. It is easy to see, as before, that the experimental result shows the assumptions (in brackets) to be superfluous. For, as the ends of each copper wire are at the same temperature, the circuit acts as one of iron-gold alone, its junctions also having the temperatures t„ ^,.j Thus if, by the process of § 1 90, we form the lines repre- senting the positions of iron and of gold, each with regard to copper taken as the standard, the lines so drawn will show the position of. iron and gold relatively to one XI.] THERMO-ELECTRICITY. 167 another. Hence the possibility of constructing a Thermo- electric Diagram containing a single line for each metal. The idea of such a representation was suggested by W. Thomson in 1855, and he gave a rough preliminary sketch of it. A "first approximation" to an accurate diagram was given by Tait in 1873. This is repro- duced on a small scale in § 198, but it cannot be fully explained until some additional experimental facts are stated. It will then be found that a properly drawn thermo- electric diagram embodies all that is known about the subject of thermo-electricity. And thus, by the use of the diagram, we are now able to present the subject in a much more simple and connected manner than was formerly possible. 192. The electric current in a thermo-electric circuit represents a certain (usually very small) amount of energy, whose measure as before stated is the product of the E.M.F. by the strength of the current. This can have no other source than the heat which has been given (as in § 49) to a part of the circuit. And the existence of the current imphes a loss of heat by the circuit as a whole. It is a very remarkable fact in the history of science that, without any reference to the theory of energy, Peltier (1834) discovered by experiment that — When a airrent of electricity from an external source passes through a junction of iivo metals, it causes an ab- sorption or a disengagement of heat. If the direction, of the current be the same as that of the current which would be produced by heating the junction, the effect is absorption ; and vice versa. This is very easily proved by the help pf a galvanometer. Two wires (say iron and copper) are brazed together at their middle points only ; one of the free ends of the iron l68 HEAT. [chap. is maintained in connection with one pole of a single voltaic cell, the other end is in connection with one of the ends of a galvanometer coil. The copper wire is so arranged that by rocking it over to one side we close the circuit of the cell, leaving the galvanometer circuit open ; and by rocking to the other side we break the battery circuit and close the circuit of the galvanometer ; the iron- copper junction being thus alternately in one circuit and in the other. There is always a deflection of the galvano- meter after the voltaic current has traversed the junction. And it changes sign when the direction of that current is reversed. The wires used for this experiment should be stout, else the heat generated in the circuit by ordinary resistance to the battery current may be greater than the Peltier effect which is sought. When proper precautions of this kind are taken, a notable amount of heat may be produced or absorbed. Lenz, in 1838, succeeded in freezing a little water by passing a current of electricity in the proper direction through a bismuth-antimony junction, the metals themselves being surrounded by melting snow. The direct quantitative measurement of the Peltier effect, at different temperatures, presents very great experimental difficulties. These have been only partially overcome as yet, mainly by the experiments of Naccari and Bellati. The results agree, as well as could be expected, with those stated below. There can be no doubt, however, that for any one pair of metals, kept at a given temperature, the Peltier effect is directly proportional to the strength of the current employed. 193. It was Joule who first remarked that the Peltier phenomenon furnished a clue to the source of the energy XI.] THERMO-ELECTRICITY. 169 of a thermo-electric current ; and it is quite possible that there may he pairs of metals (more probably alloys) for whose circuits it is the only source of the current. [This will be seen in § 199 below.] But a phenomenon, noticed by Gumming very soon after the publication of Seebeck's discovery, shows that the Peltier effect is in general a part only of the source of the current. This discovery of Cumming's may be stated as follows : — In certain circuits, such as those of iron-copper, iron-silver, iron-gold, etc., if one junction be kept at ordinary tempera- tures, and the temperature of the other be steadily raised, the E.M.F. increases more and more slowly till it reaches a maximum, th^n gradually diminishes, and finally is reversed. W. Thomson explained this effect as follows : — At that temperature of the hot junction for which the E.M.F is a maximum (in Cumming's experiment), the two metals are neutral to one another, i.e. their thermo-electric power vanishes, and the Peltier effect also. This occurs, as the figure in § 190 shows, when the lines representing them intersect; for, after the intersection, the rectangles (from which the figure was composed) are turned the opposite way, and must therefore have the opposite algebraic sign. He proceeds to reason thus: — Suppose a copper-iron circuit, in which the hot junction is at the neutral tem- perature, and the other at any lower temperature : we cannot suppose the energy of the current to come from the heat of the hot junction : for, as the metals are there neutral to one another, the Peltier effect must be nil. Also the cold junction is heated, not cooled, by the current. Hence the energy can only come from one or other, or both, of the wires themselves, and it must come from them I70 HEAT. [CHAP. in virtue of the differences of temperatures of their ends. The assumption that the Peltier effect vanishes at the neutral temperature requires experimental proof, which has as yet been only partially furnished. We merely mention this to show that the experimental basis of the reasoning is not quite complete. Campbell {Proc. R.S.E. 1882) has to a certain extent supplied a qualitative verification, by showing that the ratio of tlie Peltier effects at two tempera- tures is consistent with that deduced from the thermo- electric diagram. It is probable that by his method the vanishing of the Peltier effect at the neutral point may be experimentally established. Thus Thomson was led to the conclusion that : An electric current in an unequally heated conductor, of one at least of two metals which have a neutral point, must produce abs'orptio7i, or disengagement, of heat, according as it passes from hot parts to cold, or vice vers A. After a series of elaborate experiments (described in the Phil. Trans, for 1855) Thomson found that: — An electric current in an unequally heated copper conductor behaves as a real fluid would do, i.e. it tends to reduce differ- ences of temperature. In iron it tends to exaggerate them. Thus, when the current passes from cold to hot in copper there is absorption of heat, and vice versct. In iron the effects are the opposite. This "Thomson effect" is some- times called Electric Convection of Heat. 194. Thus, Thomson speaks of the specific heat of vitre- ous electricity in a metal ; and regards it as positive in copper and negative in iron, in consequence of the differ- ence of behaviour just explained. Thomson gave, along with these remarkable results, the thermo-dynamical theory of the thermo-electric circuit in terms of Peltier effect and electric convection, but without assigning a special expres- 5CI.] THERMO-ELECTRICITY. 171 sion for the amount of convection or of the Peltier effect in terms of temperature. In fact, he says that the lines representing various metals, in the thermo-electric diagram, " will generally be curves." But he showed experimentally that the thermo-electric current, in a circuit of two metals, vanishes not only when the junctions have the same tem- perature (as in § 190), but also when their temperatures are equidistant from that of their neutral point. When one of the metals is represented by a straight line (as in the diagram of § 190) the other is therefore represented by a curve which crosses it at the neutral point, and has the same form and dimensions on opposite sides of that point. [The rest of the chapter is, for the most part, taken from the Rede Lecture for 1873.*] It has been found, by measurements of the E. M. F. of numerous pairs of metals through the whole range of mer- cury thermometers, that if the thermo-electric position of any one metal be represented by a straight line on the diagram, the lines for other metals are (in general) also straight. This involves no assumption other than that just named. So that whatever may ultimately be found to be the rigorous expression for electric convection of heat in terms of temperature, and therefore the . true form of the thermo-electric diagram, we may assert that if the diagram be so distorted (by shearing) that the line of any one metal becomes straight, those of the great majority of other metals will also become straight. From this it follows that the rate at which the thermo-electric power of two metals changes ■with temperature is constant. 195. Thomson's theoretical expressions, just alluded to, are based on the supposition that there is no reversible * Nature, vol. viii. pp. 86 and 122. 172 HEAT. [CHAP. thermal effect in the circuit but the Pehier and Thomson effects. If this be so, it follows from the theory of heat and the result just given (as will be shown in Chapter XXI.), that the Peltier effect is proportional to the thermo-electric power and to the absolute temperature. Also the (alge- braic) difference of the specific heats of electricity in the two metals is proportional to the absolute temperature. Hence, if there be a metal which has no specific heat of electricity at any temperature, it follows by theory from the above experimental result (§ 194) that : The specific heat of electricity in a metal is in general directly proportional to the absolute temperature, Le Roux, in 1867, made a valuable series of direct measurements of the electric convection of heat, and showed that it is null, or at least exceedingly small, in lead. 196. We are now in a position to explain the indications of the thermo-electric diagram, and to use it for the calcu- lation of the Peltier and Thomson effects, showing fully whence the energy of the thermo-electric current is derived. Let the cut represent a part of the diagram, two metals (copper and iron, suppose) being represented by the straight lines CC and //'. The vertical axis O E corresponds to the absolute zero of temperature. Draw lines parallel to O E, corresponding to t and /, the (absolute) temperatures of the two junctions, and cutting the lines of the metals in C, I, and C", /', respectively. Also (for simplicity) suppose both of these temperatures to be lower than that of the neutral point N. Then the E.M.F. of the copper-iron circuit is (§ 190) represented by the area C C I' I; and, as the figure is drawn, it produces a current going round the circuit in the positive direction as shown by the arrows, passing from copper to iron at the hotter junction. [The XI.] THERMOELECTRICITY. 173 reason why the line of copper has been drawn so as to rise towards the right, is the fact of its specific heat of electricity being positive (§ 194).] This area C C /' /will represent the amount of energy of unit current passing round the circuit. Now (by § 19s) the areaZX ^ C /' represents the Peltier effect (absorption of heat) at the hot junction if uiiit current were passing through it ; and DICE the Peltier effect (development of heat) at the cold junction under the. same D Hi ^ K' ^^^ T ^>^ T* ^ \ \..-- B' •e'"^ -^\ ^,^---'' ^ — C _^^j^- — £ \ i ' ' condition. And therefore the area £ B C C' will represent under the same condition the Thomson effect in the copper, D £l r /that in the iron. Both of these represent absorp- tion, for the current passes from cold to hot in copper, and from hot to cold in iron» It will be easily seen (by Euclid) that rc) I, + b{c+a) I^ bc+ca + ab In practice the resistance c includes that of the galvano- meter, which is usually large compared with either a or b; N 2 i8o HEAT. [chap. and then tlie above formula takes the sufficiently approxi- mate and exceedingly simple form al« + J>Ti a + b By the use of this artifice we can fill up all the gaps in the diagram. 201. To conclude this part of the subject, we give a brief table of approximate thermo-electric data for some common metals. These show how to draw the line on the diagram, between the limits o° and 350° C. Different speci- mens of the same metal do not agree very closely, so that the numbers are given on the same terms as those in §§ 104, 153, etc. Neutral Points with Lead. Sb - 156° Pb .... Rd + 132 Ir 00 Bi (-580), [The numbers in parentheses are not, of course, temperatures. ^ Specific Heat of Electricity. Ni (-424°) Ag -144° Sn + 75° Au (-276) Cu -132 Ru +136 •Arg -238 Za - 95 Al +212 Co -228 Cd - 59 Mg +239 Pd -172 Pt - 56 Fe +356 Fe - -00247 Pt (soft) . . . - -00056 Ir — -00000 Mg - -00048 Arg — '00260 Cd + -00218 Zn + -00122 Ag + -00076 Ru - -00106 Sb +'0XI40 Co - -00585 Au + -00052 _ Cu + -00048 Pb -00000 Sn + -00028 Pd --00182 Ni (to 175° C.) . .--00260 Ni (250°— 310° C.) .+ -01225 Ni (from 340° C.) . - -00180 Rd - -00058 Bi -'OOSS Al + -0002 The quantities in this last table are to be multiplied by the absolute temperature C. The unit employed corresponds XI.] THERMO-ELECTRICITY. i8i to about io~5 of the E.M.F. of a single Grove's cell. This is adopted because it enables the reader to form an idea of the extreme feebleness of the thermo-electric cur- rents in general. The mode of calculating the E.M.F. of a circuit of two metals will be given later, when we discuss the thermo-dynamic theory of the circuit. 202. So far, we have considered the wires employed to be each of the same material and section throughout. Magnus long ago showed that differences of section in the circuit, or in any parts of it, had no effect on the E.M.F. But it has been found that hammering, knotting, twisting, and stress in general, applied to one part of a circuit of one metal only, makes it capable of giving thermo-electric cur- rents when irregularly heated. The altered parts of the circuit behave to the unaltered parts as if they were of a different metal. In iron, nickel, &c., such effects can be produced by magnetising part of the wires. Also it has been found that if the two ends of a wire are at different temperatures, and be brought into contact, a current (of very short duration) is produced. 203. Rksume of §§ 189-202. General experimental pro- perties of thermo-electric circuits. Consequences. Thermo- electric power defined. Peltier Effect. Neutral Point. Thomson Effect. Specific heat of Electricity in a metal. Thermo-electric diagram. How it represents E.M.F., and the Peltier and Thomson effects. Mode of constructing the diagram experimentally. Anomalous behaviour of Nickel and Iron. Circuits in which there is (a) no Thomson effect, (p) no Peltier effect. Null effect of unequal thickness of wires. Effects of irregular stress in a wire. CHAPTER XII. OTHER EFFECTS OF HEAT. 204. In this chapter we propose to collect a few odd remarks on some of the effects of heat, the discussion of which would have unduly interrupted the continuity of the more important divisions of the subject already treated ; and to take advantage of the opportunity to introduce others which cannot as yet be strictly classified. We shall thus be able, for instance, to say a few words about some of the chemical aspects of the subject, other than the heat of combination to which the next chapter is devoted. 205. A curious example of mechanical motion, due directly to expansion, is afforded by what is commonly called the Trevelyan Experiment. A piece of heated metal is laid on a cold block, usually of lead, so as to touch it at a place which has been recently scraped or filed (to remove the oxide). If the heated mass be of such a form and so placed that it can rock easily on its supports, the rocking (once started) becomes very much more rapid, so much so as to produce a quasi-musical sound which often continues until the two bodies have acquired almost the same tempera- ture. The explanation of this phenomenon, given by Leshe and confirmed by Faraday, depends merely upon the CH. XII.] OTHER EFFECTS OF HEAT. 183 fact that at the point where the hot metal touches it the cold metal is so suddenly heated that it swells up, thus canting the rocker over to meet with similar treatment on the other side. Before it has come back again to its first position the swelling on the cold block has subsided, the heat having penetrated into the block by conduction, and thus the process is repeated until the temperatures of the two bodies are nearly equal. The effectiveness of the arrangement is often much increased by filing a notch in the lead, so that the rocker touches the block alternately on opposite sides of the notch. The rapidity of the vibrations, and therefore the pitch of the note produced, is usually much altered by pressing the rocker more or less forcibly on the cold block j and, when this process is skilfully conducted, sounds of the most extraordinary character, sometimes almost vocal, are produced. 206. An interesting example of the effect of heat, upon what are usually called "molecular" forces, is supplied by the modifications produced on capillary phenomena. In general the curvature of the surface of water and other liquids in a capillary tube becomes less as the tem- perature is raised; and at the same time the surface tension also becomes less; so that, on both accounts, the height to which the liquid rises, or the depth to which it is depressed, in a capillary tube, becomes less as the temperature rises. The former phenomenon is easily seen by careful in- spection, and the proof of the latter is established by a multitude of simple experiments. Thus if a thin layer of water of considerable surface be placed on a large metal plate, and a ^mall lamp flame be 1 84 HEAT. [chap. applied under the middle of the plate, the effect of the lamp is of exactly the same kind as that which it would produce on a tightly stretched sheet of India-rubber. The greater surface tension of the colder water enables it to tear asunder the film at the surface of the hotter water, and thus we have the same sort of result as we get by putting a single drop of alcohol or ether on the middle of the water surface. In the course of a little time the water is dragged away on all sides from the heated part, in the one case, as it is in the other case from the point at which the alcohol was applied. Thus we see why a drop of solder seems to be repelled from the hotter to the colder parts of a soldering iron. 207. Another curious fact, closely connected with this part of the subject, is the effect of the curvature of a liquid surface upon the pressure of saturated vapour which is in equilibrium in contact with it. When a number of narrow tubes, open at each end, dip into water in a receiver (free of air, let us suppose) the water surface is in each raised in proportion as its curva- ture is greater. Also the vapour pressure is less as the surface is more raised. Hence, the more concave is a water- surface, the less is the pressure of saturated vapour in contact with it. If the tubes be now closed at the bottom, with a little water in each, evaporation or conden- sation will take place according as the water level is now higher or lower than before, and will proceed until the heights above the external water-surface become the same as when the lower ends were open. W. Thomson, who first called attention to this phenomenon, considers that it accounts for the great amount of water absorbed from damp air by bodies with fine pores or interstices, as cotton-wool, &c. Clerk-Maxwell pointed out that it also accounts for the growth of the larger drops in a cloud at the expense of XII.] OTHER EFFECTS OF HEAT. 185 the smaller ones ; and, at least in part, for that collapse of small bubbles of steam, which gives rise to the so-called " singing " of a kettle of water just before boiling commences.. 208. This leads us to make a few more remarks on the. subject of boiling. Water boils at a lower temperature in a metallic vessel than in one made of glass. If water be carefully deprived of air, it is found that with care its temperature may be raised in a smooth vessel many degrees above the boiling point corresponding to the pres- sure to which the water is subjected. But if some chips or filings of metal, or any solid with sharp angular points or edges, be dropped in, it suddenly boils with violence. It is possible that this property may have something to do with boiler-explosions, at least when no fresh supply of water has been put in for some time before their occurrence. These facts have not yet been thoroughly explained, but it would appear that a nucleus of some kind is required for boiling, just as (§ 176) it is required for condensation. 209. A very singular example of explosive boiling is furnished by the geysers, or volcanic hot springs, of Iceland. The phenomena exhibited by these were carefully examined by Bunsen and Descloiseaux (Liebig's Annalen, 1847). They determined the temperature of the water at different depths in the shaft of the Great Geyser, at various intervals before and after a grand eruption ; and on these observa- tions Bunsen founded his theory of the action which takes place. Between two eruptions it was found that the temperature of the water in the shaft increased from the surface down- wards, but in no place rose so high as to reach the boiUng- point corresponding to the sum of the pressures of the atmo- sphere and of the superincumbent water. The temperature 1 86 HEAT. [chap. steadily rose at every point, steam and very hot water being supplied from below and probably also at the sides of the shaft until a small upward displacement of the water column into the basin lowered the pressure to the boiling-point at one part of the shaft. This usually occurred at a depth of somewhere about seventy feet under the surface. Then there was violent boiling, blowing out the Upper part of the column of water ; and thus relieving the lower part from pressure, so that it also boiled explosively. Then colder water, from the geyser basin, ran back into the shaft, and the gradual heating from below recommenced. It is easy to exhibit the phenomena on a small scale, by artificially producing the observed state of temperatures, and thus to fully verify the sufficiency of Bunsen's theory. 210. In §§ 60, 134, while defining the fixed points of the Newtonian thermometer scale, we used the terms pure ice and pure water. This implies, of course, that the freezing and boiling points of water (and of other liquids), are affected by impurities held in solution. [Mere mechanical suspension of impurities, as in the case of muddy water, produces in general no measurable effect] The most important case, so far as the freezing point is concerned, is when water contains common salt in solution. It is found that the temperature of the freezing point is then notably lowered. Walker (in the Fox Expedition) found that the freezing temperature of ordinary sea-water is below 28°"S F. He observed, however, that a great deal of salt is extruded from the ice. So he melted the ice^ and re- froze it, severaHimes in the hope of thus obtaining drinkable water. Although the freezing-point was raised by each operation, the water was still brackish after four repetitions of the process. The lowest specific gravity of the liquid thus obtained was from i"oo25 to i'oo2. Healso tried the XII.] OTHER EFFECTS OF HEAT. 187 converse process, removing in succession the various ice- crusts formed at lower and lower temperatures from a tub full of sea water, in order to find how strong a brine could thus be procured. 211. Very extensive and careful measurements have been made of the boiling-points of aqueous solutions of various salts, of different degrees of concentration. In all cases the boiling-point is above 100° C, and the amount of rise is approximately proportional to the percentage of salt in solution : — the co-efficient of proportionality being dif- ferent for different salts. Such solutions, when boiling, give off almost pure water-vapour; and it is sometimes stated that, however high be the boiling-point of the solution, the vapour comes off at the temperature corresponding (in Regnault's table) to the pressure. There can be no doubt, from special experiments made by Regnault and others, that a thermometer placed in the issuing vapour does indicate very nearly this temperature corresponding to the pressure. But it is to be remarked that the aqueous vapour just before it leaves the liquid is certainly superheated ; and that superheated vapour leaving the liquid freely in a par- tially closed vessel soon becomes saturated vapour, which deposits a layer of water on the bulb of the thermometer. The temperature of the bulb is therefore that at which pure water is in equilibrium with water-vapour at the pressure to which they are exposed, so thatthe method of observation is fallacious. We do not yet know, by any certain method, what is the exact temperature of vapour leaving a saline solution, boiling under ordinary pressure at a temperature above 100° C. 212. The rise of the boiling-point produced by salts in solution is attributed to the molecular attraction between the salt and the water. Hence, when the circumstances of i88 HEAT. [chap. the experiment are altered, so that pure steam at ioo°C. is made to pass into an aqueous solution of a salt, condensa- tion of steam (with its accompanying disengagement of latent heat) goes on until the mixture rises to its boiling- point. Thus, with steam at loo'^C. such solutions maybe raised to temperatures very considerably higher. In Leshe's mode of freezing water, the greater part of the heat developed in the sulphuric acid is the latent heat of the vapour absorbed. In the next chapter we shall see how to account for the rest. 213. What is called the spheroidal state of liquids is a phenomenon which, in one at least of its many forms, has been known from remote times. When a laundress wishes to test whether a flat-iron is sufficiently hot, she dips her finger in water and allows a drop to fall gently on the iron. When the iron is not much above 100° C. in temperature, the drop spreads over the surface, and rapidly boils away. But if it be considerably hotter, the drop glides off from the surface without wetting it, and without suffering much evaporation. [See Phil. Mag. 1850, I. 319.] The phenomenon is easily studied by dropping water cautiously from a pipette into a shallow platinum dish heated from below by a Bunsen lamp. When the dish is sufficiently hot, the water appears to behave on it very much as it does on a cabbage-leaf or on an oiled or greasy surface. The water is, of course, not in contact with the metal. This can be verified in many ways. For instance, if the dish be slightly convex instead of concave, a ray of light can be made to pass between it and the drop. Poggendorff con- nected one pole of a battery with the water and the other with the dish, and found that no current passed. The temperature of water in the spheroidal state is found to be about 95" C. only. XII.] OTHER EFFECTS OF HEAT. 189 The force required to support the drop is easily calculated. Thus, suppose it to be a square inch in lower surface, and a quarter of an inch thick. A water barometer stands at about thirty-three and a half feet at the mean atmospheric pressure. Hence the additional pressure required to support the drop is only about i/33"5 X 12 x 4, or roughly, 1/1600 of the at- mospheric pressure. This is supplied, as will be easily seen when we are dealing with the Kinetic theory of gases, by the momentum acquired by air and vapour particles which have come in contact with the hot surface. On leaving it they move in directions more nearly perpendicular to the surface than those in which they impinged ; i.e. more nearly verti- cal : — and thus, in the very thin layer between the water and the metal, the gaseous medium exerts a somewhat greater pressure in a vertical than in a horizontal direction. To produce a similar result in a thicker layer the gas or vapour must be rarefied, so that the mean free path (Chap. XXII.) may be much increased in length. 2 14. This simple consideration gives the explanation of the chief phenomena shown by the Radiometer. This instru- ment consists essentially of a set of very light vanes attached to an axle, about which they can freely turn. One side of each vane is blackened (so as to absorb heat), the other side is polished. The whole is fixed in a partial vacuum. When it is exposed to radiation there is greater gaseous pressure on the blackened than on the poHshed sides of the vanes, and the apparatus consequently rotates.^ The first experimental results which seem to have had a direct bearing on this explanation are due to Fresnel, but they were left unnoticed. 215. With care water may be kept in the spheroidal state in a glass vessel, such as a watch-glass. But to insure success ^ Dewar and Tait, Nature, xii. 217. jgo HEAT. [CHAP. in this experiment the water must be nearly at its boiUng- point when it is dropped on the glass. We may even have a spheroid of water above the surface of very hot oil. But here great caution is requisite, as explosions often occur, scattering the hot oil in all directions. 216. Another striking form of the same experiment is to dip under water a solid ball of silver heated almost to its melting-point. The ball is seen to- glow in the middle of the water for a few seconds (cooling, however, more rapidly than it would have done in air), when suddenly the water comes in contact with it, a slight explosion occurs, and all is dark. The experiment is considerably facilitated by previously adding some ammonia to the water. 217. Brewster long ago discovered that in many speci- mens of topaz and other crystals there are cavities, usually inicroscopic, which are partially filled with liquid. Sang observed that little bubbles of gas in the liquid in the cavities of Iceland spar seem to move towards the side of the cavity at which the temperature of the crystal is slightly raised. If part of the walls of the bubble be formed by the solid, the explanation of this curious fact probably involves the capillary phenomena described in § 206 above. But when the bubble is wholly surrounded by liquid, it would seem that the motion is only apparent, the liquid distilling- from the warmer side of the bubble and condensing on the colder. As the liquid is usually carbonic acid, and under considerable pressure, this explanation seems to accord with known facts. Very similar phenomena can be produced on a comparatively large scale by holding a warm body near one end of a small free bubble in a sealed tube containing liquid sulphurous acid. 218. Davy's ingenious safety-lamp is merely an ordinary lamp surrounded by wire-gauze. He found by trial that XII.] OTHER EFFECTS OF HEAT. 191 what is called flame cannot, except in: extreme cases, pass through such gauze. If we lower a piece of wire-gauze, whose meshes are not too wide, over the flame of a Bunsen lamp, the flame is arrested by the gauze, although the un- consumed gaseous mixture which passes through the meshes is highly inflammable. This may be proved by applying a 192 HEAT. [CH. XII. lighted match to it. Or we may operate with the lamp unlit, when it will be found that the mixed gases can be inflamed above the gauze without igniting the explosive mixture below it. It will be seen, in the next chapter, that the ignition of such a mixture begins at a definite tempera- ture. The conducting power of the gauze (Chap. XIV. below) prevents the heated part from being raised to this temperature. Our further remarks on flame will come more suitably in next chapter. 219. Resume of §§ 204-218. Trevelyan experiment. Effects of heat on molecular forces. Vapour pressure over curved surface of liquid. Boiling under abnormal condi- tions. Geysers. Freezing and boiling points of aqueous solutions. Spheroidal state. Radiometer. Motion of bubbles. Safety-lamp. CHAPTER XIII. COMBINATION AND DISSOCIATION. , 220. Refer again to §§ 47, 51, 155 — 7. As we have already remarked (§ 38), we know almost nothing as to the state in which energy exists in a body. We can measure, in general, the amount which goes in, or which comes out ; and we can tell in what forms it does so. But we have seen that molecular changes such as melting, solution, solidification, crystallisation, &c., are usually associated with absorption or evolution of heat. We attribute these thermal phenomena to the work done against, or by, the so-called molecular forces. This is, of course, merely an hypothesis ; so framed, however, as to fit in with the law of conservation of energy. The actual nature of the process is wholly unknown to us. It is not to be expected, then, that in the present state of science we should have arrived at anything more satis- factory than this in connection with the profound changes which take place in what is called chemical combination. Still, the little we do know is of great importance from the physical, as well as from the more purely chemical, point of view. And, from the purely practical point of view, it is one of the most important parts of our subject. For it is directly concerned with almost all our artificial processes for producing heat. 194 HEAT. [chap. 221. It might at first sight appear, from some well-known experiments, that no very definite information is to be had on such subjects. For we can make the same two bodies combine in many different ways. Thus, to take a simple example, we may burn a jet of hydrogen in an atmosphere of oxygen, both originally at any one ordinary temperature ; or we may mix together, once for all, in the proper proportions, and at the same temperature, our oxygen and hydrogen, and apply a lighted match or an electric spark to the mixture. The one process may be made to take place almost as slowly and as quietly as we please ; the other (in a vessel of moderate size) is practically instantaneous,^ and is ac- companied by all the physical concomitants of a violent explosion. But, if we consider the two processes in the light of Carnot's fundamental principle (§ 85), we see that when the water produced in each case is of the same amount, and has reached the same final state, the amounts of energy set free must also be the same. In the case of slow combustion it is given out almost entirely in the form of heat ; in the explosion a great part appears at first as sound and ordinary mechanical energy : but both of these ultimately become heat. And the quantity of heat thus pro- duced must be the same in the two cases, when the products have arrived at the same final state. So far, well. But suppose that the mixed gases, just before the explosion, had been raised to a higher temperature than that at which the slow combustion took place, would the whole amount of energy involved in the explosion be the same as in the former case ? We simply mention this difficulty for the moment. 222. It would take us far beyond our imposed limits to ' There is a definite rate at which a surface of combination runs along in each explosive mixture of gases, but it is usually considerable in comparison with the dimensions of any ordinary apparatus. XIII.] COMBINATION AND -DISSOCIATION. 195 give even a complete sketch, without details, of a subject like this, which has been developed in many directions by careful experimenters. We therefore content ourselves with a mere mention (in addition to what has been said in §§ 69, 155, 156 above) of such facts as the following : — ■ Heat is developed, often in considerable amount, when gases such as chlorine, hydrochloric or hydriodic acid, &c., are dissolved in water. Part of this heat is, of course, due to the change to the liquid from the gaseous form. But the rest is to be accounted for by the same process as are the facts which follow. When liquids, e.g^. sulphuric acid of commerce, alcohol, &c., are diluted with water, there is generally evolution of heat accompanied by diminution of volume. But it is' not always the case that there is evolution of heat in the mere mixture of liquids. Thus, although when 27 parts (by weight) of water and 23 of alcohol are mixed at the same ordinary temperature, the temperature of the mixture rises about 8°'3 C; if we mix 31 parts (by weight) of bisulphide of carbon with 19 of alcohol at the same temperature, the temperature of the mixture is lowered by 5°'9 C. Now it has been found by direct experiment that, in the former case, the water-equivalent (§ 183) of the mix- ture is about -^th greater than the sum of those of the con- stituents ; while in the latter case the excess is only about iVth. On the other hand, the volume of the first mixture is 3-6 per cent, less than the sum of the component volumes; with the second mixture it is 17 per cent. greater. The first of these supplementary facts would tell against the observed results, the second in favour of them. The subject is thus easily seen to be one of considerable complexity, and we cannot farther develop it here. We may mention, however, that experiments have been o 2 196 HEAT. [CHAP. carried out with mixtures of the same pair of liquids at various (common) initial temperatures. And it appears probable (according to Berthelot) that the specific heats of mixtures vary as much with temperature as do those of simple liquids — so much so, in fact, that the mixture of bisulphide of carbon and alcohol mentioned above would give rise to evolution of heat if both components were originally at a temperature lower than 0° C. 223. The application of the great laws of thermodynamics (§§ 37) 82) to chemical combination is, in its elements at least, a matter of no great difficulty. The experimental part of the work has been carried out with great ability by many investigators, prominent among whom are Andrews, Favre and Silbermann, and (more recently) Berthelot, Deville, and Thomsen. We must confine ourselves to the more salient features of the subject. It follows from the first law that, if any mixture undergo a definite chemical change without taking in or giving out work, the heat developed or absorbed depends solely upon the initial and final states of the system, and in no way upon the intermediate transformations. This is, in one respect, a great gain ; for it much simplifies the reasoning from the results of experiment ; but it is also a great loss, inasmuch as it pravents our obtaining (by this means alone) any in- formation as to the nature and order of the successive steps of a complex reaction. Again, in any transformation which takes place without the application or the giving out of work, the heat developed is the equivalent of the excess of the original over the final potential energy due to the chemical affinities involved. [We have spoken here only of heat developed. The reason for this restriction will appear at once from the next statement] XIII.] COMBINATION AND DISSOCIATION. 197 In accordance with the second law of thermodynamics (or its consequence, the degradation of energy), the final state of every combination is that in which the potential energy of chemical affinity is a minimum ; or, what comes prac- tically to the same thing, the reactions which take place are such as to develop the greatest amount of heat. And, conversely, compounds which are formed in this way require for their decomposition a supply of energy from without, 224. Thus, to take a simple case, suppose equal weights of hydrogen to enter into combination with oxygen — no matter how. There are but two compounds which (so far as we know) can be formed, water and peroxide of hydro- gen. The heat of combination is found to be nearly 50 per cent, greater for the former product than- for the latter. Hence, when hydrogen and oxygen act directly on one another, water alone is formed, and the excess of either gas is simply unacted on. And, by the first of the three statements of § 223, if we wished to cause water to be oxidised into peroxide of hydrogen, we should have to supply energy equal to the excess of the heat of combina- tion of two atoms of hydrogen with one of oxygen, over that of two atoms of hydrogen with two of oxygen. Hence we should expect to find, as in fact we do find, that peroxide of hydrogen is essentially an unstable com- pound. The energy required for the formation of such compounds is usually supplied from some direct reaction (forming a stable compound) which takes place at the same time. Thus peroxide of hydrogen is obtained during the formation of chloride of barium from peroxide of barium and dilute hydrochloric acid. 225. We have seen that a compound, which is formed by the direct action of its components with the evolution of heat, is essentially stable. To decompose it, external energy 198 HEAT. [chap. is required. Thus, in the case just cited, peroxide of barium and hydrochloric acid are each essentially stable, for heat is given out when either is formed directly from its elements. But a mixture of the two is not stable, for a rearrangement of elements is possible by which farther heat can be developed. Here the energy required for the decom- positions is supplied by a chemical process. But there are many other ways in which it can be supplied. Thus heat itself, as in the formation of quick- lime from limestone, may be directly the agent. Light also is effective under certain conditions, as in the decom- position of carbonic acid in the leaves of plants, or of salts of silver on a photographic plate. And the discovery of the alkaline and earthy metals, Davy's grandest contribution to chemistry, was effected by the electric decomposition of their oxides. On the other hand, compounds which are formed, with absorption of heat are often liable to spontaneous decom- position. This is the case with many highly explosive bodies, such as chloride of nitrogen and the oxides of ' chlorine. 226. The displacement of one element by another from a compound gives an excellent instance of dissipation of energy. Thus the heat of combination of iodine with hydrogen or other metal is less than that of chlorine with the same metal — and chlorine has, in consequence, the power of decomposing such iodides, setting the iodine free. But the whole of this part of the subject is much more easily studied by measuring the energy, given out during a transformation, in its electrical forms than in the form of heat ; and therefore, so far as it is not chemical, it belongs more properly to Electricity than to Heat. Although all combinations take place in accordance with xiiLJ COMBINATION AND DISSOCIATION. 199 tbe laws of energy, these laws alone do not enable us to determine the result in any particular case. We must have special data with reference to each pair of the sub- stances involved. These, of course, can be found only by experiment. As they belong much more directly to Chemistry than to Heat, we cannot enter minutely into what is known of them, but will quote, as a specimen, a few general results of one branch of this extensive subject as given by Berthelot. Strong acids and strong bases, when made to combine in equivalent proportions, each being previously dissolved in a sufficient quantity of water, evolve nearly equal amounts of heat in the formation of stable neutral salts, whatever be the acid or the base. The heat disengaged is but slightly altered by the presence of greater quantities of water, or of additional quantities either of the same, or of another, base. Examples of this are furnished by the neutral salts of soda, potash, baryta, &a, with hydrochloric, nitric, sulphuric, &c., acids. Even in the formation of the alkaline salts of strong acids more heat is disengaged than in the combinations of feeble acids with the same alkaline bases. The feeble acids form, even with the strongest bases, salts which are decomposable by water to an extent in- creasing with the amount of water, but less as there is greater excess of base or acid present. The increase of decomposition goes on when water is added, without limit in the case of some feeble acids, but tends to a definite limit with others. But there are instances in which a moderate amount of water wholly effects the de- composition, so that the effect on the thermometer is almost exactly the reverse of that due to the formation of the salt. 200 HEAT. [CHAP. 227. To illustrate some of the above, and a few further, remarks, we give a little table of roughly approximate values of Heat of Combination. H" with O . . . . 68,000 . . 34,000 H= with 0° . . . 45,000 . . . 22, SCO C with O . .30,000. . . 2,500 C withO' ... . 97,000 . . . 8,100 CO with O 67,000 . 2,400 The first column indicates the substances combining, and the proportions in which they combine. Here H, O, C, stand for "atoms" (not "molecules") of hydrogen, oxygen, and carbon, in the chemical sense of the term ; so that their relative masses are as i, 16, and 12. The second column gives the number of units of heat (§ 179) produced when 2 lbs. of hydrogen, iz lbs. of carbon, or 28 lbs of carbonic oxide, respectively, enter into the oxygen combination indi- cated. The numbers in the third column give the units of heat set free per lb. of the substance to be oxidised. The processes by which these numbers were obtained were all essentially such as to reduce the compound to the same (ordinary) temperature as that of the components, and to measure the amount of heat given out during the reduction of temperature. Thus, when a pound of hydrogen is oxidised into 9 lbs. of water, 34,000 units of heat are given out. If it could be directly peroxidised, only 22,500 units would be given out. Hence, to effect the farther oxidation of gibs of water, energy equivalent to the difference of these, or 11,500 units of heat, must be supplied. And we also see that the number of units of heat set free by oxidising a quantity of carbon at once into carbonic acid, is the sum of those corresponding to the two stages — Xiii.] COMBINATION AND DISSOCIATION. 201 formation of carbonic oxide, and its subsequent farther oxidation. 228. Many combiriations, even of the most vigorous kind, do not take place at ordinary temperatures. Even the ex- plosive mixture of one volume of oxygen with two of hydro- ' gen requires a start, as it were. If the smallest portion of the mixture be raised to the requisite temperature (as by a match or an electric spark) the heat developed by the combination suffices to raise other portions, in turn, to the proper temperature. It is not yet certainly known how a jet of hydrogen is inflamed by contact ,(in air) with spongy platinum. It is possible that this may be due to the heat developed by its sudden condensation, in which case this would be a phenomenon of the same class as that just mentioned. But it may be due to the mere approximation of oxygen and hydrogen particles by surface condensation. 229. We are now prepared to consider some of the chief phenomena of ordinary combustion. For the table, meagre as it is, shows how very large are the quantities of heat developed in some of the commonest cases of combination. When the gaseous products of a combination are raised to a sufficiently high temperature to become self-luminous, they form what is called a flame. We will take one case with a little detail. 230. An excellent and typical instance, which has been very carefully studied by Deville, is that of a blowpipe flame fed with a mixture of oxygen and carbonic oxide, escaping from an orifice of i^^j^th of a square inch in area under a pressure of from half an inch to an inch of water. The flame proper consists of an exterior cone, three to four inches long, vividly blue in its lower parts and of a very faint yellow at th^ apex. Within this there is an interior cone, barely half an inch long, which consists of as 302 HEAT. [chap. yet unignited gas. Just at the apex of this interior cone, where the gases have just begun to burn, the flame has the highest temperature, and it is there capable of readily melt- ing even a stoutish platinum wire. The temperature of the flame falls off rapidly from this point upwards. [The velocity of propagation of the combining temperature (foot- note to §221 ) is obviously that of the jet of gas at the apex of the interior cone.] To find the composition of the luminous matter at any assigned point of the axis of the flame, Deville employed a thin tube of silver, about i^ths of an inch in diameter, which was kept cool by a current of cold water passing through it. This tube was fixed transversely in the flame, in such a way that a small hole (xiirth of a inch in diameter) in its side occupied the assigned position in the axis of the flame, and faced the orifice of the jet. The cold water was made to pass so rapidly as to produce suction through this small hole. Thus a sample of the gases of the flame was carried along in the current of water. This was led, after escaping from the water, through caustic potash to remove the undissolved carbonic acid, and then carefully analysed. In order to determine from what amount of the unignited mixture this combustion product had been furnished, the experimenter had previously added to the mixture a known (small) percentage of nitrogen. This gas, of course, passed almost entirely to the collecting vessel, the only loss being due to slight diffusion through the flame. -A. great number of concordant determinations made by this method showed that at the apex of the flame there is practically nothing but carbonic acid (provided, of course, that the original combustibles were mixed in the proper pro- portion). But, from the apex downwards, the proportion of XIII.] COMBINATION AND DISSOCIATION. 203 uncombined to combined gases steadily rose, till at the apex of the interior cone (where the temperature is highest) the uncombined gases formed about -Ird of the whole. This result is, at first sight, somewhat surprising, but its explana- tion is not far to seek. We must take account of the dis- sociation (§ 47) due to the high temperature. 23 1. On the theoretical aspects of dissociation we intend to make some remarks later. Meanwhile we may, roughly, assimilate its laws to those of evaporation. We have seen that a liquid, at any definite temperature, is in kinetic equilibrium at its free surface with its own vapour at a definite pressure (§ 162), so that, at the common boundary, there is constant condensation accompanied by an equal amount of evaporation, with equal continuous disengage- ment and absorption of latent heat. Thus, in a compound gas, at least at any one temperature within certain limits (which now probably depend on the pressure), there is kinetic equiUbrium maintained by a constant amount of dissociation, accompanied by an equal amount of recombination, and also with equal continuous disengagement and absorption of heat, the heat of combina- tion. The percentage of the whole, which is dissociated, rises from zero at the lower limit of temperature (at, and below, which there is no dissociation) faster and faster till about half way, and then slower and slower till the upper limit (at, and above, which the whole is dissociated) is reached. This half-way temperature, at which one half of the gas is dissociated, is called the temperature of dis- sociation. When the mixed gases are gradually heated, they cannot begin to combine till a certain temperature, depending on the constituents and their relative proportions (in a way not yet ascertained), is reached. Thus we see houi the explosive 204 HEAT. [CH. XIII. mixture (§228) requires a "start." And, if the constituents be such as to combine directly with evolution of heat, this com- mencement of combination is sufficient of itself to raise the temperature, with the effect of producing farther and farther combination. This goes on (provided the gas lose no energy to external bodies) until kinetic equilibrium is established, i.e. until the percentage of the mixtui-e which is in the combined state corresponds to the temperature reached by the whole in consequence of the heat of com- bination set free. If, however, the gas be permitted to lose energy, as by contact with a cold vessel or by expanding and doing work, the process of combination will go on farther and may become complete, in which case the final temperature must be under the lower limit. Similar consequences may be traced if we suppose that the gases be first individually heated above the higher limit, then mixed, and the temperature gradually reduced. 232. R'esumi of §§ 220-231. Different modes of com- bination. Heat or cold produced by mixing. Direct con- sequences of the Laws of Thermodynamics. Stability of compounds. Displacement. Heat of combination. Com- bustion. Composition of flame. Dissociation, CHAPTER XIV. CONDUCTION OF HEAT. 233. Refer to §§ 4, 73. We have already seen that bodies in contact ultimately acquire the same temperature. This experimental fact is the necessary basis of almost all our methods of measuring temperature. The same must, of course, be true of contiguous parts of the same body. This can only occur by transference of heat from part to part of the body. For the present we need not inquire whether this equalisation is effected by the mere passage of heat from the warmer to the colder parts, or whether there is a mutual interchange in which the warmer part gives more to the colder part than it receives from it. So far, indeed, as the. ultimate result is concerned, it does not matter which of these is the correct view, for we cannot identify any portion of energy; though the question is theoretically of great interest and importance, as we shall find when we come to Radiation. To this latter head also we will refer all cases in which the transfer of heat in a body appears to take place directly between parts at a finite distance from one another. 234. That bodies differ very greatly in conducting- power for heat, is matter of common observation. We hold without inconvenience one end of a short piece of 206 HEAT. [chap. glass, even wlien the other end is melting in a blowpipe flame. No one would try the same experiment with a similar piece of copper or even of iron. The pieces of wood or bone which are inserted between the handle and the body of a metal tea-pot owe their presence to a recog- nition of the same fact. So does the packing of the interior of a Norwegian cooking-stove. Two of the ordinary forms of lecture experiment may also be mentioned here. First, that of Ingenhouz, in which a metallic trough, from the front of which a number of rods of the same diameter, but of different materials, project. These rods are covered with a thin film of bees-wax, which melts whenever it is aliiiiiiii raised to the proper temperature. Suppose the whole to be at the temperature of the air, and hot water to be suddenly poured into the trough. By watching the limits of the melted wax we can judge of the relative thermometric conducting powers of the various materials. A less objectionable form of the experiment consists in placing end to end two bars of iron and copper, of equal sectional area, having small bullets attached by bees-wax to their lower sides, and placing a lamp flame impartially between them. The temperature at which the wax softens sufficiently to let the bullets fall is found to travel much xiv] CONDUCTION OF HEAT. 207 faster along the copper than along tlie iron. And here it does really indicate superior conducting power on the part of the copper, for the thermal capacities (§ 183) of iron and copper are not very different. 235. Some creditable work was done on this subject last century by Lambert, But the first to give a thoroughly satisfactory definition of conducting power, or conductivity {not, in English at least, conductibility or conducibility) was Fourier. The entire subject of the present chapter, with all but its later experimental developments, may be said to have been created by his Thkorie Analytique de la Chaleur, first published in a complete form in 1822 ; and these later developments may be said to have been suggested, or ren- dered possible, by Fourier's work. Its exquisitely original methods have been the source of inspiration of some of the greatest mathematicians ; and the mere application of one of its simplest portions, to the conduction of electricity, has made the name of Ohm famous. And more, in this work of Fourier's, attention was first specially called to the extremely important subject of Dimensions of physical quantities in terms of the fundamental units. [See Chapter XIX.] 236. Conducting power must, ceteris paribus, be mea- sured by the quantity of heat which passes. We must therefore find how to estimate the amount of heat passing 2o8 HEAT. [chap. from part to part of any one body, and what are the cir- cumstances which have influence on it. We can then place two different bodies under precisely the same determining circumstances, and thus compare their conductivities by comparing the quantities of heat transferred. 237. The simplest mode of passage is that in which the transfer of heat takes place, throughout a body, in parallel lines. Thus, for instance, when a lake is covered with a uniform sheet of ice, the ice grows gradually thicker if the temperature of the air be below the freezing point. This is caused by the passage of heat through the ice from the water immediately below it, the upper surface of the ice being kept colder than the lower. Thus the layer of water next the ice freezes. And as the temperature is the same throughout any horizontal layer, whether of water or ice, the transference of heat is wholly perpendicular to such layers, i.e. it is vertically upwards. Hence we choose, as our typical form of conducting solid, a plate of uniform thickness and of (practically) infinite surface, whose sides are kept permanently at two different temperatures. After a lapse of time, theoretically infinite but in general practically short, a permanent state of distribution of temperature is arrived at throughout the slab, and therefore every- layer parts with as much heat as it receives. If, under these circumstances, we measure the quantity of heat which passes through a square inch, foot, or yard of any layer of the plate in a minute, we may take this as representing numerically the conducting power. To compare plates of different substances, we must of course take them all of the same thickness, and use the same pair of temperatures for their sides. Thus we are led to define as follows : — The thermal con- ductivity of a body at any temperature is the number of units XIV.] CONDUCTION OF HEAT. 209 of heat which pass, per unit of time, per unit of surface, through an infinite plate (or layer) of the substance, of unit thickness, when its sides are kepi at temperatures respectively half a degree above, and half a degree below, that temperature. The units which we employ are, as hitherto, the foot, the minute, the pound, and the degree centigrade. In a special chapter we will show how to pass ro any other system of units. [It will be observed that conductivity may, so far as the above definition goes, depend upon the temperature of the. body. This will be considered later.] 238. The methods chiefly employed for measuring thermal conductivity depend ultimately upon observations oif tem- perature of the conducting body at different parts of its mass. Thus the amount of heat which passes across any surface in the body, one of the quantities iri terms of which the conductivity is defined, is deduced from the change of temperature which it produces in the body itself. Now the temperature effects of a given quantity of heat are inversely as the thermal capacity (§ 183) of the body. Hence what we directly deduce from such experiments is not the con- ductivity itself, as defined above, but its ratio to the thermal capacity. [This ratio is called by Maxwell the Thermometric Conductivity, and by Thomson the Thermal Diffusivity.'] Thus it follows that the determination of conductivity re- quires that these methods be' supplemented by a separate set of experiments for the determination of specific gravity and of specific heat. 239. Let us now think of the state of things in the interior of the slab while a steady passage of heat is going on. The only reason why it goes on is that every layer differs in temperature from those adjacent to it. And, because it goes on to the same amount across each (thin) layer, the p 2IO HEAT. [chap. difference of temperatures ot the sides of such layer must be proportional to its thickness. [This is easily seen by sup- posing the layer to be divided into a great number of sub- layers of equal thickness, and then considering two, three, or more adjacent ones as a single layer.] Thus the amount of heat .passing across any layer depends ceteris paribus on the gradient of temperature across that layer. This may usefully be substituted for the difference of temperatures of the sides of a layer of unit thickness. For we thus have the means of calculating the amount of heat which passes across any layer, even when there is a variable state of temperature : — as for instance when one side of the slab is kept at a constant temperature while the other is alternately heated and cooled. 240. AH experimental results hitherto obtained are con- sistent with the assumption, which must evidently be true for small gradients, that the amount of heat which passes across any layer, kept at a given temperature, is directly proportional to the gradient of temperature perpendicular to that layer. [It is necessary to point out that the exten- sion to high gradients is an assumption found to be consistent with facts, because conductivity is generally measured under circumstances in which the gradient is very much greater than i°C. per foot, which is that implied in our definition. And we cannot yet positively assert that the quantity of heat which would pass through a slab a millionth of a foot in thickness is exactly a million times as great as that passing through a slab one foot thick, when the surface tempera- tures are the same, even if the conductivity of the material were the same at all temperatures. The corresponding prt)- position in electric conduction (which is called Ohm's Law, though directly based on the splendid work of Fourier) in- volves the same assumption, but here we can miich more XIV.] CONDUCTION OF HEAT. 21 r easily verify it, and the verification has been carried to extreme lengths by Maxwell and Chrystal.] 241. It is practically impossible to realise experimentally the simple conditions which are assumed in the above definition of conductivity. Hence, in order to measure the value of this quantity in different substances, experi- mental arrangements of a somewhat more complex character must be adopted. Two chief principles have been employed as the bases of experiment. The first requires a steady state of tempera- ture, and a consequent steady flow of heat. The second requires a variable, but essentially periodic, state of tem- perature, maintained by external causes in one part of a body and thence propagated as a species of wave. In both we calculate, in terms of the (known) gradients of temperature and the (unknown) conductivity, how much more heat enters an assigned portion of a body by con- duction than leaves it by the same process. If the tem- perature remain constant, this part of the body must lose heat otherwise than by conduction ; if it do not so lose heat, its temperature must rise. In the first method, as each part of the body maintains a steady temperature, we must measure by some independent process how much heat the assigned portion loses otherwise than by conduc- tion. In the second, we must equate the gain of heat by conduction to the quantity of heat indicated by the thermal capacity of the element and its change of temperature* A third method of some celebrity involves both of these considerations. 242. The first method was employed by Lambert, in an imperfect form: but was afterwards greatly improved by Forbes, who obtained by means of it the first absolute determination of conductivity of any real value. The p 2 2IZ HEAT. [chap. principle of the method is extremely simple, but the experimental work and the details of the calculation from it are both very tedious. A long bar of uniform cross section has one end raised to, and maintained at, a definite high temperature, while the rest is exposed directly to the air of the laboratory. Small holes are bored in the bar at regular intervals ; these (when the bar is of any other metal than iron) are lined with thin iron shells, and in them the cylindrical bulbs of accurate thermometers are inserted, each surrounded by a few drops of mercury. [It is found by calculation based on the results of the experiment itself, that these holes do not, to any perceptible extent, interfere with the transference of heat along the bar.] A second bar, of the same material and of equal cross section, but of much smaller length, also pr6vided with -an inserted thermometer, is placed near to the first. The bars have been left all night, let us say, in the laboratory, no heat being applied. They will then be of the same temperature throughout, viz., that of the air, and all XIV.] CONDUCTION OF HEAT. 213 the thermometers will show that temperature. Now suppose^ heat to be applied to one end of the long bar. [This ig usually done by inserting it in a bath of melted solder, or lead, which is maintained by a proper gas-regulator at a steady temperature.] After a short time the thermometer nearest to the source of heat begins . to rise, then rises faster and faster. Meanwhile, the next thermometer, in turn, begins to rise. And so on. In Forbes's iron bar, which is 8 feet long by about i J inch square section, the final steady state of temperatures was not reached till after a lapse of from six to eight hours. When this steady state was reached {i.e. when, for half an hour or jnore, no change of reading was observed in any of the thermometers) it was found that even the thermometer farthest from the source indicated a temperature perceptibly higher than that of the air of the room, as shown by the thermometer in the short bar. The others showed each a rise of temperature, increas- ing more and more rapidly as they were placed nearer the source. [The conductivity of copper is so much greater than that of iron, that the farther end of the eight-foot bar has to be kept cool by immersion in a vessel, intq which a, steady supply of water is admitted from below.] From this experiment, repeated if necessary with different tempera- tures of the source, it is obvious that we can obtain a very accurate determination of the permanent distribution of temperature along the axis of the bar. [And calculation, by Fourier's methods, shows that the temperature is practically the same throughout any cross section of the bar, unless the section be very great, or the conducavity pf the material very small.] 243. So far, the investigation has dealt with temperatures only : we must now consider heat. The quantity of heat which passes, per minute, across any transverse section of 214 HEAT. ' [chap. the bar, must obviously be the product of the area of the section, the conductivity (§ 237), and the gradient of temperature (§ 239) at that section. The gradient can, of course, be calculated from the observed distribution of temperature, and the area of the section is known. Hence, the heat passing is expressed by a definite multiple of the unknown conductivity. But that heat does not raise the temperature of the part of the bar to which it passes, for we have supposed that the stationary state has been reached. Hence the rest of the bar, beyond the section in question, must lose by cooling in the air (and, in the case of copper, the water-bath at the end) precisely as much as it gains by conduction. It is only necessary, then, to find what amount of heat is thus lost, and we have at once the determination of the conductivity of the bar at the tempera- ture of the particular cross section considered. [The words in italics are of special importance, as they indicate one of the most valuable consequences of the improvements due to Forbes, which we now give.] 244. Starting afresh, we once for all heat the short bar to a high temperature, insert its thermometer, and leave it to cool in the neighbourhood of the long bar (which is, in this experiment, left at the temperature of the air). The thermometer in the short bar is now read at exactly equal intervals of time, say every half minute while it is very hot and cooling fast, then every minute, every five minutes, and finally every half hour, till it has acquired practically the temperature of the air. Some of the thermometers in the long bar are read at intervals during this process. From what we have said above it will be obvious that, from this form of experiment, we can calculate how much heat is lost by the short bar, per minute, per unit of length, at each temperature within the range employed; XIV.] CONDUCTION OF HEAT. 215 and hence we can calculate what was lost in the former experiment at any assigned part of the long bar, since we know what was the stationary distribution of temperature along it. 245. The loss of heat by cooling dejlends mainly upon the excess of temperature of the bar above that of the air, and it is for the purpose of finding this that we use the two bars in each part of the experiment. But it also depends, to some extent, upon the actual temperature and pressure of the air ; and thus the two parts of the experi- ment should be conducted as nearly as possible at the same air-temperature and pressure.- (See § 336.) 246. The unit of heat, in terms of which the conductivity is given by direct comparison of the results of §§ 242, 244, is of course that which raises by 1° C. the temperature of unit volume of the substance of the bar. The results of one of Forbes's experiments on iron, in terms of this unit, are as follows : — Thermometric Conductivity of Iron. Temperature C. o" loo" 200° 0-01505 . . . o'oii40 . . o'co987 Thus it would at first sight appear that the conduc- tivity of iron for heat, like its electric conductivity, is diminished by rise of temperature. Forbes had expected to find it so, having remarked that the order of the metals as conductors of heat is the same as their order as electric conductors, while the electric conductivity of every metal is diminished by rise of temperature. This remark was fully confirmed by the well-planned experiments of Wiedemann and Franz, but Forbes's farther expectation was not, as we shall see, realised. The following results, in terms of the same or similar 2i5 HEAT. [CHAP. units, were obtained by Tait in a repetition and extension of Forbes's experiments (made, like those of Forbes, with the assistance of the British Association). The iron bar employed was that of Forbes. Thermometric Conductivity. Temperature C. o" loo" 200° 300° Iron o'oi4g ... o'oi28 ... o'oii4 ... o'oios 52*9 (1+0*00140 Copper, electrically good ... o"076 ... o'oyg ... o'o82 ... o'oSs .53'4 (i+o"ooo880 Copper, electrically bad o"C54 ... o'o57 ...0060 ... o'o63 52.5 (i+o"oocgO German Silver o'oo88 ... o'cog ... 0*0092 ... 00094 46 '6(1 +0*0009?) In this list, all the substances but iron seem to improve in thermometric conductivity by rise of temperature. To convert these numbers into thermal conductivities as defined above (§ 237) they must each be multiplied by the numerical value of the water equivalent of a cubic foot of the corresponding substance, i.e. by its thermal capacity (§ 183). These water equivalents are given in the last column of the table. The temperature changes indicated by the factors in brackets are very uncertain. The following table contains the results of Forbes, with some more recent determinations. Thermal Conductivity of Iron and Copper. To foot, minute, and degree C. Iron. Copper. 0-835 (1-0*00147;) . ■ . .Forbes. .0-795 (1-0-00287.) {^:^'j;:°r^lfj| *Angstr6m. 0-788(1-0-00002,. j^l^K-^-g } Tait. 0-666 (I-O-O0O23;) . 2*88 (1 + 0-00004.) Lorenz. 0*677 (1-0-002/) . . 2*04 (1 + 0-0057.) ■ Kirchhoff. 247. It is to be noted, however, that in deducing his final numbers from the data of experiment, Forbes did not allow for the increase of specific heat with rise of XIV.] CONDUCTION OF HEAT. 217 temperature (§ 184). Angstrom unfortunately says expressly that this cause can affect the change of conductivity with tem- perature only to an inconsiderable extent, so that his num- bers also are not corrected for this cause. But the specific heat of iron increases by about i per cent, for every 7° C, and the introduction of this consideration would reduce to about |-th of their amount the changes of conductivity given by Forbes, and they would then be (in a matter of such delicacy and difficulty) within the limits of experimental error. This change of specific heat has been taken into account by the other experimenters whose results are given above. Thus, it would appear that, though the order of the metals as heat conductors is practically the same as their order in electric conductivity, the farther analogy sought by Forbes does not exist. The dimunition of elec- tric conductivity by rise of temperature holds for all metals, and nearly to the same extent in all. On the contrary, the thermal conductivity seems (at least in the majority of metals) to improve with rise of temperature. But it has been shown that copper of good electric quality has higher thermal conductivity than that of bad electric quality. The whole subject, so far as experimental details are concerned, is still in a very crude state, as may be judged from the preceding tables. Of course, a good deal of the discrepancy arises from the fact that the materials operated on differed not only in chemical constitution, but also physically, — i.e. having been cast, rolled, drawn, annealed, &c. 248. Angstrom's process is the mixed method referred to in § 241. It consists in alternately heating and cooling, for fixed periods and to a fixed amount, one end of a bar, which need not be nearly so long as that employed in Forbes's method. This periodic change is steadily carried 2i8 HEAT. [chap. out until the indications of each of the thermometers at different parts of the bar have also become strictly periodic, or at least practically so. Fourier's method, applied to this problem,' shows that (at least if the conductivity and specific heat do not vary with temperature) the conductivity can be calculated directly from the rate of diminution of ranges of the successive thermometers, g,nd the postpone- ment of their dates of maximum temperature, per unit of length along the bar ; altogether independently of the rate of loss of heat by the surface (provided this loss be every- where proportional to the excess of temperature over that of the air). The details of the calculation cannot be given here, but a general idea of its nature will be gleaned from the solution of a more restricted problem to be given in the next section, the mathematical part of which may be omitted by any reader without interruption of the continuity of our exposition. 249. Two extremely important practical questions con- nected with this part of our subject are — (i.) How does the internal heat of the earth affect (by conduction) the temperature near the surface ? (2.) How far, and according to what law, do the fluctuations of surface temperature, from day to night or from summer to winter,, penetrate into the crust of the earth ? The second of these questions obviously involves a problem somewhat similar to that presented by Angstrom's method which we have just discussed. And both questions afford us very simple instances of the application of Fourier's beautiful method. So long as we confine ourselves to the upper strata of the earth's crust, we may treat them as planes, the temperature throughout depending on the depth only, and the passage XIV.] CONDUCTION OF HEAT. 219 of heat beiiig therefore in parallel (vertical) lines. This greatly simplifies the investigation. Let V be the temperature at a depth x under the earth's surface. Then the temperature gradient is dv\dx. Thus the rate at which heat passes upwards through a horizontal area of one square foot, at depth x, is kdv/dx, where k is the conductivity at temperature v. For a depth x + ^x under the surface this becomes dx dx ^ dx' The excess of the latter over the former denotes the rate at which heat has been (on the whole) communicated to a slab of crust a square foot in surface and of thickness Ix. The rate at which its temperature rises is dvjdt ; whence, if rbe the thermal capacity (§ 183) of the crust, we have as another expression for the rate at which heat is communi- cated to the slab, per square foot of surface, and thick- ness Sx, dv 5, c — hx. dt Equating these independently obtained expressions for the same quantity of heat, we have finally dv _ d I ,_dv\ dt dx \ dx) This equation is merely the translation into symbols of the statement of § 241. 250. It is entirely inconsistent with our plan to attempt a full discussion of the varied possible consequences of this equation of Fourier's. We will therefore restrict ourselves to the two particular questions given above. And in each case we will suppose that the conductivity and thermal capacity are constant throughout the range of temperature involved- 220 ^ HEAT. [chap. 251. When there is a steady state of temperature we have whence dv d-r°' ax ^ dx' Here A is the surface temperature, and B is the rate at which temperature rises per foot as we descend. This might have been obtained at once from the statements of § 239- Hence a stationary state of temperature near the earth's surface, maintained by the internal heat, i7nplies a uniforni rise of temperature per foot of descent, if the strata are all of the same conducting power. If the conducting power be different in different strata, the rise of temperature per foot is greater in each in pro- portion as the conductivity is less. It is carefully to be observed that, as we are dealing with a steady state of temperature, the thermal capacity of the crust does not appear in our equation. The quantity of heat which, in one minute, passes through a square foot of area of any stratum, is numerically the product of the conductivity by the rate of increase of temperature per foot of depth. Very numerous observations connected with this subject have been collected in late years by a committee of the British Association. The results vary greatly in different localities, biit on the average the rise of temperature in mines, artesian bores, &c., is somewhere about o°'oi C. per foot. The average value of k maj' be taken as about o'o24, so that the yearly supply of heat to a square foot of surface from the interior, averages somewhere about 126 units of XIV.] CONDUCTION OF HEAT. ' 321 heat. This is of no consequence in tropical or temperate regions, but towards the poles it forms a minute one among the causes which prevent the formation of an ice-crust of more than some 400 or 500 feet in thickness. 252. Particular stationary periodic solutions of the equa- tion, such as are due to surface conditions (any number of which solutions may exist simultaneously because the equation is linear), are necessarily of the form € COS (^/-Ar + ^), where T is the period of the disturbance, and zv^ its range at the surface. It we substitute this value in the equation of § 249 we find the condition Here, of course, we assume h, &c., to be constant ; i.e. independent alike of temperature and of depth. To introduce temperature changes of these quantities is un- necessary, because the range of temperature in any of these periodic solutions is usually small. And the problem becomes too complex for such a work as this when the strata are supposed to vary (with depth) in conductivity and capacity, 253. Hence, when the earth's surface is subjected for a sufficiently long time to a simple harmonic change of tem- perature, all layers of the crust have simple harmonic changes of temperature of the same period. But the ranges of temperature at successive equal increments of depth diminish in geometrical progression. The factor for tinit of depth is 6 ' or e ^*r 222. ' HEAT. [CHAP.- Thus the range diminishes more slowly with depth if the conductivity be higher, or if the period be longer ; more quickly if the thermal capacity of the crust be greater. And; in a precise form, the depth at which the surface disturbance is reduced to a definite fraction of its amount, is directly as the square roots of the conductivity and of the period, and inversely as the square root of the thermal capacity. Thus, as the square root of 365 is about 19, a surface disturbance of a year period will be sensible at nineteen times as great a depth as a disturbance of a day period — provided their original ranges are the same. Again, the date at which the maximum temperature arrives at any depth is later in simple proportion as the depth is greater. The rate at which the crest of the periodic heat wave travels is PT ^ cT It is therefore directly as the square root of the conductivity, and inversely as the square roots of the period and of the thermal capacity. A rough representation of what takes place in this case may be obtained by imagining waves at sea to run from shallower to deeper water, i.e. in the opposite direction to that in which we usually see them coming up a shelving beach, so that their heights shall steadily diminish, instead of increasing, as they progress.. 254. Observations of temperature at different depths have been made in various places. The most extensive series is that which, commenced in 1837 by FoSrbes, has been carried on ever since (with the exception of the years 1876 — 79, which were occupied in replacing the instruments, which had been destroyed by a madman). These observa- tions are made in the grounds of the Edinburgh Observatory, XIV.] CONDUCTION OF HEAT. 223 where four gigantic thermometers, with their scales above ground, have their bulbs sunk to depths of 3, 6, 12, and 24 French feet respectively, in the porphyritic rock of the Calton Hill. It is found sufficient, so slow are the usual changes, to read these thermometers once a week. From the sketch, just given, of the theoretical results, we should expect to find the indications of the four thermo- meters very different in general character : — that nearest the surface being considerably affected by disturbances of short period, which do not penetrate far into the crust; while those of the 24 feet thermometer should depend only upon the surface disturbances of longer periods. This is found to be the case. 255. Fourier has given us a method of decomposing any strictly periodical disturbance, however complicated, into its separate simple harmonic elements. By the aid of this method we can separate from the recorded indications of each thermometer that part which belongs to the yearly change from winter, through summer, to winter again ; and to this, as by far the most important, we will confine our attention. We give the approximate average numbers for the years 1837 — 42. The mean temperature is about 7 "-5 C. at the highest thermometer, and rises steadily to about 8°'i at the lowest (§251). The ranges of the annual wave are, for the various instruments in order — 8°-2, 5°-6, 2°7, and o°7 C. These do not very accurately follow the law of geometrical progression (§ 253), but from them we obtain, a year being our unit of time, the average value V —- = 0-115 psr French foot, nearly. k 224 HEAT. [CHAP. The epochs of maximum temperature are, in order — August 19, September 8, October 19, January 6. These follow very closely the arithmetical law (§ 253), and giye for the velocity of propagation of the heat-wave 2 A/ — = 54-8 in French feet per annum. Multiplying together the right hand sides, and the left hand sides, of these two equations, we have, as a test of their accordance, 2 7r= 6"302 ; which, being within |rd per cent, of the truth, gives us some confidence in the general accuracy of the work. Taking, therefore, the quotients of corresponding sides of the equations, we have — = ■ = 238 nearly. °36 nearly. That is, a body falling under [constant] gravity acquires, in each hour, an additional speed of 79,036 miles per hour. 342. Look on these two examples from another point of view. Speed is greater as the space passed over is greater, and less as the time employed is greater. Hence u 2 392 HEAT. [CHAP. it involves length directly, and time inversely; or as it is the custom to write it, - But acceleration is greater as the additional speed produced is greater, and less as the time employed in producing it is greater : thus M = [|] - [J.]- Now, when a unit is increased in any proportion, a con- crete quantity, expressed in terms of it has its numerical value diminished in the same proportion. [Thus when we increase the unit twentyfold, as in passing from shillings to pounds, we find 4S0S. = ^V 480^- = 24/.] Every definite quantity is homogeneous in terms of each fundamental unit it involves. Thus in the expression for the space described under constant acceleration in the line of motion, we have s = a + l:^ + i ct\ Here s and a are each of dimensions [Z] h, a speed, is • v^ and c, an acceleration Mr^ Thus each tsrm of the value of s is,- like .f itself, of the dimensions [Z]. Hence it appears that^ to determine the numerical factor required to pass from any one system of units to another, all that is required is to find the dimensions of the quantity uie are measuring, in terms of the fundamental quantities, to one or more of which every other measurable -quantity can,. XIX.] UNITS AND DIMENSIONS. 293 always be referred. As already remarked, the theory of dimensions is due to Fourier. 343. The fundamental quantities are length [Z], mass [M], and time [T], and it is a matter of mere convention what amounts of these we shall assume as our units. Thus we may employ, as we have hitherto done, a foot- pound-minute system; we might adopt a mile-ton-day system ; or, what seems to be in a fair way towards adoption in science, a centimetre-gramme-second (C. G. S.) system. The student must remember that the choice of units is in no way whatever of scientific importance, being a matter mainly of convenience : what is really wanted for science is a general system, even if it be inconvenient for mere busi- ness purposes. (See again, § 56.) 344. But the question of dimensions is of the utmost scien- tific importance, and is too apt to be lost sight of in the contest about units. It would be, perhaps, too much to say that an ill-chosen system of units, which should force a man to Mnk, would be preferable to a well-chosen system, likely to cause error by inspiring a blind, mechanical confidence. Still, there is some force in such arguments. Spelling and composition are altogether independent of the form of handwriting one employs. But one must know spelling and grammar before he can write correctly, even in the best of "hands." And the unfortunate advocating of the C. G. S. system, under the specious denomination of Aiso- lute Units, is very apt to mislead the beginner, by giving him the impressioh that this choice of units has some mysterious connection with the truths of science. If a pro- posed system of units were more handy for general purposes than those which they are designed to supersede, every one would cry out for a change. It is undoubtedly on the score of general convenience that our present standards 294 HEAT. [CHAP. have come into use. Every one knows what is meant by a man of five feet eight, who " scales " twelve stone. In the C. G. S. system this advantage is wholly sacrificed for the sake of another advantage (felt especially in electrical measurements). The average height of a man, and his ordinary walking pace, are here each expressed by about 170 of the proper units; his mass is somewhere about 70,000 units; while his weight, also in the proper unit {dyne), approaches the gigantic figure of 70,000,000. A horse-power is about 7,5000,00,000 ergs per second. Thus the system is not likely to be employed for any but strictly scientific purposes. [A much more imperative want, of the same kind, is a common language of science. If original works were now, as they used to be, written in Latin — how much more rapidly would not science progress ? No doubt most of the new and valuable scientific work of the day is published in English, French, or German ; or at least given in abstract in some one of these languages. But there is much of it left imbedded in Czech, Danish, Dutch, Italian, Russian, Swedish, &c., and thus practically lost to the great majority of those to whom it might be of the utmost importance. The wise conservatism of the botanists, much as it is ridi- culed by " advanced " science, has preserved to them this invaluable system of freemasonry.] 345. The following statements of dimensions are self- evident : — ■ Volume [p] = [Z3]. Speed [F] = \-'\ Density \p\ = [^J. Momentum [^u.] = f— ]• ML- Force [i^J = [-^] = [|i:j XIX.] UNITS AND DIMENSIONS. 295 Pressure [/] = force per unit surface = f ^H = f T. Energy [E] = [i^Z] = [JfF=] =. [/z,] = [^'J. Power [/>] = [J] = YFV] = [^]. 346. We now come to complex dimensions specially con- nected, with our subject. And here a new fundamental unit, temperature [0], is required. This, again, is optional, but we may take it for illustration as i°C. on the absolute scale (§96). A few instances, fully worked out and explained, will show the reader the principles of this subject; and he may then easily work out the other cases for himself. The co-efficient of dilatation (§§ 100, 113) is the ratio of the percentage change of length or volume to the corre- sponding change of temperature. The percentage change is a mere number, so that the dimensions of expansibility are simply Heat, [H], as we have seen, may be measured in many ways. If it be measured in dynamical units, its dimensions are those of [£] above. If it be measured in thermal units, i.e. by the rise of temperature it produces in a mass of some standard sub- stance (as of water, § 55), its quantity is proportional to the mass and to the rise of temperature, and its dimensions are therefore {M®\. But if it be measured in thermometric units (§ 246) it is of the order 296 HEAT. [CHAP. Now by the definition of thermal conductivity (§ 237), we see that Conductivity x Gradient of temperature x Surface x Time is the measure of the heat which has passed. Thus if k denote thermal conductivity ® [/.] [^] [ZT [7] = \ir\. or H = f-^ — I in thermal units, = -— f m therraometric units. Similarly we find that Rate of Emission = \ | = r 77^ I in thermal units, = -^rl in thermometric units. 347. Thus, to turn the results of § 246, which are in thermal foot-pound-minute units, into the corresponding expressions in C. G. S. units, we have as above XIX.] UNITS AND DIMENSIONS. 297 But One pound = 45 3 '6 grammes, nearly, One foot = 30-48 centimetres, „ One minute = 60 seconds. Hence the factor required is — . ^^ ^= nearly. 60 X 30-48 4-03 Thus in C. G. S. units the thermal conductivity of iron is about 0*2, and that of copper from i to 0-5. Again for rate of emission, as in § 337, we have as the reducing factor rM®-i IL' TJ Here [0] is a centigrade degree in both systems, so that the requisite factor is 453'6 I 1 =t^ _ _^ . nearly. 60 X (30-48) 123 Similarly, and with equal ease, the reducing factor from any one system to another can be found for the other experimental data of our subject. 348. Resume of §§ 340-347. Change of units is a mere arithmetical operation. The real difficulty lies in ascertaining how the various units are involved. Dimen- sions of Volume, Pressure, Energy, &c. Of Heat, Con- ductivity, Rate of Emission, &c. Examples of reduction to C. G. S. units. CHAPTER XX. WATTS INDICATOR DIAGRAM. 349. Besides the many capital improvements which Watt introduced into the steam engine [some, such as the separate condenser or the expansive action, being appli- cations of physical knowledge, others, such as the parallel motion, being applications of mechanical ingenuity] we owe to him what is called the Indicator Diagram, which is of the utmost importance to the elementary exposition of the fundamental principles of Thermodynamics. Watt devised it for the purpose of determining the work done by a steam engine ; and it is still employed for that and similar pur- poses. But in the hands of Clapeyron, and more recently of Rankine, its properties have been so fully developed that we can represent by means of it not merely the work done by an engine, but the various stages of the process ; the therrnal properties of the working substance itself; and- their connection with the laws of Thermodynamics. The germs of the method indeed are to be found in various parts of the Principia, wherever Newton had to represent graphically what we now call an integral. 350. It would be inconsistent with our plan to enter into the practical details of construction of the Indicator itself, of which many ingenious forms are in use. These, as well as the details of construction of steam engines, &c., belong CH. XX.] WATT'S INDICATOR DIAGRAM. 299 rather to engineering than to physics proper ; and can be far more successfully studied by careful examination of the working instrument than by reading descriptions ever so minute, or inspecting drawings ever so accurate. All we need do is to explain the principle involved. And it is simply this : A pencil is so attached to the piston-rod of the engine, that it shares the to-and-fro motion of the piston, and its consequent position at any instant thus indicates the volume of the contents of the cylinder. The pencil, how- ever, has another motion, in a direction perpendicular to the first, such that its displacement in the new direction is, at every instant, proportional to the pressure of the contents of the cylinder. Thus, as the pencil-point moves over a fixed sheet of paper, it traces a line, every point of which represents a pair of simultaneous values of volume and pressure of the working substance. [In some forms of the instrument, the pencil has one of the two motions, and the paper the other. Also, it is usual to make the adjustments so that the volumes and pressures are represented on a conveniently reduced scale. But the final result is the same : the mode of attainment being mere matter of ingenuity or convenience.] 351. The figure /If" (2' (2 represents one of these dia- grams. The various values of OM represent the volume of the working substance, the corresponding values of MP its pressure. From Watt's point of view, the diagram gives the work done during a stroke of the engine. In fact if S be the area of the piston, and p the pressure (understood as pressure on unit surface) the whole force exerted is pS. If then the piston move under the action of this force through a space h, the work done is (§ 13) pS. h or p. Sh. 300 HEAT. [chap. But Sh is the increase of volume of the working substance. Hence, when the pressure is constant, the work done is the product of the pressure by the increase of volume. This would be the case in the figure, if FF' were a straight line parallel to Ov, and then the work during the expansion from OM to OM' would be represented by the area of the rectangle MFF'M'. When the pressure is (as in the j figure) not uniform, we must break up the expansion into separate stages, each corresponding to an infinitesimal change of volume. We thus obtain (as in § 1 90) a number of narrow rectangles, the sum of whose areas is ultimately tlie curvilinear space MFF'M' . Similar constructions must be made for the expansion from P' to Q[ and the contractions from Q[ to Q, and from Q to F. If the volume diminish instead of increasing, the work, estimated as before, must be regarded as spent upon the working substance, and therefore reckoned as negative. XX.] WATT'S INDICATOR DIAGRAM. 301 Bearing this in mind, we see at once that if the closed figure FP'QQ be the diagram of an engine, its area repre- sents the work given out during a complete cycle. For the work is positive from P to P', and from P' to Q ; but negative from Q to Q, and from <2 to P. The work done on the whole, in any such cycle, is there- fore positive if the pencil run round the diagram in the direction of the hands of a watch, negative if in the opposite direction. If the diagram mtersect itself, some parts of its area will be positive, others negative ; but the statement above applies separately to each of the parts. 352. Were this all that the diagram affords, its value (great as it is) would be mainly practical, as Watt originally designed it to be. But we must now examine it from a higher point of view. We assume for the sake of reasoning that there is a definite amount, say unit of mass, of the working substance, and that it does not leave the cylinder ; also that it has, through- out (at each instant), the same temperature and also the same hydrostatic pressure. By this last consideration our reasoning is practically restricted to fluids, whether they be liquids, vapours, or gases, or even a complex arrangement such as a liquid in the presence of its saturated vapour. In the last of these cases there is, between the limits of volume at which the whole is liquid, or the whole is vapour, a definite relation between temperature and pressure alone. The volume, when assigned, gives us in this case the farther information how much of the substance is in the liquid state. But in the first three of these cases we have seen (§§ 121, 124) that there is a definite relation between the volijme, pressure,, and temperature j a relation whose form 302 HEAT. [chap. depends upon the particular substance treated, but which is sufficient to determine any one of the three quantities above when the other two are assigned. That relation we assume to have been experimentally obtained for the par- ticular substance whose behaviour we are for the time discussing. 353. But this is not all. The physical state of the sub- stance is entirely defined when any two of these quantities are assigned. [The reader must be reminded that we are dealing with a definite quantity of the substance.] Hence, as a particular case, when the volume and pressure are assigned, the temperature can be definitely calculated. Every point, P, on the diagram thus corresponds to one temperature ; and, by drawing lines, each through all the points corresponding to one particular temperature, we may cover the diagram with Isothermals, or lines of equal tem- perature. Portions of two of these, PP' and QQ, are roughly indicated in the diagram of section 351. Each of these lines gives a graphic representation of the relation between the pressure and volume of the substance so long as its temperature is unchanged. This corresponds to the second zxiA fourth of the operations in Carnot's Cycle (§ 86). 'Y\v& first and third operations of that cycle involve the behaviour of the working substance when it is sur- rounded by non-conducting bodies, and therefore cannot gain or lose heat directly. From any point in the diagram (which, as we have seen, represents a definite state of the body) we may suppose a line drawn representing the relation between pressure and volume under this .new limitation. Thus the whole diagram may be covered. with a new set of curves, called by Rankine Adiabatic lines. In the rough diagram above, PQ and P Q represent portions of two such lines. Any other definite condition will, in XXi] WATT'S INDICATOR DIAGRAM. 303 general, give rise to its own particular class of lines ; but the two classes we have mentioned are by far the most important for our present purpose. We must discuss their properties with some care. 354. So far, temperature may be considered as being measured on any scale, no matter how defined. But one of the great results to be developed in this chapter is the absolute measurement suggested to Thomson by the remark- able investigation of Carnot. Once we have got this mode of measurement, every other method must give place to it. 355. Meanwhile we make the general remark that any class of lines on the plane diagram may be regarded as suc- cessive parallel sections of a surface, which represents the general relation between volume, pressure, and the quantity characteristic of the class of lines. Thus the lines of equal temperature are sections, perpendicular to the axis on which temperature is measured, of the surface which gives the relation between volume, pressure, and temperature. Sec- tions of this surface perpendicular to the axis of volumes would be a set of curves of equal volume in terms of pres- sure and temperature as co-ordinates. The surfaces them- selves may in fact be regarded as portions of a hilly country, while the parallel sections play the part of con- tour lines. And all the properties of contour lines find here new and interesting applications. Any number of such surfaces and corresponding curves of section can be devised ; we will refer to those only which are of paramount importance. 356. The isothermals or lines of equal temperature might be conceived as being drawn by the indicator it- self, the contents of the cylinder being kept successively at temperatures rising step by step, while at each tempera- te the piston is made to go to and fro in the cylinder. 3'o4 HEAT. [CHAP. We may for the moment assume these steps to be each 1° C, such as we have hitherto employed, or a definite mul- tiple or fraction of such a. degree. But one of the great objects which we have now in view is the absolute measure- ment of temperature. When we have secured this; we shall have an obviously appropriate rule suggested to us for drawing the successive isothermals. 357. The isothermals of the ideal perfect gas (§126) may be very briefly treated. Since the product of the pressure and volume is constant at any one temperature, the lines PP, Q Q', in the fig. of § 351 are equilateral hyperbolas of which the lines Ov and Op are the asymptotes. These ' curves are all similar, and similarly situated, and the linear dimensions of each are as the square root of the correspond- ing absolute temperature. Thus, if drawn at successive equal intervals of temperature, they approach more and more closely to one another as the temperature is higher. 358. The isothermals of the more permanent gases, such as hydrogen, air and its constituents, &c., do not, within ordinary ranges of temperature and pressure, differ much from equilateral hyperbolas. For pressures less than about 140 atmospheres, the air isothermals lie a very little below the hyperbolas of the ideal perfect gas. What happens at exceedingly small pressures is not cer- tainly known. In fact, if the kinetic gas theory be true, ! a gas whose volume is immensely increased, cannot in any strict sense be said to have one definite pressure through-' out. At any instant there would be here and there isolated impacts on widely different portions of the walls of the con- taining vessel ; instead of that close and continuous bom- bardment which (to our coarse senses) appears as uniform and constant pressure. At ordinary temperatures, and about 140 atmospheres, the XX.] WATT'S INDICATOR DIAGRAM. 305 air isothermals cross the hyperbolas, and for higher pres- sures show volumes in constantly increasing ratio to those of the ideal gas under the same pressure. They appear, in fact, to have an asymptote parallel to the line of no volume {Op in the fig. of §351), but at a finite distance from it. This, of course, we are^ prepared to expect — for we have absolutely no reason to think that any finite portion of matter can be deprived altogether of volume — be the pressure what it may. 359. The isothermals of several gases, within ordinary ranges of temperature, have been recently determined up to very high pressures with great care by Amagat, The dis- tinctive merit of his process lay in measuring the pressure directly by means of a column of mercury ; which some- times exceeded 1,000 feet in height. His results are of very great value, but they are only remotely connected with the main features of our present inquiry. 360. The distinction between a true gas and a vapour appears very clearly from the isothermals of carbonic acid, obtained by Andrews in the classical investigation already mentioned (§ 174). Andrews' figure is reproduced on next page, with some slight modifications which he pointed out and accounted for in his paper. According to a suggestion of Clerk-Maxwell, one new line (dotted) has been introduced. Part of this line is to some extent conjectural, from the want of experimental data : — but, imperfect as it is, it gives much novel and valuable information. As the reader is now supposed to understand fully the immediate teachings of the indicator diagram, he is referred again to § 175, which he should peruse attentively with the aid of the figure on the next page. X CH. XX.] WATT'S INDICATOR DIAGRAM. 307 This figure contains two groups of curved lines which are the isothermals for carbonic acid and air, respectively, at temperatures between 13°.! C. and 48°.! C. [The masses of the two quantities of gas were not equal, so that in that respect the comparison is not of the kind which has been hitherto assumed in this chapter. For the air was used merely as a manometer, to give (by its changes of volume) the pressure of the carbonic acid at each stage of the process. The isothermals, therefore, are those of a mass of air which at i atmosphere and 13°. i C. had the same volume as the carbonic acid.] The new dotted line is drawn so as to pass through all the pairs of points on each isothermal (under the critical tem- perature) at which the carbonic acid either just ceases to be wholly vapour, or just becomes wholly liquid. The region included by it is therefore that in which the Dapour can be in thermal equilibrium with the liquid. A glance at the figure shows that the limits of volume corresponding to this region graduallyapproach one another as the temperature is raised ; the left-hand branch of the dotted curve leaning towards the right, and the right-hand branch towards the left. Thus, as the temperature is gradually raised, the smallest volume at which the carbonic acid is wholly vaporous becomes less, while the greatest volume at which it is wholly liquid becomes greater. Close to the critical temperature (but under it) these volumes are practically equal. Also, as is obvious from the figure, the compressibility of the carbonic acid just before it is partially liquefied becomes less and less as the temperature is raised, while that of the liquid (when just completely formed) becomes greater and greater. Thus in volume, and in compressibility, at one tempera- ture, these two states gradually approach one another, until X 2 3o8 HEAT. [chap. at and above the critical temperature they can no longer be distinguished from one another. Thilorier's result (§ 122) as to the great expansibility of liquid carbonic acid, is obvious from the figure : — as is also its great compressibility (discovered by Andrews). If we imagine a broken line to be drawn in the figure, formed of the left-hand branch of the dotted curve, and the critical isothermal (for the higher range of pressures) it is obvious that, for any condition represented by a point to the left of this line, the carbonic acid is wholly liquid. Another broken line, consisting of the right-hand branch of the dotted curve and the same portion of thecritical isothermal, has to the right oi it all points expressing conditions at which the substance is wholly non-liquid. Now, by properly applying heat and pressure, we can bring the substance by any path we choose (on the diagram) from one of these states to the other. Choose two such paths, one (A) wholly free of the dotted curve, the other (B) intersecting it (twice). Operate on the gas according to the B path, and we see it at one part of the course partly vapour and partly liquid. Return from the undoubtedly liquid state to the undoubtedly non- liquid state, by the path A. At no stage of the operation is there any indication that the substance is partly in one molecular state, partly in another. This, however, is on the supposition, which we have hitherto made for the indicator diagram, viz., that the pressure and temperature shall be uniform at every instant throughout the whole mass of the substance operated on. If carbonic acid be in a state represented by a point near the apex of the dotted curve, very slight differences of tem- perature or pressure at different points of the mass give rise to extraordinary differences of optical properties ; and the whole presents, in an exaggerated form, the appearances XX.] WATT'S INDICATOR DIAGRAM. 309 seen when we look through a column of air ascending from a hot body, or through a vessel in which water and strong brine have been suddenly mixed. 361. The relative densities of the liquid carbonic acid and its saturated vapour are, as the diagram shows, about 57 : 1 at i3.°i C. At 21. °s C. the ratio is only about half as great. Thus there is no difficulty in representing the relative volumes graphically. But when we deal with water and saturated steam at any ordinary temperature, the ratio of densities is (roughly) 1600 : I. To give anything approaching this we must take carbonic acid at very low temperatures. Hence no diagram of moderate size can be constructed so as to represent fully the isothermals of water-substance, at the temperatures for. which Andrews has given us those of carbonic acid. On the other hand, a diagram, somewhat resembling that of Andrews, would represent the isothermals of water for temperatures over 400° C. 362. It will be observed that in all these cases, as a rule, the isothermal lines are inclined downwards towards the right, i.e., when the substance is kept at constant tempera- ture, increase of volume implies diminution of pressure ; or increase of pressure implies diminution of volume. This merely signifies that every known iluid, in whatever state it be, is compressed by the application of greater pressure, its temperature being kept unaltered. Apparent exceptions are necessarily collapsible or explosive bodies, which suddenly and abruptly change Volume when pressure is applied. Such exceptions are only apparent be- cause in them the isothermal condition is necessarily violated. A real exception is in the case of saturated vapour in presence of the liquid, for here the pressure remains un- changed as the volume varies, whether by diminution or by 3IO HEAT. [chap. increase. This, of course, is due to the fact that part of the substance undergoes a change of molecular state, involving abrupt change of volume with considerable absorption or evolution of heat, and the proper realisation of the iso- thermal requires that the volume be made to alter so slowly that the change of temperature which would thus be caused can be guarded against by external applications. 363. Tlie adiabatic lines cannot conveniently be drawn for any substance as the result of direct experiment, simply because it is impossible to make an absolutely non- conducting vessel in which to conduct the experiments. It is, however, possible to calculate their form for any class of substances by the help of theory, from the results of experi- ments which can be carried ,out. This point must be deferred for the present. 364. Meanwhile we may, but only for the purpose of reasoning (§ 86), suppose that we have the substance in- closed in a cylinder which can be made, at will, either a perfect conductor of heat or an absolute non-conductor. Let this cylinder be supposed to be surrounded by a mass of perfectly conducting liquid, whose specific heat is so great that its temperature remains practically unaltered by any transference of heat, either way, between it and the contents of the cylinder. Then, if the indicator be attached to the piston, it will trace the isothermal, or the adiabatic, according as the walls of the cylinder conduct heat or not. 365. Now suppose the substance to give out heat under compression, Let the piston be forced inwards. If the cylinder conduct, the heat developed is at once removed ; but for all that the pressure in general rises, as we saw in § 362. But, if the cyUnder do not conduct, the effect will be the same as if the substance, with its pressure already increased by (isothermal) compression, had farther heat XX.] WATT'S INDICATOR DIAGRAM. 311 supplied to it without being permitted to change its volume. In such bodies as we are now considering, the effect will be to still further increase the pressure. Thus the adiabatic line through any point of the diagram is more inclined to the axis of volume than is the corresponding isothermal. 366. The same thing is true if the substance be, like water between 0° C. and 4° C, one of those which are cooled t)y the application of pressure, because they contract when heated. For if the cylinder be a conductor, heat passes through it into the substance, and thus the pressure becomes less than if, as in the adiabatic, heat be not allowed to enter. The statement is still true when we are dealing with a liquid, and its saturated vapour, in presence of one another. For compression liquefies some of the vapour, and sets free its latent heat. When this is allowed to escape as it is developed, the pressure remains unchanged. But in the adiabatic the pressure must, in consequence of the heating, increase with diminution of volume. It holds also when we are dealing with a mixture of ice and water. For here it is the ice which melts, because it is bulkier than the water produced from it, and the whole becomes colder in consequence of the latent heat required for the water which is formed. If heat be allowed to enter, so as to restore the original temperature, more ice is melted, and the pressure sinks in consequence. 367. These phenomena are instances of a general law which has been formulated independently by different physicists. Thus Helmholtz says, as to the effect of pressure on a mixture of ice and water : " Here mechanical pressure, as happens in the majority of cases of interaction of different natural forces, favours the production of the change, melting, which is favourable to the development of its own action." 312 HEAT. [CHAP. Clerk-Maxwell, speaking of the greater steepness of the adiabatics than of the isothermals, says : " " This is an illustration of the general principle that, when the state of a body is changed in any way by the application of force in any form, and if in one case the body is subjected to some constraint, while in another case it is free from this con- straint but similarly circumstanced in all other respects, then if during the change the body takes advantage of this freedom, less force will be required to produce the change than when the body is subjected to constraint." 368. In general, any one point of the diagram cor- responds to one perfectly definite state of the working substance, and therefore there can be drawn through it only one isothermal and one adiabatic The adiabatic (as we see by §§ 365, 366) crosses the isothermal from above downwards, and towards the right of the diagram. But, in certain special cases, a point of the diagram may correspond to more than one essentially different state of the substance. Each of these states has its own isothermal and its own adiabatic ; and thus we find it sometimes said that two or more isothermals, or adiabatics, may intersect one another. The proper view to take of such cases is to look on them as instances in which part of the diagram overlies another part, so that (as in the contour lines of an overhanging cliff), though designated in the diagram by the same rectangular co-ordinates, they lie in regions which must be regarded as perfectly distinct from one another. This will appear clearly enough if we consider the rela- tion between volume and temperature in water at the ordinary atmospheric pressure. Thus (§ 121) we know that the volume of water is the same at 2° C. and at about 6° C. Hence the isothermals for water at 2° C. and at 6° C. XX.] WATT'S INDICATOR DIAGRAM. 313 intersect in a point given by one atmosphere pressure and volume I "00003. But if we think of a water thermometer, we see that the scale of such an instrument wOuld be^ as it were, doubled back on itself, the lowest point being at 4° C. (the maximum density point), and the scale reading upwards from this point to 5°, 6°, &c., for increase of tem- perature, but also upwards to 3°, 2°, &c., for diminution of temperature. These are not to be regarded as one scale, but as two distinct parts of a scale doubled back on itself. And in a similar way we must regard the corresponding part of the indicator diagram above mentioned. 369. Hence, in the reasonings which follow, we shall consider the systems of isothermals and adiabatics as in themselves groups of non-intersecting curves ; but such that each curve of one group intersects once, and once only, each curve of the other group. Two of the same group which appear to intersect will be regarded as lying in essentially different regions, though depicted on the same part of our diagram. 370. Another remark must be made here. We have seen (§ 358) that our direct knowledge of the form of iso- thermals is limited to their middle regions, where the volume of the substance is neither very great nor very small. The same is true as regards the adiabatics, of which our know- ledge is considerably less complete because less direct. But we must presently consider the area (which will be proved to be finite), included between a finite portion of one isothermal and two adiabatics passing towards the right through its extremities. Remark that all that is wanted is a mode of completing our diagram, however imperfectly or even (as subsequent experiments may show) erroneously, provided that it can lead us into no error in the special reasoning for which it is devised, and for which a:lone it is 314 HEAT. [chap. to be employed. Clerk-Maxwell suggests the following method which, while convenient and sufificient for our solitary object, is so obviously incorrect as to details that no one can run any risk of being misled by it. Let HR'R" (in the fig. of § 374 below) be the isothermal of lowest temperature whose form we know, QJi, QR' , Q'R", . . . adiabatic lines. Draw any line SSS" and call it (for our temporary purpose) the isothermal of absolute zero. Then it is clear that we may draw (each in an infinite number of different ways) lines RS, R'S. R''S", . . . such that the curvilinear areas RR'S'S, R'R"S"S' &c., shall have any assignable finite values. 371. We now recur to Carnot's Reversible Cycle, in order that we may be able to interpret the diagram in the light of the two Laws of Thermodynamics. Let PP be the isothermal of the working substance at the temperature /„ of the hot body; QQ' that at the tempera- ture 4 of the cold body; while QFanA. F'Q are the adia- batics of the first and third operations respectively. Also XX.] WATT'S INDICATOR DIAGRAM. 315 let H^, H^ be the quantities of heat taken in and given out respectively in the direct working of the cycle. Then, by the definition of absolute temperature (§ 95), we have As the cycle is reversible, no heat-transaction takes place except these, and therefore the work done is the ■ equivalent of the excess of the heat supplied frqm the source over that given out to the condenser. If . we choose, for convenience, to measure heat in dynamical units, we must use the word ^"^ equal" instead of '■^equiva- lent." [Joule's equivalent, as originally given (§ 37), was 772 foot lbs. for the unit of heat to the Fahrenheit degree. This is, of course, about 1,390 foot lbs. for the centigrade degree.] Hence, with this convention and the proposition of § 351, we have H,- H,^ area FF'Q'Q 372. As the extent of the isothermal expansion, P to /*", may be what we choose, we will for simplicity suppose it so taken that we have numerically (the unit of heat being now the foot-pound) ZT, ^ /.. The definition of absolute temperature then gives -^o = 4 j so that, in our present system of units we have 373. Now we are prepared to choose scientifically our system of isothermals and adiabatics, and thus to settle the values of the several degrees of the scale of absolute temperature. 3i6 HEAT. [chap. Proceeding along the isothermal PP' in the diagram opposite, let us mark successive additional points P", P'", ike, so that A heat units are taken in from P' to P", P" to P'", &c., as well as from P to /". Through the points P", P'", &c., draw adiabatics toeeting QQ in Q', Q'", &c. Then it is clear that the heat given out in passing from Q" to Q', or from Q'" to Q", &c., is (like that from Q' to Q) represented by 4. Thus, 4 -/„ = area PP'Q'Q ^--zrea. P'P'Q"Q' = area. P"P"'Q"'Q" = &c. And this holds whatever be the value of 4- Hence, if we assume 4 to be one degree lower than 4, so that K-t,= I, each of the areas PP'Q'Q, P'P"Q'Q', &c., will be one unit. And a third isothermal, two degrees under 4, will be one degree undef 4< and will thus cut off a new set of unit areas from the series of adiabatics. The whole explored part of the field may thus be divided into unit areas by the system of adiabatics just described, and a set of isothermals for successive degrees of tempera- ture. But the length of a degree, so far, is perfectly arbitrary, though when its value is assigned at any part of the scale the whole becomes definite. 374. The area contained between two successive adi- abatics of this series, and any two isothermals, has therefore as its measure the number expressed by the difference of the absolute temperatures of the isothermals. [Here we see, at once, one of the great merits of Carnct's process. For the statement just made is altogether inde- pendent of the nature of the working substance.] This area also represents, as we have seen, the excess of the heat supplied over that given out. Its utmost valiie, therefore, when the lower isothermal is that of absolute zero, is a finite quantity representing the whole heat sup-' plied. Thus if RR'R"R"'. ... be the lowest isothermal XX.] WATT'S INDICATOR DIAGRAM. 317 whose form is known, we may (as in § 370) take any line ^^ >i ^ .... as the isothermal of absolute zero ; and the lines RS, R'S, R"S" &c., must be so drawn that the several areas FPQ'R'S'SRQ, F'P'Q'R"S'S'R'Q, .... may each be equal io t^. From what has been already said we see at once that whatever isothermal is represented by (2<2'<2"<2"' the areas QQ'R'S'SR, QQ'R'S'S'R', must also be V equal to one another. Thus the absolute temperatures in any two isothermals FF and QQ' are to one another as the com- plete areas /'^' (2'-^'^ 5i? (2 and QQ'R'S'SR. The determina- tion of the ratio of these areas, when FF' and QQ' belong to any two definite temperatures, such as those of water boiling, and of ice melting, under one atmosphere of pressure, is a matter entirely for experiment. (How such 3i8 HEAT. [chap. experiments have been conducted we will afterwards show.) The result for these two temperatures was found by Joule .and Thomson to be nearly i'36s : i. (^Phil. Trans. 1854.) . 375. Hence, if we adopt the centigrade scale, but merely in so far as to divide the interval between the freezing and boiling points into 100 degrees, whether these be the same degrees as those of our earlier scale (§ 61) or not ; and if .r be, on our new scale, the absolute temperature of melting ice, a + 100 will be that of boiUng water : — so that ijc+ioo : X :: i'36s : i. s Thus we have, as already stated, a:= 274 very nearly. " 376. This may be stated in the easily intelligible form : — Jf a reversible engine work between the boiling and the freezing points of water, it gives to the condenser 274 out of every 374 units of heat which it takes from the boiler. Or, to introduce a practical term, Efficiency, whose mea- sure is the fraction of the whole heat taken in which i*i converted into work : — The efficiency of a perfect engine, workhig between the boiling point and the freezing point . of water is \^. The efficiency, of course, rises with the ratio of the higher to the lower absolute temperature. In practice, the very best engines fall far short of this. Joule gives as a fair instance of the data for a good high- pressure steam-engine the following : — ■ If it work at 3|- atmospheres' pressure (about 53 lbs. weight per square inch) the temperature of the boiler must be about 300° F., and it is found practically impossible to keep the condenser at a lower temperature than about iio° F. nearly. Absolute zero on the Fahrenheit scale is - 274 i§^ -1- 32 = — 461° F. nearly. Hence even the theo- retical efficiency is only yf^, very nearly \. The' actual efficiency is rarely more than about half as much. XX.] WATT'S INDICATOR DIAGRAM. 319 377. To recur to our diagram, § 374. The absolute temperature, t, completely defines a particular isothermal, when we know the working substance. Let, now, ^ be the corresponding characteristic of an adiabatic, i.e. the quantity which has the same value at all points of such a line. Rankine originally called ,it the Thermodynamic Function, and Clausius has since called it Entropy. It is obvious that (^ depends in some way on the heat given to or taken from the substance, for it is constant only when there is no direct gain or loss of heat. And we see at once from the equation which is true for all values of /, and 4, that the amount by which ^ increases, in passing from one adiabatic to another along an isothermal, may be defined as simply the common value of these equal quantities. Thus since in our standard method of drawing a group of adiabatics (§ 372) we took H^ numerically equal to /„ the value of ^ increases by unity from any one to the next of the group. 378. By working backwards through the group of adia- batics, along the isothermal /, we remove t units of heat for each unit by which ^ diminishes. This suggests the measure- ment of ^ from a zero at which the substance has no heat, to part with. Practically, however, we measure ^ (as, in dynamics, we measure a potential) from some assumed origin. For it is with its changes alone, and not with its actual value, that we are mainly concerned. Suppose, then, that we assume for this purpose a definite point in the diagram as the origin. Draw the corresponding isothermal, say 4 , and produce it to cut the adiabatic for which ^ is to be measured. Let the substance expand or 32P HEAT. [CHA?. contract adiabatically till its temperature is t^, and let its volume then change isothermally till its state is that of the assumed origin. If H be the heat given out during this last operation, we have Here H may be negative, in which case is also negative. It follows that if a substance change its state isothermally at temperature t, from ^^ to <^, it takes in an amount of heat denoted by If it be restored to (^» along the isothermal 4, it gives out heat to the amount 4 (<^ - <^o)- In a Carnot's cycle, bounded by t, , 4, ^o> the excess of heat taken in over that given out, i.e. the work done, or the diagram area of the cycle, is thus (^-4) ( (thermodynamic function, or entropy). But we can also easily see that we might have expressed the position of a point {i.e. the state of' the substance) by any other pair of these four quantities : — i.e. in terms of V and t, V and ^, p and t, or / and ^. And we also saw that (E) the whole energy (to a constant prh) could be expressed in terms of / and v, and therefore in terms of any other pair of the four quantities. Hence CH. XXI.] ELEMENTS OF THERMODYNAMICS. 325 we see that there exist three necessary relations among the five quantities /, V, t, (ji, E, and therefore that any three of them can be expressed in terms of the remaining two. We proceed to develop; analytically, the more ele- mentary results of the application of the Laws of Thermodynamics : — and we will choose sometimes one pair, sometimes another, of these five quantities (as shall best suit our immediate purpose) for the independent variables in terms of which the others are to be expressed. 384. We commence with the expressions for the Energy. This subject was attacked by W. Thomson in 1851. If, under pressure /, the volume of the substance change from V to V - represented (§§ 371, 378) in dynamical units. Thus we have, for the change of energy, dE = tdif, - pdv .... (1) Hence i: may be regarded as a function of the two independent variables v and <^, such that its partial differential coefficients are (f)=-Mf ) = '■■■■■« 326 HEAT. [CHAP. 385. These equations tell us, respectively, that (a) The loss of energy per unit increase of volume in adiabatic expansion is measured by the pressure. (^) If the substance be kept at constant volume, the gain of energy, per unit increase of entropy, is measured by the absolute temperature. This is merely the same as : — At constant volume, the increase of energy is measured by the heat supplied. 386. From equations (2), by -partial differentiation, we find _ ( dp \ r_ d (dE\ _ d fdE^~i _ / dt \ \'d4>) L~" li^lvJ " ~dv V^/J ~ \dv)' To interpret this equation, multiply and divide the left hand member by t, and .we have : — The fall of temperature per unit increase of volume in adiabatic, expansion, is equal to the increase of pressure per (dynamical) unit of heat taken in at constant volume, multiplied by the absolute temperature. 387. The expression for dE [(i) of § 384] consists of two parts, neither of which is (separately) a complete differen- tial, though their (algebraic) sum is necessarily so. Com- pare § 379, where the reason is made obvious. Both the work done, and the heat supplied, depend on the form of the path {i-e- on the succession of states through which the substance passes) and this is wholly arbitrary. The change of energy, on the other hand, depends only on the initial and final states of the substance. 388. By adding various complete differentials to both sides of (i), we may change the independent variables to any of the other pairs v and t, p and <^, / and t. And from the results we can, as before, draw definite conclusions as to relations between the thermal properties of the substance. XXT.] ELEMENTS OF THERMODYNAMICS. 327 389. Thus, for V and t as independent variables, d {E - t^)= ~ <^di - pdv. This gives, by a process precisely similar to that employed in § 386, Multipljing both sides by t, this reads : — The latent heat of (isothermal) expansion is measured by the product of the absolute temperature and the increase of pressure per unit rise of temperature at constant volume. Thus if we have steam in presence of water, or water in presence of ice {i.e. the same substance in two states in which the first differs thermally from the second by the latent heat, L, per unit mass) ; let Va, Vi be the volumes of unit mass of each aX p, t ; e : 1 - e the ratio in which the unit of substance operated on is made up of these forms, V ^=eva + ( I - «) Vi. Hence (^^ = ^ (^\ \ dt ' V, - v,\ de y The first member is the rate at which pressure must change with temperature so that there shall be no change of volume, i.e. no alteration of the relative proportions of the parts of the substance in the two different states. The second, multiplied by t, is the heat which must be supplied per unit increase of volume, the temperature remaining unchanged. Hence Lde, the latent heat required for the change of state represented by de, must be equal to td<^, the heat supplied. Thus the preceding equation takes the form dp \ _ L \dt) 328 HEAT. [CHAP. or, for small simultaneous changes of pressure and temperature, under the assigned conditions, 8t= ^i"^-"') Sj>. Thus, when »„ > v, (as with steam and water) the tem- perature is raised by increase of pressure : but when v^ < »i (as with water and ice) the temperature of the mixture falls when the pressure is raised. 390. The formula above is equivalent to that given by J.Thomson in 1849 (§ 142) from Carnot's principle alone. Let us deduce a numerical result. A cubic foot of water at o°C. weighs about 62 -5 pounds. Hence the volume of a pound of water is about o-oi6 cubic feet. The density of ice is (§ 145) 0-92, so that the volume of a pound of ice is [o-oi6 -f o'92 = ] o'oi74 cubic feet. The latent heat of water is (§ 151) 79"2S units of heat, which must be multiplied by Joule's equivalent, 1390 (§ 37 1), to reduce it to foot-pounds. One atmosphere of pressure is about 2117 pounds weight per square foot. Hence, calculating from the formula, we find that the freezing point is lowered by o°'oo74 C. for each atmosphere of pressure. 391. We cannot directly apply this process to a numerical calculation in the case of water and steam, for experimental difficulties of a formidable character lie in the way of the determination of the density of saturated steam at different temperatures. But, if we take the corresponding values of 8/ and 8/ from Regnault's table (§ 164), and L from § 166, we may use the formula to calculate the volume of one pound of saturated steam at any temperature. But a curious result, due to Rankine and Clausius, XXI.] ELEMENTS OF THERMODYNAMICS. 329 may easily be deduced. Let c„, r, (functions of t alone) be the specific heats of any substance in its two states of saturated vapour,-and liquid, in the same vessel. Then remembering that the condition of unit mass of the sub- stance is fully characterised by the quantities t and e, we have for the heat required to change these to ^ + dt, and e + de, respectively, td(fi = {c„e + c^{i - e)) dt + L de . . . . (i). Hence d' [chap. Let the pressures before and after passing the plug be/ and /', the corresponding volumes of unit mass v and v. Then pv is the work done on unit mass of the gas as it passes a cross section of the tube before reaching the plug, while J>'v' is the work it gives out as it passes a section after leaving the plug. Their difference is the gain of energy, provided no heat be supplied from without, and no energy lost as sound. Hence, sirxe the motion is regarded as uniform, if £ and £' be the intrinsic energy of unit mass before and after passing the plug, E' - E = p V - p'v', or the conditions of the experiment are such that E -V f V is constant. Therefore (§ 392), /S<^ + vlp = o, or, taking/ and / as independent variables, Af + ^^Sp + vhp = o. dt dp ^ ^ Now ^ — r- IS h^re evidently the specific heat at constant pressure, which (§ 395) we called k. Also (§ 393) we have dii> dv dp dt if e be the expansibility at constant pressure. With these values our equation becomes kht = - v{i - et) Sp . . . . (i) Here ^represents absolute temperature. To compare this with the scale of a gas-thermometer, let J" be the tern- XXI.] ELEMENTS OF THERMODYNAMICS. 339 perature Centigrade on such a thermometer, corresponding to the absolute temperature t ; the lengths of the degrees on the two being assumed as equal throughout the (very small) range of the experiment. Then we have, by § 124, pv = C(i + ar) as an approximate expression, to be rectified. Thus, as the degrees are practically equal on the two scales, throughout the small range of change, 1 dv a. ^ " vdT~ 1 + a.T and therefore >" = -c{^~o.{t-T)y-l- ....(.) Thus, finally, In the experiments of Joule and Thomson, the changes of pressure and temperature are not infinitesimal. To adapt the formula to such a case we may lawfully integrate (2), through the small range of the experiment, neglecting the variation oi t — T. Thus if the observed change of tempera- ture be ^, and the pressures/ and j/, the formula becomes. I kQ T + a A*(log.j» - log./)' 404. The first two terms of this expression give the re- sult of § 126, directly suggested by Charles' -Law. The third term was found by experiment to be small for bodies at temperatures above their critical points, i.e. for true z 2 34° HEAT. [chap. gases ; and considerably larger for vapours. All the true gases, except hydrogen, were slightly colder after than before passing the plug j hydrogen slightly warmer. It was by these experiments that the temperature of absolute zero was determined (§ 374) to be very nearly - 273°7 on the Centigrade scale. 405, The following little table exhibits the main features of Joule's and Thomson's results, so far as the air-thermo- meter is concerned. The numbers in the first column (eacli with 2Tz°'i added) are absolute temperatures. Those in the second column are the quantities to' be added to the corresponding absolute temperatures to give temperatures by the air-thermometer, when the air is kept at the density corresponding to 0° C. and i atmosphere of pressure. The third column gives the corresponding numbers for the air- thermometer when the pressure is maintained throughout at one atmosphere. {Phil. Trans. 1854.) Difference between Air-Thermometers and Absolute Scale. Constant Volume. . -H O- . + 0-0298 . . +0-0403 + o'o366 . + 0-0223 • + o'oooo . - 0-0284 . - 0-0615 ■ - 0-0983 . - 0-1382 . - o"i798 . - 0-2232 . - 0-2663 • - 0-3141 . - 0-3610 . -0-4085 . Absolute Scale. 273°-7-l-o . . . -1-20 ! .■ -t-40 . + 60 . . , -t-80 .' + 100 . . . 4- 120 . . • + 140 . . + 160 . + '80 . . -l-,2oa .' . . + 220 . + 240 . . . -1-260 ,' . . -1- 280 . . . + 300 ■ . . . Constant Pressure. . H-o- + 0-0404 + 0-0477 + 0-0467 -1- 0-0277 -1- 0-0000 - 0-0339 . - 0-0721 -O-I134 - OT571 . - 0-2018 - 0-2478 . - 0-2932 - 0-3420 - 0-3897 - o"4377 XXI.] ELEMENTS OF THERMODYNAMICS. 341 406. Joule's earliest determinations of the value of the dynamical equivalent of heat were made by working a porous plug or piston up and down in a cylinder full of water, and measuring the change of temperature produced for a given difference of pressure above and below the plug. Equation (i ) of section 403 gives us the means of making the necessary calculation. For 8 ^ is measured ; 8/ is known ; v is the volume of water present multiplied by the number of up and down strokes made by the piston ; and the remaining quantities, e and t, may be estimated with sufficient accuracy by the air-thermometer. The result is k in dynamical units ; i.e. the value of Joule's equivalent itself, because the va;lue of k in thermal units is simply unity. 407. In §§ 78-81 a first notion was given as to the dissi- pation of energy. Closely connected with this is the Restoration of energy, a question also first treated by W. Thomson. The tendency of heat (whether by conduction, radiation, or convection) towards equalisation Of tempera- ture, i.e. to loss of availability^ gave the first hint of dissipa- tion or degradation. It becomes, then, an interesting problem to seek what amount of work can be obtained, by perfect engines, from an assigned distribution of heat. 408. If we continue to measure heat in dynamical units, the dyiiamical value of the quantity ^ is simply .ff" itself, whatever be the temperature of the body which contains it. But the utmost realisable value. Unless we have a bbdy at absolute zero to act as the condenser of a perfect engine, is always less. In fact, if t be the absolute temperature of the hot body, and 4 the lowest available temperature of the condenser, the realisable value is (§ 376) only t - t 342 • HEAT. [chap. H-t.^. Suppose that we operate upon a number of bodies at different temperatures ; some being used to supply heat, others to have heat supplied to them ; then the work will be simply the excess of the heat taken from some of the bodies over that given to others. This must always, except when per- fect engines are employed, b? less than the realisable value Hence we see that the expression the supply of H to the mass at / necessitates the taking of — from the body at /„• The form of the expression is therefore the same in either case; and the distinction betvyeen giving and taking is provided for by the fixed order of the limits in the integral, which is taken from a lower to a higher temperature in the former case, and from a higher to a lower in the latter:] ' From this /„ can be calculated, and the expression for the motivity then takes the very simple form 2 m r cdt. J to 411. One very curious consequence of this is, that if the system consist of two equal masses, m, of the same substance, at temperatures /, and 4. and if we assume the specific heat to be independent of the temperature, the common temperature when the internal motivity has been entirely realised is and the motivity itself- is Thus the internal motivity of a system consisting of a pound of ice-cold water and a pound of boiling water is, in foot-pounds, 1390 (i9'339- 16-553)°= 1390 x7'76- XXI.] ELEMENTS OF THERMODYNAMICS. 345 The absolute temperature to which the system is reduced, when its internal motivity is thus exhausted, is 320"i2, corresponding to 46°'i 2 C. But, if the two parts of the system had been simply mixed, the resulting temperature would have been 324, or 50° C. Hence the energy would have been greater by that of 2 lbs. of water raised 3°'88, i.e. by the quantity of work 1390x776 obtained in the former process. But this excess of energy is only in part available. With an un- limited external system at temperature 46°'i2 C. we can realise only about 1390 X o'048 foot-pounds. Thus when the water is, in each case, brought to the uniform temperature of 46°"i2 C, we realise more than 160 times as much work by the first process as by the second. 412. The entropy of a system changes along with its motivity; but the two things are quite distinct, as the following simple case shows. When the element, H, of heat is in a body at tempera- ture /„ its motivity is (as we have seen) where t„ is the lowest available temperature. Hence, if this heat be transferred to another body at a lower tempera- ture 4, the loss of motivity is On the other hand, when the element H of heat passes fron; a body at temperature /„ to another at temperature t„ the first 34'> HEAT. [chap. TJ body loses entropy to the amount — , and the second gains TT to the amount — (§ 378); so that the whole entropy of the system increases by the amount (- - -)^- Thus the loss of motivity is simultaneous with gain of entropy. But the loss of motivity by the passage of heat from a warmer to a colder body is less as the lowest avail- able temperature is lower ; while the corresponding gain of entropy is the^same whatever be this lowest temperature. 413. The only effect of a limit of temperature, on the entropy, is to limit its final amount. But if the external universe were at the temperature of absolute zero, there could be (theoretically) no loss of motivity, i.e. no dissipation of thermal energy ; while the entropy would go on increasing without limit as the heat gradually passed to colder bodies. Thus we see that Clausius' theorem, '^ The entropy of the universe tends to a maximum," is by no means iden- tical with, though it is closely connected with, Thomson's previously published theory of dissipation. 414. In § 195 we promised to give Thomson's investiga- tion of the phenomena of the Thermo-electric circuit. We can now do so, by the help of the formulae of this chapter. Let 4, ''be the absolute temperatures of the junctions in a circuit of two metals, in which the specific heat of elec- tricity has the values o-i, o-j respectively. Let T be the temperature of the neutral point, and 11 the Peltier effect of unit current passing through the junction at temperature t. Then the change of E, the electromotive force, caused by XXI.] ELEMENTS OF THERMODYNAMICS. 3-47 raising the temperature of the hot junction from ^ to i+Sf, will be for it must be remembered that the direction from hot to cold is necessarily reversed in passing from one to the other of the two metals. But the change of the expression ^(f) which (§ 408) is always zero for a set of reversible opera- tions, such as this is (§ 195) supposed to be, gives dt^ t I t dt Without any assumption as to the expression for o-i — 0-3, we may eliminate it from these equations, and we obtain the very interesting result dE ^ 'a_ dt t ' [Hence we see at once that 11 vanishes at the neutral point, for then (§ 193) ^ is a maximum, and therefore dE T — =0. dt ^ This equation, shows that the value of 11 may be com- pletely deterinined from measurements of the current in the circuit as depending on the temperature of the hotter junction, the colder being kept at constant temperature. Conversely the comparison of the observed currents with corresponding measures of the Peltier effect, would enable 348 HEAT. [chap. us to test the admissibility of the assumptions we have made. 415. So far, we have been following Thomson. But if we now introduce the experimental result (§ 197) that the specific heat of electricity is proportional to the absolute temperature, we have