THE UNIYERSITY .Jv OF ILLINOIS LIBRARY The person charging this material is re¬ sponsible for its return on or before the Latest Date stamped below. Theft, mutilation, and underlinipo#'pf books are reasons for disciplincy^^^w^^nd may result in dismissai|rf^ln \heUniversity. Illinois Library CARNEGIE INSTITUTION OF WASHINGTON Publication No. 408 1930 THERMODYNAMIC RELATIONS IN MULTI-COMPONENT SYSTEMS BY ROY W. GORANSON Physicist, Geophysical Laboratory, Carnegie Institution of Washington Published by Carnegie Institution of Washington July, 1930 WASHINGTON TYPOGRAPHERS, INC. WASHINGTON, D. C. 5 3C,7 PREFACE A large part of the work at the Geophysical Laboratory consists of experimental thermodynamics. This means that we are con¬ tinually evaluating explicitly the thermodynamic relations which exist between the variables of particular systems, these variables including temperature, pressure, and amounts of the substances that determine their composition. It is therefore very desirable, and for this reason work was begun, to evaluate as many as possible of the relations existing between the variable quantities of multi- component systems in terms of quantities that can be readily obtained from experiment, and tabulate them in a compact and easily accessible form. In order to do this it was found necessary to make a skeleton outline of the whole structure of thermodynamics. Gibbs de¬ veloped this subject but his treatment was couched in the mathe¬ matical language of his day which had not then been developed as a tool for the physicist and consequently readers found difficulty in deciphering it. Later writers went to the other extreme in avoiding mathematical language as much as possible. Now state¬ ments by such writers must necessarily be incomplete or ambiguous for physics has progressed to such an extent that physicists find ordinary language too poor to express the precise delicate shades of meaning that are found to be necessary. Mathematics was there¬ fore developed to serve this purpose. Hence it was suggested that the outline that was constructed and used here should be completed very fully and made to serve as text for the tables. I am indebted to George Tunell of this Laboratory who is in a large measure responsible for the existence of the book by giving lavishly of his time and effort in constructing the framework of this thesis, and in particular for his work in chapters one to three. I also wish to thank P. W. Bridgman of Harvard University and L. H. Adams of this Laboratory for reading and criticizing the manuscript. V July, 1929. Geophysical Laboratory Washington, D. C. ROY W. GORANSON . • • . lloll r |1 ^H HI CONTENTS PAGE Preface .'. v Introductory . xi Nomenclature . xv CHAPTER I. FUNDAMENTAL IDEAS Undefined concepts or directly measurable quantities ART. 1. Length. 1 2. Time. 4 3. Mass. 6 4. Force. 8 5. Temperature. 9 6. General note regarding the physical quantities. 10 7. Property: extensive and intensive properties, state, homo¬ geneous AND HETEROGENEOUS SYSTEMS, EQUILIBRIUM STATE . . 12 r CHAPTER II. SIMPLE HOMOGENEOUS SYSTEMS Unit mass systems 8. Equation of state or characteristic equation. 14 9. Definitions of work and heat. 15 10. Reversible vs. irreversible processes. 16 11. Transformation of heat and work integrals. 17 12. Differentials and derivatives of heat and work. 18 13. Definitions of the heat capacities per unit mass. 20 14. The first law of thermodynamics. 22 15. Geometric interpretation of the first law. 22 16. The necessary and sufficient conditions for a line in¬ tegral, EXPRESSED AS THE ENERGY INTEGRAL, TO BE INDE¬ PENDENT OF THE PATH. 23 17. An analysis of some incomplete statements of the first LAW THAT HAVE BEEN USED. 28 18. The SECOND LAW OF THERMODYNAMICS. 31 19. Differential and partial derivatives of the entropy. 31 20. Relations between energy derivatives and heat capacities, BETWEEN ENTROPY DERIVATIVES AND HEAT CAPACITIES, AND BETWEEN DERIVATIVES OF HEAT CAPACITIES (ALL PER UNIT MASS). 33 vii Vlll CONTENTS art. CHAPTER II— Continued page Homogeneous one-component systems of variable mass 21. Definitions of component and phase. 35 22. Definitions of heat and work. 39 23. The first law of thermodynamics. 40 24. The second law of thermodynamics. 41 25. Definitions of Gibbs’ thermodynamic functions. 42 CHAPTER III. HOMOGENEOUS BINARY SYSTEMS OF VARIABLE MASS AND COMPOSITION 26. Definitions of specific volume and mass fraction. 43 27. Equation of state. 43 28. Definitions of work and heat. 44 29. Definitions of c p , 1 p , l mi — lm*. 45 30. Transformation of the heat and work integrals. 46 31. Definitions of c v , 1 v> lwi — lw 2 . 47 32. The first law of thermodynamics. 48 33. Transformation of the energy integral. 50 34. The second law of thermodynamics. 52 35. Transformation of the entropy integral. 54 36. Relations between unit mass and variable mass binary SYSTEMS. 55 CHAPTER IV. HOMOGENEOUS N-COMPONENT SYSTEMS 37. Definitions of specific volume and mass fraction. 57 38. Equation of state . 57 39. Definitions of work and heat. 58 40. Transformation of the heat integral. 59 41. The first law of thermodynamics . 60 42. Transformation of the energy integral. 62 43. The second law of thermodynamics. 64 44. Transformation of the entropy integral . 66 45. Derivation of Gibbs’ equation 12. 68 46. Definitions of Gibbs’ thermodynamic functions. 70 47. Differential and partial derivatives of Gibbs’ Zeta. 70 48. Differential and partial derivatives of enthalphy or Gibbs’ Chi . 72 49. Differential and partial derivatives of Gibbs’ Psi . 73 50. Derivation of Gibbs’ equation 97. 74 CHAPTER V. STRAIN 51. Definition of strain . 76 52. Strain transformations. 78 CONTENTS IX art. CHAPTER V — Continued page 53. The functions associated with strain. 80 54. A PHYSICAL INTERPRETATION OF THE QUANTITIES ei, e2, e 3 . 81 55. The angle between two curves altered by strain. A PHYSICAL INTERPRETATION OF THE QUANTITIES e4, 65, e6. 82 56. Linear dilatation at a point of the system. 84 57. Homogeneous strain. 85 58. Pure strain. 87 59. Strain tangent at a point. 90 60. Very small deformations. 91 61. Conditions of compatibility for strain-components. 92 CHAPTER VI. STRESS 62. Concept of stress. 95 63. Interior and exterior forces. Six necessary conditions FOR EQUILIBRIUM OF A RIGID BODY. 97 64. Equations of equilibrium. 99 65. Specification of stress at a point. 101 CHAPTER VII. THERMODYNAMIC TREATMENT OF SYSTEMS HOMOGENEOUSLY STRAINED 66. Definitions of work and heat. 107 67. Definition of c x , 1 X1 , . . , 1 X6 . 109 68. Transformation of the heat and work integrals. 109 69. The first law of thermodynamics . Ill 70. Transformation of the energy integral. 112 71. The second law of thermodynamics. 113 72. Transformation of the entropy integral. 114 73. Derivation of Gibbs’ equation 12 for strained systems. ... 115 74. Differential and partial derivatives of Gibbs’ Zeta. 116 75. Differential and partial derivatives of enthalpy or Gibbs’Chi. 117 76. Differential and partial derivatives of Gibbs’ Psi. 118 CHAPTER VIII. THE STRESS-STRAIN RELATIONS FOR ISOTHERMAL CHANGES OF STATE 77. Generalized Hooke’s Law of the proportionality of stress AND STRAIN. 120 78. “Strain-energy function” for isothermal changes of STATE. 120 79. “Strain-energy function” for adiabatic changes of state . 121 80. Static vs. dynamic methods of determining the stress- strain RELATIONS. RELATION BETWEEN WORK FOR ISOTHER¬ MAL AND WORK FOR ADIABATIC CHANGES OF STATE. 122 X CONTENTS art. CHAPTER VIII—Continued page 81. The elastic coefficients or “elastic constants” of the SYSTEM. 122 82. The stress-strain relations for isotropic bodies. 123 83. Moduli OF ELASTICITY FOR ISOTROPIC SUBSTANCES. 125 84. Anisotropic character of homogeneous crystalline sub¬ stances . 127 85. Transformations of the strain-energy function. 130 86. Moduli of elasticity for anisotropic homogeneous sub¬ stances . 132 CHAPTER IX. SYSTEMS NOT IN EQUILIBRIUM. IRREVERSIBLE PROCESSES 87. Displacement transformations. 134 88. Definitions of work and heat received. 136 89. The first law of thermodynamics. 137 90. The second law of thermodynamics. 138 91. Relations of non-homogeneous to homogeneous systems ... 139 92. “Reversible processes”. 142 93. Justification for defining the entropy and energy of a SIMPLE SUBSTANCE AT SOME ARBITRARY STATE AS ZERO . 143 CHAPTER X. INTRODUCTION TO THE TABLES OF THERMODYNAMIC RELATIONS 94. The variable properties or quantities of the tables and THEIR RELATIONS .,. 146 95. The standard derivatives for Table 1. 148 96. Transformations of standard derivatives of Table I to FUNDAMENTAL DERIVATIVES . 149 97. Abbreviations and special notation introduced in the TABLES . 149 98. Transformations necessary to convert the tables for VARIABLE MASS SYSTEMS TO TABLES FOR UNIT MASS SYSTEMS .... 151 99. Experimental determination of the standard derivatives. 152 100. Fundamental equation for a binary system of variable mass. 155 101. Electromotive force measurements. 159 102. The second derivatives (Table II). 161 Part 2 Key to Tables. 167 Table I. First derivatives. 170 Table II. Second derivatives. 263 Appendix. Dilute solution laws; ideal solutions; definitions OF FUGACITY AND ACTIVITY . 319 Index. 327 INTRODUCTORY The science of thermodynamics deals with work and heat. Since all physical and chemical processes, which thereby include all natural phenomena, involve work and heat, it is apparent that thermodynamics is a fundamental and far-reaching science. It was therefore thought desirable to have as many thermo¬ dynamic relations as possible in multi-component systems readily available, especially since a large part of the work done at the Geophysical Laboratory consists of experimental thermodynamics. This idea grew out of a study of the condensed collection of thermo¬ dynamic formulas derived by Professor Bridgman 1 for one-com¬ ponent systems of one and two phases and constant mass (the three phase one-component system not being variable). Bridgman also indicates the extension of his tables to more complex cases of one- component systems in which electrical forces, surface tension, and other forces are present in addition to hydrostatic pressure. r Bridgman’s tables refer to systems of constant mass, but it is well known that all of the functions for the one-component system of variable mass may be computed from the unit mass functions. It is proved herein that the variable mass functions for multi- component systems may also be readily computed from the unit mass functions alone without the necessity of any additional experimental measurements. For the purposes of experimental thermodynamics it is highly desirable if not essential that the quantities necessary to be directly measured in order to formulate a fundamental equation be known. Bridgman has already stated the quantities necessary to be meas¬ ured for formulating the fundamental equation for the one-com¬ ponent single phase system. I have done this for the multi- component systems in my introduction to the tables. How complete it is desired to make a mathematical treatment, i. e., of what length to make the steps between the equations, 1 P. W. Bridgman, A Condensed Collection of Thermodynamic Formulas (Harvard University Press), 1925. xi Xll INTRODUCTORY depends both on the use to which it is to be put and upon the mathematical facility and intuition of the user. For example, in publishing his papers on Heterogeneous Equilibria, Gibbs omitted so many of the intervening steps between his equations that many of his readers have found difficulty in following his development of the subject. For this reason it was suggested that the derivations given here be treated very fully and completely. Thermodynamics begins with certain undefined physical con¬ cepts (directly measurable quantities) and certain unproved hypothetical relations between them (physical hypotheses). All of the other concepts (variable quantities) of the science are defined in terms of the initial undefined concepts and all of the theorems of the science are deduced from the definitions and the initial physical hypotheses. 1 The physical hypotheses of thermodynamics can of course be stated in words without the use of mathematical symbols. In order to deduce theorems from the physical hypotheses it is necessary to express the hypotheses symbolically by means of mathematical equations. Furthermore they can be stated with the same clarity and much more briefly by means of the mathe¬ matical equations of partial derivatives and line integrals. On this point Poincare 2 says: “All laws are deduced from experiment; but to enumerate them, a special language is needful; ordinary language is too poor, it is besides too vague, to express relations so delicate, so rich, and so precise. This therefore is one reason why the physicist can not do without mathematics; it furnishes him the only language he can speak.” In most of the problems independent variables of two orders are present and for dealing with independent variables of two orders 3 the methods of partial derivatives and line integrals were especially designed. As Professor W. F. Osgood has said, “In thermodynamics a thoroughgoing appreciation of what the independent variables are (in order that, when the letters expressing the variables of the two classes overlap the meaning of 1 See Warren Weaver, American Math. Monthly 36, 1929, p. 39. 2 H. Poincare, The Foundations of Science, 1921, p. 281. 3 This is the terminology used by W. F. Osgood, Advanced Calculus, 1925, pp. 115, 140. Another phraseology used by mathematicians is “functions of functions.” INTRODUCTORY Xlll the partial derivatives may be clear) and the ability to think in terms of line integrals, are indispensable.” While this treatment is an attempt to begin with the undefined concepts, definitions and physical hypotheses and from them develop the subject in a mathematically rigorous manner, mathe¬ matical rigor has not been pursued for its own sake. On the broad subject of the relationships between physics and mathematics Courant 1 has written: “From time immemorial mathematics has derived powerful impulses from the close relationships which exist between the problems and methods of analysis and the perceptual concepts of physics. For the first time in the last decade a crumbling away of this connection has taken place in that mathematical investigation in many cases cut loose from its perceptual starting points and especially in analysis often concerned itself all too exclusively with refinement of its methods and sharpening of its concepts. Many students of analysis have thus lost a full knowledge of the close connection of their science with physics and other subjects while on the other hand the physicists have lost the understanding of the problems and methods of the mathematicians, in fact even of the mathematicians’ entire sphere of interest and language. Without doubt in this tendency lies an important threat for science; the current of scientific development is in danger of seeping out further and further to ooze away and dry up. If it shall avoid this fate we must direct a good part of our forces toward again uniting the separated parts in developing clearly by means of collective points of view the internal connections of multi¬ farious facts. Only thus will a real mastery of the materials be possible for the student and the ground be prepared for the in¬ vestigator leading to a further organic development.” In any case the question is merely whether rigorous mathematics constitutes a more useful tool than “non-rigorous” mathematics and this question probably can not be answered from a 'priori consider¬ ations. At the same time the term “non-rigorous” can only mean incomplete or incorrect. The degree of completeness desirable is 1 Author’s translation from the Preface of Methoden der Mathematischen Physik I, R. Courant und D. Hilbert, Julius Springer, Berlin, 1924. XIV INTRODUCTORY of course a matter of judgment. As to the second interpretation of the term “non-rigorous,” while the whole question must of course be left to the empirical test, it is hard to see how incorrect mathematics could be more useful than correct mathematics. Of course no physical treatment can be more accurate than the initial hypotheses. Hence no treatment of science can be made rigorously accurate. For example, all physical hypotheses and definitions in this treatment presuppose that our physical opera¬ tions on systems are carried out on a certain scale. It is believed that all matter can be subdivided into electrons and protons. Now suppose that we are able to observe and wish to treat the behavior of these electrons and protons in a system. All we can say is that this treatment does not apply to such dynamic systems since the variables used in this treatment would not uniquely define the internal conditions of such a system. We therefore definitely limit ourselves to systems composed of such a large aggregate of these electrons and protons that when any such system is con¬ sidered as a whole, for we do not concern ourselves with the in¬ ternal condition of the system, the state of this system is uniquely determined by the set of operations we limit ourselves to, which is, for any one system, the characteristic equation or equation of state we set down for it. As a resume, it may be stated that the following have been the principal aims of this treatment: (1) Assuming the above limitations of the subject, to begin with the undefined concepts and physical assumptions and present the science of classical thermodynamics in a logical and mathe¬ matically rigorous manner. (2) To fill in the gaps existing in the present literature by de¬ ducing the theorems necessary for this development. (3) To evaluate the mathematical functions in terms of directly measurable quantities. (4) To compute the mathematical relationships obtainable be¬ tween the variables. NOMENCLATURE t 6 V m k m k v V £ e n V ( = e + pv- 6 { = e + p v — 6 2C = e + V v X = e + V v ijr = £ — 6 n \p = e — 6 rj W and W Q and Q temperature in degrees on the Centigrade scale, temperature in degrees on the absolute ther¬ modynamic scale. pressure in dynes per square centimeter or baryes. mass in grams of component k. mass fraction of component k, =-• mk , - mi + • • • +m n total volume in cubic centimeters, specific volume, volume per unit mass, in cubic centimeters per gram, total internal energy in dyne centimeters internal energy per unit mass in dyne centi¬ meters per gram. total entropy in dyne centimeters per degree, entropy per unit mass in dyne centimeters per degree per gram. n total zeta (Lewis’ free energy F) in dyne centimeters. rj zeta per unit mass in dyne centimeters per gram. total enthalphy or chi in dyne centimeters enthalpy per unit mass in dyne centimeters per gram. total psi in dyne centimeters psi per unit mass in dyne centimeters per gram, total work and work per unit mass respective¬ ly, received by the system in dyne centi¬ meters and dyne centimeters per gram, total heat and heat per unit mass respectively, received by the system in dyne centimeters and dyne centimeters per gram. XV XVI NOMENCLATURE h = -8 if) \Ol/ p,mi, m n C v C p -f~ Ip ^ ^ t, mi, m n ‘'Mk ^Wk Irak "l - Ip dp ,5m Mk P E q c <7 (x, y, z), (£, 7?, r) U, V, w Cl, . . . , Gq X, Y, Z X ! = X x X 2 = Yy X 3 = Z z X 4 = Z y = Y z x 5 = x z = z x X 6 = 7 X = X y c x u. k/2, », mi heat capacity per unit mass at constant pressure and concentration, latent heat of change of pressure per unit mass at constant temperature and con¬ centration, heat capacity per unit mass at constant Vt . .m n volume and concentration. latent heat of change of volume per unit mass at constant temperature and concentration, “reversible” heat of change of mass of com¬ ponent k where temperature, pressure, and the other component masses are constant, reversible” heat of change of mass of component k where temperature, vol¬ ume, and the other component masses are constant. “chemical potential” of component k in the phase. a parameter; used as curve or path, or time, density electromotive force quantity of electricity velocity of light acceleration of gravity coordinates projections of displacement vectors on the x, y, z axes respectively, strain components, (see sections 53, 60) projections of body force vectors on the x, y, z, axes respectively Stress components (e. g. Z y = traction along the z-axis across the y-plane). Considered positive when they are tensions and negative when pressures, (section 65) heat capacity per unit volume at constant stress. NOMENCLATURE XVII c e heat capacity per unit volume at constant strain. latent heat of change of stress parallel to the dd Jx x-axis per unit volume where temperature and the other stresses are constant. Similarly for l Xl , i = 2, ..., 6 dXA Latent heat of change of strain along the dei / t , e2 .e 6 x-axis per unit volume where tempera¬ ture and the other strains are constant. Similarly for l ei , i = 2, . . ., 6 dh, (i, h) 1, . . ., 6 “Elastic constants” (section 81). K Bulk modulus or modulus of compression compressibility E a R A e t (t, p, mi) Young’s modulus Poisson’s ratio modulus of rigidity cubical dilatation (section 86) is a notation for Tables I and II — m a denotes all the component masses, i.e. mi, • • •, m n ; mi all except m k ; mj all except m h ; m g all except m k and m h ; m b all except m k , m h and m y . Table II —Subscripts: e = e; n = n; x = y = z = C. ' . THERMODYNAMIC RELATIONS IN MULTI-COMPONENT SYSTEMS By Roy W. Goranson PART I CHAPTER I Fundamental Ideas UNDEFINED CONCEPTS OR DIRECTLY MEASURABLE QUANTITIES Let us begin our inquiry by considering what the undefined concepts or directly measurable quantities of thermodynamics are. We take them to be: length, time, mass, force, and temperature. The point of view adopted here in considering the physical concepts is that of Bridgman . 1 In treating these physical concepts there is no intent here to make the investigation an exhaustive one since that is not possible in a work of this size. However, the analysis of any theory must begin with such a treatment since these concepts are the starting point and hence an integral part of the theory. r 1. Length: Bridgman says: “Our task is to find the operations by which we measure the length of any concrete physical object. We begin with objects of our commonest experience, such as a house or a house lot. What we do is sufficiently indicated by the following rough description. We start with a measuring rod, lay it on the object so that one of its ends coincides with one end of the object, mark on the object the position of the other end of the rod, then move the rod along in a straight line extension of its previous position until the first end coincides with the previous position of the second end, repeat this process as often as we can, and call the length the total number of times the rod was applied. This procedure, apparently so simple, is in practice exceedingly com¬ plicated, and doubtless a full description of all the precautions that must be taken would fill a large treatise. We must, for example, be sure that the temperature of the rod is the standard temperature at w T hich its length is defined, or else we must make a 1 P. W. Bridgman, The Logic of Modern Physics (Macmillan, New York), 1927 . 1 2 THERMODYNAMICS §1 correction for it; or we must correct for the gravitational distortion of the rod if we measure a vertical length; or we must be sure that the rod is not a magnet or is not subject to electrical forces. . . . Practically of course precautions such as these are not mentioned, but the justification is in our experience that vari¬ ations or procedure of this kind are without effect on the final result. But we always have to recognize that all our experience is subject to error, and that at some time in the future we may have to specify more carefully the acceleration, for example, of the rod in moving from one position to another if experimental accuracy should be so increased as to show a measurable effect. In principle the operations by which length is measured should be uniquely specified. . .” sf: Hs “We . . . are also compelled to modify our procedures when we go to small distances. Down to the scale of microscopic dimen¬ sions a fairly straightforward extension of the ordinary measuring procedure is sufficient, as when we measure a length in a microm¬ eter eyepiece of a microscope. This is of course a combination of tactual and optical measurements, and certain assumptions, justified as far as possible by experience, have to be made about the behavior of light beams. These assumptions are of a quite different character from those which give us concern on the as¬ tronomical scale, because here we meet difficulty from interference effects due to the finite scale of the structure of light, and are not concerned with a possible curvature of light beams in the long reaches of space. Apart from the matter of convenience, we might also measure small distances by the tactual method. “As the dimensions become smaller, certain difficulties become increasingly important that were negligible on a larger scale. In carrying out physically the operations equivalent to our concepts, there are a host of practical precautions to be taken which could be explicitly enumerated with difficulty, but of which nevertheless any practical physicist is conscious. Suppose, for example, we measure length tactually by a combination of Johanssen gauges. In piling these together, we must be sure that they are clean, and FUNDAMENTAL IDEAS 3 § 1 are thus in actual contact. Particles of mechanical dirt first engage our attention. Then as we go to smaller dimensions we perhaps have to pay attention to adsorbed films of moisture, then at still smaller dimensions to adsorbed films of gas, until finally we have to work in a vacuum, which must be the more nearly complete the smaller the dimensions. About the time that we discover the necessity for a complete vacuum, we discover that the gauges themselves are atomic in structure, that they have no definite boundaries, and therefore no definite length, but the length is a hazy thing, varying rapidly in time between certain limits. We treat this situation as best we can by taking a time average of the apparent positions of the boundaries, assuming that along with the decrease of dimensions we have acquired a cor¬ responding extravagant increase in nimbleness. But as the dimen¬ sions get smaller continually, the difficulties due to this haziness increase indefinitely in percentage effect, and we are eventually driven to give up altogether. We have made the discovery that there are essential physical limitations to the operations which defined the concept of length. ... At the same time that we have r come to the end of our rope with our Johanssen gauge procedure, our companion with the microscope has been encountering diffi¬ culties due to the finite wave length of light; this difficulty he has been able to minimize by using light of progressively shorter wave lengths, but he has eventually had to stop on reaching X-rays. Of course this optical procedure with the microscope is more convenient and is therefore adopted in practice.” When using more than one operation we must so choose them that they give, within the experimental error, the same numerical results in the domain in which the two sets of operations may be both applied; but we must recognize in principle that in changing the operations we have really changed the concept and that to use the same name for these different concepts over the entire range is dictated only by considerations of convenience. We must therefore always be prepared some day to find that an increase in experimental accuracy may show that the two sets of operations which give the same results in the more ordinary part of the domain of experience lead to measurably different results in the more 4 THERMODYNAMICS §2 unfamiliar parts of the domain and thus must keep aware of the joints in our conceptual structure if we hope to render unnecessary the services of the unborn Einsteins. We choose as our concept the tactual and assume that the optical operation gives identical results. This is verified by our past experience but must remain subject to the possible modi¬ fication stated above. The concept of volume is an extension of that of length. The concept arises from the fundamental sensations in which all geometrical notions are founded. If we are in a room the presence of rigid bodies, as furniture, may prevent us from moving our limbs in certain ways and as a natural consequence we might say that the rigid bodies which prevent certain motions occupy space corresponding to the motions which they render impossible and which may be called motor space . 1 Further, before the estab¬ lishing of a quantitative method of measurement, we conclude from our vague senses that the amount of muscular motion excluded by a body indicates its volume. To obtain the general notion of a difference between different positions in space that does not depend upon their occupation by one material system rather than another and to find means of measuring magnitudes to replace the vague and subjective mus¬ cular sensations, a system of coordinates must be introduced. We accept as a fact that to every position in space defined by muscular motion there corresponds (if the system of coordinates is rightly chosen) one and only one set of coordinates. 2. Time: All ideas of time depend on the immediate judg¬ ments of “before,” “after,” and “simultaneous with” at the place of the observer. These relations are such that events can be arranged in a numerical order and thus physical judgments can be rendered in respect of time. The measurement of “time” is effected by means of periods which are properties of systems or individual bodies. The reali- 1 Poincare says if there were no solid bodies in nature there would be no geometry. §2 FUNDAMENTAL IDEAS 5 zation of a standard series is obtained by the use of isoperiodic systems, namely such as are characterized by a series of events A, B, C, D, which may be as long as we please, the periods AB, BC, CD, . . . being all equal as tested against some other period. Accordingly in such systems the periods AB, AC, AD provide the integral members of the standard series. The fractional members are provided by other isoperiodic systems, one event of these periods being made simultaneous with a member of the integral series. Pendulums and clocks are examples of isoperiodic systems. (For our limited purposes the ultimate isoperiodic system pro¬ viding an unending series of events is the rotating Earth.) With the standard system of periods established, it is then pos¬ sible to measure magnitudes, called time-intervals. The concept of time is determined by the operations by which it is measured, but the physical operations at the basis of the measurement of time have never been subjected to the critical examination which seems to be required. We might seek to specify the measurement of time in purely mechanical terms, as for in¬ stance in terms of the vibration of a tuning fork, or the rotation of r a flywheel. But here we encounter difficulties. As Bridgman says, “We want to use the clock as a physical instrument in determining the laws of mechanics, which of course are not determined until we can measure time, and we find that the laws of mechanics enter into the operation of the clock. “The dilemma which confronts us here is not an impossible one, and is in fact of the same nature as that which confronted the first physicist who had to discover simultaneously the approximate laws of mechanics and geometry with a string which stretched when he pulled it. We must first guess at what the laws are ap¬ proximately, then design an experiment so that, in accordance wdth this guess, the effect of motion on some phenomenon is much greater than the expected effect on the clock, then from measure¬ ments with uncorrected clock time find an approximate expression for the effect of motion on mass or length, with which we correct the clock, and so on ad infinitum. However, so far as I know, the possibility of such a procedure has not been analyzed, and until the analysis is given, our complacency is troubled by a real dis- THERMODYNAMICS 6 quietude, the intensity of which depends on the natural skepticism of our temperament. * * * “This discussion of the concept of time will doubtless be felt by some to be superficial in that it makes no mention of the properties of the physical time to which the concept is designed to apply. For instance, we do not discuss the one dimensional flow of time, or the irrevocability of the past. Such a discussion, however, is beyond our present purpose, and would take us deeper than I feel competent to go, and perhaps beyond the verge of meaning itself. Our discussion here is from the point of view of operations: we assume the operations to be given, and do not attempt to ask why precisely these operations were chosen, or whether others might not be more suitable. Such properties of time as its irrevoca¬ bility are implicitly contained in the operations themselves, and the physical essence of time is buried in that long physical ex¬ perience that taught us what operations are adapted to describing and correlating nature. We may digress, however, to consider one question. It is quite common to talk about a reversal of the direction of flow of time. Particularly, for example, in discussing the equations of mechanics, it is shown that if the direction of flow of time is reversed, the whole history of the system is retraced. The statement is sometimes added that such a reversal is actually impossible, because it is one of the properties of physical time to flow always forward. If this last statement is subjected to an operational analysis, I believe that it will be found not to be a statement about nature at all, but merely a statement about operations. It is meaningless to talk about time moving backward: by definition, forward is the direction in which time flows.” 1 3. Mass: The notion of mass is also sufficiently familiar, pro¬ vided we take it just as it comes, in its naive original form. We may try to convey the idea by saying that mass is the quantity of matter which a body contains, but this is not a defi¬ nition of mass; it is merely a statement of the concept in different words. 1 P. W. Bridgman, The Logic of Modern Physics, pp. 71, 78-79. §3 FUNDAMENTAL IDEAS 7 We agree that the unit of mass shall be the mass of a standard platinum-iridium cylinder known as the International Prototype kilogram. Measuring the mass of a body consists in comparing its mass with that of the standard cylinder. In order that this might be done conveniently, it was first necessary to construct bodies of the same mass as that of the standard cylinder, and then to make a whole series of bodies whose masses were 1/2, 1/10, 1/100, 1/1000, etc., of the standard mass; in other words to construct a set of standard masses. This we are able to do quite easily since the mass of a compound body formed by uniting two or more bodies is, by the nature of the concept, equal to the sum of the masses of the separate bodies. Thus two equal masses which are together equal to the standard unit mass will each be one-half the mass of the unit; four equal masses which together are equal to the standard unit mass will each be one-quarter of the mass of the unit; and so on. With the aid of such a set of standard masses the determination of the mass of any unknown body is made by first placing the body upon one pan of a balance and counterpoising with shot, paper, etc., and then replacing the unknown body by as many of the standard masses as are required to bring the pointer back to zero again. The mass of the body is equal to the sum of these standard masses. 1 We have assumed that the gravitational field of the earth acts equally on equal masses. For our purposes this method of de¬ termination of mass is sufficient. If we, however, go to high velocities and to extended space we are in difficulties since we do not know whether the force of gravity is independent of velocity at high velocities. Here the concepts of force and mass lose their definiteness and become partially fused since there are no oper¬ ations by which force can be obtained as a function of velocity without knowing the mass or any operation by which mass can be measured without knowing the force. 1 This method of determination of mass is called the method of substitu¬ tion. See R. A. Millikan and H. G. Gale, A First Course in Physics, 1913, 8 THERMODYNAMICS §4 4. Force: The idea of force is also familiar in the form of a push or pull. 1 To find how hard we are pulling when we hold a kite string it is only necessary to tie a spring balance to the end of the string and note how far the spring is stretched. The unit of force is perfectly arbitrary, being simply the force which causes any agreed-upon amount of stretch in a standard spring; and a complete scale of multiples and submultiples of the unit is readily established by simply opposing one or more un-, marked springs, in various combinations, against the standard, or unit, spring, and marking the positions reached by the pointers. 2 By means of a portable, graduated spring balance thus con¬ structed we are able theoretically to measure forces for systems in equilibrium in terms of the arbitrarily chosen unit of force repre¬ sented by our originally chosen standard spring. “We next extend the force concept to systems not in equilibrium, in which there are accelerations, and we must conceive that at first all our experiments are made in an isolated laboratory far out in empty space, where there is no gravitational field. We here encounter a new concept, that of mass, which as it is originally met is en¬ tangled with the force concept, but may later be disentangled by a process of successive approximations. The details of the various steps in the process of approximation are very instructive as typical of all methods in physics, but need not be elaborated here. Suffice it to say that we are eventually able to give to each rigid material body a numerical tag characteristic of the body, such that the product of this number and the acceleration it receives under the action of any given force applied to it by a spring balance is numerically equal to the force, the force being defined, except for a correction, in terms of the deformation of the balance, exactly as it was in the static case. In particular, the relation found between mass, force, and acceleration applies to the spring balance itself by which the force is applied, so that a correction has to be applied for a diminution of the force exerted by the balance arising from its own acceleration. 1 W. F. Osgood, Introduction to the Calculus, 1922, p. 348. 2 This process involves no knowledge of Hooke’s Law; it merely assumes that the elastic properties of the spring do not vary with the time. §5 FUNDAMENTAL IDEAS 9 “We now extend the scope of our measurements by bringing our laboratory into the gravitational field of the earth, and immediately our experience is extended, in that we continually see bodies accelerated with no spring balance (that is, no force) acting on them. We extend the concept of force, and say that any body accelerated is acted on by a force, and the magnitude of this force is defined as that which would have been necessary to produce in the same body the same acceleration with a spring balance in empty space. There is physical justification for this extension in that we find we can remove the acceleration which a body acquires in a gravitational field by exerting on it with a spring balance a force of exactly the specified amount in the opposite direction.” 1 In extending the notion of force to systems not in equilibrium (moving in force fields) we have changed the character of the concept—the force acting on the body is now measured in terms of mass-acceleration. The hypothesis is made that these two operations measure the same thing. 5. Temperature: The concept “which sets thermodynamics off apart from the simple subjects is probably that of temperature. In origin this concept was without question physiological in much the same way as the mechanical concept of force was physiological. But just as the force concept was made more precise, so the tem¬ perature concept may be more or less divorced from its crude significance in terms of immediate sensation and be given a more precise meaning. This precision may be obtained through the notion of equilibrium states.” 2 The idea of temperature is obtained from our sensation of how hot or how cold a body is. By touching a number of bodies with the hand, we can arrange them roughly in order, from the coldest to the hottest. Such an order, however, is likely to be modified by a second series of observations. By touching the bodies with a glass tube enlarged at the bottom and partly filled with mercury, and determining the place of any body in the series according to 1 P. W. Bridgman, The Logic of Modern Physics, pp. 102-103. 2 P. W. Bridgman, op. cit., pp. 117, 118. 10 THERMODYNAMICS §6 the amount by which the level of the mercury rises when the tube is placed in contact with it, we secure an order which is modified only in the cases of a few bodies when a second series of obser¬ vations is made. If, finally, we use an apparatus which we call a hydrogen gas thermometer, we can repeat the observations many times for all the bodies, each of which is kept under constant conditions, without finding it necessary to change the order. The reading of this hydrogen gas thermometer, when placed in contact with a body, we agree to take as the temperature of the body. This will be true only if the mass of the thermometer is infinitely small compared with the masses of the bodies whose temperatures we are measuring. We find that the temperature reading will depend on the relative masses of the systems measured and the thermometer, and on the kind of material used as container for the hydrogen gas. Thus, since in actual practice the thermometer must be of finite size with respect to the system measured, suitable corrections must be made for the mass of the thermometer and kind of material used in its construction. By means of this thermometer we can measure temperature just as we can measure force by means of the spring balance. Our concept is that the reading of this thermometer corresponds to a physical quantity. 1 6. General Note regarding the physical quantities: Among the reasons which justify the introduction of the quantities enumer¬ ated as physical quantities is the empirical observation of certain rather obvious relations. In the case of temperature the first of these relations is as follows: Let A, B and C be three bodies. If A undergoes no change of 1 For purposes of thermodynamics as we have limited them temperature can certainly not be defined in terms of flow of heat. Thus the statement by G. N. Lewis and M. Randall on page 57 of “Thermodynamics and the free Energy of Chemical Substances,” for example, that “if there can be no thermal flow from one body to another, the two bodies are at the same temperature; but if one can lose energy to the other by thermal flow, the temperature of the former is the greater” would be incorrect as a definition of temperature. On this subject P. W. Bridgman, Logic of Modern Physics, p. 125, says: “The essential fact that a quantity of heat can by itself be defined only in terms of a drop of temperature is somewhat obscured by the usual method of thermodynamic analysis.” §6 FUNDAMENTAL IDEAS 11 volume on being placed in contact with B and if further B under¬ goes no change of volume on being placed in contact with C, then it is a matter of experience that A will undergo no change of volume on being placed in contact with C. (Thus if A and B are in thermal equilibrium, and B and C are in thermal equilibrium, then A and C are in thermal equilibrium.) The other relation is: If the volume of A decreases and that of B increases when they are placed in contact, if further the volume of B decreases and the volume of C increases when they are placed in contact, then we usually find that the volume of A will decrease and that of C will increase if they are placed in contact. (Thus if the temperature of A is greater than that of B and the temperature of B is greater than that of C then the temperature of A is greater than that of C.) In the case of force analogous relations are observed. Let A, B and C be three spring balances each with a pointer attached to the spring. If A is extended until its pointer is opposite an arbi¬ trary notch on its frame when opposed by B and a notch is then cut opposite the pointer of B on its frame, if further B is extended until its pointer is opposite the notch on its frame when opposed by C and a notch is then cut opposite the pointer of C on its frame, then it is found that if A is extended until its pointer is opposite the notch on its frame when opposed by C the pointer of C will be opposite the notch on its frame. A statement regarding inequalities analogous to that made in the case of temperature could also be made in the case of force. Similarly for the other physical quantities. These, then, are our fundamental quantities: length, as measured by a meter bar; time, by a chronometer; mass, by a balance; force, by a spring balance; and temperature, as measured by a hydrogen gas thermometer. Thermodynamics, like the other branches of physics, deals with two kinds of quantities: those directly measurable experimentally 1 and those which are defined in terms of the directly measurable quantities by means of mathematical equations. In addition to the equations of definition of the quantities not directly measurable two other kinds of relations are present in the subject. We assume 1 Called undefined concepts. 12 THERMODYNAMICS §7 certain equations between various quantities of the two classes to be true as physical hypotheses: our knowledge of the truth of these equations may be directly or indirectly obtained from ex¬ periment; they can not, however, be derived mathematically from any relations previously obtained by definition and physical hypothesis. Lastly we have equations derived mathematically from equations already obtained by definition or physical hypo¬ thesis. According to Bridgman equations derived mathematically are never definitely known to be true physically until they are proved so experimentally. Poincare reaches the same conclusion and adds that the function of mathematics is not to produce new truth from old physical truth but to suggest new significant experiments. Moreover the mathematically derived consequences of a physical hypothesis may be useful in testing the hypothesis in case the hypothesis can not be verified directly by experiment. If the mathematically derived consequences are found to be true ex¬ perimentally we conclude that the probability that the hypothesis is also true is thereby increased. 1 The directly measurable quantities were first known quali- tativety and roughly quantitatively by physical methods. They are accurately defined qualitatively and quantitatively, as stated by Bridgman, in terms of physical operations. 7. Property : 2 By stating the properties of a system we describe its condition at a given instant. When a piece of steel is subjected to mechanical treatment its final volume is a property of the steel at the end of the process. These properties are either length, force, temperature, or mass, or quantities expressed in terms of the fundamental four quantities. Properties can be divided into two classes. The mass or volume of two identical systems, say two kilogram weights of brass or two exactly similar balloons of hydrogen, is double that of each one. 1 F. C. S. Schiller (Studies in the History and Method of Science, Edited by Charles Singer, Clarendon Press, 1917, p. 268) states that the logicians have pointed out for 2000 3 'ears that this involves a breach of formal logic. Schiller agrees that it does but concludes not that this fundamental method of science is wrong but rather that formal logic is worse than useless in science. 2 G. N. Lewis and M. Randall, Thermodynamics, 1923, pp. 12, 13. §7 FUNDAMENTAL IDEAS 13 Such properties are called extensive. The temperature of the two identical objects, on the other hand, is the same as that of either one. Properties of this kind are called intensive. Intensive properties are in many cases derived from the extensive properties. Thus while mass and volume are both extensive, the density and specific volume are intensive properties. STATE 1 In thermodynamic considerations we sav that the state of a •» ** system is given when all of its intensive properties are fixed. In thermodynamics we first consider systems in which the m •» properties vary continuously from point to point. A system, the properties of which are the same at all points, we define, following Gibbs,- as a homogeneous system. A heterogeneous system is defined as one consisting of two or more homogeneous parts. EQUILIBRIUM STATE If the properties of a system undergo no change after the lapse of a period of time, no matter how greatly it is extended, the system is said to be in a state of equilibrium. Properties here refer to the external conditions of the system. If we were to examine the internal conditions of, say, a gas on a small enough scale we would then observe inhomogeneities re¬ sulting from Brownian movement and other phenomena. All that we are concerned with, however, is that, for an equilibrium state, all such interactions in the system will be subject to the condition that the external properties remain constant as measured by our scale of operations. Our first physical hypothesis is that a homogeneous system is in an equilibrium state. 5 : G. X. Lewis and M. Randall, op. cit., p. 12. : The Collected Works of J. Willard Gibbs. YoL 1, p. 63. •' The assumption that one of the conditions for the state of a homogeneous system to be a state of equilibrium is that the pressure be the same at all points :s identical with one of the postulates of statics. CHAPTER II Simple homogeneous systems UNIT MASS SYSTEMS In thermodynamics we consider first simple homogeneous systems in which the composition, density, and temperature are the same at all points, and which are subject to a pressure which is the same at all points and which is the same in all directions at any given point. 8. Equation of State or characteristic equation: The specific volume of a system of this type is defined as the total volume divided by the total mass, v v = — m We assume as a physical hypothesis that when any two of the properties pressure, specific volume, and temperature of a par¬ ticular sj^stem are given, the third is determined, or that 9 (P, v, t) = 0, where p denotes the pressure in dynes per square centimeter, v the specific volume in cubic centimeters per gram, and t the temper¬ ature in centigrade degrees on the scale of the hydrogen gas thermometer. Each pair of values of the two properties which can be varied independently may be represented by a point in a Cartesian co¬ ordinate plane, the abscissa and ordinate of which are proportional to the values of the properties. Thus we may “say” that each point represents a state of the system. A series of states in which the properties vary continuously may be spoken of as a continuous series of states and may be represented geometrically by a continuous curve in the coordinate plane. A hypothetical “reversible process” of expansion is then a 14 §9 SIMPLE HOMOGENEOUS SYSTEMS 15 continuous series of equilibrium states which can be represented by a continuous curve in the ( t , p) coordinate plane or analytically by the equation t = f(p). Following the usage of Gibbs, 1 “For the sake of brevity it will be convenient to use language which at¬ tributes to the diagram properties which belong to the associated states of the body. Thus it can give rise to no ambiguity, if we speak of the volume or temperature of a point in the diagram instead of the volume or temperature of the body in the state associated with the point. ... In like manner also we may speak of the body moving along a line in the diagram, instead of passing through the series of states represented by the line.” 9. Definitions of work and heat: Let us take as the two inde¬ pendent variables t and p, and let us represent t by the ordinate and p by the abscissa of a point in a Cartesian coordinate plane. Then the work received by the system during the “reversible” expansion, i.e. in passing through a series of states represented by points of the continuous curve or simply work of the path, 2 W, in ergs per gram is defined geometrically as the area of the cylinder erected on the curve or analytically as the line integral, 3 t, V f dv At i A -J v Jt At + v dv dv (1) to, po t, p f dv At I dV A w = — P Jt dt + p-dp (2) J to, po 1 J. Willard Gibbs, Collected Works, Vol. 1, footnote, p. 3. 2 The terms “work of the path” and “heat of the path” are adopted from Gibbs, Scientific Papers, Vol. 1 (1906), p. 3. He says “W and H (= our Q) are not functions of the state of the body (or functions of any of the quantities, v, p, t, e and i?), but are determined by the whole series of states through which the body is supposed to pass . . . Suppose the body to change its state, the points associated with the states through which the body passes will form a line, which we may call the path of the body. The conception of a path must include the idea of direction, to express the order in which the body passes through the series of states.” 3 For a definition of line integral see W. F. Osgood, Lehrbuch der Funk- tionentheorie, p. 125. THERMODYNAMICS 16 § 10 and the heat received or heat of the path, Q, in ergs per gram is defined as the line integral, t, p r / l p ( t , p) dp + c p (t , p) d l, (3) J to, pc t, p r / c p (t , p) d t + Ip ( t , p) dp (4) J to, P 0 where c p and l p are some continuous functions of A t and Ap. 10. Reversible vs. irreversible processes: A system can not actually pass through a continuous series of equilibrium states, since according to the hypothesis made by all authors an equilibrium state is one in which the properties of the system do not change with time. 1 In reality the consideration of the “‘irreversible” processes, i.e. continuous series of non-equilibrium states, must therefore logi¬ cally precede the consideration of “reversible” processes. 2 The quantity customarily called the heat received in the re¬ versible process, but correctly named by the phrase of Gibbs “work or heat of a line” can in reality only be evaluated as the limit of the quadruple integral which is measured physically and which is the heat received in an irreversible process. Following the customary methods in treating mathematical and physical subjects, we shall, however, first take up the simpler part of the subject although it must logically be preceded by the more general case which is more complex and therefore is left until the end. 1 See § 91 2 J. Willard Gibbs, Collected Works, Vol. 1, p. 55. Gibbs has said: “As the difference of the values of the energy for any two states represents the combined amount of work and heat received or yielded by the system when it is brought from one state to the other, and the difference of entropy is the limit (my italics) of all the possible values of the integral J *(dQ denoting the element of the heat received from external sources, and t the temperature of the part of the system receiving it), the varying values of the energy and entropy characterize in all that is essential the effects producible by the system in passing from one state to another.” §11 SIMPLE HOMOGENEOUS SYSTEMS 17 There is no objection to doing this provided that in the portion left over for the time being we do not make use of any of the theorems brought out in the part treated first. In fact, the part left over is really so treated that it is logically first and anyone desiring a logical treatment may secure this by reading it first. It may also be pointed out that the treatment of so-called reversible processes does not make use of theorems deduced in the treatment of irreversible processes, and thus the mathematical treatment of each of these two parts is complete in itself. The necessity for the logical precedence of the irreversible processes is due to the physical, not the mathematical, nature of the situation. For the convenience of those readers who do not wish to spend time going through the treatment of irreversible processes the customary language of reversible processes, which Gibbs also has made use of (although as has been pointed out above he clearly recognized that a “reversible” process in reality must be treated as the limit of “irreversible” processes), will be used in the first part, and thus we shall speak of the heat received and work done in reversible processes. 11 . Transformation of heat and work integrals: Now by hypothesis p can be expressed as a function of t and v, Thus t I V = U (t, v ) t, V t,v c p d£ -j-Ip dp = J [c p + Ip dp dt d t+lpp^ dy dv to, V o Let us define and to, VO 11 dp _ Cp “ h Lp dt ~ Cv then t I l — = l h dv v ’ t, p t, v Cp dt + l P dp = to, p 0 to. VO ( 1 ) ( 2 ) (3) (4) c„ d( + l„ dy (5) 18 THERMODYNAMICS § 12 Similarly for the work integral we can make the transformation, t, p l, V - I p^dt + p^dp = - / pdv to, p o to, VO ( 6 ) 12. Differentials and derivatives of heat and work: Now the coordinates of the curve t — f (p) can be expressed as functions of the distance along the regular curve, s, measured from an arbitrary point, t = ¥>(s), p = £ (s), (1 i s S L), and is a notation for c p d£ + l p dp to, po n — 1 lim n > co 2j°p V'k) A4 + 1 P (^k, p'k) Ap k , k = 0 where the interval of the regular curve from (4, po) to (t, p) is divided into n parts by the points s 0 = 1, Si, s 2 ,., s n _i, s n = L, and L — 1 is the length of s. A4 and Ap k denote the differences 4+1 — 4 and p k+ i — p k , and (t' k , p' k ) is an arbitrary point of the kth arc, (s k _i, s k ). To evaluate the limit we may write the summand in the form: c„ (4, p'O + l p (4, P' t ) ^ As. Now since lim As —> At lim n . = cos a, 0 As As Ap a n —r~ — C°S j8, 0 As it follows, by Duhamel’s theorem, that * n — 1 C P (l'k, p'k) A4 + l p (t\, p'k) Ap k k = 0 §12 SIMPLE HOMOGENEOUS SYSTEMS 19 n — 1 iim n —> °o j^Cp (Z k, p k) COS a + l p (Z' k, p'k) cos /3 As. k = 0 The limit of the latter sum is (L ),4> (L) V (s) f ds diHL) dL Since L is the length of arc from an arbitrary point we can now replace it by s dQ _ dZ , dp ds Cp ds ”ds’ or dQ = Cp dZ + Zp dp, where A s is the independent variable and A Z and A p depend upon A s. 20 THERMODYNAMICS § 13 Similarly dw = p f tAt+p ^ dp where A s is the independent variable and A t and A p depend upon As. 13. Definitions of the heat capacities per unit mass: Let us suppose that t and p are the properties which can be varied independently. Along any definite path s and hence and V = k (s), t = l(s), dQ dp ds p ds +p ds dW ds dv d£ _ dv dp ^ dt ds ^ dp ds* ( 1 ) ( 2 ) (3) In particular the path may be defined by the equations t = s, p = K, where K denotes a constant. Along this path dQ d Q ds dt per unit mass at constant pressure. Then which we define as the heat capacity d Q d£ c p and along this path dW d W dv ds d t V df Similarly, the path may be defined by the equations t = K f p = s, where K denotes a constant. Along this path —■ = ^ as dp which we define as the latent heat of change of pressure per unit mass at constant temperature. § 13 Then SIMPLE HOMOGENEOUS SYSTEMS 21 d Q_, dp p and along this path dW = dW = _ dv ds dp ^ dp' ( 6 ) (7) Let us now suppose that t and v are the properties which can be varied independently. Along any definite path v = j (s), t = h(s), where s denotes the distance along the curve, and hence dQ _ dt 1 dv ds Cv ds v ds and dlT _ dv ds ^ ds* ( 8 ) (9) In particular the path may be defined by the equations t — s, v = K, where K denotes a constant. Along this path ^ ^ which we define as the heat capacity per unit mass at constant volume. Then dQ dt C y ( 10 ) and along this path dW _ dW ds d£ ( 11 ) Similarly, the path may be defined by the equations t — K, v = s, where K denotes a constant. Along this path ^ = — which we define as the latent heat of change of volume per unit mass at constant temperature. 22 THERMODYNAMICS § 14 Then d Q _ 7 dv v ’ and along this path dW = dW ds dz; ( 12 ) (13) 14. The first law of thermodynamics: The FIRST LAW of thermodynamics for homogeneous systems of unit mass states that t, v (t, V) ~ e (to, po) = to, V o dv dv L c * “ p aij dt T 1 ' 1 ^ a I 1 dp (1) the line integral being extended along any path connecting the coordinates (to, po) and (t, p), where e (t, p) denotes a function of the temperature and pressure of the system. This function defined by the preceding equation is called the internal energy of the system at the temperature t and pressure p of the coordinates (t, p). We further complete the definition of this function by defining e (t 0 , p Q ) as zero, e (to, po) = 0. 1 15. Geometric interpretation of the first law: The geometric in¬ terpretation of the line integral dv dp_ ds extended along a particular path c is this: Let a cylinder be con¬ structed on c as a generatrix, 2 its elements being perpendicular to the (t, p)-plane, and let the values of the function F (s) = c P - V dv dt Ip p dv dp_ dp ds be laid off along the elements of this cylinder. Then the area of the 1 For justification of this definition see § 93 . 2 1 use the terms cylinder and generatrix in their usual geometric meaning. See W. F. Osgood and W. C. Graustein, Plane and Solid Analytic Geometry (Macmillan Co.) 1922, p. 532. SIMPLE HOMOGENEOUS SYSTEMS 23 § 16 cylinder bounded by this curve and the generatrix represents the line integral. The first law of thermodynamics is the assumption that the areas of the cylinders erected on any two curves Ci and c 2 connecting the coordinates (to, po) and (t, p) are equal. 16. The necessary and sufficient conditions for a line integral, expressed as the energy integral, to be independent of the path 1 : Let c v — V ?? and l v — Pi~ be two continuous functions in a dt dp simply connected region S. 2 Let; exist and be continuous in S. The line integral dv i d , dv and — dt c dv 7 dv l e ’- p dt\ dt -j- _ lp ~ p ep_ dp extended over a curve C lying in S and made up of a finite number of smooth pieces of curve, i.e. over a regular curve, is then and only then independent of the path if over the whole interior of S c v -p dv dt d_ dt 7 dv . ” ~ P dP J For a fixed initial point the integral is then given as'a continuous function e (t, p) of the coordinates (t, p) of the endpoint of C, the first derivatives of e being continuous and given byj the equations /. n _ dv e t (l P) G> V dp e p 0 t,p ) = l P - We shall prove first the necessity of our condition. 1 The proof given here can be readily extended up to and including the n-component variable mass case with n + 2 independent variables. The proof by Green’s theorem, although shorter, makes use of a great deal of geometric intuition, and, furthermore, it can not be extended to the other cases we deal with where more than two independent variables are present. For a further treatment of line integrals see W. F. Osgood, Lehrbuch der Funktionentheorie (B. G. Teubner, Leipzig), 1912, pp. 123-145; and A. Hur- witz and R. Courant, Allgemeine Funktionentheorie und Elliptische Funk- tionen (Julius Springer, Berlin), 1925, pp. 267-268. 2 By a simply connected region is meant a region such that no closed curve drawn in the region contains in its interior a boundary point of the region. All other regions are called multiply connected. 24 THERMODYNAMICS § 16 For this we shall assume that our line integral is independent of the path, then it is given as a function e (t, p) of the endpoint alone. We are to prove that this function is differentiable and that equations (3) hold. Now since the line integral is assumed independent of what path we choose between the limits of integration provided that the path be a regular curve lying inside the simply connected region in which the line integral is defined, we may suppose that the path of integration goes from (a, b) to (t, p), and then from ( t, p) to (l + A t, p) along a line parallel to the £-axis, i.e. along which p = constant. € (t + A t, p) — e ( t, p) = integral from (a, b) to ( t, p) + integral from ( t , p) to (t + At, p) — integral from (a, b) to ( t , p). =/ + p,P)-P d -^]dt t, v Applying the law of the mean, we write t(t + At,p) - t (t, p) = At [c, (t + a A t, p) - p dv (t + d a ' At ’ P) j where 0 < a < 1, 0 < a' < 1. Taking the limit when A t approaches zero of e (t + At, p) - e ( t, p) At u , dv gives ( t, p) = c p - p — • Similarly ( t, p) = l P -p~ ot But d 2 € dpdt and O') SIMPLE HOMOGENEOUS SYSTEMS 25 § 16 d dv , d 7 dv since — dp _ Cp ~ P and Jt y ~ v dj>_ exist and are con¬ tinuous by hypothesis, we have d 2 a 2 dp dt dt dp Therefore dp dv~\ d f dv (2') is a necessary condition for the line integral to be independent of the path. To prove that condition (2) is sufficient that the line integral (1) be independent of the path, i.e. zero for all closed paths, we shall set up a function e (t , p) such that e t (t, p) *= c p — p ^ and e p (t, p) = l p — p If such a function exists the line integral is independent of the path, for then f[c P -pf^di + [l p -pl^dp C V) d< + (t, p) dp = fdt (i, v) a* = (tl, Pi) t =

z P, 4 a, t, a 2 Diagram 1 € (t lt pi) dv 7 dv L c * - p a^J d t -f- l l ’- p r P J p dv (t, bi) dt d t -f- V i J Ip (fi, v) bi V dv(tl, P) dp 1 The assumption that the region S is a simply connected region is essential in our proof. From this assumption we make use of the fact that the integral extended over a simple closed stepped path can be represented as the sum of integrals over rectangles. Since this assumption checks the physical hypoth¬ esis we need not concern ourselves here with more complex regions. § 16 Thus SIMPLE HOMOGENEOUS SYSTEMS 27 de (t h pi) dt x c P (t x , bi) - p dv (t h bi) dt + Similarly Vi _d dhi l P (h, V) - V bi dv(t x , v) dp dp \ dv (< 1 , bi) = C„ ( Po) = 0 where 6 = V (t 0 ) = 0. 2 19. Differential and partial derivatives of the entropy: Since the line integral is independent of the path we can connect the two limits of integration by any regular curve provided that this curve lies wholly in the simply connected region in which the line integral is defined. Let us integrate over some regular curve from (a, b) to (t, p), then along the straight line from (t, p) to (t + At, p), p = constant. 1 This limit is assumed, as a physical hypothesis, to be zero for crystalline substances. Some others believe it to be zero also for liquids. 2 For justification of this definition see § 93. 32 THERMODYNAMICS § 19 7 i(t + At, p) — y ft V) — integral from (a, b) to ft p) -f integral from ( t, p) to (t + At, p) — integral from (a, b) to ( t, p) t -f At, p [ c p ( t, p) d t J 0 / t, p Taking the limit, when AZ approaches zero, of V (t + At, p) - rj (t, p) At gives rj t ( t, p) = Similarly t? p ( t, p) = Thus dT, = C fdt + l jd P , where At and A p are the independent variables. Along any definite path s Id Q 6 ds Integrating we have the line integral being extended along the given path s. For simple systems of constant mass, homogeneous composition, and having the same temperature and the same pressure at all points and in all directions at a given point we assume as a physical hypothesis that there is a functional relationship between t, p and v which may be expressed as

(P, 9) = 0, where 0 represents some function of p, r, 20. Relations between energy derivatives and heat capacities, between entropy derivatives and heat capacities, and between derivatives of heat capacities (all per unit mass): From the first law we have r (0,p) = Cp-P^j (1) and From the characteristic equation p — g(v, 6), according to the first law 6, v 0, v € (6, v) — e (6q, Vo) f c, 00, to d0 + l v dv — / p dt; -/> 00, to 34 THERMODYNAMICS §20 Hence Thus e e (0, v) = c v and e v ( 6 , v) = l v — p dc v \ _ / dl v \ / dp ,dv )e \dd Jv \dd J v From the second law we have Thus V* (9, v) = j and ij„ (0, P) = l f dp dd or / dc p \ _ /dZ p \ _ Ip \dv)e~\ddjp 0 According to the second law Hence 9, v v - v (K vo) = J -gde + l fdv do, VO V 0 (9, v) = ~ and ij„ (0, v) = ^ Thus or dv Je \ dd dcA _ / dl v \ _ ly dv Je \d0/ v 6 Combining (20.6) and (20.12) we have dp' l v = 6 dd J v (4, 5) ( 6 ) (7,8) (9) ( 10 , 11 ) ( 12 ) (13) §21 SIMPLE HOMOGENEOUS SYSTEMS 35 Substituting the value of l v in (20.12) we have ( dc v \ = Q (d 2 p dv ) e \ dd' 2 / v Similarly from (20.3) and (20.9) Substituting the value of l p in (20.9) we have _ _ n (Vv\ dp )e \de 2 J T (14) (15) (16) DIFFERENCES OF HEAT CAPACITIES PER UNIT MASS From (20.5) From (20.13) Hence lv = 0 yp c v = 6 dp dp dv dd. ,dd / „ p Substituting the values of c v and l v in (13.8) we have dQ _ /de\ dv /de\ dd dv \dv)e ds \dd/ v ds ^ds ds Substituting the values of c v and l p in (13.2) we have dQ (de\ dd (de\ dp , / ds \d0/pds \dpje ds ^ dA d0 / d A dp K dO/p ds \dp)e ds_ (17) (18) (19) HOMOGENEOUS ONE-COMPONENT SYSTEMS OF VARIABLE MASS 21. Definitions of component and phase: For the substances or components of which we consider the mass composed we shall choose chemical species or combinations of chemical species. 1 1 It would do no good, as we have seen, to choose a dynamic system con¬ sisting of electrons and protons as our system for we could not treat it thermo¬ dynamically here since our variables would not define the state of such a system. 36 THERMODYNAMICS §21 Furthermore these substances need not have any relation to the internal constitution of the system. They must, however, be so chosen that the masses of each of them in the system, that is, mi, . . ., m n , where the number of components is n, are all independent of each other and such that they will express the composition of the homogeneous masses ( = phases), over the whole range of states through which we wish the system to pass. Gibbs calls a substance an actual component of a phase when the substance is capable of a continuous increase or decrease in amount in that phase. He calls a substance a possible component of a phase if the sub¬ stance, though not initially present in this phase, exists in some other phase which is in equilibrium with the first phase and from which the first phase might abstract the substance by a continuous change of concentration. A substance would then be capable only of a continuous increase in amount in the phase in which it exists as a possible component. The actual components need give us no difficulty, but I am not able to cite any actual examples to illustrate his possible com¬ ponents. In fact some people believe that no such examples of possible components exist. Such a case would occur if we had, say, anhydrous sulfuric acid in equilibrium with some other phase from which it might abstract water. Then water would be a possible component of the phase anhydrous sulfuric acid. A possible confusion of actual components w T hich might occur is in considering dilute solutions (see Appendix). We have here two situations which depend on our choice of components. For example if we choose FeCl 3 .6H 2 0 (represented on diagram 2 as a vertical dot-dash line), and water as our components, then at a temperature of 310 degrees on the absolute thermodynamic scale and a pressure of one atmosphere a solution composed solely of FeCl 3 .6H 2 0 (i.e. the mass of the water component = 0), will be in equilibrium with crystals of the same composition. Now whether we add water to or subtract water from this liquid phase, keeping temperature and pressure constant, crystals of FeCl 3 .6H 2 0 will go into solution. Here water, for m H2 o = 0, is capable of §21 SIMPLE HOMOGENEOUS SYSTEMS 37 positive and negative values. This situation will occur when we have a maximum as illustrated in Diag. 2. On the other hand we may so choose our components that the mass of the water com¬ ponent, when zero, is capable only of positive values, i.e. can only increase in amount. Such would be the case if, for example, we chose anhydrous sulfuric acid or chloroform and water as our components, for here water can not decrease below the value zero. Diagram 2 In neither of these situations does water come under the classi¬ fication of possible component since we have either added or removed it from the whole system. For water to be a possible component of an anhydrous phase we must have the anhydrous phase in equilibrium at some state with the phase containing water, assuming of course the water can exist as a component of the anhydrous phase. This would be true if the anhydrous phase 38 THERMODYNAMICS §21 were miscible with water over the range of temperature, pressure and concentrations we are considering. We do not have to consider the internal changes in the system and thus may disregard dissociation in the phase. For example 1 let us consider a potassium chloride-water solution where part of the KC1 is dissociated, KC1^K++C1- (1) Let n" be the chemical potential of the solid potassium chloride, i u' that of the potassium chloride in solution, [x , that of the undissociated potassium chloride in the solu¬ tion, ju',, that of the potassium ion in the solution, and [i!,,, that of the chlorine ion in the solution. Now if we have the potassium chloride-water solution in equilib¬ rium with solid potassium chloride then / = **" ( 2 ) but 74.56 m', = 39.1 + 35.46 (3) and thus 74.56 p! = 74.56 (1 - a) /, +39.1 a+ 35.46 a (4) where a denotes the degree of dissociation. If we multiply (3) by a and subtract (3) and (4) we have / / M = M , or, from (2), 1 This discussion belongs here but since we have not yet defined /x it may be passed over for the time being. §22 SIMPLE HOMOGENEOUS SYSTEMS 39 The dissociation constant has thus dropped out of the equations. Furthermore we can not obtain the amount of dissociation from this kind of reasoning. Thus in order to determine the potentials of the ions one of the things we must have given us is the degree of dissociation. Each homogeneous part of a mass we shall call a phase and thus the number of phases will be determined by the number of parts of the mass considered that differ in composition or state or both. Crystalline FeCl3.6H 2 0 in equilibrium with crystalline FeCl 3 .3JH 2 0 is an illustration of two phases that differ in composition; ice in equilibrium with water is an illustration of two phases that differ in state; a water solution of sodium chloride in equilibrium with water vapor is an illustration of two phases that differ both in composition and state. Thus a phase is a geometrically connected subdivision of a physical system in which each one of the components is physically distinguishable from the same component in the other phases. 22. Definitions of heat and work: The work, W, done on the system we define by the equation t, p, m t, p, m W = — J pdv = — J mp-dt-t-mp-^dp-j-pv dm to, V o, mo to, po, mo and the heat, Q, received by the equation t, p, m Q = j m c p dt + m l p dp + / dm to, po, mo where c p , l p and / denote functions of t and p. This is really an extension of the ordinary definition of heat. To a physicist / here has meaning in terms of heat received measured as such only when there is an interchange of heat between different parts of the system; for example if we had two phases present, one increasing and the other decreasing. The physical significance of / is indicated in §24. 40 THERMODYNAMICS §23 23. The first law of thermodynamics: Then the first law of thermodynamics is expressed by the equation t 9 j) f m e ( t, p, m) - c (t 0 , p 0 , m 0 ) = J m ^c p - p^jdt to, po, mo + m (l P -p dp + 0 + / - pv) dm (1) where n denotes a function of the temperature and pressure. In the special case where the mass, m, remains constant we have t, p c (t, p, m) - e (t Q , po, m) = m J (c p - p^j d t to, po + (u - V g) d P- Now by definition e = me, then Thus and hence Now £ (to, Po, m) = m e (to, Po) = 0. E (to, Po, 0) = 0, e (t, p, 0) = 0. t, p, m t(t,p,m) - t(t,p,0) = J (jt+f-pv) dm, t, p, 0 or * ft Pi m) = (m +/- pv) m, and by definition ( 2 ) (3) (4) (5) m' then t = n +/— pv ( 6 ) §24 SIMPLE HOMOGENEOUS SYSTEMS 41 24. The second law of thermodynamics: The second law is expressed by the equation n (t, p, m) — n (to, p 0 , m 0 ) = t, p, m J mjdt-\-m~dp -f^dm to, p o, mo ( 1 ) where 6 denotes some function of t, 6 = r (t), which is the same for all systems. In the’special case where the mass, m, remains constant we have t, p n (t, p, m) - n (to, po, m) =m J ^dt + ^dp (2) < 0 , Po = mrj(t,p) - my (to, Po). Now we have defined the entropy of a unit mass simple crystal¬ line substance as zero at 6 = T (to) = 0 and p = po, V (to, Vo) = 0 and since, by definition, r n = m y we have n (to, po, m) = m y (t 0 , p 0 ) = 0 Thus n (to, po, 0 ) = 0 and hence « (t, p, 0) = 0. (3) Now or n (t, p, m) n (t,p, 0 ) t, p, m t, p, 0 n ( t, p, m) / ( t , P) m, e (4) 42 THERMODYNAMICS §25 and by definition then 77 = f(t, y) e But we have e = \x + / — pv, from (23.6), then from the relation we obtain or e = /i + 6 77 — pv H = e — 6 77 + pv. 25. Definitions of Gibbs’ thermodynamic functions: Gibbs has defined three additional thermodynamic functions £, x and 1 (r by the equations £ = £ + pv — On X = £ + pv i]r = £ — On We denote the values of the functions for unit mass by f, x and \p, that is, by definition f = m X = ~~y and m i m* Therefore we have from (24.6) for the one-component single phase system V = f- CHAPTER III Homogeneous binary systems of variable mass and composition Let us now consider systems variable in mass and composition in successive states but homogeneous in each state, that is, in each state the composition, density and temperature are the same at all points, the pressure is the same at all points and is the same in all directions at any given point. Let us consider first a homo¬ geneous system composed of two components. 26. Definitions of specific volume and mass fraction: The specific volume, v, is defined by the equation v mi -f m 2 ’ 0 < v < oo 0 ^ mi < co 0 ^ m 2 < mi + m 2 ^ 0. where v denotes the total volume of the system, mi the mass of substance Si, and m 2 the mass of substance s 2 . We define the mass fraction of the first component, m h by the equation mi mi =-j-, mi + m 2 0 ^ mi < co 0 ^ m 2 < oo mi + m 2 ^ 0. and similarly the mass fraction ra 2 of the second component by m 2 m 2 mi + m 2 ’ Hence mi + m 2 = 1. 27. Equation of state: We assume as a physical hypothesis that when any three of the properties, pressure, specific volume, 43 44 THERMODYNAMICS §28 temperature, fraction of substance si, of a particular system are given the fourth is uniquely determined, or that

- p at + l *dt- p V P dt_\ to, vo, m i 0 dt -j - , dp dvdp _ v dv ^ dp dv_ dv + 7 7 dv 7 dp dv dp imi — Inn — p 7- H Ml ~ M2 + Ip ZZ - P TZ dm i drai dp dmi_ dwi £, v, mi = c v dt ( l v — p) du 4~ (^tfi — Iwi *4* Mi — M 2 ) d?Mi ( 1 ) to, Vo, mi 0 since, by definition, § 30, we have Hence Thus C +l d -P = c t/p “ Vp ^ IsV l — = l V7) ~ l"V dv Imi Inn + h dp dni\ Iwi — Iwi de dt ~ de dv de dull Uvi — lw 2 T Mi — M 2 dc v _ dl v dp dv ” dt dt d Cy dlwi dlw 2 + dpi dp2 dmi dt dt dt dt dl v dp dlwi dlwi 4_ dpi dp2 dmi dmi dv dv dv dv 52 THERMODYNAMICS § 34 34. The second law of thermodynamics: The second law is expressed by the equation t, p, mi, m2 n {t, p, mi, m 2 ) - n (Z 0 , p 0 , mi 0 , m 2o ) = J (mi + m 2 ) ^ d t to, pc, mi 0 , mj 0 + (mi + m 2 ) j dp + dmi + dm 2 (2) where 6 is a function of t only, 6 = T (£), the same for all systems. SPECIAL CASES (1) In the special case where mi and m 2 are constant the equation reduces to the following t, p, mi, m2 f C l n ( t , v, mi, m 2 ) - n (t 0 , p 0 , m x , m 2 ) = (mi + m 2 ) / dt + ^ dp (3) to, po, mi, m2 Now n (t , p, mi, 0) = n' (t , p , m) and n' ( t , p, 0) =0 by (24.3). Then n (t, P, o, 0) =0 (4) (2) In the special case where t , p and mi are constant n (i!, p, mi, m 2 ) - n (<, p, 0, 0) = mi + m 2 l -j- (5) Hence n (t, p, mi, m 2 ) = mi l ~~ + m 2 1 -~ (6) Let 7] =-;-by definition mi T* m.2 Then rj = mi + lmt m *T = n ( l, p, mi) ( 7 ) § 34 HOMOGENEOUS BINARY SYSTEMS 53 Thus n {t, p, mi, m 2 ) = (mi + m 2 ) 77 ( t , p, mi) (8) Further we have e — m\ l mi + m 2 l m2 — pv + Pimi -f- p^m^ = Orj —pv + pimi + /x 2 w 2 (1) (9) (3) In the special case where mi + m 2 = 1, i.e. in the unit mass case, the second law of thermodynamics for binary systems reduces to t, p, mi V (t, p, mi) - 17 {to, Po, raio) = J ~ dt + ^ dp + d m x (10) to, po, mi 0 Further 77 ( t 0 , p 0 , 0) = 77 ' (t Q , p 0 ) = 0 and 77 (t 0 , p 0 , 1) = 77 " (/ 0 , Po) = 0 since we have already defined 77 ' ( t 0 , p 0 ) = v" (^ 0 , Po) = 0. Hence <977 _ Cp dt~~d dt] _ l p dp 6 dr] dnh •mi v m2 ~o ~e Thus and therefore (id ( 12 ) (13) ( 14 ) 1 This is Gibbs’ equation 93 for a binary system of unit mass. 54 THERMODYNAMICS §35 and therefore dpi dp2 Imi Imz > dt dt ~ d d ’ (15) 1 dip 1 d (Imi Imo) d dm,i d dp and therefore S’ 1 a Ps II rH r© (16) 35. Transformation of the entropy integral: By hypothesis V = V (t, v, m i), Thus t, p, mi C-n i . i l ifd< +%dp + - dm, U U U to, po, mi 0 t, v, mi Cp Ip dp d t+Yjdv+ d dv lm\ Imz | dp 0 0 drai dmi (1) to, vo, mi 0 and since, by definition, we have I / dp _ Cp "T" t/p « . C v dl l — = l bp r, by dv and dp jp dm,i Imi l m2 + Ip ^ l\Vi l Wo Then r] (t, v, mi) — rj (t 0 , Vq, m u ) = t, v, mi Si di _i_ h _i_ ^ Wi ~ ^ w * e e 6 d mi (2) to, vo, m i 0 Hence §36 and HOMOGENEOUS BINARY SYSTEMS 55 Thus and therefore and therefore dr} _ I wi — Iwj dmi 6 1 dc v _ 1 dl v l v ~e ~dv ~ ~e et ~ J 2 ’ 1 dc v 6 dirii 1 d (Iwi ~~ Iwi) Iwi Uv 2 ~6 dt ¥ ’ dfJ-i _ dji2 _ _ Iwi — l w i . ~dt ~dt ~ 6 5 1 dl v _ 1 d (Jwi — l\Vt) ~6 dmi 6 dv ’ and therefore d/j ,i d/j .2 _ dp dv dv dmi 36. Relations between unit mass and variable mass binary sys¬ tems : We shall now prove that if the functions for a binary system of one phase and unit mass are known, the functions for a binary system of one phase and variable mass may be calculated from them alone, without any additional experimental measurements. We have ■wi T ) _ d (mi + m 2 ) 77 ' t, p, rr .2 dm.! t, p, m 2 = V + (mi + m2) dr} v dm.i /1, p, m 2 = r} 4 - m 2 = 77 + 7??2 dr} dm\) t , p lm 1 ^m 2# Q 56 THERMODYNAMICS §36 and Irm _ ( dll \ e ~ \dm 2 J t , d (mi + m 2 ) v = 7 ] — m 1 P> mi ( drj \dm dm 2 — V ( m i + m 2 ) t, p (*l) \dm 2 /t, P , mi Also 7 ] — m\ de \ l-mi Inn e Ml = l T— I ~ Irm + VV + pm 2 \dmi/t, p, m 2 _ ( d (mi -f m 2 ) e \ _ \ dmi / 1, p, m 2 (tt) dmi = € + (mi + m 2 ) u+pv+pm (£^\ V de m 2 — L Jmj t ,p (mi + m 2 ) 2 + pv + pm 2 dv .dmi p = e + m-2 (1 -1 - dv . V dm, +m M2 ' lm ‘ + pv + pm 2 (—) \dmi/ 1 , p — e — 6 tj + pv + m 2 (mi “ M 2 ) Similarly M2 = (—) VSmj/,. p , — lm, + pv — pm, m2 (*L) \dm\/t, p = e + (mi + m 2 ) de - mi ^dmij \(mi -f m 2 ) 2 , - I rm + pv — pmi (—) \dmi/t,p dv e mi y l mi Imt P Qyyi ^ M 2 ) ^ rm + pv — pmi ' dv \ ,dmi/ 1 , P = e — 6 tj -f- pv — mi (mi — M 2 ) CHAPTER IV Homogeneous, n-component systems We shall now consider the general homogeneous system con¬ sisting of n components. We assume the system to be variable in mass and composition for different states but homogeneous in each state, i.e. in each state the composition, density, and temper¬ ature are the same at all points and the pressure is the same at all points and is the same in all directions at any given point. 37 . Definitions of specific volume and mass fraction: The specific volume, v, is defined by the equation v mi -f- m2 -f* m n ’ 0 < v < 00 0 ^ m k < oo, 1 ^ k (integer) n mi -f-+ m n ^ 0, r where v denotes the total volume of the system, m k the mass of the substance s k . We define the mass fraction of the k-th component, m k , as m k = _ r~r -7—> 1 ^ k ^ n, k being an integer mi -f- m2 ~r * * * * r ni n Thus mi + * * * * + m n — 1. 38. Equation of state: We assume as a physical hypothesis that

0, k = 1, • • •, n mi + • • + ni n 57 58 THERMODYNAMICS § 39 39 . Definitions of work and heat: The work, W, done on the system is defined by the integral t, mi, , n>n W = — J pdv , where v = (mi + m 2 + • • • • + m n ) to, po, mi 0 , •••■ ,ninQ v(t,pmi, -,m n _i) Expanding we have t, J) } Dll, •••• , mn W = — J (mi + • • • to, po> ••••, m no + m n ) p ~ t dt + (mi + n—1 + m n) P ^ dP + P | f + dv V' dv - - Zjm i-— dm i am\_ i= 1 dmi + + p v — n—1 s i=l m i dv dm i_ dm n . In the special case where mi + m 2 H-+ m n = 1 (total mass constant) mi = m h • • •, m n = m n , and the integral reduces to t, p, mi, •••• 1 TRn -1 w= -J p ft At+p Z Ap+p ^ dmi+ to, POf Wh lg, •••• , Hl/Ti — 1 q + V dv dm n _] dm n _! The heat, Q, received by the system is defined by the integral t, p , mi, ••••, nin Q = J (mi -f m 2 +-+ m n ) [c v dt + l p dp] + Imi dmi to, po, mi,,.IT„ 0 + • • * * + l ma dm n In the special case where mi + m 2 + • • • • + m n = 1 (total mass constant) that is mi = mi, • •, m n = m n , this integral reduces to t, p, mi, Win —1 Q ~ j Cp dt ~f~ Ip dp “j - (Jmi ^m n ) dmi H - ~}~ (Jm n -i ^»i n ) dm n —1 to, PO, TTIIq, •••• , "fTln— §40 HOMOGENEOUS N-COMPONENT SYSTEMS 59 40. Transformation of the heat integral: By hypothesis p = p (t, v, mi, • • • •, m n _i) where m k = m k Thus mi+-+ m n l^k^n — 1, k = integer. ty nilr •••• t m n (mi + •••• + m n ) c p dt + (mi + • • • • + m n ) l p dp top pOf mi 0 , •••• y mn, t f v y m t .... | n?n + l m , dmi + + l mn dm n — J (nii +-+ m n ) | c p -\rl p J dt + l dp dv dz; tOy V 0 » •••• » Hln 0 + Ina H~ lp dp -l h " m ‘ dm i dm\ i=l dmi + n—1 + 7 7 V dp iffin lp 7Yl\ dni{_ dm n . i=l ty Vy mil •••• 9 n? n == J (mi + • • • + m n ) c v dt + (mi + • • • + m n ) l v dzz toy V0y mi 0 , •••• 9 nin 0 -f* Iwi dnii + 4- lw n dm n since by definition dp Cp ~f* lp c v l d -2 = l tp ~ tv dv dp Imk + (2 m a ) lp -— = l Wk , 1 ^ k ^ n, k = integer. diUk In the special case where mi 4- • • • • + m n = 1, i. e. in the constant total mass case, this integral reduces to ty Vy TJX\y •••• 9 TJX n—1 J c v d£ T l v di> + {Iwi — l\v n ) dmi + • • • • + (£w„_i — lw D ) dm n -i t, v, mi 0 , ...., m a - 1 0 60 THERMODYNAMICS §41 41. The first law of thermodynamics: The first law of thermo¬ dynamics is expressed by the equation £ (t, p , nii,., m n ) £ (^o, Po) mio, , nino) tf Pi mi, • ••• | llln mi -{-•••• + m n dt + n—1 u - pv dm„. (1) i== 1 where ah, • • • •, Mn denote continuous single-valued functions of t, p, m h -, m n - 1 . SPECIAL CASES (1) In the special case where mi, • • • •, m n are all constant, the first law reduces to the following: e (t, p, mi,-, m n ) - e (to, p 0 , mi,-, m n ) = tf p. Dll, •••• , Win (mi H-+ m„) J (c p - p^j dt + (l T - p^j dp (2) (o, po, mi, ...., m n Now e (t, p, mi, 0, • • • •, 0) = z (t, p, m) and £' (t, p, 0) =0 by (23.3). Hence z(t,p, 0, • • • •, 0) =0 (3) (2) In the special case where t , p, mi, • • • •, m n _i are constant the first law reduces to t (<, p, mi,-, m„) - £ ((, p, 0,-, 0) = ( lm , — p + Ml) mi + ■ • 4- Qmn — pv Mn) m. (4) §41 HOMOGENEOUS N-COMPONENT SYSTEMS 61 e ft p, mi,-, m n ) = mi l mi +-+ m n l mn — pv (mi -f- • • • • + ni n ) + mi mi + ••••+ m n m n (5) £ Let e = -r-;- by definition mi -f--f- m n Then 6 = Wll l mi + * * * ‘ lm n ~ PV ”h Ml 77l\ + * * • * + Mn 7Yl n (6) = e 0 l>P,mi ,-,m n _i). Hence e ft p, mi,-, m n ) = (m x +- + m n ) € ft p, mi, -, m n _i) (7) (3) In the special case where mi + m 2 + • ■ • • + m n = 1, i.e. in the constant total mass or unit mass case, mi = mi, • • •, m n = m n , and the first law reduces to € ft p, mi, ., m n _i) — e ft, p 0 , m u , -, m n - 1 0 ) = tf Pf 771 lj •••• f 771 n —1 dt + l l P - v ^) dp + / ( c ’- - p i) to t pOj 771 Iqi •••• f 771 n—1 q + 2 mi ^ + w ) ~ ( u ~ pv + 2 i=l i=l lmi — PV — P dv dmi dv . Wli “ *4“ Mn dmi dmi + i=i n—1 + pv v dm„_! + 2 m ' ami + Mn ~ I ) i=l n—1 — [lmn — pv + 2 dTOn-! ( 8 ) 1=1 <, p, 7»1, .... , TOn-l = / («, - P £) d < + (l, - V g) dp + [*-. - U - V ^ ft>*fP0, 7711q 9 •••• , TTIu—Iq + Ml — Mn dmi + + lmn -1 Ln p dv ton-1 — f~ Mn—1 Mn dm n _i. 62 THERMODYNAMICS §42 Hence (t, p, mi,-, m n _i) = c p — p dv dt, P, mi . . —i e p (t,p,m h -= b ~ p(i^) \°P/ t, mi, .... , Wln-l € mk (J'j P> Wl\, ' , 772 n _i) = lm n P “l~ Mk Mn where l^k^n — 1, k = integer and e (t 0 , po, mi 0 , ., m n -i 0 ) = e' (J 0 , p 0 ) = 0 by definition where m ko = 1, 1 ^ k ^ n, i.e. all m io zero except m ko . Further, from the second derivatives of e, we have dc p dv _ dip dp~dt~~dt ’ dCp dl m k dl mn d/z k dfJ. n dm k ~~di dt ^ ~dt ~dt’ dl v dl OTk dl m n dm k dp dp dp dp dm k where l^kfSn — 1, k = integer. ( 10 ) ( 11 ) ( 12 ) (13) (14) (15) (16) 42. Transformation of the energy integral. By hypothesis we have p = p (t, v, ith, -, m„_i) Thus ^ _ Cj> P dv dt dt + ( mi +-+ m n n—1 7 dv , 'V dv , l -- pv - p e^ + p 2j mi e^+^ i= 1 n—1 dmi -f- • • • • -f- % -p V + p^ /m ^+ tla dm n i=l §42 HOMOGENEOUS N-COMPONENT SYSTEMS 63 t, v, mii •••• y nin = j ^mi +-+ m n to, Vo, mi 0 , ••••• y ninQ dv . 7 dp dv dp Cp ~ v dt + lp ~dt ~~ v &p Tt_ d t + 1 mi +-+ m n J | l P — p ^ dp dv dv n—1 + dv dv l ™-v v -Vd^ + v2j m 'e^ + Mi i = 1 n—1 + ( Zp-p S)(£~2 TOi £)] dm > i = 1 n—1 +.+ Ln -/>« - + M i = 1 d- [lp — p dv dp , n—1 ~2 mi i=l dp dm-J _ dm„ t, v, mil •••• i mn = J (mi + • • • + m n ) c v d£ + (m x + • • • + m n ) (k ~ p) dv to, Vo, mi 0 , ••••• i mn 0 dr [Iwi + Mi] dmi -(-••'• + [lw n + Mn] dm n since by definition , 7 dp _ Cp "r tji ^ C v l — = l I'P ~ v'D dv dp l my + (2 m a ) l p —— = l Wk , 1 ^ k ^ n, k = integer. om k In the special case where mi +••••+ m n = 1, then mi = mi, •, m n = m n , and the integral reduces to t, V, mi .. TOn-l c v dt -f- ( l v — p) dv + \lwi — l\v n ~h Mi — Mn] dmi gdt, + r-r‘ + ~ ¥rr -kyimi dp 6 6 dni\ 6 dm x _ i = 1 dmi +•••• + tf Vf mii •••• y m n l n—1 9 e Cy ^ mi dp dm x _ dm n i—1 Id nil + * * * -f~ m n ) — dt + ( nil ~b • • • H~ ni n J -z dv 6 Q toy VQf mi 0 , •••• y m no + dmi + + l f dm„ In the special case where mi + • • • • + m n = 1, i.e. where the total mass remains constant the integral reduces to T\ (tj V , TYli, ’ ‘ ' j TYlxi—l) V (to) VO) mi 0 > ’ y VYln— lo) = t, v, mi, ...., m n_i llVi l\V n C f dt + L e dv + — tOy Vq 9 1TL\ Q, •••• y dmi + . Wn-l ~ l\V n A + ----dm n _i Hence Vt (t, v,m h -, m n _i) = 0 " Vv (tf V) Wily * * * * j ^n—l) = T) mk (t, v, mi, -, m n _i) = - ;k —, l^k^n — l,k = integer and from the second derivatives of 77 we have or 1 dc v _ 1 dl v l v “ ~e~dt ~ J 2 1_ dc v _ 1 d (l\V k - lw n ) _ lw\c - Iwn ~e dm k ~ ~e dt o 2 68 % THERMODYNAMICS or d/z k d/i n _ l\Vk — lw n dt dt " Q and 1 dl v _ 1 d (ZiFt ^w„) 6 dm k 0 dv or d/x k dju n _ dp dv dv dm k §45 45. Derivation of Gibbs’ equation 12: Assuming that we can solve for 6 as a single valued continuous function of rj, v, mi, ., m n _i and obtain e as a continuous function of 77 , v , mi, • • • •, m n _i, e = e (77, v, mi,-, m n _ 1) we obtain e v (77, »,mi, •••, m n -i) € v (77, v, mi, s. de ded 0 , m n _i) = ” + = — V dv SO dv (3) v ’ m ”-‘) = £ + | where 1 ^ k g n - 1 Mk Mn (4) Thus de = e dr] — p dv + (mi — Mn) dm x +-+ On-1 — Mn) dm n -i (5) Now e = (mi +-+ m n ) e (77, v, mi,-, m n _i) where n 7? = -1-j- mi + • • • + m n v mi + • • • + m n m k mi + • • • + m„ , 1 ^ k ^ n - 1 m k = §45 Thus HOMOGENEOUS N-COMPONENT SYSTEMS 69 E = e(n,v,mi, •••*,m n ) Hence £„ (n, v, mi,-, m„) = (mi + • • ■ + m„) ^ = $ t, («, v,mi, • • •,m„) = (mi H-- p c m , (n,v,m„ ■-,«,) = (mi + • • • +mj jS^S + ^i \ 1 = 1 ■ de dv } dv dmi ( = (mi — M n ) (1 — m 0 — (m 2 — Mn) rrh —- “ (Mn—1 ~ Mn) Win-1 ~ 077 + pV + € = Ml { Wll Ml +.+ Win Mn} — Of) + + e = Ml ~ (^ + py — 01?) — 07? + pV + €. = Ml *- v, m„ • • •, m„) = (mi + • • • + m„) £+ £ ^ \ 1 = 1 , de dv) + 3nto„j +e = — (mi — Mn) mi — .— (Mn—1 Mn) Wln -1 — 07? + PV + € = Mn (Wil + * * ’ + Wln_l) — Ml Wll “ — Mn— 1 Wln_l — 07? + PV + C = Mn (1 Win) Ml ^1 ' Mn—1 Win—1 — 07? + + e = Mn Thus de = 0dn — pdv + mi dmi +.+ Mn dm n (1) 1 This is Gibbs’ equation 12. 70 THERMODYNAMICS §46 46. Definitions of Gibbs’ thermodynamic functions: These functions he defines 1 as £ = t + pv — 6n = n imi + X = e + pv = 0n + /x imi-f-" ijr = e — 6n = — pv + /xi mj + Let us define C “f - Mn ID-n “f" Mn ID n ~I - /x n m„. r = X = mi + • • • -f m n _ X mi + • • • + m n mi H-h m n = e -j- pv — dr] ~ Ml Wll + ' * • + Mn ^n = e + pv — Oy + Mi 'mi ~b * * *' + Mn win — e — 6rj = — pw + mi rrti +-+ Mn m n 47. Differential and partial derivatives of Gibbs’ Zeta: By definition < = c (0, p, mi, • • •, m n ) + pv - 6n ( 6, p, mi, • • •, m n ) where v = (mi + • • • + m n ) v (6, p, mi, • • •, m n _i) = v (0, p, mi, • • •, m n ) Thus C is a function of 6, p, mi, • • •, m n C = { (6, p, mi, • • •, m n ) Hence to (0, p, mi, • • •, m n ) = l ^ p, mi,.., m n = ( m, + • • • + m n dv dn + P d0- n - e d8 dv P d6 + p (^mi + • • • + m n j — - n — 0 |^mi + • • • + m n j — — n 1 Gibbs, J. W., Collected Papers, vol. 1, p. 87. §47 HOMOGENEOUS N-COMPONENT SYSTEMS 71 . N dz dv dn KAe,P,m h ---,m n ) =- + v + p^-e- = nil + • • • + rn n Ip p dv dv dp_ + v + l m H-H m » IP 6 ni] + • • • + m n Ip ~e = V ds. Cmu (9, p, mi, • • •, m„) = — + pv + n—1 p , _, , _ 'V dv drtii (mi + ■ • • + m„) 2 /^ - 6 dn dnii dm k dm k i = 1 n—1 7 dv , 'V dv i = l ™- pv ~ p ^ + p Z m ^ + ^ i = 1 n—1 + pv + P dv — V^j w>i dV . lr Q ^ mk dm k L ^d ' 1 drrii 6 i=l = H k , l^k^n — 1, k = integer n—1 Cmn (0,p,mi, • • - ,m n ) =L-F + p2/mj^4-Mn + F- 1=1 n—1 dt> l p l^ m ^- e T i=l = Mn Thus d< = — nd0 + vdp + mi dmi +--f Mn dm n (1) In the special case where mi H-+ m n = 1 this equation reduces to d£ = — rjdd + vdp + (mi — Mn) dwi + • • • + (Mn— i Mn) dmn_i. 1 This is Gibbs’ equation 92. 72 THERMODYNAMICS §48 From the second derivatives of < we have dn 9, mi, .., m n dv ~dd. p, mi, .., m n dn \ dv \ v dm k / e, p, mi,.., p, mi, .., mk-i, mk+i, .., m n nik-ij mk+i, ••» mn or dr] K dp/0 t mit " , rna-i drj_\ >dm k /d t p t mi) .. f mk-i, m+_i, ..., m n -i dv djUk \ dd /p, mi, ..., m n d/fk\ dp / 0, mi.m n dv ,dm 9 t p, wii, ..., mk-i, mk+i, .., mn -i P. mi, .., m n . djUn dd , /* (3 1 dp A Pt ••• » Wln-1 48. Differential and partial derivatives of Enthalpy or Gibbs* Chi: By definition X = e (n, v, mi,-, m n ) + pv where v = (mi + • • • + m n ) n v ( 77 ,p, wi, • • •, m n _i) and rj = mi + • • • + m n = X («, p, mi, • • • •, m n ) Thus ?-(*) +(?) (: dn \dn/ Vf m , ( .. f m „ \dv/ n , mi, .., m n \ \dvdr] mi + • • • + m n 1— — /dr] dn + p ^mi + ••• + m n J^ \dv dr] ) dr] dn = e dx = de / dp dv \ — ( mi + • • * + m n ^ — + v + P ^ m i + • • • + m a ^ — = v dx de de dv + dm k dm k dv dm k + P dv dm k = Mk §49 Thus HOMOGENEOUS N-COMPONENT SYSTEMS 73 dx = d dn + v dp + Mi dmi +-+ Mn dm n (1) In the special case where mi + • • • • + m n = 1 the equation reduces to dx = d drj + v dp + Oi — Mn) dmi + • • • + (jun-i — Mn) dm n _i and 49. Differentialand partial derivatives of Gibbs’Psi: By definition t|r = e (0, v, • • •, m n ) — On (6, v, mi, • • •, m n ) Hence = tjr (0, v, mi, • • •, m n ) dtjr _ (-) dd ~ \dd/ v, mi, .., m n = — n dt{r _ dv \dV/ g, mi, .., mn = - V dt{f _ de dn dm k dm k dm k -n-d m n m n = Mk 1 This is Gibbs’ equation 90. 74 THERMODYNAMICS §50 Thus d So, the displace¬ ment being from P to P 1 . Let x 0 + Ax, y 0 + Ay, z 0 + Az be the coordinates of a neigh¬ boring point Q at a time s 0 and Xi + Axi, yi + Ayi, Z] + Az 3 be the coordinates of the point Q 1 at a time Sj, the displacement being from Qto Q 1 ; i.e. a point in the neighborhood of P transforms into a point in the neighborhood of P 1 . Then xi, yi, Zi are uniform and continuous functions of x 0 , yo, z 0 ; Si in the volume R. Thus Xi = f (x 0 ,y 0 , z 0 ; Si) yi =

0 As ds 82 THERMODYNAMICS §55 thus the extensions or stretch of the components, which are parallel to the axes of the coordinates in the unstrained state, are re¬ spectively Vl-f 2ei - 1 y / 1 H - 2 62 1 ( 1 ) y /1 ~t- 2 63 — 1 where the positive values of the square roots are taken. We thus obtain a physical interpretation of the quantities ei, e 2 , 63 . 55. The angle between two curves altered by strain. A physical interpretation of the quantities e 4 , e 5 , e G : Let 1, m, n, and l 1 , m 1 , n 1 be the direction cosines of two vectors ; issuing from the point (x, y, z) in the unstrained state and let 0 be the angle between them. Let h, mi, ni and h 1 , md, nd be the direction cosines of the corresponding lines in the strained state and 0 i the angle between them. Thus 1 = x s , m = y s , n = z 8 . . d (u + x) d (y + v) d (z + w) dsi ’ 1 dsi ’ 1 dsi ( 1 ) then 1 (1 + u x ) + mu y + nu z ( 2 ) 1 similarly for mi, ni. Now cos 0i = L Id + mi md + ni nd §55 STRAIN 83 ds ds 1 dsi dsd _ ll 1 + ll 1 u x + • • + mm 1 + + nn 1 + substituting cos 0 = ll 1 + mm 1 + nn 1 , and the values e*, • • • •, e 6 in the above expression we have cos 0i = ds ds 1 dsi dsi 1 cos 0 + 2 (ei ll 1 + e 2 mm 1 + £3 nn 1 ) + e 4 (mn 1 + mhi) -f e 5 (nl 1 + n 1 !) + e& (lm 1 + Pm) Now let the two given directions be the positive directions of the axes of y and z. Thus 1 = l 1 = 0, and either m = n 1 =0, m 1 = n = 1, or m 1 = n = 0, m = n 1 = 1. Let us arbitrarily choose m = n 1 = 1 then cos 0 ] = ds ds 3 dsi dsi ] 1 + C4 (1+ 0) or e 4 = cos 0i dsi dsi ] ds ds 3 and since dsi ds dsPV ds — 0 —(1 " T ~ 2 62) + 0 — 0 -j- 0 + (1 T 2 63) + 0 thus 04 = cos 0i V(l+2e*) (1+2 e 3 ) which gives us an interpretation of the quantity e 4 . Similarly with e 5 , e 6 . 84 THERMODYNAMICS §56 56. Linear dilatation at a point of the system; ELLIPSOID OF DILATATION at this point: Let the displacement be from P to Pi. Let P 1 be a point in the neighborhood of P, and Pp the dis¬ placement of this point. From our assumption of continuity Pp will be in the neighborhood of Pi. In the strain PP 1 = As is displaced to P] Pp = Asi which is, in general, a change in length and orientation. Let us consider changes of length only, where lim Pi PP — PP 1 dsi PP 1 -*!) PP 1 “ ds 1 > 5 =5 0 , 8 ^ — 1 . Let 1, m, n, be the direction cosines of PP 1 , i.e. 1 dX 4 . 1 = -T-, etc. ds ( 1 ) ( 2 ) Now from equation (53.4) we have dsp = (1 + 2 ei) dx 2 + (1 + 2 a) dy 2 + therefore (£)' - (1 -f* 2 6 i) l 2 —{— (1 —f— 2 € 2 ) m 2 -f- (1 -f- 2 63 ) n 2 -f- 2 e\ mn + 2 e 5 nl + 2 e 6 lm. dsi ( 3 ) We want to represent the change in about a point P. To do this let us take on PP 1 a length PQ where H dsx 1 + 5* and let the vector PQ revolve an angle of 2 t about P as center. Let Px, Py, Pz represent the system of rectangular coordinates. Thus in the unstrained state we have x 2 + y 2 + z 2 = 1 (4) §57 and STRAIN 85 y-„(PQ)-mg -W-C then the equation (56.3) reduces to 1 = (ds^) = (1 + 2 ei) x 2 + (1 + 2 € 2 ) y 2 + (1 + 2 e 3 ) z 2 + 2 e 4 yz + 2 e 5 zx + 2 e 6 xy = \J/ (x, y, z) ( 5 ) and this quadratic is an ellipsoid about P as center. The points of no change must satisfy equations (56.4) and (56.5) simultaneously (assuming that the sphere touches or cuts the ellipsoid, otherwise no such points exist). r Thus 1 = (1 + 2 ei) x 2 +(l+2 e 2 ) y 2 +(l +2 e 3) z 2 -j-2 e 4 yz+2 e 5 zx-f-2 e§ xy 1 = x 2 + y 2 + z 2 0= 0. §58 Then STRAIN 87 £i — an + 2 , 1 02 — 9 - 22+2 ,1 03 — 933 + o 9 n 2 + 92 i ? + a 3 i 2 9l2 2 + a 2 2 2 + 9 3 2 2 9is 2 + 9 2 3 2 + 933 2 04 = a 3 2 + a 2 3 + a 12 ai3 + a 2 2 a 2 3 + a 3 2 a 3 3 05 = a 3 i + a 13 + an an + a 2 i a 23 + 931 a 3 3 06 = 921 + an + an an + a 2 i a 22 + a 3 i a 32 Let us simplify these formulae by the aid of a translation. Since aw, a 2 o, a 30 do not enter into ei they can be made zero without modifying the deformation. Let £ = + 9io V — v' a 2 o r = + a 3 o then £' = (1 + an) x + a i2 y + an z V = a 2 ix + (1 + a 22 ) y + a 2 3 z (5) = a 3 ix + a 32 y + (1 + 933 ) z In these formulae the origin goes into itself since when 0 = x = y = z we have £' = tj' = f' = 0. In the following treatment we shall assume that an, a 2 o, a 3 o have been eliminated by the above transformation and drop the primes. 58. Pure strain: In homogeneous strain there is one set of three orthogonal lines in the unstrained state which remain orthogonal after the strain, the direction of these lines being in general altered by the strain. A homogeneous strain is defined as a PURE STRAIN when there exists in the first state three orthogonal lines which remain THERMODYNAMICS 88 §58 unaltered in direction by the strain. These directions are called the principal directions of the pure strain. Let OXYZ be the principal directions. Let x, y, z be the coordinates of the point P before deformation, and £, 77, f be the coordinates of the corresponding point Pi after deformation. The deformation being homogeneous, the expressions for £, 77, £ as functions of x, y, z are given by £ = (1 + an) x + ai2 y + a ]3 z 77 = a2i x + (1 + a 2 2) y + a23 z (1) r = a 3 i x -f- a 32 y + (1 + a 33 ) z where aij are constants. By hypothesis, when P is on OX then Pi is on OX; thus, when y and z are zero, 77 and must be zero whatever value x may have, i.e. asi and a 3 i must be zero identically. Similarly all the other coefficients except an, 2^2, a 33 are zero. Thus we have the pure deformation referred to its principal directions given by £ = (1 + an) x 77 = (1 + a 22 ) y (2) f = (1 + a 33 ) z. Now in order that a homogeneous deformation be a pure deformation it is necessary and sufficient that the field of vectors PPi be derived from a function of the vectors, i.e. that u, v, w be the derivatives of a function of second degree in x, y, z. The condition indicated is necessary for, if the deformation is pure, then on taking for axes the principal directions OXYZ we have the coordinates of Pi defined as functions of those of P by £ = (1 + an) x 77 = (1 + a2 2 ) y t — (1 + a 33 ) z, from which, for the projections u, v, w of the vector PPi on the axes OXYZ, we have u' = £ — x = an x, v' = a^ y, w' = a 33 z §58 which one can write STRAIN 89 on setting dF dz’ an X 2 + a 22 y 2 + a 33 z 2 It is sufficient: for suppose that, in the general formulae defining u,v,w for any homogeneous deformation with respect to the axes OXYZ, u, v, w be the partial derivatives of a similar function F(x, y, z), or, what amounts to the same thing, suppose that du _ dv dx “ dx’.’ i.e. a i2 = a 2i , a 23 = a 32 , a 3 i — ai 3 . The function F is then F 1 2 an x 2 + a 22 y 2 + a 33 z 2 + 2 a^ yz + 2 a 3i zx + 2 a^ xy , which gives u dF dF dF t-, v = — , w = — . dx dy dz The surfaces of second degree given by F (x, y, z) = const. play with respect to the vector u, v, w the role of plane surfaces. These surfaces have the same center 0 and the same directions of the principal axes OXYZ. Let us revolve the axes so that we have as our new axes OXYZ and call x', y', z' the new coordinates of a point P, and u', v', w' the new projections of the vector PPij the function F will take on the reduced form an x' 2 + a 22 y' 2 + a 33 z /2 90 THERMODYNAMICS §59 which gives an x', a2 2 y', a 3 3 z'. The deformation is then pure, being equivalent to three simple extensions in three directions mutually at right angles. 59. Strain tangent at a point: Let P be any point in the system before the strain and have as coordinates x, y, z; and let p represent an infinitesimal portion of the system around this point P, i.e ., let p represent the neighborhood of P. Let P 1 (x + Ax, y -f- Ay, z + Az) be a point in p. Let P be displaced to Pi in the strain, then the coordinates of Pi will be £ = x + u, ?? = y + v, £* = z + w, and if P 1 is displaced to Pd in the strain then the coordinates of Pp are given by £ + A£ = x + u +A(x + u) ^7 + A ?7 = y + v + A(y + v) (1) £ + A£* = z + w + A(z + w) which, on expanding A (x + u), etc., can be written in the form . . . , du du du dx dy dz y + v+Ay + gAx+gAy + ^Az (2) dw dw dw z + w + Az + Ax + ~ Ay + ^ Az dx dy dz when squares and products of Ax, Ay, Az are neglected. (When the displacement is sufficiently small the approximation involved in this simplification will lie within the experimental error and thus will give an accurate expression of the physical facts.) Now let us change our coordinates so that P(x, y, z) is the origin and Px 1 , Py 1 , Pz 1 the coordinate axes. §60 STRAIN 91 Let P 1 (x 1 , y 1 , z 1 ) and Pp (U, 77 1 , £ l ) be the coordinates of P 1 and Pi 1 respectively with respect to this coordinate system; we then have the coordinates of P 1 given as x 1 = Ax, y 1 = Ay, z 1 = Az, and those of PP given by U = u + (1 + u x ) x 1 + u y y x + u z z 1 rj l = v + VxX 1 + (1 +v y )y 1 + VzZ 1 (3) £-1 = w + Wx x 1 + w y y 1 + (1 + w z ) z 1 Formulae (59.3) giving the coordinates of the point PP as functions of the coordinates of P 1 define the deformation of the region m around P and mi around Pi. As they are linear in x 1 , y 1 , z 1 they define a homogeneous deformation, and we can ac¬ cordingly say that, in a sufficiently small neighborhood of a point of the system, the relative displacements are linear functions of the coordinates, i.e. THE STRAIN ABOUT ANY POINT IS SENSIBLY HOMOGENEOUS. In identifying with formulae (58.1) it is necessary to make an u x , ai 2 Uy, ai 3 u z , a 2 i Vx, a 22 Vy, a23 v z , a 3] = w x , a 32 = w y , a 33 = w z . (4) In order that the homogeneous deformation defined by (59.3) be a pure deformation it is necessary and sufficient that a 3 2 = a 2 3 , a ]3 = a 3 i, ai 2 = a 2 i (see theorem back) i.e. that w y — v z = 0, u z — w x = 0 , v x — u y = 0 . 60. Very small deformations: Suppose that a continuous me¬ dium is deformed in a continuous manner and so that the displace¬ ment PPi of each of its points P be sufficiently small such that the scalar squares and products of these displacements can be neglected, i.e. the approximation involved would lie within the experimental error. Such a strain has also been called infinitely small. The six characteristic functions referred to the general equations (53.5) neglecting the squares of the partial derivatives of u, v, w with respect to x, y, z give us 61 = u x , e 2 = v y , e 3 = w z , e 4 = w y + v z , e 5 = w x + u z , e 6 = Vx + u y . (i) 92 THERMODYNAMICS §61 Then u, v, w can be written u = u x x + u y y + u z z = 61 x + i (e 6 - v x + u y ) y + ^ (e 6 - w x + u z ) z = ei x + i e 6 y + ^ e 5 z - co z y + z (2) similar y v = ic 6 x + e2y + ^e 4 z - co x z+co z x (3) w = i e 5 x + ^ e 4 y + c 3 z - co y x + w x y (4) where 2 C0 X Wy Vz, 2 COy U Z W X , 2 C0 Z V X Uy. 61. Conditions of compatibility for strain-components: The fact that all six of the strain-components can be expressed in terms of the three component displacements indicates that these six quantities must be, to some extent, interconnected, i.e. if we assign an arbitrary expression to each strain component we shall not in general obtain a possible distribution of strain since the conditions for continuity of the system after strain will as a rule be violated. The values of the components of strain e\ as functions of x, y, z must satisfy the relations e\ = u x , etc. If we introduce the three relations 2 co x = w y — v z , etc., then all the partials of u, v, w with respect to x, y, z can be expressed in terms of e\, ., e 6 , u x , co y , co z . We have m . 1 1 Uy + o V x — S Vj = \ (V* + Uy) - | (Vx - Uy) 1 2 e 6 ^z ( 1 ) §61 STRAIN 93 , 1 1 Vx = Vx + ^Uy - 2 Uy = \ (Vx + Uy) + i (Vx - U y ) = 2 + w z (2) Similarly for u z , w x , w y , v z . The conditions that these be compatible with the equations Ux = 6l) . are given by the 9 equations % dei _ 1 de^ dco z dy 2 dx dx’ which express the partials of co x , w y , co z , in terms of e x . Now the equations containing o> x are a dto x de 5 dfi6 (3) 2 — = dx ~ dy dz 2 dUl - dei 2 ^? (4) dy dy dz 2 dw = 2 — d^4 (5) dz dy dz Differentiating (61.3) with respect to y and (61.4) with respect to x, and subtracting we have o d 2 e 2 = d_ dx dz dy dei de 5 , de 6 dx dy dz ( 6 ) Differentiating (61.4) with respect to z and (61.5) with respect to y, and subtracting we have d 2 e\ _ d 2 e Xi > xj — 5 ji + v>yi>yi-y Zi + € > Zi > Zi — e where i = 1, • • • •, n respectively. Consider an arbitrary neighborhood of mass m and coordinates (x, y, z,). We can divide the forces applied at this neighborhood into two parts. (1) Those which make up the interior forces acting on m; we shall call Xi, Y i, Z x the projections of one of these forces. (2) Those which make up the exterior forces acting on the same m; we shall call X e , Y e , Z e the projections of one of these forces. Isolate one of these neighborhoods. If the system is in equilib¬ rium, then each neighborhood or particle of the system is in equilibrium. In order that such a particle be in equilibrium it is necessary and sufficient that the resultant of all the forces acting on the particle be zero. Projecting them on the coordinate axes we have then, for the equilibrium of the particle m, the three equations 2li + 2l e = 0 27i + 27 e = 0 (1) 2Zi + 2Z e = 0 98 THERMODYNAMICS §63 where 2 denotes the sum of the projections of all the interior forces applied at m. Summing up for all particles of the system and letting the number of particles increase without limit, i.e. the sizes of the particles approach zero as a limit, we have “ n n 1 1 and since the system is in equilibrium therefore similarly with the other two. In equations (63.1) multiply the first by — y, the second by x and add, this gives us 2 (x Fi — y Xi) + 2 (x Y e — y X e ) = 0; Summing up these equations for all the particles m n and letting n increase without limit we have lim n 00 Now 1 = 0 . n §64 STRESS 99 represents the sum of the moments of all the interior forces with respect to the O z axis; this expression is then zero by hypothesis and therefore n^Tco 22 (xy e -yXe) = °, etc. This gives us Xidv = 0, Y idv = 0, Zidv = 0, (x Y i - yXO dv =0, (yZi - z FO dv = 0, (3) (z Xi — x Z i) dv = 0, or X e dv = 0, / / / Y e dv = 0, Z e dv = 0, (x Fe - yXe) dv = 0, (yZe-z Ye) dv = 0, (4) (z X e — X Z e ) dv = 0. i.e. For any system to be in equilibrium it is necessary that the sum of the projections of the interior or exterior forces on the three coordinate axes and the sum of their moments with respect to each of the three axes be zero. 64. Equations of equilibrium: The body forces applied to any portion of a system are statically equivalent to a single force applied at one point together with a couple. The components, parallel to the coordinate axes, of the force can be written ///»-«-. ///pi-dv, ///pZO, (1) 100 THERMODYNAMICS §64 where p, in general a function of x, y, z, is the density, and where the limits of integration are the bounding surfaces of the system. The moments about the origin of these components will be P [yX - x Y] dv, p[zX — xZ] dv, (2) p[zY-yZ] dv Similarly the tractions applied at the surfaces, d, of the system are equivalent to a resultant force and a couple. The resultant force can be written X„dd, Y„dd, / / Z v dd ( 3 ) and the moments of these components about the origin [y X v — x Y v ] dd, [zX y — x Z v ] dd, [z Y„ - y ZJ dd. (4) Thus for equilibrium we have six equations, from (63.3), X dv + / / X„ dd = 0 P 7dv+ / F„ dd = 0 p Z dv + / / Z v dd = 0 ( 5 ) [X y - x Y] dv + / / [y X v - x Y v ] dd = 0 p [z X — x Z\ dv + / / [z — x Z„] dd = 0 (6) p [z Y - y Z] dv + / / [z F„ - y ZJ dd = 0 §65 STRESS 101 65. Specification of stress at a point: Through any point O in a body there passes an oo 2 system of planes and the complete specification of the stress at 0 involves the knowledge of the trac¬ tion at 0 across all these planes. We can express all these tractions in terms of the component tractions across planes parallel to the coordinate planes, and to obtain relations between these components. Let X x , F x , Z x denote the vector components of the traction across the plane x = con¬ stant, and a similar notation for the traction across the planes y = constant and z = constant. The capital letter denotes the direction of the component traction and the subscript the plane across which it acts. The sense is such that X x is positive when it is a tension, negative when it is a pressure. Consider the equilibrium of a tetrahedral portion of the body, having one vertex at 0 (x, y, z), and the three edges that meet at this vertex parallel to the axis of coordinates. The remaining vertices are the intersections of these edges with a plane near to 0. Denote the normal to this plane drawn away from O by v, so that its direction cosines are cos (x, v), cos (y, v), cos (z, v). Denote the area of this plane by Ad. (1) The projections of the external forces acting on the volume Av (body forces), are approximately — p X Av, — p Y Av, — p Z Av where p denotes the density at 0. [The approximation will be¬ come more accurate as the volume of the tetrahedron becomes less.] (2) The projections of the stresses on the surfaces of the tetra¬ hedron will be given approximately by — X x Ad cos (x, v), — F x Ad cos (x, v), — Z x Ad cos (x, v) — X y Ad cos (y, v), — Y y Ad cos (y, v), — Z y Ad cos (y, v ) — X z Ad cos (z, v), — Y z Ad cos (z, v), — Z z Ad cos (z, v) X , Ad Y v Ad Z v Ad where X„ Ad, Y y Ad, Z v Ad are the resultant tractions across the face. 102 THERMODYNAMICS § 65 [This approximation will become more accurate as the volume of the tetrahedron becomes less.] The sum of these projections on any axis must be zero since by hypothesis the body is in equilibrium; thus on 0 x we have — pX Av — X x Ad cos (x, v) — X y Ad cos (y, v) — X z Ad cos (z, v) -{- X v Ad = 0 Dividing by Ad, and letting Ad approach zero as a limit, we have X v = X x cos (x, v ) -f X y cos (y, v) + X z cos (z, v) „ lim Av + pA Ad^0Ad But Av Ad • Op 4 where O v is the height of the tetrahedron from the vertex 0. Thus « Av _ Ad Op Ad ~ Ad 4 Op T and therefore lim Av _ lim Op A d—>0Ad - Ad—>0~4 Hence we have X„ = X x cos (x, p) + X y cos (y, p) + X z cos (z, p) (1) similarly Y v = 7 X cos (x, p) + Yy cos (y, v) + Y z cos (z, p) (2) Z v = Z x cos (x, p) + Z y cos (y, v) + Z z cos (z, p) (3) By these equations the traction across any plane through a point 0 is expressed in terms of the tractions across planes parallel to the coordinate planes. On substituting the values of X„, Y y , Z v from equations (65.1, 2, 3) in equations (64.5), (64.6) we have the necessary §65 STRESS 103 conditions for equilibrium with respect to the body forces and surface tractions given as p(yZ - z F) dv + p (x Z — z X) dv + p(xF-yl) dv + cos (x, v) yZx-zYx + cos (y, v) yZ y -zYy + cos (z, v) y Z z - z Y z | d> 1 w X 1_1 | dd = 0. ( 4 ) = 0 » xir+ JJl x x cos (x, v) + X y cos (y, v) + X z cos (z, v) J dd Ilf" ri - + IF. F x cos (x, v) + Y y cos (y, v) + Y z cos (z, v) J dd = 0 (5) III p Z dv -f- Z x COS (x, v) + Zy cos (y, v) + Z % cos (z, v) J dd = 0 104 THERMODYNAMICS §65 According to Green’s Theorem we can write X x cos (x, v) + X y cos (y, v) + X z cos (z, v) dd d X x , d X y , d X z dx + ^ + dy dz dv ( 6 ) Equations (65.5) can thus be written pl + d X x , d X y , d X z dx dy dz dv = 0, similarly for the other two. Now this region v was wholly arbitrary. We can understand by v then any subregion of the original region v and this equation will still hold for this new subregion. Therefore the integral must vanish at every point of v, or Similarly and v dX x dX y dX z dx dy dz Y + d i d Yy i d . X z ^ dx ^ dy ^ dz = 0 z + ~+ —+ —= 0 dx dy dz Again, we have from Green’s Theorem ( 7 ) / (y Z x - z Y x ) cos (x, v) + (y Z y - z Y y ) cos (y, v) + (y Z z — z Y z ) cos (z, v) dd §65 STRESS 105 d(yZ y - zYy) dy d (y Z z — z Y z ) dz dv or simplifying, Thus equations (65.4) can be written /(' " z ( z+ d Z x , d Z y , d Z j dx + dy + 7.[pY + a -^ + dY dx dy y + dz d_F z dz ) ) + Z y - Y z dv = 0 and similarly for the other two. But the coefficients of y and z are, from equations (65.7), identically zero, and the whole integrand is identically zero since the region v was wholly arbitrary, thus similarly Z x = X z , and X y = F x . In order to simplify the writing of equations we shall use the following abbreviations: X 2 = Fy X 3 = Zz X 4 = Zy = Y z X 5 = X z = Z x X 6 = F x = X y THERMODYNAMICS 106 §65 which are then analogous to e\, • • •, e% which for very small de¬ formations are given by the expressions dv 03 dw dz 04 dw dy dw du dx ‘ dz dv . du 06 = T-r T— dx dy CHAPTER VII Thermodynamic treatment of systems homogeneously strained In this chapter we shall limit ourselves to systems homo¬ geneously strained and leave the treatment of systems in which the strain is not homogeneous to Chapter IX. 66. Definitions of work and heat: Now a homogeneously strained system has been defined as one the properties of which are the same at all points of the system, i.e. are constant with respect to the coordinates x, y, z. For the systems treated previously we found that the work received by the systems could be defined by the integral — J p dv. Now we discover that for stressed systems this expression will not define the work received by the systems. We must therefore extend the definition of work, such that it will express the work received by the stressed system and such that, when the stresses can be represented as pressures which are the same at all points and in all directions at any one point of the system, the expression will reduce to the integral —Ip dv. Hence we define the work per unit mass, W, received by the system homogeneously strained or work of the path by the equation 6 so i = 1 where ei, • • •, e 6 are functions of t , X 1} • • •, X 6 ; t, Xj • • •, X 6 depend upon the path S. p is the density in the state of reference. Thus 6 6 6 w = 1 dt ds dei dt dei dX k dX k ds Ids so i = 1 k = 1 1 = 1 107 108 THERMODYNAMICS §66 Now when X\ = X 2 = X 3 = — p, X A = X 5 = X 6 = 0 we have SO SO Hence .our definition satisfies the boundary conditions, namely that it will reduce to the previous definition of work received when the stresses reduce to pressures which are the same at all points and the same in all directions at any point of the system. Similarly the heat per unit mass, Q , received by the system homogeneously strained or heat of the path is defined by the equation 6 i = 1 so where c x , 1 X1) • • •, l Xe are functions of t, X-\, • •, X 6 and t,X h • •, Xg depend upon the path S. p is the density in the state of reference. Now when Xi = X 2 = X 3 = — p, X 4 = X 5 = X 6 = 0 this ex¬ pression reduces to So since here §68 HOMOGENEOUSLY STRAINED SYSTEMS 109 Hence our definition reduces to the previous definition of heat received when the stresses reduce to pressures which are the same at all points and the same in all directions at any point of the system. 67. Definition of c x , l Xl , • • •, l Xt : Along the path t ~ S, Xi = Ki, • • •, X 6 = K 6 where Ki, • • •, K 6 are constants, dQ = dQ dS dt which is by definition the heat capacity per unit mass at constant stress. Thus ^ = - c x \ and along this path at p 6 dW dW 1 de { ds dt p £J Xi dt' i = 1 Along the path t = K, X x = S, X 2 = K 2 , • • •, X 6 = K 6 where K, K 2 , • • •, K 6 are constants, dQ _ dQ dS - dX] which is by definition the latent heat of change of stress per unit mass parallel to the x-axis, where temperature and the other stresses are constant. Thus = - l Xl and along this path a A i p 6 dW dW 1V Y dei dS dXi P ^J Ai dX, i = i Similarly for the other stresses. 68. Transformation of the heat and] work integrals: Now by hypothesis Xi, i = 1, • • •, 6 can be expressed as functions of ei, t , X i fi (#1, ' f @6) 0 y i 1> * * >6. 110 THERMODYNAMICS §68 Thus Let us define Ce Then tf X\f ••• t A 5 tf ••• « 6$ The transformation of the work integral, as we have seen, gives tf -X"ly ••• y Xq tf 6lf • ••f €0 Along the path t = S, e- x = Kd, i = 1, • • •, 6 where Kd are con¬ stants, ^ which is by definition the heat capacity per unit mass at constant strain, that is, where e\ y i = 1, • • •, 6 are constants. Thus ~ = - c e . And along this path = 0. Along the §69 HOMOGENEOUSLY STRAINED SYSTEMS 111 path t = K, e\ = S, e 2 = K 2 1 , • • •, e& = Ke 1 , y§ = which is by Cl b Q61 definition the latent heat of change of strain along the x-axis per unit mass where temperature and the other strains are constants. Thus dW d Q 1 j d* ~ p iei ' And along this path dS AW 1 = - Xi. Similarly for the other strains. 69. The first law of thermodynamics : The first law of thermo¬ dynamics, for a unit volume system homogeneously strained, is expressed by thejequation P € ( t, X-], * f Po € (to, * * , X 6 0 ) tf A. I* ••• f A6 We complete the definition of pe(t, X 1 , ■ • •, X 6 ) by defining the energy per unit volume at to, X u , • • •, X 6o as zero, po c (to, Xi 0 , • • •, Xo 0 ) = 0, (3) where po denotes the density in the state of reference. Hence 6 p*(4,X„... > X,)-«.+ 2 Xi S (4) i = 1 p (i, X„ • • •, X 6 ) = +2 X = (5) And from the second derivatives of p e we have 112 THERMODYNAMICS §70 Similarly or S C x n SCn 1 n - —, n = 1, . • - ,6. SX, SX n dt St dXil l X2 +2XA i = i i = 1 Six 1 \ / Sl X 2 \ v ai 2 / A. 3 » ••• f Ag \(9 Xi/f, x 2 ,..., y 6 a^i \ / Se 2 \ k SX 2 / tj X^l> ^ 3 * ) -A.6 \s xj tf A.2* ••• i As Similarly az Xm Shn S &m a^n ai n ax m ai n ax m ’ , (m, n) = 1, • • •, 6 (6) ( 7 ) 70. Transformation of the energy integral: By hypothesis Xi = fi {e-[, • • •, e&, t), i = 1, • • •, 6. Thus ty Aij ••• y Ag 6 +2 Zi Hi _ d( + S _^+2 Zi Ir k _ dXt i = 1 tot Xi 0 , ... i YgQ k = 1 i = 1 tf 6 i 9 ... , €$ 6 +S Zi Hi + 2 |>+2 Zi Hr k . i = 1 k = 1 i = 1 ax k i st dt ^o» ^io» ••• * 6 +2 ^+2 Zi A ax k a«i dei k = 1 i = 1 + ••• +2[^+2 Zl aYi_ ax k Se 6 de 6 k = 1 ^9 @lf ••• 9 €6 = / C ( dt +2/ le\ + Xi dei ( 1 ) §71 HOMOGENEOUSLY STRAINED SYSTEMS since, by definition 113 Hence c x +2 k = 1 6 2 k = 1 dXi 'Xk dt Cp 'Xk dX k dCi — l ei , i — 1, • • •, 6. P tt (t, 6], • • •, e 6 ) — c e P *ei (t, 6i, • • • , = lei "T Xi, i = 1, • • • , 6. ( 2 ) ( 3 ) And from the second derivatives of e we have dc e dl e\ \d 6]/ 1, ez, ..., ee \ dt + ^1» • • • » ^6 aii dt ei, ..., e6 or dC e _ dlg D . dX n de n ~ ~dt ~^~dt , n = 1, • • - ,6. ( 4 ) Similarly dl em dle a dX n dX m dCjx d6 m dc m d& n , (m,n) = 1, • • -,6. ( 5 ) 71. The second law of thermodynamics: The second law of thermodynamics, for the unit volume system homogeneously strained , is expressed by the equation P v (l) Xi, • • •, Xq) — po rj (to, X u , • • •, X6 0 ) ^ dXi, where 6^0, u and where p 0 denotes the density in the state of reference. 114 THERMODYNAMICS §72 We assume, as a physical hypothesis, that for systems in stable equilibrium lim c x 6-^0 6 (3) We thus extend the definition of the entropy function by de- fining the entropy at 6 = T ( t 0 ) = 0, X ]o • • •, X 6o , as zero, i.e. Po 7] (to, Xi oi • • • , X6 0 ) — 0. (4) Now pr U (t,X 1 ,..-,X ls ) = c -‘ (5) P Vxi (t, Xi, • • •, X.) = -^, i = 1, • • •, 6. (6) And from the second derivatives of prj we have 1 R 1 R II o (7) and TT ~l¥ =0 ’ (m,n) = V-->6. u -Xx m U -A. n (8) Thus from (69.6) and (71.7) and from (69.7) and (71.8) d@xi dCm dX m dX n 72. Transformation of the entropy integral: By hypothesis Xi = fi (ei, • i = 1, •• *,6. Thus tf 1 , ••• 1 \ 6 ••• t ffl k = 1 i =» 1 §73 HOMOGENEOUSLY STRAINED SYSTEMS 115 ^0* 61 q, ••• • ^6q since by definition 6 Hence Q PVt (t, e h • • •, ee) = j PVe a (t,e i, • • •, e 6 ) = y And from the second derivatives of p rj we have d Ce dl en le a de n dO 6 and die n __ 56m dCn Thus from (70.4) and (72.4) we have l ea = - Q ain aa , n = 1, • • -,6. £ 11 ••• * ^6 And from (70.5) and (72.5) we have a In _ ax m a^rn a^n ( 2 ) (3) (4) (5) 73. Derivation of Gibbs’ equation 12 for strained systems: Assuming that we can solve for a as a single valued continuous THERMODYNAMICS 116 §74 function of 77 , ej, - • • ,e& and obtain f as a continuous function of rjj eij • • •, e& we obtain Thus or ••,*) = (f-;) e . dd ee \&V/ ei, ... , ee = i c , • p 6 - = e P C e . v de de dd e h ■■■,*)=-+-- = -Ze,+ + P P P \ 6 Ce) — -Xi, i = 1, • • •, 6. P dt = d drj -j— Xi dgi i = 1 de = 6 dn + v X* d^i i = 1 where v denotes the total volume in the state of reference. 74. Differential and partial derivatives of Gibbs* Zeta: By definition f = € — 6 7] — - Xi 6i 1 = 1 Hence Thus p f = p r (#> Xi, • • •, x 6 ) §75 HOMOGENEOUSLY STRAINED SYSTEMS 117 6 6 = — PV p Xi, • • •, Xq) = p — p o -2 i = 1 Xi de x dTn — — e n , n — 1, • • •, 6. Hence and dt] dX~n lden p dd , n — 1, • • •, 6. Thus 6 dC = — n dd — v e x d Xi i = i where v denotes the total volume in the state of reference. 75. Differential and partial derivatives of enthalpy or Gibbs* Chi: By definition 6 • = 1 — x Xi, • • •, X 6 ) 118 THERMODYNAMICS §76 Thus xAv,X u ---,X e ) =(p) 1 de de x 1 'V'* £ dei dy/eu ...,« P ^ dei dr] p ^ 1 dr] i = l i = l = * + ;;S*S-!S*‘£ p dr] p dr] i = l i = l =9 Xxn ( V, Xh • * * > Xs) — Thus and Hence de dei 1 1 y deiJTv - ~p e " ~ ~p 'dT a i = 1 i = 1 1 'S'' v dei 1 1 'V' V dei pZj Xi dX 0 P e ° „Zj Xl ax, i = 1 1 p *■» dX n i = 1 6 n - P dx = 6 dr] - - ^ e i d i = 1 dd _ 1 de n dX n p 6)) dx = 6 dn — v e x d Xi i = 1 where v denotes the total volume in the state of reference. 76. Differential and partial derivatives of Gibbs’ Psi: By definition \p = e — 6 rj Hence pxP = p\f/(d,e i, • • - ,e 6 ) §76 Thus HOMOGENEOUSLY STRAINED SYSTEMS 119 p \J/ e (By e h • pie n (By e h • Hence and Thus * * > Ce) — P e6 le n *T X n le n — X n , n = 1, • • •, 6. d\p = — 77 d 0 + - Xidei p i = 1 dr] _ 1 d X n de n p dd i n I? * * ■; 6. dt{r = — n d0 + v Xi dej i = 1 where v denotes the total volume in the state of reference. CHAPTER VIII The stress-strain relations for isothermal changes of state 77. Generalized Hooke’s Law of the proportionality of stress and [strain: We shall now consider the special case where the changes of state take place at constant temperature. Then Ah fj (cij 62, 63, 64, 65, e$), 1 1* * * * > 6, (1) X\, • • •, Xa being zero in the state of reference, i.e. when 61 , • • •, e 6 are zero. Developing these according to the McLaurin expansion we obtain series proceeding according to increasing positive powers of ei, • • •, 6 e. If we neglect terms containing products and powers of 6 ], • • •, 66 we shall have for X h • • • ,X 6 linear homogeneous ex¬ pressions in 61 , • • •, e 6 . In the following discussion we shall assume as a physical hypothesis that the GENERALIZED HOOKE’S LAW OF THE PROPORTIONALITY OF STRESS AND STRAIN holds, i.e. that each of the six components of stress at any point of a body is a linear function of the six components of strain at the point. W 78. Strain-energy function, —, for isothermal changes of state: From the second law we know, for any closed path consisting of a continuous series of equilibrium states in which the temper¬ ature remains constant, the heat of this path must be zero. Vi Qs = J 0i dn = 01 (til — n 0 ) VO where Qs is the heat of the path S from the state in which n = n 0 to the state in which n = «i. VO Qa = J 0i dn = 0i (n 0 — ni) >71 120 §79 ISOTHERMAL STRESS-STRAIN RELATIONS 121 where Q ff is the heat of the path a from the state in which n = tii to the state in which n = n 0 . Thus Qs + Qa = [( n i — Ho) + (no — nj)] = 0. Hence the heat of the path for a continuous series of isothermal equilibrium states is independent of the path and thus, for a fixed initial state, is a function of the final state only. Therefore this must also be true for the work of the path and so for an isothermal change of state we have, for the work received by the system, the line integral being extended along any path connecting the points (ei 0 , • • •, e 6o ) and (e h • • •, e 6 ). W 79. Strain-energy function, —, for adiabatic changes of state: Now for changes of state that take place adiabatically, i.e. where no heat is gained or lost by any part of the system, we have for the W work — received by the system of unit volume, W v which, from the first law of thermodynamics, is independent of the path for, in this case. - (ti, • • * > e%) — - (to, e\ 0 , • • •, e 6o ) = — Now since we have assumed the stresses to be linear functions d /W N of the strains, and since •— ( — ) — X\. i = 1, * * •, 6, the strain- de, \ v / W . energy function —- is therefore a homogeneous quadratic function of the strains. 122 THERMODYNAMICS §80 80. Static vs. dynamic methods of determining the stress-strain relations. Relation between W for isothermal and W for adi¬ abatic changes of state. Now Xi are in general functions of W t,e i, • • •, e 6 . Therefore the strain-energy function — will not as a rule be the same for the adiabatic and the isothermal changes of state. Thus from a theoretical point of view Xi, (i = 1, • • •, 6), as determined experimentally by statical methods (involving isotherma] changes of state) will differ from Xi, i = 1, • • •, 6, as determined experimentally by dynamical methods (involving adi¬ abatic changes of state). This has experimentally been shown to be the case although the measured differences were not very large. Now stress-strain relations, as determined by statical methods, have been applied, in some cases it seems rather indiscriminately, to adiabatic changes of state. Therefore it might be well to stress the point that, when using these two strain-energy functions as interchangeable, the burden of proving that the discrepancies between the two are negligible for the problem under consideration rests with the user. 81. The elastic coefficients or “elastic constants” of the system: From §77, we have Xi, i = 1, • • •, 6 given as linear homogeneous expressions in h = 1, • • •, 6. Hence we can write 6 Cih ^h> 1 1 > h = 1 In these equations the coefficients Ci h of e h number 36. These coefficients depend on the constitution of the body at the point P (x, y, z) in question; for a body whose constitution varies from point to point these coefficients will be functions of x, y, z. Assume that the system is homogeneous in the reference state, i.e. its constitution is the same at each point. The density p, and the 36 coefficients are then constant for the system under obser¬ vation. If we substitute the values of Xi, • • •, X 6 from (81.1) in 123 § 82 ISOTHERMAL STRESS-STRAIN RELATIONS we find that the “Elastic Constants’’ Ci h are the coefficients of a 2 W W homogeneous quadratic function —where — is the strain-energy function; they are therefore connected by relations which insure the existence of the function. These relations are of the form Cih = c h i, (h, i) = 1, • • •, 6 (2) and the number of constants is reduced from 36 to 21. 2 W v 22 i = 1 h = 1 Cih e\ e h where c ih = c h i. 82. The stress-strain relations for isotropic bodies: In iso¬ tropic solids every plane is a plane of symmetry and every axis is an axis of symmetry, and the corresponding rotation may be of any amount. Hence the equations connecting stress components are independent of the axes of coordinates, i.e. of direction. Thus W . . — is invariant for all transformations from one set of orthogonal axes to another. We shall assume the theorem 1 that in the transformation of the quadratic expression the following are the only invariants with respect to transformations from one set of rectangular axes to another ei + e ci2 + 2A)J _ T (ci2 + R) _ Cl2 2 (C]2 + R) (15) Whatever the stress system may be, the shearing strain cor¬ responding with a pair of rectangular axes and the shearing stress on the pair of planes at right angles to those axes are given by Xq — R e e, X±. = R 64 , X$ = R e 5 (lb) and are independent of the directions of the axes. The quantity R is defined as the MODULUS OF RIGIDITY. For convenience the relations between these elastic moduli for isotropic substances are summed up in the table on page 128. 84. Anisotropic character of homogeneous crystalline sub¬ stances : The most important examples we have of non-isotropic homogeneous bodies are crystalline substances. We assume as a physical hypothesis that the symmetry possessed by the crystallographic form of the substance applies also to every physical characteristic of the substance. The substance may, however, possess some physical characteristics that belong to a higher order of symmetry. An example of this is the optical iso¬ tropy of isometric crystals. In general the stress-strain relations will be dependent on the rectangular set of coordinate axes chosen; however, the crystallo¬ graphic symmetry relations make possible certain transformations of the coordinate axes for which the quadratic expression remains unaltered. The restrictions imposed on the strain components by 128 THERMODYNAMICS §84 such transformations for which the quadratic expression remains unaltered result in a simplification of the elastic constants, the amount of simplification depending on the invariants of the trans¬ formation. 1 ELASTIC MODULI OF ISOTROPIC SUBSTANCES o ►—* H-* II > C 12 = B R O II CQ K R E cr A A —- B + 2 R 3K+4K 3 E (1 — < 7 ) 1 - or - 2 C 23 > C 26 > 033 , C 36 , cm, 045 ? C 55 ,C 66 are all fundamental, the others being identically zero. 130 THERMODYNAMICS §85 Similarly for (9) we have: Cn, Cl2, Ci 3 , C22, C 23 , C33, C44, C55, C 66 for (7a): Cn = C22, C12, C ]3 = C23, C14 = — C24 = C56, C15 — — C25 — — • C 46 ? C33, C44 — C55, C 66 — 2 ( Cl1 ~ Cl2 ) for (6a): Cll = C22, C12, C13 = C23, C15 = — C25 = — C46, C33, C44 = C55, C 66 = 2 ( C ll ~ C12). for (5): Cn = C22, C12, C13 = C23, C33, C44 = C55, C 66 = 2 ( c n — C12) for (7b): Cll = C2?, Cl2, C13 = C23, Ci 6 = — C 26 , C33, C44 = C55, C 66 - for (6b): Cll = C22, C12, C13 = C23, C 33 j C44 = C55, C 66 . and for (3): Cll = C22 = C33, C12 = C13 = C 23 ) C44 = C55 = C66- In the following discussion the stress-strain relations shall be developed for the generalized strain-energy function. Then if we desire the strain-energy function for a certain crystalline body we merely have to make the substitutions according to the table just given. 85. Transformations of the strain-energy function: Solving the equations, 6 X{ = / j Cih e h , i = 1, • • •, 6 h = 1 § 85 ISOTHERMAL STRESS-STRAIN RELATIONS of § 81 for ei we have 131 X \ Ci 2 Ci 3 Ci 4 Ci 5 Ci 6 X % C22 C23 C24 C25 C26 x 3 C32 C33 C34 C35 C36 X4 C42 C43 C44 C45 C46 Xq C52 C53 C54 C55 C56 x 6 C62 C63 C64 C65 C66 Cll C12 C13 Ci4 C15 C]6 C21 C22 C25 C24 C25 C26 C 31 C32 C33 C34 C35 C36 C41 C42 C43 C44 C45 C46 C5I C52 C53 C54 C55 C56 C61 C62 C63 C64 C65 C66 Similarly we can solve for e^, • • •, e 6 . Now let J denote the determinant that corresponds with the denominator of the expression for e\ and let J rs denote the minor determinant that corresponds with c rs , then J 61 — X\ J 11 — X2 J21 X3 J31 — X4 J41 + Xq J51 — Xq J6i J 62 = — Xi J12 4" X2 J22 — X3 J32 + X4 J42 — Xq J52 H“ Xq J62 J €q — — X\ J]6 “h X 2 J 26 — X 3 J 36 4“ X 4 J 46 — Xq J 56 H - Xq J 66 where J rs = J sr , (s, r) = 1 , • • •, 6 . The quantities i Cn, Cih, (i, h) = 1, • • •, 6 , i 5 ^ h, are the coeffi- A cients of a homogeneous quadratic function of e k) k = 1, • • •, 6. This function is the strain-energy-function expressed in terms of the strain components. Likewise the quantities ^ ^p, are the 132 THERMODYNAMICS §86 coefficients of a homogeneous quadratic function of X k . This function is the strain-energy-function expressed in terms of stress components. 86. Moduli of elasticity for anisotropic homogeneous sub¬ stances : Let X 1 = X 2 = X 3 = - P X 4 = X 5 = X 6 = 0 then the corresponding is the cubical dilatation: A = e\ + &2. + 03 and, from § 85 , J €i = — P (J11 — J21 H - J31) J 62 = — P (— J12 ~f- J22 — J32) J &3 = P (Jl 3 — J 23 + J33) Thus we have, since J rs = J sr , (s, r) = 1 , • • •, 6 , — J A = P (J11 -f* J22 “ 1 “ J33 4“2Ji3 — 2 Ji 2 — 2J23) Hence the bulk modulus or modulus of compression, K, is K = — — =_-_— A J11 4 “ J22 T - J33 — 2 J12 + 2 J13 — 2 J23 If we let Xi = X 2 = X 3 = X 5 = X 6 = 0 , and X4 be the shearing stress applied, J e 4 = X 4 J 44 ; then R yz , the modulus of rigidity cor¬ responding to the pair of directions y, z, is . J 4 4 Similarly for the rigidity corresponding to other pairs of directions. If we let X 2 = X 3 = X 4 = X b = X 6 = 0 , then J Ci = Xi J11 J e^ — Xi J ] 2 J C3 = Xi J13 §86 ISOTHERMAL STRESS-STRAIN RELATIONS 133 and thus E x , Young’s Modulus corresponding to the direction x J Similarly for Young’s moduli corresponding to other directions. Poisson’s ratio in the y direction is then and in the z direction is 63 _ Jl 3 61 Jn The value of Poisson’s ratio for anisotropic substances thus depends on the direction of the contracted linear elements as well as on the direction of the extended longitudinal ones. CHAPTER IX Systems not in equilibrium. Irreversible processes 87. Displacement ^transformations: If the system is not in equilibrium under the action of the external forces it will be moving from one configuration to another. Each of these configurations can be represented by its displacements. PRINCIPLE OF SUPERPOSITION We assume as a physical hypothesis that each component stress is accompanied by the same strains whether it acts alone or in conjunction with other stresses. Now the body in motion will possess kinetic energy which depends on the distribution of mass and velocity. We shall consider the case of very small displacements (see §60 for defi¬ nition). Let u, v, w be the coefficients of the displacement by which the body passes from its state of reference to the strained state. Assume that the kinetic energy per unit volume can be expressed by 1 2 P .£)'+ (£j+(S a) where p is the density of the body in the state of reference and u, v, w are functions of x, y, z, s; x, y, z denoting the coordinates and s the time. The rate at which work is done by the body forces is V where v represents the volume of the body in the state of reference. The rate at which work is done by the surface tractions is du dv y dw ^ V *V "d * V ~ l” ^ V rv ds ds ds dcr ( 3 ) 134 §87 SYSTEMS NOT IN EQUILIBRIUM 135 where a represents the surface of the body in the state of reference. Transforming by Green’s Theorem and using the relations of (65.1, 2, 3) and of (65.8), and the notation of (60.1) for ei, • • • , e 6 , we have, for the rate at which work is done by the surface tractions h = dX x . dX y dZ x \ du / dX y dY y d7A dv dx'~ h ‘dy + ^hT/ds \~dx~ + dy "^llz/ds dZ x d7 z dZA dw dx dy dz / ds _ dv + ^ dei v de 2 -Yx ~ ~r y ^ ds ds ^7 d&s | d ^4 | /y de 5 | -^ 7 - d 66 TT - "r z FT - i ^x "T “F -^y V ds ds ds ds dv (4) where v represents the volume of the body in the state of reference. The rate of increase of kinetic energy, obtained by differen¬ tiating (87.1) with respect to s, is d 2 u du .ds 2 ds d 2 v dv ds 2 ds d 2 w dw ds 2 ds dv Substituting the equations of motion, i.e. dXx dx d7 x dx dZ x dx + + + dXy dy dYy dy dZy dy dX z dz dTz dz + pX = p + p 7 = p d 2 u ds 2 d 2 V ds 2 dZ z dz + P Z = p d 2 W ds 2 in (87.5) we have Is = v , dX x , dXy , dX z \ du , ,x + -d7 + ^7 + -drM + p7 + d 7 X , d 7 y d 7 Z \ dV dx + ^ + - i dy dz / ds + ( 7 dZ x dZ pZ + —— -f- dx _y dZA dw dy dz / ds _ dx dy dz (7) 136 THERMODYNAMICS §88 Thus we have Ii + I 2 — I 3 y d6\ | y 0&2 I y dC3 | "T "T" I ^z ~ l dS dS dS y de 6 , v de 4 de 5 -^-y ~ I -* z « T" ^x ~ ds dS dv ( 8 ) which is the excess of the rate at which work is done by the external forces above the rate of increase of the kinetic energy, i.e. gives us the rate of increase in internal energy due to deformation. 88. Definitions of work and heat received: Thus the work, W, done on the system is given by the equation y , dXi dX 6 6 X 5 px + 777 + 777 + dx dy dz _ y dX6 , dX2 , dX4 pf T 777 + 777 + dx dy dZ ^ . dXs dX4 aX3 P & + “7 -r t: + dx dy dz du ds dv ds dw ds + + dv ds + ••• +X„f^dvds. ds ds (i) where Xi, • • •, X 6 , are functions of ei, • • •, e 6 , t; ei, • • •, e 6 ,t are func¬ tions of x, y, z, and the time s. v represents the volume of the body in the state of reference. Similarly we have for the heat, Q, received by the body (die 1 , diet . die* \ du / dl e , dl e2 . dl ei \ dv _ \ dx ' dy dz / ds \ dx dy dz / ds diet . dle< . d?e 3 \ dW dx dy dz / ds _ dv ds + c M _l 1 4 . ds ds + 1 - + e ’ ds dv ds. ( 2 ) §89 SYSTEMS NOT IN EQUILIBRIUM 137 where c e , l ei , • • • , l e6 are functions of e h • • •, e 6 , t; ei, • • •, e*, t are functions of x, y, z and the time s. v denotes the volume of the body in the state of reference. Since a force always acts equally in opposite directions when the point of application moves, an amount of work is received by one system and an equal amount given up by the other system. Similarly for heat the amount received by one system is equal to the amount given up by the other system. 1 As we readily perceive, quantity of heat is a fundamental con¬ cept of thermodynamics. Heat is, however, a term so universally applied by all of us that we may be inclined to consider it as a rather straightforward concept given immediately in terms of everyday experience, but an analysis of the operation by which we measure quantity of heat will show us that the situation is not simple but extremely complicated. In fact no physical significance can be given directly to heat flow because there are no operations by which we can measure it as such. All we can measure are temperature changes and distributions and rates of increase or decrease of temperature. One of the justifications for the treatment of heat as a quantity is that if two systems A and B are in contact and both otherwise isolated from the rest of the universe, no work being received by either of the systems, and if A then undergoes a change a while B undergoes a change 0 the heat received by A is defined as an integ¬ ral which we shall call I and that lost by B is defined by a second integral, II, made equal to the first, I = II. Secondly if B and C are in contact and both otherwise isolated from the rest of the universe and if A undergoes a change a while C undergoes a change 7 the heat received by A has been defined by integral I and that lost by C is defined by a third integral, III, made equal to I, I = III. Now if B and C are in contact but otherwise isolated from the rest of the universe and if B then undergoes a change /3 then our physical hypothesis is that C will undergo a change 7. 89. The first law of thermodynamics: Let K represent the kinetic and e the internal energy of the system per unit mass. 1 The statement that the energy of an isolated system is constant is not the first law of thermodynamics. THERMODYNAMICS 138 §90 Then the first law of thermodynamics is expressed by the equation (ei + Ki) p dv (e 0 + K 0 ) p dv = SO SO So so dZ 7 dei 7 dee Ce T- + hr + ' * • + ds d s d s dvds + v dei + x, de& ds dv ds + dlei , die* , dl e6 + ^ + _ dx dy d z du + diet , dZ 62 | diet "T ~T~ I ds L dx dy dz _ dv ds + diet] , dZ e4 , dl + ^ + v es _ dx dy dz J ds dwl dvds + v dX\ dXe , dX 5 P -A H- - "r "TIT + dx dy dz du ds + P Y + dXe . dX 2 , dX 4 + ^ + dx dy dz - + ds „ dXe . dX 4 , dXz p % H—~ H—— + dx dy dz dwl ds dv ds where v denotes the volume and p the density of the body in the state of reference. 90. The second law of thermodynamics: The second law of thermodynamics, where 77 denotes the entropy of the system per unit mass, is expressed by the inequality r?ipdv — J JJri 0 pdv> V V §91 SYSTEMS NOT IN EQUILIBRIUM 139 die 1 . dl + ^ + dl, es _ dx d y dz _ du ds + dies , dl ei . dx dy + die 4 dz es dig | die 4 , _ dx dy dz _ dw ds dv ds dV ds + where v denotes the volume and p the density of the body in the state of reference. 91. Relations of non-homogeneous to homogeneous systems: By definition a system is said to be in a state of equilibrium if the properties of the system by which the state is defined under¬ go no change after the lapse of a period of time no matter how greatly it is extended. Hence in an equilibrium state the proper¬ ties defining the state of the system are constants with 'respect to time. Now we can set up a system acted on by external fields of force such that the stresses would be functions of the coordinates of the system and yet constant with respect to time. Such fields of forces we call body forces and the most common example of such a field of force is the gravitational field of the Earth. The work received by the system from the gravitational field, if we assume gravity acts in the negative direction of the z-axis, will be Jg p dv dz zo where g denotes the force of gravity, p and v the density and volume respectively of the system in the state of reference. But a necessary assumption we make is that such variable properties be continuous functions of the coordinates of the 140 THERMODYNAMICS §91 system. This being the case it is possible to choose a neighborhood about any point of the system such that in this neighborhood the properties of the system will differ from the properties at the point about which the neighborhood was drawn by less than any previously assigned small positive quantity K. In other words, the system in this neighborhood will remain sensibly homogeneous. Hence we can divide the system up into a finite number of neighborhoods or regions each of which is sensibly homogeneous and thus treat such a system as if it were made up of a finite number of homogeneous parts. Now if we expand our neighborhood to include the whole system which we then define as a homogeneous system, t,e 1, • • •, e& will be constants with respect to the coordinates of the system for any one state. Hence as t,e 1, • • •, e 6 approach constant values with respect to time and the coordinates x, y, z for any one state the expression for the work received by the system approaches as a limit the integral s 6 so i = 1 where e 1} • • •, e 6 depend upon the path s, v denoting the total volume of the system in the reference state. Similarly the expression for the heat received approaches as a limit the integral s 6 so i = 1 where c e , Z e „ • • •, l ee are functions of t, e i, • • • , e 6 ; and t, e i, • • •, e 6 depend'on the path s; v is the volume of the system in the state of reference. Now as and simultaneously approach zero the change in kinetic energy approaches zero as a limit and we have §91 SYSTEMS NOT IN EQUILIBRIUM 141 lim du dv dw\ _ ds’ ds’ ds / ~* (. K 1 -K 0 ) P dv = 0 K being a function of the density and velocity only. Hence since ^ and its partial derivatives are by hypothesis £ continuous functions of t, e i, • • •, e 6 only, - (t, e i} • • •, e 6 ), the first law of thermodynamics reduces to £ £ - (t, eij • • •, e 6 ) — - (t 0 , e lo , • • •, e 6o ) t, ei, .... 66 6 j c e dt + [l ei + Xi] de. t()i ^1q* •••* i 1 where v is the volume in the state of reference. If the properties defining the state reduce to temperature and volume the first law can be expressed by the equation t, V t ( t , v) — £ (to, v 0 ) = m / c v dt + (l v — p) dv to, VO where c v and l v are functions of temperature and the specific volume, v = —. m We assume as a physical hypothesis that as the series of states through which the system passes in a process become more and more nearly equal to a series of equilibrium states the total entropy of the system approaches as a limit the value of the integral t, ei, Co 6 C e j , J dt +2 0 dei tOf & 1 q 9 •••» 1 1 n where 0 , a function of t only, is the same for all systems, and - with THERMODYNAMICS 142 §92 its partial derivatives are continuous functions of t,e 1, • • •, e 6 only, v denoting the volume in the state of reference. Thus tf &lf •••• 66 6 ^Of 61 qj • • • f 66q 1 3 1 If the properties defining the state reduce to temperature and volume the second law can be expressed by the equation t, V n (t , v) - n ( t Q , v 0 ) = mJ^dt + jdv to, Vo where c v and l v are functions of temperature and the specific volume v = —. m 92. “Reversible ProcessesNow in treating homogeneous systems we speak of a series of equilibrium states in which each property of the system varies continuously as a continuous series of states. But since, by definition, the system can not pass through a continuous series of equilibrium states it would' be absurd to speak of “the work or heat received by the system in passing through a continuous series of equilibrium states.” Now we have defined a quantity W for the continuous series of equilibrium states t, ei, ee 6 W = v J 2 Xide > t()i 6lg, •••» 6gg i = 1 and this is the limit of the expression for work in a process as the series of states the system goes through approaches a series of equilibrium states. Similarly we have defined a quantity Q for the continuous series of equilibrium states §93 SYSTEMS NOT IN EQUILIBRIUM 143 Now we can make our process go through a series of states which differ from a series of equilibrium states by an amount that is less than some previously assigned small positive quantity 5 , and thus have our process approach as near as we please to a series of equilibrium states. This is what we mean to imply when we speak of a reversible process. Such a series of states has also been called a quasi-static process and such states quasi-static states. Hence Q for a continuous series of equilibrium states is determined by the limit that Q for a continuous series of non-equilbrium states approaches as these states become more and more nearly equal to the equilibrium states. Similarly for W of a continuous series of equilibrium states. Thus it is the physics of the situation that demands the treat¬ ment of systems not in equilibrium should logically precede the treatment of systems in equilibrium. 93. Justification for defining the entropy and energy of a simple substance at some arbitrary state as zero: Now we have seen how we may obtain the difference in energy and .entropy of a system between any two equilibrium states. Furthermore it is only differences in energy and entropy that we can measure and therefore that have a physical meaning in classical thermo¬ dynamics. As Gibbs 1 says, “the values of these quantities are so far arbitrary, that we may choose independently for each simple substance the state in which its energy and entropy are both zero. The values of the energy and entropy of any compound body will then be fixed.” Furthermore, the state in which the entropy of the simple substance is chosen as zero need not be the state in which its energy is chosen as zero. Therefore we can define the energy and entropy of a simple substance as zero at some convenient state. However, suppose we wish to coordinate thermodynamics with other fields of science. Now when a moving particle is acted omby a conservative 2 force 1 J. Willard Gibbs, Collected Works, vol. 1 , p. 85 . 2 A conservative force is one such that any work done by displacing a system against it would be completely regained if the motion of the system was reversed. 144 THERMODYNAMICS §93 we assume as a physical hypothesis that its kinetic energy has been transformed into potential energy. The increase in the potential energy of the particle is then equal to the kinetic energy which has been destroyed and hence equal to the work done by the particle against the force. When an isolated particle is set in motion we have from the theory of relativity where c denotes the velocity of light and m the mass of the particle, and at zero velocity K 0 is defined as zero. Thus the kinetic energy of the particle in ergs is equal to the mass in grams multi¬ plied by the square of the velocity of light. Furthermore, Tolman 1 says “when a moving particle is brought to rest and thus loses both its kinetic energy and its extra (‘kinetic’) mass, there seems to be every reason for believing that this mass and energy which are associated together when the particle is in motion and leave the particle when it is brought to rest will still remain always associated together. For example, if the particle is brought to rest by collision with another particle, it is an evident consequence of our considerations that the energy and the mass corresponding to it do remain associated together since they are both passed on to the new particle. On the other hand, if the particle is brought to rest by the action of a conservative force, say for example that exerted by an elastic spring, the kinetic energy which leaves the particle will be transformed into the potential energy of the stretched spring, and since the mass which has undoubtedly left the particle must still be in existence, we shall believe that this mass is now associated with the potential energy of the stretched spring. “Such considerations have led us to believe that matter and energy may be best regarded as different names for the same fundamental entity: matter, the name which has been applied 1 R. C. Tolman, The Theory of the Relativity of Motion (Univ. of Calif. Press) 1917, pp. 83, 84. §93 SYSTEMS NOT IN EQUILIBRIUM 145 when we have been interested in the property of mass or inertia possessed by the entity, and energy , the name applied when we have been interested in the part taken by the entity in the produc¬ tion of motion and other changes in the physical universe.” But from the equation above we have for one gram of matter Energy = c 2 = ( 2.9986 x 10 10 ) 2 = approx. 9 x 10 20 ergs Hence to measure the energy change to an accuracy of plus or minus one erg the change in mass must be measured to an accuracy of plus or minus 10 -21 grams. Therefore, except for systems such as electrons moving at high velocities, this method of measuring energy is of little value to us. Again we know that the loss in energy from say room temper¬ ature to absolute zero on the thermodynamic scale of temperature will be inappreciable compared with its total energy. However, if we wish to define energy in conformity with the theory of relativity we could define the energy of the system at some arbitrary state (t 0 , ei 0 , • • •, C6o) as me 2 and then use our ordinary methods for r obtaining the energy change between this and any other state. The accuracy with which the energy is determined will depend then on the accuracy with which the mass was determined at the state (t 0 , e lo , • • •, e 6o ) and the accuracy of our measurements of the heat and work of a process. We assume that for our measure¬ ments of the heat and work we are dealing only with systems for which the velocity of the process can be made as small as we please and that we are dealing only with local space. CHAPTER X. Introduction to the Tables of Thermodynamic Relations 94. The variable properties or quantities of the tables and their relations: In the following tables thermodynamic relation¬ ships between the derivatives of the following 10 + n variable properties or quantities are given. 8 = temperature in degrees on the absolute thermo¬ dynamic scale. ( 6 B — 8{ — 100° where 8 a denotes the temperature of steam and di that of ice at one atmosphere pressure.) p = pressure in dynes per square centimeter or baryes. nik = mass in grams of component k, k = 1, • • •, n. v = volume in cubic centimeters of the system (phase). e = energy in dyne centimeters of the system (phase), n = entropy in dyne centimeters per degree of the system (phase). £ = e + pv — 8n 2 C = t + pv x\ = t — 8n W = work, in dyne centimeters, received by the system. Q = heat, in dyne centimeters, received by the system. The relations between the 10 + n quantities, with their first and second derivatives, are connected by various relations. The relations between the quantities themselves are rather simple and chiefly of the nature of definitions and hypotheses. But the relations between the first and second derivatives are more compli¬ cated and it is these in which we are now interested. Let us consider the first derivatives. The physical hypotheses made, which include the characteristic equation of the system and the first and second laws of thermodynamics together with the definitions of the secondary quantities <, x, i|r, give us the functional ' 146 §94 INTRODUCTION TO THE TABLES 147 relations necessary for the existence and continuity of the deriva¬ tives. In each derivative of the type ( dt dV/0, p, m 2 ,..., m n there are n + 3 independent variables, n + 1 of them being held fast. If we then separate these derivatives into groups according (n 8) ' to the n + 1 variables which are held fast, we shall have j ; groups in the table of first derivatives and 42 derivatives in each group. Furthermore there will be 14 derivatives of the type dW dv where W = f (S), v = S, 6 = K', p = K", mi = E/", i = 2, n E', K", Ki"' being constants, in each group. (0 + 8) ' This makes a total of 7— , ■ w * ■ first derivatives. (n + 1) : 90 (n + 8) ' Now these 7— , + derivatives, 18,480 for a ternary system, are connected by various functional relations, and in general there is an equation connecting any 4 +3n of them and certain of the 10 + n variable properties. There are therefore (n + 8) ! L(n + 1) : 90J (4 + 3n) l ' (n + 8): (n + 1) : 90 - (4 + 3n) such relations between the first derivatives. For a ternary system u 18480 : this would be i3 ; 1 ' 8 4g 7 ~; • Such a tabulation is out of the question. However, a table can (n + 8) ' be derived in which the 7 v N ‘ - derivatives can be obtained (n + 1) . 90 in terms of the same 3n + 3 standard derivatives. Then to obtain any desired one of the numerous relations between any 3n + 4 of 148 THERMODYNAMICS §95 / n I s') ' the derivatives in Table I we merely have to elimi- (n + 1) . 90 nate the 3n + 3 standard derivatives between the 3n + 4 equa¬ tions for the 3n + 4 derivatives. It should be remarked that the (n + 8) ' method used here of tabulating the ^ ' qq derivatives in terms of the same standard 3n + 3 will largely do away with the necessity for determining the other relations by an elimination. For if in any special problem every quantity of interest is kept in terms of the same standard 3n + 3 one may be sure that at the end of the discussion there are no essential relations not brought to light. 95. The Standard Derivatives for Table I: The 3n + 3 standard derivatives may be chosen in many ways. The 3n + 3 chosen here are the isothermal compressibility, the isopiestic dilatation, ... t the change of volume with change in mass of component K, () , mi denoting all the component masses except mkj \dmk/0, p , mi the heat capacity at constant pressure, (mi + • • • + m n ) c p ; the heat of change of mass of component k at constant temperature and pressure, l mk , k = 1, • • • , n; and ju k , k = 1, • • •, n. These functions tabulated as the standard derivatives are not fundamental in the sense that thev are the minimum number of quantities left after elimination by means of all available quan- (n + 8) : tities. However, if it is desired to express the (n + 1) : 90 deriva¬ tives of the tables in terms of the fundamental 3n derivatives this can be done by means of the transformations of Table A in which the standard derivatives are expressed in terms of the fundamental derivatives, which are intensive quantities, and the masses. §97 INTRODUCTION TO THE TABLES 149 96. Transformations of Standard Derivatives of Table I to Fundamental Derivatives: Table A. dv\ K dp/e, mi, ..., m n \dd/ Pt m j, ... t mn (mi + • • • + m n ) ( \PV/ 0. Wll, ... , Win—1 (mi + • • • + m n ) ( ~) pv / p, mu ..., Win—l \dmk/0, p, m , t .. = V + i nik-i, mk+t, ••• i m n dv 9, p, mi ,... , wik-i, Wlk+l, ... , Win—1 n — 1 = i n — 1 2 ro ‘£;- ifk = n lwlk — i = 1 @ V “f~ (^w»k ^Wln) ??2i (Imi ^TO n ) Mk = e + pv — 6 7 ] + (Mk — Mn) — 2 mi 0*1 — Mn) These 3n fundamental derivatives have been chosen rather than some other set of 3n derivatives because they are basic intensive quantities in the development of the theory of thermodynamics and, as will be shown later, are readily obtainable from experiment. 97. Abbreviations and Special Notation introduced in the Tables: Now it is possible, by introducing some abbreviations and special notations, which do not sacrifice clarity by so doing, to reduce the number of tabulations still further. Since all the partial derivatives with respect to the component masses are symmetrical it is only necessary to write the derivative THERMODYNAMICS 150 §97 with respect to one of the masses, the others being obtainable by symmetry. Thus dn l .dm k/ 0 1 mi i •••} mk—1» mk+i, •••» mn mk d where k can have the value 1 ^ k ^ n, k being an integer. Furthermore, instead of listing each derivative separately, functions of temperature, pressure, and the masses or mass fractions are assigned to each of the n + 10 variable quantities such that when the quotient of the two corresponding functions is taken it will express the derivative. At first glance this might seem as though w T e had doubled the size of the tables, but we find the functional relationships to be such that each variable quantity needs to be tabulated only once in each group. Thus it is necessary to tabulate only 9 of these functions in each group, making a total (n + 8) ! 9 of for Table I of the first derivatives. (n + 1) '.7 Hence the quotient of any two of the same group expresses the derivative, the variables held constant being given at the top of each group. Thus in the first group (dv)® _ ( dv and (dmi) ® ydnii/fl, Pt TO 2,..., m n (dW)® _ dW (dmi) ® dim = —■, where W = f (S), 6 = K', p = K", mi = S, mi = Ki'" where i = 2, • • •, n, K', K", Ki'" being constants. ydnii/fl, m2> ..., mn This notation was first used by Bridgman in 1914 1 in listing derivatives for the single phase one component system of constant mass. His table of first derivatives was reproduced by Lewis and 1 P. W. Bridgman, Physical Review, 1914. §98 INTRODUCTION TO THE TABLES 151 Randall. 1 Bridgman republished 2 his tables, correcting the typographical errors, and added a table for two phase one com¬ ponent systems of constant mass. It is therefore thought that this notation will be familiar to most readers. However, in order to avoid any misapprehension of this notation, it might be well to state that these expressions are not increments nor partial differentials but merely have such functional relationships that if the quotient is taken of any two in the same group this quotient expresses the desired derivative. Table I contains all the first derivatives thought to be of practical value. 98. Transformations necessary to convert the tables for variable mass systems to tables for unit mass systems: The 10 + n quantities of an n-component variable mass system, of which all except 6 and p are extensive quantities, become 10 + n — 1 = 9 + n intensive quantities in an n-component unit mass system. The reason we have one less variable in the unit mass system is because only n — 1 of the mass fractions are independent, The nth mass fraction being given by the equation mi +.+ m n = 1. Thus to make Tables I and II applicable to systems of unit mass we must make the following substitutions in the tables: v, e, n, C, 2C, W, Q replaced respectively by v, e, 77, f, x, ^ I V,Q; mi, • • •, m n replaced by m h ■ • •, m n _i; pi by m — p n) and l mi by l mi — where 1 can be replaced by 2, • • •, n — 1; and (mi + • • • + m n ) c p by c p . Thus (d nin y and from §43, dlm k \ dp / d, mi, ... , m n am k \ dd) e, p, mi, mk—if mk+i, ... ,m n The most widely applicable method for obtaining is the freezing point method (see §100). This method, and the others listed below, for determining fx k presuppose a knowledge of the thermodynamic relations between phases. Gibbs 1 has shown that n\, the thermodynamic potential of 1 J. Willard Gibbs, Collected Works, vol. 1, p. 65. 154 THERMODYNAMICS §99 constituent k in the solid state is equal to /x k , the thermodynamic potential of constituent k in the solution, when the solid phase of k is in equilibrium with the solution. Thus, knowing the // k of the solid phase k, we can obtain the ju k of component k in the solution along the equilibrium line of solid k and solution. Other methods have been found useful in special systems. For example electromotive force measurements on reversible cells (see §101), can be used in binary systems to obtain /x 2 over the range of applicability provided we know the value at some one point in this range. Again osmotic pressure determinations have been used with success in special cases and may become more widely useful in the future, for Townend 1 has extended the appli¬ cability of this method to aqueous solutions of electrolytes and solutions in organic liquids, his methods being particularly appli¬ cable to dilute solutions. l mk may be obtained directly where reversible cells can be used by measuring the temperature coefficient of the electromotive force. However, in general, l mk can not be obtained directly. By measuring the heats of mixing in a constant volume calori¬ meter under non-equilibrium conditions we can get e as a function of the component masses, further e must.be measured over the same temperature interval in which the known values of hi, ■ ■ •, n n lie, otherwise we must obtain e as a function of temperature and the component masses. Then l mk can be obtained as a function of the <9e mass fractions at the temperature and pressure at which -— and dm k Mk were obtained, <3s \ dm k / e, v, mi, f fhk —ly nik+lj ••• t Again by measuring the heats of mixing in a constant pressure calorimeter under non-equilibrium conditions we can get x as a 1 R. V. Townend, J. Am. Chem. Soc., vol. 50, 1928, p. 2958. 155 § 100 INTRODUCTION TO THE TABLES function of the component masses. This is sufficient provided is given for the same temperature as hi was obtained where 1 = 1 , • ■ n, otherwise we must obtain x as a function of tempera¬ ture and the component masses. Then l mk can be obtained as a function of the mass fractions at the temperature and pressure <9 v at which -—■ and ju k were obtained, dm k dm k / e, p, mi, Mk* , mk-i, mk-Hi, •••» m n 100. Fundamental equation for a binary system of variable mass: “Fundamental equation” is used in the same sense as Gibbs (Collected Works, vol. 1, p. 88) defined it. For any homogeneous mass whatever, considered in general as having n independently variable components, to which the subscript numerals refer (but not excluding the case in which n = 1 and the composition of the body is invariable), there are relations between certain of the thermodynamic quantities from which, if the relations are known explicitly, with the aid only of general principles and relations, we may deduce all the relations subsisting for such a mass between all the thermodynamic quantities for the homogeneous mass. Thus if we can evaluate explicitly, for a homogeneous system or phase, the functional relationship between the quantities or or or or c, n, v, mi, • • • , m n <,0, p,m h • • -,m n V > mi, ■, m.n tb 0, v, m x , • • •, m n @1 V > Ml) ’ ) Mn we have a fundamental equation for the phase in question. 156 THERMODYNAMICS § 100 As an example we shall indicate how a fundamental equation may be obtained over all or part of a region of a binary system of variable mass. ► The method illustrated will be one which is readily capable of being extended to systems containing more than two com¬ ponents. H 2 0 / 9 .6 .754 7 .6 .5 KQ 0 / .2 .246 .3 .4 .5 Mass Fractions Diagram 4 The particular system chosen is the system, or phase, potassium « chloride-water. Since we are not able to obtain a fundamental equation from a consideration of this single phase we must also know the thermodynamic properties of potassium chloride and of water and use the relations existing between phases in equilib¬ rium with each other. INTRODUCTION TO THE TABLES 157 § 100 The density of KC1-H 2 0 solutions must be measured as a func¬ tion of the temperature, pressure, and concentration of KC1. This gives us the characteristic equation for the phase KC1-H 2 0. Next calorimetric measurements must be made to obtain the heat capacity per unit mass at constant pressure, c p , as a function of the temperature and concentration of KC1 at some one pressure, let us say atmospheric pressure. Let _ c, ___ / Mice Sice Ml MH2O in solution Ml Mkci = Tkci = M2' MKC 1 in solution M 2 Along AB of diagram 4 1 , which is the equilibrium line between Ice and KC1-H 2 0 at p = one atmosphere = 1.013 X 10 6 baryes, Mi = Mi- But Now Ml' = Tice = e + pv — d 77 e, p {6, p) - € ( 0 O , p 0 ) = J (c p - p d 0 - [e do, Po dv . dv d8 + P Vp, dp where e ( 6 0 , p 0 ) = 0 by definition 2 and e, p n (0, V) - v (0, p) = / J d$ Cj e do = 0 , p where 17 ( 0 , p) = 0 by definition . 2 Hence we know ix\ and therefore mi along the line AB. Similarly we know /x 2 ' and therefore ju 2 along the line BC. Now in order to determine mi at any point S in the region ABG of diagram 4, we must know e (§A \dd J Pt mi m n Coordinates (0,273) should be denoted by A. 2 See § 93 for justification of these definitions. THERMODYNAMICS 158 § 100 As we found in §99 it is sufficient to know l mi as a function of the concentration at some one temperature and pressure. To do this we shall measure the heats of mixing under non¬ equilibrium conditions in a constant pressure calorimeter to obtain x as a function of the temperatuie range CHi, (1) i.e. the temperature range in which the known values of mi and m lie, and the component masses mi and m 2 at atmospheric pressure. Thus we get / -( d *\ Lmi — l ~ ) Ml \dmi/e, P , m 2 dm 2 / o, p, mi — M2 Hence we have l mi and l m2 as known functions of 0, p, m\ since, from §99, d_ flmi\ dd \ 0 / d dmi and^ dp Thus we now have mi in the region GBA and M 2 in the region GBC. We still have to determine M 2 in GBA and mi in GBC. At B, the triple point, the three phases ice, KC1, and solution are in equilibrium, hence Mi' (262.6°, p) = M 2 ' (262.6°, p) = mi (262.6°, p, 0.754) where p = 1.013 X 10 6 baryes. 1 If we know 2C as a function of 0, mi, m 2 then we have directly Thus X = < + 0n Mi (02, Pi, mi) mi (0i, Ph mi) 02 0i #2, PU m l, m 2 8 2 \dmje, p. m, 0i, pu mi, m 2 d0 § 101 INTRODUCTION TO THE TABLES But we know mi as a function of temperature, 159 And from §50 we have mi d/u + m 2 d ^2 = 0 where temperature and pressure are constant. Therefore M 2 (0, V: Wl) ~ M 2 (0, v, Wlo) P, Wll (9, p, mi 0 which gives us m 2 in the region GBA. Similarly we get mi in the region GBC. Hence we now have v, c p , m 1 , M 2 , Imi and l m2 as known functions of 0 , p, m\ which is all we need to know in order to formulate a fundamental equation for a binary system. 101. Electromotive force measurements: If our binary solution is an electrolyte, which is the case for our KCI-H 2 O 1 solution except at infinite dilution, it is sometimes possible, if we know the value of H 2 at one point in the region in which this method is applicable, to obtain ja 2 and l m2 rather readily by experiment. Accurate measurements of electromotive force, however, become increasingly difficult at high dilution and therefore of increasingly doubtful value. We shall set up a double cell in which the process is reversible, i.e. can be made to go forward and backward through the same series of states or path Furthermore our cell is one in which the chemical reaction or “transference” is known. The electrolyte in the cells is a potassium chloride solution, KCl(aq). In cell I the concentration of potassium chloride is m 2 and in cell II is m 2o . A g + KC1 (aq)- > K + Ag Cl + aq-> A g + KC1 (aq) The net result is to transfer a certain amount, m 2 grams, of KC1 from the solution I to solution II. 1 Duncan A. Maclnnes and Karr Parker, J. A. C. S., vol. 37, 1915, pp. 1449-1455. 163 THERMODYNAMICS § 101 We shall consider this double cell as a whole and to simplify nomenclature write a prime, as (' where we mean £ for the com¬ bined cells (£ of I + < of II). Now since the process in the cell is a reversible one 1 taking place at constant temperature and pressure we can write n' Q' = J 6 dn = 0 (n' — n/) n'o where n' 0 and n' denote the total entropy of the combined cells at the initial and final states respectively. The mechanical and electrical work W' and W/ respectively v', q =q W' + W e '=-^ pdv' + £dq v'o, q = 0 = — p (v' — VoO — E q where E denotes the electromotive force and q the quantity of electricity that flows. The total energy change is then e' - t 0 ' = d (n' — n 0 ') — p (v' — v 0 ') — E q 1 See §92, for definition of “reversible process.” INTRODUCTION TO THE TABLES 161 § 102 or C'-Co'= -E q But C' - Co' = m 2 M 2 - m 2 M 2 0 where m 2 is the mass of KC1 transferred from cell I to cell II, the w T ater remaining constant. Hence M2 — M 2o = Eg m 2 which gives us M 2 (t, p , mi) if we know M 2 0 ( t , p, m io ). From the temperature coefficient of the electromotive force we obtain l m2 since d/i 2 dd This method, however, lacks the general applicability of the previous method discussed. V> mi, m 2 J fTL2 T m 102. The second derivatives (Table II): The second derivatives may be divided into two groups. Any second derivative formed from the first n + 8 variable uantities is of the type d_/dv\ _dd \dp/e, mi, ... , m„ _ _dp \dd) Pt mi , ir.n Any second derivative which includes the work and heat is of the type ~ d^ / dW\ _(ld \ dp / 0 , mi, ... , m n 7^ ^/dW _d p \ dd p, mi, ..., m n _ The number of combinations of second derivatives is so great that it can not be reduced to a reasonable number, as could the number of combinations of first derivatives. 162 THERMODYNAMICS § 102 All that has been attempted here is to tabulate a number of the standard second derivatives so that the relations thought to be of most use may be found readily by ordinary formal differentiation and algebraic elimination. It can easily be shown (see §99) that there are 4 + 3n + 3n 2 standard second derivatives, d 2 V d 2 V d 2 v d 2 V d 2 V d 2 V , dd 2 ’ dd dp’ dd dm k ’ dp 2 ’ dpd m k ’ dm h dm k ’ v + m n ) dc p ' d dd ’ dm k (nil + • • • + m n ) c p . d Mk . ’ dm h ’ and d Lk . dm h ’ where (h, k) = 1, ...n. As an example we shall evaluate d_fdn _dp \ddj Pi mii ... , ir n _ d, mi, ..., n?n by means of the tables. - ( m l + * * • + m n) -J \d v/ p, mi, ... , m n " from Table I Group 2. ~±(dn\ _dp \dd/ p t mi.m n _ 6, mi, ..., m n L A{ (mi + ... +mn) |; d, mi, ..., m n - i (mi + + m„) 1 dCp dp / e, mi, ..., m n d 2 v\ p, mi,..., m n from Table II Group 1. In dealing with the second derivatives of Q and W care must be used since, in general, if the order of differentiation is changed the value of the second derivative is changed, for Q and W are not functions of the state of the system but of the path traversed in reaching that state. INTRODUCTION TO THE TABLES 163 § 102 For example d / dW\ _d P \ dd /p, mi,..., m n _ Of mi, • •• , ITn dv\ dd/ p f mi, ..., m n - V d_(dv\ _dp \d0/ Vt mi,..., m n _ 8, mi, ... , m n d / dW\ _d0 \ dp / e, mi, ..., m n _ p, mi, ..., m n = ~ V d_ /dv\ _dd \dp/ d, mi, ..., m n J p * nil, ••• 9 nin Table II has not been completed as far as Table I because the expressions become very clumsy and furthermore the extension of this table can be made rather readily by formal differentiation which gives derivatives that can be evaluated by Tables I and II. For example suppose we wish second derivatives where m g , v, t, n are held constant. Let us choose one of the derivatives which may be involved, namely, d_ _dp dv ,dm k / p 1 m;_ m g , v, e, n Now by formal differentiation we have d 2 v dp dm k d_ / «3v \ _dp \dm k /e, p, mi. "a ( dv ^ _dd \dmjg, p, mi _ d 2 v \ / dm k \ dm k 2 / e, p, m\ dp J d ( dv\ _dm h \dm k /e, P , mi _ + 9, mi, ... , m n p, mi, ..., m n 'dd\ K dp/ m g , v, e, n + + m g , v, e, n 8, p, mj dm h dp / m gl v, e, n The values of the first and second derivatives on the right hand side of the equation are given in group 73 of Table I, and in groups 1, 2, and 9 of Table II respectively. ' ■ ■ ' . j . 4 ’;- . PART II 165 KEY TO TABLES The numeral denotes the group number, the letters following are the variables held constant in the group. All the component masses constant. Thus the groups in this section include all the thermodynamic relations existing for one component unit mass systems (constant total mass). 1. 0, mi ..m n 4. mi, ..., m n , t 7. mi, ..., m n , x 2. p, mi, ..., m n 5. mi, ..., m n , n 8. mi, ..., m n , ifc 3. mi, ..., m n , v 6. mi, ..., m n , £ All the component masses but one constant. Thus the groups in this section, together with the above groups, include all the thermodynamic relations for one component variable mass systems or two component unit mass systems (total mass constant). 9. 0, p, mi* 19. V, m i, < 29. m^ 2C * 10. e, mi, v 20. P, mi, 30. mi, L til 11. e, mi, e 21. P, m i, 31. mi, ", £ 12. e, m^ n 22. mi, v, £ 32. mi, ", x 13. e, mi, £ 23. mi, V, n 33. mi, ", 14. e, mi, 24. m^ V, £ 34. mi, £, X 15. e, mi, * 25. mi, V, 2C 35. mi, £, ik 16. V, mi, v 26. mi,- V, 4 36. mi, 2C, 4 17. v , m^ £ 27. mi, £> n 18. P> mi, n 28. m i? £ All the component masses but two constant. Thus the groups in this section, together with the above, include all thermodynamic relations existing for two component variable mass systems or three component unit mass systems (constant total mass). 37. 0, p, m g , v* 40. 0, p, m g , £ 43. 0, m g , v, e 38. 0, p, m g , z 41. 0, p, m g , x 44. 0, m g , v, n 39. 0, p, m g , n 42. 0, p, m g , t|j 45. 0, m g , v, £ * mi denotes all the component masses except mkj m g all except mt and mh. 167 168 THERMODYNAMICS 46. o, m g , V, JC 62. V, m g , V, c 80. m g , V, c, X 49. e, m g , £> C 65. P, m g , X 81. m g , V, c, 50. o, m g , X 66. P, m g , 82. m g , V, x, * 51. e, m g , 67. P, m g , ", c 83. m g , £ , ", C 52. e, m g , 68. P, m g , ", X 84. m g , £ , ", X 53. e, m g , n > X 69. P, m g , ", 4 85. m g , £ , ", 54. e, m g , 70. P, m g , C, X 86. m g , £ > c, X 55. e, m g , C, x 71. P, m g , C, 4 87. m g , £ , c, 56. o, m g , C, ife 72. P, m g , X, 88. m g , £ , X, * 57. e, m g , X, ^ 73. m g, v, £ , n 89. m g , ", c, X 58. P, m g , V, £ 74. m g, v, £ , c 90. m g , ", c, 59. V, m g , v, n 75. m g, V, £ , X 91. m g , ", x, 60. P, m g , v, < 76. m g> V, £ , 92. m g , X, til 61. p, m g , V, X 77. m g, V, ", < All the component masses but three constant. Thus the groups in this section, together with the above, include all the thermo¬ dynamic relations existing for the three component variable mass systems or four component unit mass systems (total mass con¬ stant). This also includes all the relations thought to be of practical value for systems of more than four components. 93. 0, p, m b , v, £* 106. o, P, m b , C, it 119. e, m b , X 94. e, p, m b , V, n 107. 0, P, m b , X; . ^ 120. e, m b , <1 95. o, p> m b , V, < 108. e, m b , v, £ , n 121. o', m b , C, X 96. 0, p, m b , V, X 109. e, m b , v, £ , c 122. o, m b , C, 97. 0, p, m b , V, 110. e, m b , v, £ , X 123. o, m b , X; t!r 98. o, p, m b , £ , n 111. o, m b , v, £ , 4 124. o, m b , C, X 99. 0, p, m b , £ , < 112. e, m b , V, ", C 125. o, m b , 100. o, p, m b , £ , X 113. e, m b , V, ", X 126. o, m b , x> 101 0, p, m b , £ , 114. e, m b , V, ", + 127. o, m b , C, X, 102 e, p, m b , ", C 115. e, m b , V; C, X 128. p, m b , v, n 103. o, p, m b , ", X 116. o, m b , V, C, 129. p, m b , V, < 104. e, p» m b , ", tlr w 117. e, m b , V, X, 130. p, m b , V, X 105. o, p, m b , C, X 118. o, m b , £ , ", < 131. p, m b , V, 4 * mb denotes all the component masses except mh, mk, m y . 169 132. V , m b , v, ", c 133. P, m b , V, X 134. V, m b , V, ", 135. V, nib, V, C, X 136. P, m b , V, C, * 137. P, m b , V, X, * 138. P , m b , ", c 139. P, m b , ", X 140. P, m b , ", 141. P, m b , c, X 142. P, m b , c, KEY TO TABLES 143. P, m b , £, X, 144. P, m b , ", C, X 145. P, m b> ", c, 146. P, m b , ", 147. P , m b, C, x, * 148. m b , v, £, ", C 149. m b , v, £, ", X 150. m b , v, ", 151. m b , v, £, C, X 152. m b , v, £, c, 4 153. m b , v, £, X, 154. m b , v, ", C, X 155. m b , v, ", c, 156. m b , v, ", X, Ifc 157. m b , V, C, X, 158. m b , ", C, X 159. m b , £, ", c, 4 160. m b , ", x, 161. m b , C, x, 162. m b , ", C, x, TABLE I First Derivatives 6, mi, • •, m n constant. (dp) = - 1 (dv) dv dp' N dv . .dv {dt)=P di + e d9 (A \ <3v (an) = M 00 = - V 02C) = “ v — 6 dv Yd = p (dW) = p (dQ) = 0 dv dp dv dp dv d0 Group 1 Group 2 p, mi, • • •, m n constant. m = i ,, s dv (av) - T0 dv (dc) = (mi + • • • + nin) c p ~ V ~qq 170 TABLE I—FIRST DERIVATIVES 171 0*0 00 0x) 0« (dW) (dQ) mi • • m (dp) (dt) (dn) 00 02C) 0*fc) (dW) (dQ) Group 2 (Con.) i (mi -f • • • + m n ) c p — n (mi + • • * + m n ) c f dv - p re~ n dV -~ p re = c p ( mi + • • • + m n ) Group 3 , m n , v constant. _ dv dp dv = ~d9 = (mi + • • = 7 {(mi + . \ dv , + m n ) c p — + d (S)' 6 dv (dv + m n ) c p — + 0 l dp dd dv dv v dd~ n dp dv , v ^+ (m > + dv + m n ) c p — + 6 [ — (S)' = — n dv dp 0 (mi + dv + m n ) c p — + 0 (is)’ 172 THERMODYNAMICS Group 1+ mi, • • •, m n , e constant. m . dv dv ~ e d6~ P dp d"V (dp) = - (mi + • • • + m n ) c v + p — (dv) = - (mi + dv (dv + m „) Cp --^- v P , . . N dv (dv (3") = -f m n ) Cp — - v Up w-(*s+»g)-’[ (mi + • • • + m : dv~| »)c P P d9 J 0X) = - (mi -+ m„) c„ I p — + v / dv \ dv ( dv \ A p ^ + y )- p ee\ 0 de- y ) (d»4) -( *fs+* [• + p|c p (mi+ ••• +m n )g + e(^ (dW) = v (mi H-+ m n ) +p 6 (|| (dQ) = -p(m,H-f- m„) c„^ - p 6 Group 5 mi, • • •, m n , n, constant. (dp) = (mi +•••-{- m n ) c p Q (dv) — — - (mi + • • • + m u ) c p — P 0 $ v (mi + • • • + m n ) Cp — + 6 TABLE I—FIRST DERIVATIVES 173 Group 5 (Con.) C r)v 00 = — (mi + • • • + m n ) -j v + n — 0x) = ~ ( m i + • • • + m n ) — v 04) V e (mi + + m n ) c p dv dp + 6 + n dv dd ( d W)=|(m 1 +....+m n )c p g + P (^) 2 (dQ) = 0 Group 6 mi, • • •, m n , C constant. (dd) = V (dp) = n x dv, dv ( av)=v- + „- . . i dv . dv . . (at) = - n (e- + p-) + v (mi + • • • + m n ) c p — p C dv (dn) = (mi + • • • + rn n ) ~fv — n — 0x) = (mi + • • • + m n ) v c p + n ( v - 6 dv dd ) (a*t) ( dv dv\ --V(v-+n-)~vn fAXlT\ dV dV (dw) = — pv —- — pn — *dd y dp (dQ) = (mi + • • • + m n ) c p v — dn dv Yd > CO <"0 174 THERMODYNAMICS Group 7 mi, • m (dp) (dv) (dc) (6n) (at) , m n , 2C constant dv = v — 6 dd = - (mi + • • • + m n ) c p = v^-c p (m I+ ... + «.)§?-fl(g) = (m, + • • • + mn) c p (pf v + v) + = ^ (mi H-+ m n ) c p v = - (mi H-+ m n ) c p v - n - 6^) m = - = — n^V — Q dv d dd>~ P v ^ - (mi H-+ m n ) ( - 6 dv Jo. dV v dV (dv' v--(m I+ ...+m n ) Cp --a^- (dW) = - p (dQ) = (mi H-+ m n ) c p v mi, • • •, m n , t|r constant (dp) = P^ + n Group 8 TABLE I—FIRST DERIVATIVES 175 (6n) = (50 = (5jc) = (dW) = (dQ) = 6, V, (5m k ) = (5v) = (5e) = (an) = (50 = (5x) = (50 = (dW) = (dQ) = dv ~ n dd~ Group 8 (Con.) P (S) _p 7 P(mi+ . dv av' nv + p(v- + n- n(v-ef e ) + p dv dd - (mi + - e dv M + m n ) dv dp v dv + m n )c p - dv — p fl¬ ap -an—- 8 68 - p (mi + • • • + m n ) c p dv dp ’ Group 9 constant. \am k / e, p, mi, ..., mi-it mk+i, •••» m n Pk + ^mk P dv am k l mk J Pk Pk d - ^ik Pk — p dv - V am k dv am k 176 THERMODYNAMICS Group 10 d, mi, v constant (dp) = (dm k ) = (dt) = dv dm k dv dp dv dv dv N 1 7 dv . dv dv (an) + Q mk dp 1 a<9dm k 00 = Mk dv dp dv dm k 0&) = 0k + U) - v ) dv dm k (a *) = Mk s (dW) = 0 /•.^N 7 dv , n dv dv (dQ) - l ™dp + 9 dddm k ' Group 11 6, mi, e constant. (dp) = - (dm k ) = - (dv) = - Uk + ^mk P dv dv ^ + P~r~ dd dp . dv dm k _ , ,7 \dv ■ a dv dv (j*k + U) ap + e 39amk _ V ( dv\dv p, dv (an) - v dm J ge e U dp TABLE I—FIRST DERIVATIVES 177 Group 11 (Con.) (ac) = -M k (^ + P g) + v(^ (ax) = - G»k + U) (pg + ▼) - (*§ 0*)--lMk-p^)fl|5 + pUg (dWJ-^ + Wpg + ^g^ Mk dv - V P (AQ) -( dv \ n dv 7 dv Mk pimk a^ 8mi mi, n constant. Op) = - y Group 12 (dm k ) -- (av) - -=if- (ae) = — 00 = - dv dd Imk dV dv dv 6 dp dd dm k (uk — dv \ dv ^ dm k / dd dv ^39 --L e ‘ dv Mt aa' _ Ii 8 mk + e "‘'‘dp m --( 6v \ 6v p 7 dv Mk-p aiWae + e Uk dp SA\ir\ dv dv p dv (dW) P aeami[ + 9 U a . (dQ) = 0 178 THERMODYNAMICS Group 13 6, mi, ( constant. (dp) = — Mk (dm k ) = v (dv) = O) = (an) = av . dv av\ Mk ^ ^ + p Ijp) + v ^ Mk + lmk dv . Lk ^00 + v T (djc) = Mk 9 ^ + v l mk (34) = dv . ( dv\ MkP ^ + v V Mk_? ’^/ dv dV (dW)= MtP ^-pv — (dQ) = Mk ® ^ + v Z mk av \ P am k / 6, mi, x constant, (dp) = — ^ — i mk (dm k ) = - d + v Group Ilf. (av) - 0* k + U) ap \ dv am k (dt) = (Mk + lm k ) ^P + v j + , v dv . V 7 (an) = Mk ^ (*s - ’) P av ami dv (ac) = d Mk v l m]t TABLE I—FIRST DERIVATIVES 179 Group 14 (Con.) fN , dv . - dv\ . 7 dv . dv (a« -MklP^ + v-^j+pU^ + p^ •5 — V (dW)-o. k + U P g + P^ (dQ) = e Mk + V l mk (*s-0 Group 15 6 , mi, t|r constant. (dp) = - Mk + p \ dv (arn k ) = - p- p dV dm k (dv) = — ju k dv dp v ,,dv/ dv (a£ ) = e ae( Mk “ p am k \ dv ) P Lk dp v , dv \ dv P 7 dv (an) - ^Mk - ~ 0 dv / dv \ (aO = -MkP^-v^k-P^-J Ox) = -Mk(pT^ + v- a ||) - p Li dp dfl. dv dp dv / dv ^dm k \ dO (dW)=Mkpg /, A v _ 1 dv \ dv dv (dQ) — l Mk P dm J 9 afl P^dp p, mi, v constant. dv dm k Group 16 180 THERMODYNAMICS Group 16 (Con.) (am t ) = % (dz) = (dn) = (Pk + lm k) qq — c p (nil + • • • + m n ) Zmk dv , , | \ Cp dv ~z~ ~ (mi + • • • + m n j — d dd 6 dm k , A r \ dv _j_ dv (30 -M t ^+n — dv (d&) = (Mk + Z mk ) ^ ~ ( m i + • • * + m n ) c p dv dm k M) (dW) (dQ) dv . dv IMk Ve + n d^ k = 0 , dv . v dv + m n ) C p ^ p, mi, e constant. Group 17 dv (dd) = — L k + p — Mk (dm k ) = (mi H-+ m n ) c p - p dv dd (dv) — — (Mk + L k ) ^ + (mi + • • • + m n ) c p (dn) — — ^ — (mi + • • • + m n ) y U'k — P dv dm k _ (d£) = n l mk + p k (mi + • • • + m n ) c p - p || + n — n p dv dm k dv dv (d&) = — (Mk + l mk ) p -qq + (mi + • • • + m n ) Cp p TABLE I—FIRST DERIVATIVES 181 Group 17 (Con.) (difc) = l mk [ n + p ^ J + Mk (mi + • • • + m n ) c p + n (nil + * • • + m n ) c p + n V dv dm k (dW) = (p k + (mi + • + m n ) c pV^~ <9v (dQ) = — p Imi 00 — (mi + • • • + m n ) c % Mk — p dv dm k . Group 18 p, m i; n constant. m = - l f (dim) = (mi H-+ m n ) -J \ Imv dv Cp (mi + • • • + m n ) dv (3v) = 'iw + - I - (5t) — Jg + ( mi + ’ • • + m n) -g Mk — p dv dm k . (ac) = \ n Imv + (mi + • • • + m n ) c p n k (dx) — ~q ( m i + • • • + m n ) Cp ^k (a« = J U (n + p a ^j + (mi + • • • + m n ) c p ( Mk - p dv dm k , MW'i = J mkdv (m, + • • ■ + m„) c p dv y ’ 1 e dd e F dm k (dQ) = 0 182 THERMODYNAMICS Group 19 p, mi, < constant. (dO) = — Mk (dm k ) = — n n dv dv (a y )= — (dt) = Mk dv (mi H-+ m n ) c p - p — + n 1 e (mi + • • • + m n ) c p /xk + «l Wk (an) (ajc) = — Mk (mi + • • • + m n ) Cp — n (Z wk + /A k ) 0*) av . av Mk p + tl p dd av am k av Mkp a@ + np ^ (dW) (dQ) = [(mi • • • -f- m n ) Cp/x k d~ n l m ^\ Group 20 p, mi, 2C constant. (d6) = — Mk — Imk (am k ) = (mi + • • • + m n ) c p av (av) = — (/Ak + lm k ) + (mi + • • • + m n ) Cp av (ae) = (/Ak + Lk) 'Pqq~ ( m i + (an) = — — (mi + • • • + m n ) Cp /Ak nZ m k + np + m n ) Cp p av am k av am k (dO = /Ak [(mi + • • • + m n ) Cp + n] + n ? mk dv (adr) = (/Ak + Imv) l n ( mi + ' * * + m n ) Cp ( /Ak (dW) = (/A k + L k ) ( mi + ' ’' + m n ) Cp p (dQ) = — (mi + • • • + m n ) c p /A k av am k v av am k > TABLE I—FIRST DERIVATIVES 183 Group 21 p, mi, tjr constant. ( dd ) = — Mk + p dV dm k (dm k ) = - p Jq - n dv dv (dv) - - Mkgj - 0e) = - Mk (mi + • • • + m n ) c p + n — L k ( n + p + (nii + • • • + m n ) c p + n V dv dm k ( a «) = - ? 00 = - p (mi + • • • + m n ) dv . dv Mk ^ + n toT k j Mk - p dv dm k . L k / , dv' 0X) = - Mk (mi + • • • + m n ) Cp + n + p dv dd ^Wk n + p dv dd + (mi + ••• + m n ) Cj dv (3m k V (dW) = p dv . dv ^Ye + n ^ (dQ) = - (mi + • • • + m n ) c p Mk — p dv dm k _ ^w»k ( “I - P dv dd, Group 22 m i7 v, e constant. . -x , . , V dv n dv dv (dd) = — 0k + 0 k ) v-0 dp ddd nik (dp) = 0k + 0 k ) ^ — (mi + • • • + m n ) Cp 184 THERMODYNAMICS Group 22 (Con.) , N N av , jdv (am k ) = (mi + • • • + m n ) c p — + 6 (an) = (dO = d6 t - + m n )c P g+(g (Mk + L k ) l v 00 + n frp ) + av (mi + • • • + m n ) c p — + + av am k av nd— - (mi + • • • + m n ) Cp v d v ($rJ = m = (Mk + Lk) ^ — (mi + • • * + ni n ) Cp (Mk + ^Wk) « ^ + Mk +< + n 0 (mi + • • • + m n ) Cp av av av ap aaam k (dW) (dQ) = 0 — (mi + • • • + m n ) Cp p. k — — p k 6 Group 23 mi, v, n constant. m = - h av av av 6~ mt dp d8dm k (dp) = \ ljy e - («.+ •••+ m„) (am k ) = - e (m, + • • ■ + m„) c p ^ + (~) (ae) = Mk [g (mi + " • + m ") c ^ + (^J _ (d() = Mk (mi + • • • + m n ) Cp av i, . j \ av /avV . U av + V — — ^ d dQ e am k _ + n L t av . av av + _ o dp aaam k _ TABLE I—FIRST DERIVATIVES 185 (dx) = Mk Group 23 (Con.) (mi + • • • + m n ) c p dv /dvV d dp \d 0/ _ (mi + • * • + m n ) c p dv I ^wt dv **" d dd Q dm k (dt{r) = Mk (nil +•••-}- m n ) c p dv /dv\ n e l mk dv , dV dV dp + \d6j _ + _ddp'~ dd dm k _ (dW) = 0 (dQ) = 0 Group 24 mi, v, £ constant. <*>--*£+*& («p)-*£+•£ (dm k ) = — v dv dd n dv dp (di) = - (/*k + L k ) l Mk (mi + • • • + m n ) Cp dv dp +< dv dm k dv n(9^ - (mi +••• + m n ) Cp v (dn) = — Pk (mi + ''' +mn) 7 P £ + (S/ J _ v — 0 \ as dv\ dv dv , Cp + n d^)~d^ k l n de~ v(m ' + ••• + m ")y (djc) (Mk "f" Lk) n dv dp Mk Cp (mi H-+ m n ) ^ + 0 dv dv dv Imit V _ 5(9 dm k + m n ) Cp v — v n 186 THERMODYNAMICS m ■■ (dW) = (dQ) = mi, v, m - (dp) -- (dm k ) = (de) = (dn) = 00 = 0. dv /dv (mi + • • • + m n ) c p -—h 6 dp dd, -l mkl &V Jo , dv * dp, dv dm k dv nd — - v (mi + • • • + m n ) c p Group 25 2 c constant. f L7 V dv / dv \ dv (Mk + U) a „ (A™ v j 3mk - dp dd / i 7 \ dv , \ dv (Mk + Lk) -qq — ( m i + ■ •_ • + m n ) c v^j^ dv v dv /dv - v ^+ (mi +--- + m „)c p - + ^- dv dv - (Mk + U) v— + v (mi H-+ m n ) c P —^. — Mk (mi + (mi + • • • + m n ) —- v c„dv , e dp' 1 '' f av Yl l-m k _ dV (dd) J 0 v d0 dv dm k * (m k + ^mk) n ^ + Mk 17 dv dV + C k V TT + dd dm k Cp (m 1 + ...+m n )g + .(| dv nd—- (mi + • •• +m n )c p v-vn Mk (m I + ... + m„)c p g + S (g '. dv dv + n--V — dp dd + dv , dv Imk Tl “i" dp dm k nieg- y| . 0 Mk (m 1 + ...+m„)c p g + e (g' / — n d$ + (mi + • • • + m n ) c p v dv dm k TABLE I FIRST DERIVATIVES 187 mi, v, i{/ constant. (df>) = -wg. , dv dv Group 26 \ av (am k ) = -»^. av 0e) = — (ju k + Z mk ) n — — Mk (mi + i \ ^V + m n ) c„ + 0 (an) = — Mk av .aa. .av av — n d — a a am k ‘ d dp \dd) _ C k av — n 9 dp av av — n dd am k * av av (ac) = w v- + nv— a v (a^) = — (Mk + L k )n— - Mk (mi + • • • + m n ) Cp ^ av\ av av am k Q dv " e Te~ ny (dW) = 0 (dQ) = — Mk (m 1 + ...+m n )c p g + ,(g n -wk dv dp „dV dv n 6 — a a anik mi, e, n constant. (aa) = — (ju k — p (ap) = - (p k - p Group 27 J>v\av_ z^av am k /aa -1-29 a dp' £)(“> + -•+ m »)y Cp lm k av p e dd 188 THERMODYNAMICS (dm k ) (dv) (dO 02C) 04 ) (dW) (dQ) mi, e, m (dp) 0m k ) 0v) W / vy w (v # \ v v/ xj. • j / i i sC P dW p (mi + • • • + m n ) — —. + (“' + •••+ m ")yg_ + p(miH- /I ^TT 7 . \ Cp av ••• +m " ) J - n dd. 'mk av av + pn- dp Mk Cp av f av , . , v p to k L n ^ _v(mi+ ’" +m " ) fl + p (m, + • • • + m ») y ^ ** ^ L k av ~ pv Tae i \ • • • + m n ) — + v (mi + ■" + mn) dv L k av av av Mkn M -np T^ -np aflai£ + (mi + ...+ mn )^g_ + v p (mi + Or/,Jc “ ■ nv Tdi P Mk 0 Group 28 ( constant. -*i tf S +p g)- pv £ +v( * +w = Mk (mi + • • • + m n ) c p — p — + n “l - H ^Wl k P " av am k r > rc > dv n _ dd + p dp_ — V (mi + • • • + m n ) c p ~ Py Q Mk / i i \ av ./av’ ( m i + • • • +m„) c- + ^- + (Mk + l Wk) av av\ av v — + n—)+ — dd dp) am k [_ dd dv nd— - (mi + • • • + m n ) Cp v TABLE I—FIRST DERIVATIVES 189 Group 28 (Con.) (dn) = jLt k dvY , .d p ( dv \ dv p[- a ) +(m l +..-+m„) y ^- + vJ-n- lr I v mk dV | dV + p Tl v Te + n eij dv~^ V dv dm k _ v (mi H-+ m Q ) y — n dd (djc) = Mk p <9 ^J + V (mi H-+ m n ) c p ^ + v (m x + dv + m n ) c p + pn — + vn dp + l m k dv dv . pv^ + pn— + Vn dd dp + P dv dv am L 0 n ai _nv_v ^ miH -+m n )c p M) = — Mk P 0 + ( mi + * * * + m n) C P (^P ^ + V dv + pn — + vn dp -l mk dv . dv pv- + v„ + pn- + pv dv dm k _ (mi H-+ m n ) c p + n „ dv dv — dpn dd dm k (dW) = — p Mk / I 1 x dv , -/dv -p(Mi . av av\ av + W(v^ + n-J-p — nd dv dd - (mi H-+ m n ) Cp v (dQ) = Mk 19 P (^) + ( mi H -+ m n ) Cp (p ^ + v + „vI„ k J- P y(m 1 + ...+ nl „)c p ^--«n^ Mk — P dV dm k _ 7 dv + npLk ap 190 THERMODYNAMICS Group 29 mi, e, x constant. W-<* + U^g + v) + p£( # g 6— - v . dv dv (dp) = - (Mk + U)p^ + p^-(mi+ + m n ) c p . (dm k ) = - p dv dd Q dv 6 -v dd (mi+ • •• + m n )cj dv P7 + dp dv dv (dv) = (jUk + Imt) v— - v ^ ( m ! H-+ m n ) c p . (dn) = Mk P + (mi + • • • + m n ) -j + v + V yh* !!l - VY Jl- ( m 1_ ... _L m ^ P d dd ? dm k - ^ ^ n) d' (d 0 = “ Mk dV dv + (mi + • • • + m n ) c p I p -—h v .dd dv ■f pn- - h Vn dp -l m k dp dv . dv , pv Te + pn ^ + vn . -) - V dv dv 5m k L en ^ -Vn-V(mi + "' +m " ) C ”. (dt|r) = — Mk dV dv P ® V ( mi "f~ * * * + m n) c p l V a-h V .dd dv + P n r- + vn dp — I m k dp dv . dv . pv ee + pn Y v + Vn . - v dV dv dm k \ dxv ~dd ~ Vn “ V( ^ mi -+ m n)Cp (dW) = — (^k + U) P v ^ + p v (mi + • • • + m n ) c v . (dQ) = Mk dv dv 9 P ( ^5 ) + (m, H-+ m„) c p ( p — + v .30. dv dv + P v Z Wk — - p v -— (mi + • • • + m n ) Cp. dd dm k TABLE I—FIRST DERIVATIVES 191 Group 30 rrii, t, i|r constant. ( dv\ dv , dv m =^ k -p~Je-- P u-. (dp) = ( Mk - P dv dm k / _ n + (mi + • • • + m n ) c t + l rak " +p S)- (dm k ) = n dv dv d - T) - dd 1 dpj dv + V (dv) — (/Z k + Z m J n — + Mk + «n^ 5V 9 (S) +(m ‘+- +m - ) *g_ + m , 1 )c J) g + 0(^ dd dm k * (dn) = — ( Mk — V dv \ dv L k dv di^) n M + pn Tdi (d() = Mk p 6 dv .dd, dv + (mi + • • • + m n ) c p l p — + v , dv + P „-+V n dv . . dv pv- + v„ + „p- pv dv dm k H - ^mk (mi + • • • + m n ) c p + n (djc) = Mk p d dv ,dd, + d p n dv dv dv dd dm k + (mi + • • • + m n ) c p ( p — + v dv + pn — + vn dp + l «k dv . dv . pv a« + pn ^ + vn + p dv dm k dv dii— - v n - v (nii + • • • + m n ) c p dv (dW) (pk ~j“ P W - P Pk dp /I , X dv (m x H-+m n ) c p — dp + d dv .dd, — dp n dv dv dd dm k , dv \ dv dv (dQ) = -( Mt - p _je„- + pnU^. 192 THERMODYNAMICS Group 31 mi, n, £ constant. / dV Jm k m = - Mk _-v T (ep) = - - e (dm k ) = v (mi + • • • + m n ) Mk (mi + * * • + m n ) Cp + l mk n dV e n dd (dv) = - Mk (m , , m ,c p dv (dv (m > + ••• + — V Irr^dV e dd - (mi + • • • + m n ) C p dv Q dm k _ n Lt dv dv dV _ Q dp dd dm k _ (de) = Mk HS +(mi+ --- +m " ) n p S +v — n dV dd 1 L.. r > > re l_ dv i_ + HI L dd dpj + P *n k L n dv Jd Cp v (mi H-+ m n ) (W = — n dV l mk ^Te + V T (3t[r) = Mk P\jgl +P(mi + , m ,c p dv + m n ) — — d dp + v (mi + • • • + m n ) + l 1 dv . 1 dv mk H py de + l) pn dp , l + « nv + v dv c)mi — v (mi + • • • + m n ) ° v dd (dW) = p jUk (mi + \ c P dv j-( dv + m n -—r l 0 dp \d0, — (mi + • • • + m n ) Cp dv d dm k _ + pn d + pv kk ^ 0 dd Imv dv dv dV _ d dp dd d m k _ (dQ) = 0. TABLE I—FIRST DERIVATIVES 193 Group 32 mi, n, x constant. W = -M k ^- V T . (dp) = — Mk (mi + * * • + Din) Cp 1' (am k ) = v (mi + • • • + m n ) y. (dv) = Mk dv Je t 4- (mi + -L-m ^ C P aV + m n ) — — 6 dp J ^mk av 0 aa + V (dt) = Mk dv V ( gg ) + (mH-+ rn„) -f ■»7'(»g + ')] Lk dv dv . . , v Cp + pv y --p v am k (mi + • • • + m ”4 00 = n dV . lmk Mk a@ + V T (aO = Mk P l j + (mi + • • • + m n ) y ^ + v ) + n + l m k a av pv^ + nv av , . . N c p - P v ^r( m i+ ••• +m n )y. am k (dW) = p Mk S . 1 + (mi+ , m sC„av + m » } J Yp. ^wik dV + *'fTe dv , . N C p -pv -— (m, + • • • + m n ) — am k e (dQ) = 0. O) Qj Qi <5 194 THERMODYNAMICS Group 33 mi, n, i|r constant. m = - ov \ av „. Mk V dm Jd8 + P $dp' lm k 3v (dp) = 3v \ . sC p l mk f dv n ~ p d^J (mi++mn) j~~e\ n + p de ( dv 0V 6 v 0m k ) = -^-j - P -( mi +-.. +m n )^-„- (dv) = - Mk Cp ^ + g (ml+ ... +m „ )? ^mk 6 V — n 0 6p 6 v 6 v — n ( 6 e) = 00 = dm k “ I Mk - V dv \ dv , l mk dv + ^rn k ^ - pn — —. dm k / oo 0 dp — Mk l dv Jo. p \ tz I + (mi + • • • + m n ) j ( p ^ + v m k T > > 1_ dv pv — + P n T" + nv L 60 6p V 6m k _ dv dp dv n - v (mi + • • • -f m n ) J 02c) = “ Mk V l TZ ) + ( m ! + ,dd t + ^j( p g + v + n 0V 60. 6 v T dv p v— + n v 60 + (dW) = p Mk .S)+£<* + ■••+»■)?. + pn lm.)c 6 V 0 6p + prx dv dv 60 6 m k ‘ (dQ) = 0 . TABLE I—FIRST DERIVATIVES 195 Group 84 (> X constant. dv (aa) = - a Mk y Q - v U (dp) = — Mk [(mi + • • • + m n ) c p + ti] — n / (dm k ) = v (mi + • • • + m n ) c p + n ( v - 6 ^ m k dv (dv) = — (Mk + W n ^ — Mk c p (mi -j-+ m n ) 0 + 0 V Imk ~ « av av a a dm k o dv n0 Te — (mi + • • • + m n ) CpV - v n (at) = Mk ^ P (^/ + ( mi -+ m n) Cp yp ^ + v + "lpg + V +1 TOk av , av pv ^ +?,n ^ +vn +p av a n av am k L a a v n - v (mi + • • • + m n ) c p (dn) 04 ) = n = Mk av 1 Lk Mk a0 + v T 9 p (^) + ( m ‘ ^-+ m„) c„ (p ^ + V + "(pg + v + l mk av . av , pv M +pn ^ +vn +p dv ami dv On— - vn - v(mi + • • • +m n ) Cp dv (dW) = (Mk +* Z mk ) p n — + p Mk c p (mi + • • • + m n ) ^ , ffl av 7 av dv dv + pvl ™Te + p d^l n9 Te — (mi + •••' + m n ) Cp v — v n (dQ) = n a Mk ^ ^ “I - v i m k 196 THERMODYNAMICS mi, C, m ■■ (dp) '■ (<3m t ) < (dv) = (dt) ■■ Group 85 tjr constant. dv Mk P^; + v -pv dv 'i i v dp dv dv ^p- e ~pn- dm k (5n) + (mi H-+ m n ) c p (p + v dv . dv . J,V M + Pn ^ + T “ + n dv dv + evn Ved^ dv . dv vy Yo + vn Y v +^i( pd i +v dv dm k _ n dd 02C) = (dW) (dQ) + (mi H-+ m n ) Cp (p ~ + v dv . dv P v ^ + P n r + Vn dd dp + li •m k dv dn— - vn - v - + m n ) o u dv . dv + m -) °p (p + v dv . dv py Je + pn 3i - v (mi + • • • + m Tl ) c p TABLE I—FIRST DERIVATIVES 197 Group 86 m i, 2 C> constant. dv . . dv , 7 dv , dv dv Mk L p ^ +V ‘ - d — aaj "1" P lm k ~ 4~ P £ dp am k a— - v L aa J (dp) = - Ok + U) l n + P^)“(mH- h m n ) c p Mk - p dv am k _ 0m k ) n ( v -^H-p a( dv \ { i v — — 6 [ — ) — (mi + aa .aa, + m n ) c 3 av ap_ (dv) = av “ (Mk + lm k ) n --Mk op (mi + a v + m n ) c p -~ +e dp — v dv dd dv a dv On— — V n (dt) = Mk a p am k L dd + (mi + • • • + m n ) c p ^ dv ^ + V / dv . \ r~ > <© > <© ( P ^ +V )J ■"l - Lk L pv ae + pn ^ + vn J + V dv (a«) = Mk am k l dd 2 aa anf^ — vn — v (mi + • • • + m P \ ^ ) + (mi + • • • + m n ) -q \P v ) n) Cp^J. ■) n dv aa + i mk a av pv- + nv 3v , - p W mi + I \ + nin) J. 00 = — Mk a p / av -f~ (mi -{“••• -j - m I 0 )^(pg + v ) P dv dm k L $d } + v a«^ - vn - v (mi + • • • + m n ) Cp + dv . dv pv 30 + pn ^ + vn > co <© <© 198 THERMODYNAMICS Group 36 (Con.) <9 v (dW) = p (Mk + lm k ) n -—b V Mk op e dv .d0, \ 2 dv 1 dv l |<*> I<"0 > 1 + p am t L (mi + • • • + nin) c p — + , dv 6n—; — v n (dQ) = ju k 9 p (S) +(m 1 +--- +m n ) Cp (pg + v) + n d y 0n dd + l mk dv pv ^ + nv dv , - pv taC mi + —b IHn) Cp* Group 37 * 0, p, m g , v constant, dv (dm h ) = (dm k ) = (de) = (dn) = dm k _ dv dm h [Inn + Mh] ~ 7^7“ [U + Mk] dm k dv Z dm h mh dv l -mt dm k 6 dm h 6 ^y^ dv dv (dO = 77 “ Mh — 777- Mk dm k dv dm h dv (djc) [Mh ~b lmh\ ~ [Mk ~b U dm k /n .\ dv dv (dijr) = 777-Mh — 7Z7 - Mk dm h dm k (dW) = 0 dm h (dQ) = l mh dv dm k dV dm h *m g denotes all the component masses except m k and mh. TABLE I—FIRST DERIVATIVES 199 Group 38 * 6 , p, m g , e constant. <3v (dlllh) P - 1 ~ Mk (dm k ) = - dm k 7 dY I P I Mh (dv) — ——— [l mk + Mk] ~ T' " [Zj»h H“ Mh] dm h dm k ^mh dv ^mk dv _ y h‘ p am t J y L Mh P amJ 7 5V _ Lk P 3m k _ 7 P amJ = Mh — Mk < 9 v r)V (dx) = V zzr 0*k + L k ) ■— P v—- (Mh + Lh) dm h am k 04 ) ^m K ^Mh p ^ Imh ^Mk P dv dm k , (dW) — p —— [/Xh + U “ P V~— (Mk + ImJ dm k dm h (dQ) = l m\x dv dv L Mk p amJ tmk L Mh p amJ Group 39 6 , p, m g , n constant. (amh) = z f (am k ) = - l f h ( . _ dv l mk dV Zmh ^ V ' dm h 0 dm k 6 (a«) = y‘ dV ^mh dV L Mh p amJ y L Mk P amJ * m g denotes all the component masses except m k and mh. 200 THERMODYNAMICS Group 39 (Con.) / 3 y\ Irri k l (0() = Mh -Mk ~q Wh f \ Lh (Ox) = Mhy - Mky ^n»k dv Imh dv y P dmJ T !/ k P *nJ (difc) = ^- k /Jirr\ 0V Imh dV Imk ( d W)=p — T -p — T (dQ) = 0 6, p, m g , < constant. (0m h ) = u k Group 40 dv dV — Mk - dm h dm k Imh dv = Mk P dm h _ Mh lm]c P dv dm k _ Cth Lk T _ Mh T (On) = Mk (Ox) Mk ^mh Mh ^ Wk ■ \ 0 v Ov (Oit) = v Mh T—- V Mk 0 m k (dW) = V^-^r - PMk 0 m k (dQ) /X k ?mh Mh Ck 0 m h dv 0 m h Group If.1 6, p, m g , x constant. ( 0 m h ) Mk ~f~ ^mk ( 0 m k ) Mh ^mh Ov Ov (ov) — (^ k d - ^ mfc ) ~ (^h + Lh) 0 m h 0 m k TABLE I—FIRST DERIVATIVES 201 Group Jfl (Con.) (de) = v (Mh + L b ) — v -~r (Mk + L k ) dm k L dm h = Mk vm h 0 mk (dn) (d£) Mh Lk Mk ^ 04 ) mh <9v <9v Mh l mk — Mk? TOh H“ p vzr - (Mh + l mh) P -r— — (Mk + l mk) dm k dm h (dW) (dQ) V (Mh + L h ) — v vzr (Mk + L k ) dm k Mk Lh Mh L k dm h Group J+2 6, p, m g , tjr constant. dv (dm h ) = Mk - P dm k (dm k ) — — Mh + p dv dm h . X dv dv (dv) = Mk vz: - Mh dm h dm k _ dv \ _ \ Lh[ dv (3n) = y U-p — ) L k j 9 V . dv dv (dC) = PMk-^z-P Mh dm h dm k dv dv \ (at) - U (Mk p dm J U (^Mh p Mh “ P av dm h ) dv (djc) Mk Lh Mh L k “h P (Mk ~h Lk) P (Mh d - Lb) ,,-nn dv av ( d W)- PWl — av (dQ) = Lh l Mk - P ami ' Mh p am h ) 202 THERMODYNAMICS Group 1$ 0, m g , v, e constant. (dp) = (Mh + £m h ) — -^ZT (Mk "1“ 2»»k) dm k dm h x „0v Ov . Ov , . 7 . (aniO -e-^- + -( Mt + z„ k ) (am k ) - (Mh + L„) (3n) = dp 00 = £ Ov ^mh ^m k Ov Ov Ov Op L Mk T‘ " Mh yJ "^OO _ Mk 0m h - Mh n Om k J Op _ Mh ^mk Mk ^ Ov mh +•£ Mh dV 00 [_ 0m k — Mk 0v 0m h Ov (Mh + £ mh )_ a^ (Mk + ' mk) 02C) = (Otlr) (dW) (dQ) = 0 — v Ov (Mk + ^m k ) ~ 7^7“ (Mh + U _0m h Ov 00 [_^ h 0m k — Mk Ov 0m h _ 0m k + - dpL Mh ^m k Mk l- mh = 0 Ov Op Mk Lh Mh ^ mk +•5 Mk Ov 0 0 [_ 0m h Group 44 — Mh Ov 0m k _ mk 0, m g , v, n constant. (d ) = J?L ^_ h _ l ' ^ 0m k 0 0m h 0 \ 1 0 v 7 Ov Ov (amh) - aa P Ll + a»am t /*\ \ 1 Ov Ov Ov 0 k) 6 dp 1 ™ d$dm h (A \ 9V (dt) = 77 Ov Ov Mh —-Mk 00 L 0m k 0m h _ + Ov t'Wk Mh n Op L o ^mh M k y TABLE I—FIRST DERIVATIVES 203 Group 44 (Con.) 00 = ^ Mh dv — Mk dv dd [_ dm k r ~ K dm h _ dv l mk dv l + dv dp l vm k l “m h Mk ~T" e h _dm h d dm k 6 _ 02C) = dv dd dv dv Mh -Mk dm k dv Z ^mk dm h _ dv l mh + - ' dp _ lmk lmh Pky _dm h d dm k d _ 04) = £ dv dv Mh t~ -Mk (dW) (dQ) dd[_ dm k 0 0 dm h _ + dv dp _ lmk Mh 9 lmh Mk T Group 4& 6, m g , v, < constant. \ dv dv (dp) = Mh — Tzr Mk (dm h ) = Mk dm k dv dp dm h dv r ________ dm k ^ N dv dv 0m t ) = -Mh^ + v — (8t) - 6 a9 ( Mt dmh Mh am k ) + dp ^ Lh Mh * mk) + (/*k + lmh) ~ T—— (f»h + 2 mb ) _dm h dm k 0 n ) = dv / dv dv dd V k dm h Mh dm k v dV l mh dv L k _dm k d dmh d _ , 3v / lmh + T- I Mk -7T Mh mk "d" x _dv/ dV dV \ dv/ 7 7 \ (8x) _ 0 as ( Mk Mh aW + a^ \ Mk "" Mh ”7 dv , dv . Wh dmh ' _dm k mk 204 THERMODYNAMICS (H) (dW) (dQ) = V = 0 dv _ Mk dm h Mh Group 1+5 (Con.) dv dm k _ dm h c)m k / dp — Mh lm k ) ~ \dm k m h dV dm h l-mk^ Group 1+6 d, m g , v, x constant. (dp) = —— (uh + Imh) — T—— (pk + lm k ) dm k dim ( am fc) “ % + + ( 0 Ye ~ v ) (dm k ) = ~ (Ph + U) “ ( 0 je ~ v ) dm k \ dv dm h (de) = — v ^ (Mh + U - ^ (p k + U) X dv 0n) = dd _dm k dv dm h L Mk dm„ Ph dv dm k _ + dV d lm h ^mit Mk “7^-Mh e dv Z 'rah dv l Wk (dC) = ^ _dm k d dmh d dv dv dv Mh t -Mk dd|_ dm k dmh J 1 dpi + dv Ph Z mk p k Z 'rah dv _dm h dv rak Z (H) = (dW) = 0 £ dv d-V dd dm k mh J dv Ph L dm k “ Pk dV dm h _ +- dp[_ Mh ^rak Mk l rah (dQ) = 0 ^ dv dv Pk T—-Ph d d 1_ dmh dv _dm k dv t'Wlh dm k _ dm h ^ J + dv dp _ Pk Imh Ph l rak TABLE I—FIRST DERIVATIVES 205 Group 0, m g , v, constant. , n dv dv (dp) = Mh—-Mk (dm h ) = Mk dm k dv dp dm h (am k ) = - Mh g Mk dv dm h — Mh / n X (dti) = TTa Mk dv dv am kJ dv + dv (a<) d 0 L dm h dv MhT- _ dm k — Mh Mk dm k _ dv + dp _ dv Mk Lh Mh l- m k ^mh ^mk Mk -V-Mh -7T dp L 0 0 dm h _ (dx) = (dW) = 0 (dQ) = 0 fl dV 'd0" V Mk dv dm h dv — Mh + dv dm k J Op _ Mk Mh ^ mk dV d0 Mk dv dm h dv Mh + dv dm k J Op _ Group 48 Mk Mh ^ mk 0, m g , e, n constant. (ap) = l f (am b ) = g Mk — P Mk — P (am t ) = - g Ov dm kJ Ov 0m k _ Ov l m k Mh — P 0m h 0 - P + P Mh — P Ov 0m h _ LkOv 0 Op' Imh OV T dp‘ (dv) = ~ Ov Ov Mk — Mh + Ov 1 lr 'mh l mk Om k J 1 OpL Mk 0 0 . 00 L dm h v\ dv / dv dv \ dv / Imh ^mk\ I (ac) - p deV* a^ h - «‘a5ij + V T P V*T ~ > + dV \ dm h dm k / 0 / T7 Imh ( dV \ l n lk v T^- p a^J- v vV Mh ” p W 206 THERMODYNAMICS Group 4.8 (Con.) x dv/ dv dv \ dv f l m 0x) = V y 9 ^ n asj + v Vv - e lmk\ l Mh T/ + vh*( e V ijr is constant dV \ I'm k dv \ (dW) (dQ) dv 1 > <"D > 1_ dv Irrih Imk P d0 _ Mk dm h dm k _ ~ V T V _ Mk T ~ Mh T_ = 0 Group 49 6, m g , t, < constant. (dp) = Mh ^TOk P dv dm k — Mk ^Wh P dv dm h _ dv . dv\ (am h ) = -M k (e- + P^; + v (dm k ) (n dv = Mh Vaa dv + ^'- v dv P dm k Mk /m \ dv dm h Mh l roh (dv) = v (Mh + U ” T— — (Mk + ^wk) Mk dm k dv dm h _ + dv dp _ dm h Mh l mk Mk Z mb + 0 dV Mh dv L dm k (5n) = dv dv dV dv ?mk Imh Mh dm k - Mk „ dmhj _ Mh T ~ Mk T_ lm h / dv\ lm k / dv \ TV Mk - p ataJ + v TC''-^) (djc) = v Mh 2 TOk “ Mk Z mh p d dv dd Mh dv dm k + pv dv dv l d m h/ dv L — Mk dm h _ n**, ~ mh - n-*, *wk om k dmh dV r " + + P dp _ Mh ^wk Mk ^ mh TABLE I—FIRST DERIVATIVES 207 0 —^ Ph lmk) "I - dm k / ( V lmk \ Ph P rj ) 208 THERMODYNAMICS Group 50 (Con.) a dv ( dv dv\ dv m = pe r8V k d^ h -^d^-J + p dp ( ^ 1 V L h (u k — — vl Mh ^mk) “f - (dW) (dQ) = — p v ' Mk p am k y dv _dm k (Mh ~f - Lii) mk \ Mh P ^ I d m h / dv dm h (Mk + £m k ) dV ( ^ dv dv \ dv , , P 9 50 v h am k Mk 5m h / + p dp (Mh mt Mk m,l) dv dv \ dm k / V Imh l Mk P ^ i [ i ) “f" V l mk ^ /^h P dm h . Group 51 6, m g , t, tjr constant. _ dv \ (dp) lmk l Mh P dm h / lmk ( Mk P dv dm k . (dmh) dv ( dv \ = — 0 - I Ui. — 7 ) - I aA Mk p dmj + plmk dp /*\ \ ,dv/ dv \ dv (amt) P dm J p l ™ dp - (dv) = Q dV dd Mh dv dim Mk dV dm h _ + dV dp _ Mh lmk Mk l mh n is constant. (90 a dv y dd Mh dv dm k Mk dV dm h _ + p dv dp _ Mh 'm k Mk l mh + dv dV V lmk L Mh p amJ V Imh Mk P n L dm k J (dx) = o dv dp — dv Mh dd |_ dm k — Mk dv dm h _ + p dv dp Mh Ci k Mk ^ mh + w mk (dW) (dQ) dV - v l mh dv L Mh P dmJ L dm k J dv dv dv dv Mh dd |_ dm k — Mk dm h _ - P dp _ Mh Z mk Mk l: mk = 0. TABLE I FIRST DERIVATIVES 209 Group 52 6, m g , n, ( constant. / ~ \ ^mh (dp) = Mh ~j ~ Mk y n dv V (alllh) Mk^^ q tmk N av , v, (om k ) = Mh ^ tmt (av) = av av av av aa Mh am k Mk ~ am h j dp [_ Mh ~8 Mk TT J + av L h av l m k _am k a am h a _ /n v av av av av ^mh (ae)=p^ Mk ^ Mh , am h am k j -r P Tr¬ ap L Mk T Lh av Lk av V T Mk P n am k J — v- a Mh ~ p am h _ t'Wlk y + X is constant l Mk av am h (dW) = p mh y av Mk — p — Mh av am k . av av am k — v av + p av ^mh ^ra k Mk -7-Mh ~Q v mk a ap [_ a av + Mh — P am h _ Mk Mh p v (dQ) = 0 aaL am h am k L k av z Wh av av + vzz v mh Mk Mh "mk ap |_ a a + _ a am h a am k _ Group 58 6, m g , n. x constant £ is constant. Same as Group 52. Group 54 a, m gJ n, tjr constant, c is constant. Same as Group 48. 210 THERMODYNAMICS Group 55 6, mg, x constant, n is constant. Same as Group 52. Group 56 6, m g , ijr constant. \ dv dv (dp) = Ph p —-MkP dm k dv dm h (dm h ) = p k p— + v v dv (6m k ) = - MbP^- - v Mk — p dv dm k . Mh - p dv dm h _ (dv) = v dv Mk T L dm h (de) = d p ^ — Ph dv dv dm k dv Mk dd[_ dm h Mh dm kj dV dp Mk Znih Mh l mk + v Z h ('—■£)- \ dv (dn) = p — dv Mk dd |_ dm h Mh Z m h e Mk - p dv dm k _ dV dm k — V + p dV I mh Mk “7-Mh (djc) = dpf^ dv Mk dd|_ dm h Mh e dv dm k _ dp |_ 6 dv l m\L T + Mh — P dm h _ + (»S + ’)[ Mk Onh 1711 h Mh ^mk + pv / _£ L wk dm h ' dv Wh drS; (dW) = p v (dQ) = « dV d p — ^dd dv dV Mh -Mk L dm k dv dm h dv Mk dm h Mh dm k _ dv + p apL Mk l mh Mh Z m k + Zm h V dV dV r p amJ Zmk V L Mh p amJ TABLE I—FIRST DERIVATIVES 211 Group 57 d, m g , ^ constant. (dp) = Mh l mk - l mh - p (p k + Z mk ) + p (ju h + l mh ) dm h am k (dm h ) = Mk dv dv Pt~ + v - d — dp 36 ,dV dV + P T— On k + P 7- dp dm k ndv o -V dd (am k ) = - Mh dv dv P7 + v - d — dp d6 dV 7 dV P V lmh P T—- dp am h e d -l-v ae (dv) = e dv Ye dv _ Mh am k Mk dv dm h + d - dp Ph Imk Pk l m h (ae) = a dv dp — dv Pk dd L am h — Ph dv dm k _ + V dv dp _ Pk lmh Ph l mk + V lmh ( Mk p dmJ V Lt ( Mh '^mj (an) = p av av Mk ddi am h — Ph dv am k . dv + V-KZ v m h Mk "7-Mh 1-mk apL ^ a 4 - l dv \ v tA Mk " p ^; mh l mk dv \ v T^- p YYj (d 0 = , av a p— av Mk v 7 aa[_ am h dv — Ph dv am k _ av p ^L Mk l mh ~ Ph l mk mh ^Mk — (dW) = - p ndV a-v dd dv L Mh ami Mk dv dm h _ dv V Y V Mh lmk Pk l mh (dQ) = a p dv Pk dv dv dd L am h — Mh + p dv am k J dp _ Pk l mh - Mh I mk V l mh ^Mk — dv \ 7 ( dv\ P am J v Lk V h V am J + 212 THERMODYNAMICS p, m g , v, e constant. (58) = (Mh + L h ) — (Mk + lm k ) (dm h ) = — (pk + ^mk) ~ (nii + • • • + m n ) c p dv dv (0nik) = —- — (nh + ^h) + (mi + • • • + m n ) c p -— do onih i „ \ c p dv * * * + m n) J / r, \ dV ( Imh Imk (dn) = deV k T~ dv -) - (m, + Mt to Mh dm k _ (dK) _ dv ” dd (Mh ^mk dV Mk dm h _ X is constant. (a<-) _ dv ~ dd (Mh ^mk dv Mk dm h _ Mk Imh) (mi + • • • + m n ) c v + n r— (Mk + Imk) — t— (Mh + LJ onih dnik Mk ^wih) + n (dW) = 0 (dQ) - g (Mk L„ dV Mh dm k _ (mi + • • • + m n ) Cp ph .£ ( ^ k+w - £ (Mh +1 Mh Imit) ~ ( m l + * ' ' + m n ) C p p, m g , v, n constant. Group 59 dv l mh dm k 6 dV l mk dm h 8 (dm h ) —y — — (mi + • • • + m n ) 'p dv 8 dm k / TABLE I—FIRST DERIVATIVES 213 Group 59 (Con.) (dm k ) = Imh. j f j | ^ \ C P y^+( m 1 +--. +m D ) y — fa ^ 5v (at) = Te l Mh mk l 6 Mk mh e (nii + • • • + m n ) ~ dv L Mh to t Mk dV am h _ <*> = Te Mh 6 Mk "mh y (mi H-+ m n) y dv _ Mh dm k Mk <9v am h _ + n dv l mk av i rah am h 6 am k 0 (ax) = av aa Mh b mk T Mk "mh y - (mi + + mj ^ dv L Mh dm! Mk dv am h (ail) = dv Ye I'Ttlk m^ y — Mk "mh Q - (mi + + “■>7 Mh av am k dv Mk + n dv l mk av z mh dm h J Lam h Q am k 6 _ (dW) (dQ) = 0 = 0 p, m g , v, ( constant. dv dv (dd) = Mh ZZI-Mk am k am h x av av 0 m h ) = M k ^ + n ^ \ dV dV (dm k/ = — Group 60 214 THERMODYNAMICS Group 60 (Con.) (de) = dv Yd Ph Pk lm h dv Ph ^rak dV — (nil + • • • + m n ) Cj dv Pk - L dm h n dV (A \ dV (3n) = ae am k J |_6 m h ^mh ^mk Pk — Q - Ph (Mk + Imk) n — (Ph + l rah) 02C) = Mh av aa Mh av am k _ n 6 dv l mk am k /*-M I \ -m \ Cp dV dV l Wh _am h 6 am k 6 _ Pk f'mh dv l my. Ph ^mk dv dm k _ i|r is constant. (dW) = 0 — n - (mi + • • • + m n ) c dV dv Pk _am h (Uk + Imk) am k P L" K dm h (.Ph + ^wh) (dQ) = — V V ' dd Ph pk Lh dv dm k _ ph ^mk dv — n _am h — (mi + • • • + m n ) c p dv , dv Pk L dm h Wk am k ‘' mh _ Group 61 p, m g , v, x constant. e is constant. Same as Group 58. Group 62 p, m g , v, t|r constant. C is constant. Same as Group 60. Group 68 p , m g , e, n constant. L av ^mk dv |_ Mk p amJ T L Mh P 3m h J (30) = =* (3m h )|=;- - (mi 4-h m„) ^ Mk — P av am k _ TABLE I—FIRST DERIVATIVES 215 Group 63 (Con.) dv l (dm t ) = PjjjY + (mH- +m n )j Mb — V dV dm h _ <*> = % ^mh k M k y-Mhy - (mi + Cp + m ")y Mk dV dm h dv Mh 00 = V dm k _ dv dd Imh Imk Mk y - Mb y - p (m, + • • • + m„) Cp dv Mk 6 [_ dm h Mh dv 02C) = P dm k _ dv dd + n L Imk ylMh dV \ ^ dmh/ ^mh ( dV mh Mky Mh ‘'wik d Cp p (mi + • • • + m n ) j dV Mk T- L dm h Mh dV dm k _ (dt|r) = n ^«k ( dv Tv h ~ p aW l m h d Mk — P dv (dW)= P ^ ^mk ^mh Mh y - Mk y dm k> + p (mi + • • • + m n ) Cp Mk dV d L dm h Mh dv dm k _ (dQ) = o p, m g , e, £ constant. Group 64 (dd) = Mh t'rak p dV dm k _ Mk ^mh P dv dm h (dm h ) = n . . dv , _ U P dm k J + Mk (mi + • • • + m n ) Cp - p —+ « 216 THERMODYNAMICS (dm k ) = - tl h rn-h V Group 64 (Con.) dv dm h — Mh (mi + • • • + m n ) c p dv P dd + n (av) = dv dd Mh Imk Mk m h dV Mk (dn) = p dm h _ dv + n L dV dm h I — (mi + • • • + m n ) Cj dv (Mk + ^ Wk) — 0*h + ^ rah) dV _ Mh dm k dm k dd dv mk ^rah Mh T ~ Mk T Cp - p (mi + • • • + m n ) -f dv Mh 6 L dm k Mk (W = v dm h _ dv , lm h ( dv\ l mk ( dv\ + «|_y ^ - P^-J - T C h " P tojj dd Mh l rak - Mk l mh - p (mi + • • • + m n ) c, Mh dv dm k dv Mk (dit) = V dm h _ dv + np dv . dv (Mk + lm k ) — —— (Mh + lm h ) dm h dd Mk l mh Mh ^ mk dV , dV dm k - p (mi + • • • + m n ) c p Mk dv dm h Mh (dW) = p dm k _ dv + p n dm k dv (Mh + £ rah) rv ~~ (Mk 4~ £ mk) dd Mk l mh Mh ^ mk dm h - p (mi + • • • + m n ) Cj Mk dv dm h Mh (dQ) = p dV dm k dv + P n (Mh + ^mj ~ (Mk + lm k ) dm k dd Mh ^mk Mk ^ mh dm h - p (mi + • • • + m n ) Cp dv Mk dv dm h _ + n ^mh ( Mk P dv dm k . ^mk ( Mh P _ Mh dm k dv \ dm h /_ Group 65 p, m g , e, 2C constant, v is constant. Same as Group 58. V, m g> m = (dm h ) = (am k ) = (av) = (3n) = 00 = 02c) = (dW) = (dQ) = TABLE I-FIRST DERIVATIVES 217 Group 66 z, tj/ constant. = u(«> -Pg^) - *»*(«■ - Pg~) -l 77lk n + v fe} + ( dv \ Mk " p airj dm k/ (mi + + m n ) c p + n Uln + p m n ) c v + n dv dv \ (mi + • • • + : TT7 (Mh Imk — Pk “ (mi + • • • + m n ) C t dd Pk = n dV ^ h dm k + n 0*k + lmk) — -r— (Ph + lm h ) dm h dm k dv dm h _ l mh ( dv\ l mk ( dv\ If V k ~~ V ami/ “ T V Mh “ V dmJ} dv p ^ q (Mh Ck Pk ^mh) p (mi “I - • • • -}- m n ) Cp Ph dv dm k Pk dV dm h _ + np (Pk + lm k ) ~~ (Ph + lm h ) dm h dm k dv P ~ a ( Ph lm k Pk lm\i) P (m.i “j - • • • "t - m n ) Cp do dv Ph T L <5m k Pk + np (Pk + lm k ) — 0*h "1“ lm h ) dv dm h J 1 " r [_dm h vr "^ ' dm k dv P Ta (Mh ~ Pk lmk) + V ( m i + * • • + do m n ) Cp dv dv Ph n <3m k (Ph "i~ lmk) dV — Pk dv dm h _ — np dv dm h (Pk + lmk) dm k lmk i Pk P dm k/ U [ Mh v am J_ 218 THERMODYNAMICS Group 67 p, m g , n, ( constant. {66) = p h — - Mk y (dmh) = n + (mi + • • • + m n ) ~ Mk (dm k ) = — n-^ — (mi + • • • + m n ) 6v ( l mk l (dv) = ^ ( Mh ^- k - MkT ) - (mi + • •: + rn n ) ^ ^ ^ 66 \ h 6 6v Mk (6z) = p 6 m h _ 6v 66 + n l 6 6 V lmk dv 6v l mb. mb v m k _dm h 6 dm k 6 _ l 6v Ph dv dm k _ , lmk( dv \ + n T U-p—j Imb ( ~TV k ~ v Pk dv 6m k /_ 02C) = n (at|r) = p lm k Imb ~6 Mh ” T Mk dv 66 dv Imb lmk Mk T ” Mh T /I , v Cp dv “P(mH-h m n ) y Mk^ Mh am k _ n p lmk ^V Imh 6V mb ^)=V% Imb lmk pk t _ Mh y _ 0 dm h 6 dm k _ - p (mi + • • + m n ) dv Mk 6 L dm h Ph dv dm k _ + pn dv l mb dv l m k (dQ) = 0 (9m k 6 dm h 6 Group 68 p, m g , n, x constant. / lmk Imb {66) = p h — - Mk y (dmh) — Mk (mi + • • • + m n ) Cp J TABLE I—FIRST DERIVATIVES 219 (dm k ) = (dv) = (de) = (d<) - m = (dW) = (dQ) = V, m g> (dd) - (dm h ) = (dm k ) = Group 68 (Con.) Cp Mh (mi + • • • + m n ) ~ dV ( l m dd\ h Q Imh (mi + • • • + m n ) y Mh dv dm k Mk dv dm h _ dv ( Zmh Ck I / i V -QQ \Vk -Q - Mh y ) — P (mi + + m n)^ dv M ky— L dm h Mh ■( dv dm k _ Lh ^rnk Mk T ~ Mh T dv d0. Mh ^mh ^mk Mt T ~ Mh T + p (mi H - • • • + dv 6 _ dm k — Mk dv dm h _ dv f l mh Irrik I , / , - Mhy ) + p (mi + ^ * I ‘'mb 7 de \ k T I \ kp + m „) 0- Mh dv dm k Mk dv dm h _ = 0 Group 69 n, tjr constant. ( dv\ lm k ( dv \ l mb = ( n+p S)y + (mi + "- +m " ) f 220 THERMODYNAMICS (dv) = dv dd M h 1*1 ■'mk v mh Group 69 (Con.) I + (mi H-+ m n ) Pk dv dm h dv dm k _ + n dv l mk dv l mh _0m h 6 dm k 6 _ (de) — n l mk dv q \Vh P Q m , 00 = V dv dd lmh Ck Mby-Mky Ch{ dv - p (mi + • • • +m,)^ Mh dv dm k dv P k 0x) -( dm h _ n+ p I ^ “mk + np dv ^mk dv ^mh dv \ Cp dd. Mk _ 6 dm h 6 dm k _ L,i l mk l mh Mh T Mk T + V ( m i + • • • + dv dm h Mh dv dm k _ (dW) = p dv dd lmh lmh Mk T _ Mh T + p (mi + + m„) f Mh dv dm k Mk dv dm h _ + pn lmh dV Imv dV mk d dm k d dm h _ (dQ) = 0 Group 70 p, m g , 2C constant. (.dd) Mk lmh Mh ^mk (dmh) = — n (pb -f- l mk ) — (mi + • • • + m n ) c p p k (dm k ) = n (ph. + lmh) + ( m i + • • • + m n ) c p ph \ dv (av) = ae Mk Mk lmh - Mh l mk + (mi + • • • + m n ) c 2 Mh dv dm k dv dm h _ + n (Mh + lmh ) — TT~ (Mk + ^m k ) dm k dm h TABLE I—FIRST DERIVATIVES 221 (de) = p dv dd m n ) c Mh ^mk P Group 70 (Con.) Mk l mh + p (mi + • • dv dv + Mk T-Mh T- dm h dm k . + prx _am h (Mk + Zmk) (m h + Lb) % “ (dn) = n OH) = V Imk ^mh [_T Mh “ T Mk _ dv dd m n ) c p dv Mh l mk Mk ^ mh dv Mk n Mh ^ dm h dm k . + p (mi + • • • dv + + P n dm h (Mk + Lk) (Mh + lm h ) dW) V dv dd m n ) c. dv Mh l mk Mk l mh dv -{- p (mi d - • • • ~f* dv Mk ~ Mh ^ dm h dm k + pn (Mk + L k ) (Mh + hr*) Q) ^ [Mh ^mk Mk LJ Group 71 n g , (, i|r constant. > constant. Same as Group 60. Group 72 p, m g , ifc constant. dv dv (dd) Mh Z OTk Mk ^mh P T (Mk — l - Lk) + P T ^Mh - f' ^mh) dm h dm k dv (dmh) — ( n V ^ mk ) ( mi * ’ * H“ m n ) c p ( n- Pg~) 222 THERMODYNAMICS Group 72 (dm k ) = (3v) = dv Yd (Con.) + V (m h 4~ lm h ) — (mi + • • • lp V Mh _ V Si) Mil ^mk Mk Imh (nil — f~ + m n ) c. 4- m n ) ’L Mh dmk (Mh + Imh) ~ (Mk + lm k ) Mh ^wk 4" V (ni] + ' • * + m n ) C p /Xh dv Mk dv (3t) = p dm h . dv n Mk 50 dv Mk ^mh 5m h _ + P ” 3m k (Mh + Lh) dm h = ft ^r»k lm\x Mh-y-Mky dv + (nil + • • • + m n ) -~ Mk dv dm h Mh (dm h ) = - (dm k ) = (dO = dm k _ ( i i \ c p dv (dvY Mk (“> + ••' +m " ) y£+(S) j Mh dV . dv v aa + n v (mi + • • • + m„) dp J [_ 0 Cp J Imk Imh Mh —-Mk -y n dv Mk 7- L dm h — Mh 02C) dv v d0 m n ) ^Wk Irrih , / , Mh ~~0 -Mk + V (mi + dv dd' dv dm k _ 4- Cp dv Mk 6 [_ dm h Mh dv dm k dv ^mk ^mh dv dv dv n T“ dp _ Mh T ~ Mk T. I 1^0 e 1 L Mk 3m h Mh am J (dW) = 0 OQ) = 0 m g , v, e, C constant. Group 74 (dd) = d dv Td Mh dv dm k — Mk dv dm h _ + dv dp _ Mh Ink Mk l' mb + (Mh 4" ^Wh) “ 737 " (Mk 4" ^mk) _dm k x dv = aa Mk ^wih Mh l mk dm h + (mi + • • • + m„) c p Mh dv dm k dv Mk + n dm h J |_dm k (ph 4" Imb) — 7 _" (Mk 4" Lk) dm h 224 THERMODYNAMICS (dm h ) (dm k ) (an) 02c) 04) (dW) (dQ) mg, v, p is c< Group 74 (Con.) v — + n —^ (/ik + ^ k ) + (mi + • • • + m n ) Cj — -QQ + n ^ ) (Ph + lm h ) — (mi +•;_• + 1 > CO > CO 1_ . dv dv dv _^ k dp V dm k _ -4- d — ^ dd L Mk ^ + n amJ m n ) Cp dv dv n dv dv . dv Mh T- L dp dm h _ dd L Mh ae + n amd dv . dv v — + n T~ dd dp ^W?h 4lk I Mky-Mhy + dv "dd" v (mi H-+ m n ) dv dv dV L Mk am h Mh a dm k J = V dd m n ) c p dv Mk m h Mh TOk dv dv Mh ~_ Mk + v (mi + • • • + dv = V dm h dv dm k (Mk + ^wik) dm h _ + v n _dm k (Mh “1“ ^wh) dd m n ) c p dv Mh 1 Wk Mk ^ mh dv + v (mi + • • • + Mk dm k 0 dm h (Mh + ^wih) Mh dv dm k _ + v n dv dm h (Mk “h ^?nk) dv , dv V— + n — dd dpj Mk ^ mh - Mh l mk + ^dd“ v (mi + • • • + m n ) Cj dv Mk _ L dm h Mh dv dm k _ Group 75 t, x constant. instant. Same as Group 58. TABLE I—FIRST DERIVATIVES 225 m g , V, (dd) = (dp) = (am h ) = (am k ) ■■ (an) : 00 : 02C) (dW) (dQ) Group 76 e, ilr constant. / w --6 av aa Mh av am k av Mk + aiv Mh Z mk - Mk Z mh <9v aa Mk Mk l mh Mh lm k av am h J ap _ + (mi + • • • + ni n ) c p av Mh _ am k am h _ + n (Mh + Z OTh ) — ——— (Mk + Z mk ) 6 dd am k av am h av Mk Ye + ” 8m k j av + (mi + • • • + m n ) Cp Mk + dp n (Mk + Z mk ). dp -a — aa av av Mh 80 + " 8nuJ /i i \ a v — (mi + • • • + m n ) c p — Mh dp (Mh + Z mh ). dp n av a^ Zmh Z wk Mk y - Mh y av av av J + n 88 Mk am h Mh a am k j av v aa m n ) c dv Mk Zmh Mh Z mt P av av Mh t— -Mk v (mi -j- • • • -f- av am h av v aa nin) Cp av am k (Mk “I - Z Wk ) am h _ + v n _am k (Mh “I - Z mb ) Mk mh Mh 'mk av av Mh TT Mk + v (mi + • • • + av am h o av am k (Mk + Z mk ) am h _ + V n _am k (Mh “1“ Z W h) = n ap Mk Zmh Mh Z mk . n av + 0n 88 Mk av am h Mh av am k _ 226 THERMODYNAMICS m g , V, m = {dp) (am h ) (am k ) 0e) 0x) 04) (dW) (dQ) n, £ constant, dv = ae Mh av “ Mk dv am k am h _ Zmk av Z?nu Group 77 +^ v ^mk Z mh Mh T “ Mk T _am h d am k 6 _ dv aa • Mk av aa Mk T ~ Mh T av am h _ + n av z mh av z mk dv v mk Mk 7^ i V _am k d am h a _ l aa + (mi + v • +m n ) av am k _ av " aa dv + n a z Wk av . av av av Mk 7 a [_ ap 7- + _ a dp dd d m k _ Z av < 'w. h Mh aa + V T — (mi + • • • + m tl ) dv Mh 7- a L dp am h _ dv n l mh dv , dv dv + dv dd + n apJL _ a dp aa am h _ Zmt Z mk ‘'mh Mb y - M k y n av aa v (mi + • • • + m n ) -f av dv L Mk <3m h am k _ av Zmk Zmh av dv av n 7~ ap L Mh y Mk Tj + n a« Mh am k Mk n am h J + n v av z Wh av z mk _am k a am h a _ = v av aa Zmk Z mh Mh y - Mb y + v (mi + • • • + m„) y av _ Mk am h Mh 0 0 av am k _ + n v av z mk dv l m h _am h a am k a _ TABLE I—FIRST DERIVATIVES 227 m g> v, (dd) - (dp) (dm h ) (dm k ) (dt) (dO (< W) (dW) (dQ) Group 78 n, 2C constant. dv i > > 1 dv ^OTk L|, dd Ph Pk ^ dm k dm h _ dp _ Mh T “ Mk T. dv Imk dv l m dmh 0 dm k 6 dv Yd Pk dv Ye Lh ^mk Mk T ~ Mh T dv + (mi + • • • + m n ) dv P h d L dm k dm h _ dv . Lk- ^Te + V T + (mi + • • • + m n ) — dv Pk TT“ d [_ dp dv dm k _ dv " Ye dv . Mt ae + V T — (mi + • • • + m n ) — dv Ph t- d |_ dp dv dm h _ dV ^mk ^mh dd y ~ Mk y -f~ v (mi -f- • • • -f- dv dv Pk T Ph V dmh dm k n dv dd Pk dv dm h dv P h dm k _ , dv dp _ Pk “mh y Ph b mk d + n V dv l Wk dv Z mt _dm h d dm k d _ l dv dv v dd” n d^J L Mh n c p dv m n ) —+ n— p h ^mk y dv dm k Pk v mh d dv Pk dm h_ v (mi + • • • + dv L + nv v mk _dm h d dv L b dm k d 0 0 228 THERMODYNAMICS Group 79 m g , v, n, tjr constant. w= s P h dv dm k Pk dv dm h _ + dv ‘•'mk Ph “7-Pk *'Wl h dp [_ 0 0 (dp) = dv d0 Zwih Mk T “ Mh T + (mi + • • • + m n ) -j Ph dv dm k Pk dv dm h . + n dv Z m h dv Z mk _dm k 0 dm h 0 . (am h ) = g 0V . dv aF k + n ^ + (mi + , „ n c„av + mn) 7^ Mk + Imk d V n 0 dp (dm k ) = - dv d0 dv dv a0 Mk + n ai^. (mi + • • • + m„) — /j h Zm h dv n 0 dp (de) = n dv d0 Ph dv dm k L Pk lr, dv dm h _ + n dv ^mk ^mh Ph ^ Pk dp [_ 0 0 ^mh ‘'mk Mk ~e~ Mh T + v (mi + • • • + m„) -f dv Ph 0 l_ dm k dv Pk dm h_ + n v dv Z mh dv Z mk dm k 0 dm h 0 02C) = dv d0 dv n dpj L 0 Zmh ^mk Pk —-Ph -y + v (mi + • • • + N C P , ^V mn) y + n ^ dv Ph _ L dm k Pk dv dm h _ + n V dv Z mh _dm k 0 dV Imk dm h 0 (dW) = 0 (dQ) = 0 TABLE I—FIRST DERIVATIVES 229 v, m ■■ (dp) (3m h ) (dm k ) (a«) (dn) (34) Group 80 C, x constant. e ^ ee Mk 3v 3m h dv Mh + dv 3m k J dp _ Mk 1- mh Mh ^ mt dv j _ 3v t'rah ~ 3m k 3m h ■' mk _ 3v 30 Mh Mh ^mk Mk l' mh + (mi + • • • + m n ) c p dv L Mk 3n^ = - e dv 3m k _ dv + n _3m h dd (Mk + Imh) — -T—- (Mh + Imh) d m k + (mi + • • • + dv l mk dv - Mk "t - V- ~T~ tl- 30 Mk^ e ^ 3m k J 3v 3v 3v dv, , 7 v m n ) Cp ^ 3m k 3p_ + n L V 3m k 3p (Mk + Lk) J 0^ dd dv Imh , ^Mh+V y + „ — m n ) c p dv dv dv Mh V" = V 30 3v Mk 3m h _ dv -V 3m h dp_ Mk £ mh - Mh l mk + vn n - (mi + • • • + r 3v dv. , x L 3m^ - 3p (Mh+ mh) . 3v + v (mi H-+ m n ) c p 3v 3v Mh _ 3m k 3m k ^ h ^ _ 3mh ^ k ^ mk ^ = n dv dv Mh » ... Mk 30[_ 3m k 3m h _ dv Imh dv l + n dv l Mh mk dp [_ 0 Imh Mk T + nv “mk _3m k 0 3m h 0 _ = v 3v 30 Mk Lh Mh 1-. mk + v (mi + • • • + m n ) Cj dv Mh 3m k Mk 3v 3m h _ + vn (Mh + Imh) — n™ ’ (Mk + ^m k ) 3m k 3m h (dW) = 0 230 THERMODYNAMICS Group 80 (Con.) <3v 2C> constant. m -( ee dv M h Mk dv + ?- v (dp) = r L Mk ^ mh - Mh l mk dm k dm h J dp _ + (mi 4* * • • 4~ nin) c p Mh Z mk Mk ^ mh dv Mh dm k Mk (dm h ) dv dm h _ dv + « (Mh 4" Lh) — (Mk 4~ ^mk) J _dm k \ dd /[_ dv . dv Mk a9 + n to k j dv dm h dv + n T“ dp _ Mk + ^ mk + ] (mi + • • • + m n ) c p 7 x k — (dm k ) -('£-*)[ dv , dv Mh ^ + n dWj — n dv dp _ Mh 4" ^mh (mi 4“ • • • 4" ni n ) c v Mh dv dp \ dv ( 3 e) = V- Mh ^mk Mk l- mh + V (mi 4- • • • 4- m n ) c p 4- n dV Mk t- L dm h — Mh dv dm k _ 4- v n •'mk dv dm h mh dv dm k _ TABLE I—FIRST DERIVATIVES 231 (dn) 00 (dW) (dQ) nig, e m (dp) (dm h ) Group 82 (Con.) dv dv v- n — dd dpj ,c p dv m ^l + n V6 Imk Imh Mh T ~ T dv _ Mh dm k — Mk dv dm h _ v (mi + • • + dv l + n v mk dm h d dv l mh = V dm k 6 _ dv dd Mk ^wh Mh ^ mk + v (mi + • • • + m n ) c p dv Mh dm k Mk dv dm h _ + n v (Mh + U ~ T—— (Mk + Imk) _dm k dm h = 0 dv dv V d0 “ "dpj Mh Imk Mk l- m n ) c p + d n dv dv Mh ddj |_ dm k 'rah Mk dv dm h _ v (mi —j— • • * — dv + nv _dm h Imk dv dm k ^ J Group 83 n, ( constant. dv _ dv dv Mk ~-Mh dmh dm k _ dv \ + P dv Z, Mk dp L 0 dv Mhy + Z?nh / dV \ ^mk / dV \ T0-^)- v T0~ p aW p dv dd Imk Z mh . / i I w \ C P Mh -q — Mk ~y + V ( m i + • ‘ * + m n ) — Mk dv dm h Mh dv dm k _ + [ Zwh / T v t_ p dv dm k \ L k ( dv \ dv/ dv\ / vCpI" / dv “ IMk de\ n ~ v Ye) ~ (mi + '" + m » ) y|_^ + \ dv Imk dv . dv ) PV dm k J ~ p T L v ^ + n ^J — p n dv dv dd dm k 232 THERMODYNAMICS (dm k ) = (dv) = 02C) : M) : (dW) = (dQ) = nig, £, m - (dp) ■■ Group 83 (Con.) av ( dv \ c p f / av . Mh MV p ^ _n / + (mi+ "' +mn) »"L Mh v p ^ + v — pv dv am h _ av . av V— + n — a a ap_ + p Mk mh a l m h y av v ^ +n av ap_ . av av + Pn — aaam h Mhy mk _j_ dv n — — v (mi + • • • + aa m n ) 'V dv Mk a j L amh Ph dv am k . "l 4 v+v Mk y - Mh y + pn dv / dv _am h \ Mk aa + i mk a dv ( dv d^ k y Xh M + ' r i dv av av av ^w»k Imh pn Te Mh am k - Mk - am h j + pn ^ _ Mh T ~ Mk T_ vn ^mk i av TV ih-p toTh, l m h a" dv \ ^-^arrrjj + P dv . dv v ae + n ^j ^mk Mh y ~ Mk y + P n av aa v (mi + • • • + rn n ) y 0 Mh av am k av — Mk am h _ Group 8/f. n, 2C constant. dv ( dv dv \ dv j Z mb l m \ v eeV* to, ~ ^toj + 1 Imh ( dV T l Mk “ p to*. Imk . y l Mh - V a a / av \ am h /_ + p % ( Mh t ~ Mk y h )+p (“i + • • • + m -) f av M k T- amh Mh av am k _ TABLE I—FIRST DERIVATIVES 233 (*n h ) = - g nik dv l v Te^ + vv ~e Group 84 (Con.) — (mi -}-••• + m»)j , dv \ av M k lP^ + v )-pV—j (<3m k ) = p dv dd dV . Imh ^ Mh + V T “l - (mi \ Cp m »)f , dv ) dv ^\PV P + V )- pV ¥m h J \ dv W = y Te Im'a ^mk Mk T“ Mh y + v (mi + • • • + m n ) dv av Pk a m h _ (ao = p n av av av Mh — Pk del am k am h _ 1 dv ( l mk 1 + pn ^V Mh V _Mk ¥j + n V ^mk ( dV ir _? to h , Zmjk ( dV 6 \ k P am k/ (atjr) = pn dv n V dd dv dv Ph — Pk dv ( l Im h \ am k am h _ dv \ l +pn r P ^f~^ e/ + mk . ^ J V Mh - v am h ) m h [ dV \ 6 P am k /. (dW) = l Ph m k l 6 Pk roh y + p v (mi + • • • + \ Cp m») 7 Pk dv am h Ph dv am k _ (dQ) = 0 Group 85 m g , e, n, tjr constant. 6 is constant. Same as Group 4<§. 234 THERMODYNAMICS nig, c, m ■■ (dp) (am h ) (am k ) (dv) (an) it is Group 86 C, x constant. Mh Imk Mk l mh p 6 dv dd Mh dv am k + pv dv dv _am k v m\ 1 dV dm h "mk + Mk am h _ dv I P Qp Mh ^TOk Mk l dv v ~d~e Mk ^mh Mh l- mk + p (mi + • • • + m n ) c p mh Mh dV _ am k Mk dv dm h _ + np (Mh + Imh) — (Mk + Imk) am k am h = p 6 dv dd m n ) c p dv . l mk . dv V T + n am k J dv L\ P dp + V ^ Mk p V + (mi + dv + n ( p s +v )_ Mk + l mk am k _ — pvn + dv am k a dv — v d — F dd dv , „ L h , dV Mh aa + V T + n toT h j — (mi + • ■ • + dv \ dv m„)c P [(pg + v) H “1“ Mh + Imh + p v n dv am h = v dv dd Mh lmk Mk l Mh dv am k _ + vn mh dv + v (mi + • • • + m n ) c p dv Mk dV dm h amh l m] ^ ~ ain^ ^ / dv \ [ l mb -”(p^ + v ) L^T" v U) _ dv / d ) am k \ Mh l mk d + pn dv ( dv _to h V k Ye + dv I dv . a^:V Mh ^ + v Imh \ T/_ onstant. TABLE I—FIRST DERIVATIVES 235 (dW) = p V dV d Q m n ) c p Mk ^mh Mh Ck dv Mh a dm k Group 86 (Con.) + pv(mi + dV dm h _ + pvn dv _<3m k + (Mh “f" Imh) dV dm h 0*k + Lk) + 6 p n dv dm h dv dd + Group 87 m g , t, t|r constant. X is constant. Same as Group 86. Group 88 I m g , z ) 2C> ^ constant. £ is constant. Same as Group 86. Group 89 m g , n, x constant. 6 is constant. Same as Group 52. Group 90 m g , n, t{r constant. 236 THERMODYNAMICS Group 90 (Con.) (dp) = p dv dd Mk T l Mb mk 6 + p (mi + + m n )^ dv _ Mh dm k Mk dv dm h _ + np L h dv ^TOk dV _ 0 dm k 6 (9m h _ n dv (dm h ) = p — dv Z mk dv u k - + V- —t- Tl- ^ dd^ d ^ dm k J \ Cp m n ) ~ (dm k ) = - p 0 L\ dp dv dv \ dv PTr + vlMk-pv + (mi + • • •_ + l I'mk dd m n ) dm k _ dv + n y l V dv \ r v + V dV , Imh , Mh ae + v T + n am h j - (mi + • • • + Cp J dv . \ dv p ^ + V^- pv to h j L „ / av \ n T V aj> + 7 v dv (av) = v- ^mk ^w»h Mb + v (mi + • • • + m n ) —■ Mi dv dm h Mb dv dm k _ + n v ^mk dV Imh dV _ d dm h d dm k _ (de) = p n dv dv Mh Mk Vn d d [_ dm k ^mk dV T V Mh “ v aS"h, dv dm h dV + pn- dp v m k Mhy- ^Mh Mky + ^mh / "TV Mk " dv \ *WJ (djc) = pn dv dv Mb dd |_ dm k vn mk — Mk dv dv dm h _ + pn dv Mb ■'mk dp L o e \^~ p d^ h , Imh ( 0~ \Mk _ dv \ V dm k /_ Mk ‘'mh y + (dW) = pv^ ^mh Lk Mky-Mby p v (mi + • • • + \ Cp m »)f dv Mk _ dm h Mb dv dm k _ + p nv Imh dv lm\r dv mk _ d dm k d dm h _ (dQ) = 0 TABLE I-FIRST DERIVATIVES 237 m g , n m (dp) (dm h ) (dm k ) (dv) (dt) (dO Group 91 , x, 4 constant. = V dv Ph dv dd |_ dm k l Pk dv <9m h _ dv + Pi: Mh l m k mk I ^ u v \ Vm h I e V b v am J T V Mk v dp [_ 6 dv l mb Mk T + dV \ lm h ( ~ ¥ v k _ *" am k ^wh lm k t ~~ Mh y + p (mi + ; • • + rn \ C P nin) dv Ph d L dm k Pk dv dm h _ dv dd dV \ l Pj- e + nU k + P v- e mk + (mi + • • • + .c p ( dv \ dv mn) 7 Lv a^ + v ) Mk_?,v I „ + nv y dv dd dV \ Imh P^ + nlMh + pVy v C p ( dv \ dv mk) yU p ^ + 7 Mh " pv ^;j dm k _ — (mi + • • • + l v m h -„Vy dv dv V— — n dd dp_\ |_ d Imk l mb Mh -Mk 0 \ Cp , dv m ")y + n aa Mh dv dm k Pk dv dm h _ v (mi + • • • + dv 1 + n V v mk dm h d dv l nib dm k d _ = p n nv dv dd dv dv Ph Pk dm k 6mh_ dv ' + pn dV v mk Ph ~~r mk . ^ VV Mh_?, am h lmb ( ~dV k ~ dp L d dv \ V dm k )_ l mb pk~e 4 * — p n n V dv dd dv L dm h — Ph dv dm k _ dv + pn — l mh Wh dv e v Mk-p am" k ) l dp[_ d dv e v Mh_ p am h /j Pk-7. -Mh v mk T + mk 238 THERMODYNAMICS Group 91 (Con.) (dW) = p dV dv V dd ~ n dpJ Zmh I'm. k Mk T ~ Mh T + v v (nil ~j~ ■ ■ • H - \ i dv m “ ) 7 + n a0 Mh dv dm k Mk dv dm h _ + p n v dv l mh _dm k 0 dv Z- ■mk dm h 6 _ (dQ) = 0 Group 92 m g , C> ife constant, e is constant. Same as Group 86. Group 93 6 , p, m b , v, e constant. (dm y ) = —^ (Mh + Z mh ) — (Mk + L k ) dv dv (dmh) = (Mk + L k ) — (m y + L y ) — (Mh + l mh ) + (m> + L y ) (dm k ) = /a \ _ dv ” dm k dv dm y < a0 = & dv Zmj T i ',y Mh-y Zmh . 0 Zwh My Mh-y 'mh My Mk dV dm h Zm y T mk Mk ^ My dm y _ is constant, Zmh Mk Zm y Mh Zmk Mh dv dm h Zmk My Zm y Mk X 1S <*>=£ Zmh My Zm y Mh dv dm b Zmk My Zm y Mk TABLE I—FIRST DERIVATIVES 239 Group 93 (Con.) (dW) (dQ) = 0 dV dm k dv Mh ^Wh My dv dm h lm y Mk ^mk My 5m y L ^mk Mh Lh Mk Group 9J+ 6, p, m b , v, n constant. (dmh) = (3e) = 00 = 02C) = (3 1 dv lmy dm y 6 dm h e dv Lh lm y dv Imk dm k y My e Mh dm h T My dv Lh Lk dm y .T Mk y Mh dv Ly dv Imk dm k T My y Mh dm h y My dv Lk dm y y Mk y Mh dv lmy dv Imk dm k y My 6 Mh dm h _ Q My dv ^Wlb Imk dm y _ d Mk 6 Mh dv Imh lmy dv Imk My dm k T My y Mh dm h T dv Imb Imk dm y y Mk y Mh Imy T Mk 1% e Mk t'Wy y Mk c m y Q Mk (dW) = 0 (dQ) = 0 240 THERMODYNAMICS Group 95 0 > p, m b , v, { constant. \ dv dv (5my) = ~ aw Mk \ dv dv (dmhj = ^77- Mk — 777- My dm. dm k n dv dv (dm k ) = — —— Mh + 777- My dm y dm k (de) = dv dm k \_ Lmy dv Ly Mb “ Lh My dv dm h Imy Mk ^ mk My dm y L Lt Mh Lh Mk (dn) = dv dm k dv T Mh dm y L l mk ~e Mh (d?c) = dv dm k dv dm y L L/ly Mh ^mk Mh Lb y My Lb y Mk Lh My Lh Mk dv dm h ^my y ^mk Mk ^ My dv dm h Mk ^ mk My it is constant. (dW) = 0 (dQ) = dv dm k dv dm v Ln y Mh Lnh My Lk Mh ~ l mh Mk dv dm h _ LyMk LkMy Group 96 6 , p, m b , v, x constant, e is constant. Same as Group 93. TABLE I—FIRST DERIVATIVES 241 Group 97 6 , p, m b , v, t jr constant. C is constant. Same as Group 95. Group 98 d, p, m b , e, n constant. (dlHy) l Mk P ~ ) ^ mk ( ^h P dm k , dm h , (dm.h) lm k ( My P au. ) ^ m y \ Pk V dm. dim (dm k ) = 71 dV ) 17 ( dV "mb l My P 3 I ~l m y l Mh P dm. dm h/ (3v) = dv dm k lm,y Mh ^mh My dv dm h lm, y Mk Lk My dV dm y L ^mk Mh * - I Mh rah Mk (dC) = P dv dm k Imy Mh ^mh My - p dv dm h _ lm y Mk ^mk My p dV dm v LkMh LhMk 0k) = P dv dm k _ lm y Mh ^ mh My - p dv dm h ^ra y Mk — l mk My P dV dm y L ^mk Mh ^?«h Mk t|r is constant. (dW) = p dV dm k Lh My ^m y Mh - p dv dm h ^mk My ^m y Mk p (dQ) = 0 dV dm y L ^mh Mk ^ rak Mh 242 THERMODYNAMICS Group 99 6, p, mb, t', < constant. dv (dm y ) = Mh (dm h ) = Mk ^mk P I m y V 6m k _ 6v 6m Mk My ^mh P (6m k ) = — /x h dv v my V yJ dv v mk V 6m yJ + My ^TOh P dv 6m h _ 6v 6m k _ 6v (6v) = 6m k dv Lh My ^w y Mh 6v 6m h _ 6m h _ Lk My ^m y Mk 6 m (6n) = p ^TOh Mk ^mk Mh y L 6v V 0X) = P P (6*fc) = p P (dW) = p P (dQ) = p 6m k 6v 6m 6v 6m k 6v y L Lh ^m y yMy--y Mh Lh Cit T Mk “ T Mh 6V 4ik lm.y ~ p d^ h Lt My _ y Mk ^mh My ^m y Mh P 6v 6m h _ Cik My Imy Mk 6m y L ^TOh Mk — l mk Mh 6v 6m k 6v ^m y Mh “ Imh My - P 6v 6m h _ ^ra y Mk ^ mk My 6m y L Lik Mh ^n»b Mk 6v 6m k _ 6v lm y Mh iih My - P 6v 6m h ^m y Mk Lk My 6m y L Inik Mh LhMk 6v 6m k Lh My ^ m y Mh - P 6v 6m h _ Lk My *w y Mk TABLE I—FIRST DERIVATIVES 243 Group 100 6, p, m b , e, x constant, v is constant. Same as Group 93. Group 101 6, p, m b , £, ijr constant, n is constant. Same as Group 98. Group 102 6 , p, m b , n, £ constant. (duly) /X b 1m\i Mk 1", (6m b ) Mk 1‘my Py l (6lll k ) = Mh Irriy “f" Py l dv v m h m h (dv) = dm k _ dv Imh Py lm y Ph dv 6m h _ Lk Py l 771 y Pk dm (dt)]= p Ln Pk — l mk Ph y L dv V dm k dv 6m, Cy Ph ~ l mh Py ^mk Ph ~ l m\x Pk - V dV dm h C y Pk Lk Py L y L. X* is constant. dv (d® = V V (dW) = p V (dQ) = 0 6m k dv lm y Ph — l mix Py V dv dm h _ Imy Pk — l mk Py 6m y L Ck Ph Lb Pk dV 6m k dv lm y Ph — l mh Py - V dv 6m h lm y Pk Imk Py 6m y L ^mk Ph — l rah Mk 244 THERMODYNAMICS Group 103 6 , p, m b , n, % constant. I is constant. Same as Group 102. Group 104 6 , p, mb, n, ijr constant, e is constant. Same as Group 98. Group 105 6 , p, m b , C, 2C constant, n is constant. Same as Group 102. Group 106 6 , p, m b , Cj tfc constant, v is constant. Same as Group 95. Group 107 6 , p } m b , i|f constant. dv dv (dllly) r= Mh ^mk Mk l/tth V (Mh Lk) P ~ (Mh “f” ^mk) dm k dm h dv dv (dm b ) = Mk lm y My Lk “f" P i (Mk “f - Lk) P ~ (My + Imy) dm. dm k dv dv (dmk) = Mh Liy “1“ My ^mh P ~ (Mh + Lh) “f" P j (My “f~ ^m y ) dm. dm h (dv) = dv dm k dv (mh My ^m y Mh dv dm h Ck My (m y Mk (de) = dm P ^mh Mk ^ mk Mh y L dv dm k ^my Mh ~ l mb My P dv dm h lmy Mk — l mk My TABLE I—FIRST DERIVATIVES 245 (dn) = p V (30 = P V (dW) = p V (dQ) = p V dv dm k dv dm y L Ln y Lh y Mh ~ y My Lnk Lnh y Mh - y Mk Group 107 (Con.) dv P dm h Cy Llik y Mk - y My dv dm k dv dm y L dv dm k _ dv dm y L. dv dm k _ dv dm. ^mh My Ln y Mh ^mh Mk “ ^mk Mh ^m y Mh “ ^mh My ^mk Mh — Z »ih Mk ^m y Mh Lt My Lk Mh “ Lh Mk - p dv dm h lm k My Ln y Mk - p dv dm h _ L y Mk ~ Lk My P dv dm h lm y Mk — Z mk My Group 108 6 , mb, v, e, n constant, dv Op) = dm k dv Mh Ly My L mb dv dm h Mk Ln y My ^ Wk dm y L Mh Lit Mk 1 'mh -fl av 0 dd dv dv dm k dm h J +- dp Mh Lk Mk Lib dv dv dv • dv Mk lm y My Lnk dd Mk a L dm y My am kJ dp (arn k ) = -^ 00 = v dV dm k dv dv dv , Mh ai; _ My a5T h Mh Irriy My Lb V dv - ^ |^Mh <■»„ dv dm h Mh Ly My l Mk Ln y My L mh Wk dm y L Mh Ck Mk ^ 772 h 246 THERMODYNAMICS 02C) = V dv dm k dv Mh Imy My l Group 114 (Con.) dV mh dm h Mk lm y My l mk 5m y L_ Mh ^mk Mk 1, m h t|r is constant. (dW) = 0 (dQ) = 0 Group 109 8, m b , v, t, C constant. (dp) = dV dm k dv Irriy Mh — l mfa My dV dm h _ Irtly Mk — l mk My dm y L ^mk Mh ^mb Mk \ dV (3my) = dp L Mh ^mk Mk Lh dv + d dv dv dv Mh — Mk dd [_ dm k dm h _ + dv dm k (Mh + Ch) 3 (Mk ■+■ ^IMk) dm h (dm h ) = dv dp Mk Inly - My l dV m k + d dv dd Mk dv dm v My dv dm k _ + dv L dm y (Mk + ^mk) ^ ZT ^y + ^Wy) dm k (dm k ) = - ^ (dn) = V dv dp _ dv L dm y dv Mh lm y — My l mh d dv dd dv Mh dm. My dV dm h _ dv (Mh + ^mh) dm (My + lmy) lm y Imh dm k L M “ T “ My T — V dv dm h lmy Lk M k y-My T dv dm y L ^mk Lh Mh T _ Mk T TABLE I—FIRST DERIVATIVES 247 02C) = v dv dm k _ dv 5m y L (3<6) = v dv dm k dv 0m y L lm y Mh ^mh My Mh l mh M k Lh My Imy Mh Ck Mh l mh Mk Group 109 (Con.) dv — v dm h _ ^m y Mk — l wik My — V 0V dm h Lk My ~ l m y Mk + (dW) (dQ) = 0 = v dV dm k _ dv mh dm y L Mh ^m y My ^ Mh l mk - Mk l mh — v dv dm h Mk ^m y My ^ mk Group 110 6, m b , v, e, 2C constant, p is constant. Same as Group 93. Group 111 6, m b , v, e, tj; constant, n is constant. Same as Group 108. Group 112 6, m b , v, n, ( constant. (dp) (<3m y ) dv dm k L 0 l rriy Mh ‘'mh T My dv dm h v m y 1 Mk dv dm v dv dd l mk 0 Mh Wh Mh dv dm k e Mk Mk dv + dv dm h J dp [_ 0 l Mh mk dv dv Imh dm h 0 dm k 0 v mk e My 'mh 248 THERMODYNAMICS Group 112 (Con.) (3m h ) = g dv Mk dm. My dv +- dm k J dp _ Mk l my 0 My t'mk e dv l m y dv l Wk _dm k 6 dm y 6 _ (dm k ) = - dv dv dv dv lm y ^mh dd _ Mh dm y dm h _ dp L Mh y My T_ + dv l m y dv l Wlh (de) = v _dm h 6 dm y 6 _ dv dm k dV Imh 1 Imh Imy y/v-yw. — V dv l wk v m y <3m h lT My “ ~e Mk . ^m y L_ X is constant, dv rah 7 Mk ■mk 7 Mh (dil) = v Lh lm y dm k LT " “7 Mh . — v dv dm h ^rak 0 My lm y ~6 Mk dv dm y L ‘'mh 7 Mk I'mk 7 Mh (dW) = 0 (dQ) = 0 Group 113 6, m b , v, n, x constant. < is constant. Same as Group 112. Group 11 4 6, nib, v, n, tjr constant, e is constant. Same as Group 108 . Group 115 6 , m b , v, x constant, n is constant. Same as Group 112. TABLE I—FIRST DERIVATIVES 249 Group 116 d, mb, v, C, it constant. p is constant. Same as Group 95. Group 117 6 , m b , v, it constant, dv (dp) = 3m k dv dm lm y Mh ~ l rah My ^rak Mh — l rah Mk dv dm h lm.y Mk Lk My (dm y ) = (dm h ) = (dm k ) = (de) = v y L ndV Mh 9^-v Mk dd ^ dv _ L dm k dv dm v — Mk My dv dv dm h _ dp _ dv Mh Lk Mk l mh dm k . +- ap Mk ^ra y My £ mk a dV v -v dd dv Mh dv dm k _ dv dm y My l mh Mh ^TOy — My dV dm h dV dv dm h dp r" My Lk Mk ^Wly Mh ^m y My l- mh dm y L Mk ~ Mh I m k (dn) = v dv Lih ^m y am k L My T ~ Mh T — V dv dm h ^mk ^m v My T “ Mk T dV dm (30 = - v y L_ dv ^mh ^mk Mky-Mhy dm k My ^Ih Mh l m y + V dv dm h . My Lk Mk l. m y + dv dm y L Mk ^wh Mh 1-. mk (dW) = 0 (dQ) = v dv dm k _ dv dm. My ^mh Mh ^m y Mk Mh Lk — V dV dm h _ My ^mk “ Mk £ ray 250 THERMODYNAMICS Group 118 d, m b , e, n, C constant dv (dp) = p V (dm y ) = v dm k dv dm v L y Mh Irnh My Lnk Mh l mh Mk - V dv am h Ln y Mk 4ik My Mh Lt Mk £ mh . 0v av + pv dv dv _ a y 7 tmh T * _am k + Mh — Mk am k am h _ av + v T- dp am h ^ Mh l mk - Mk l mh / dv \ VTp + V Mk Ln y My L«k + p v av av L am y ^ am k Wy p 6 dv d9 dv Mk My dv am y am k . (am k ) = v~ + v dp Mh Ln y My ^ mh — pv dv 1 dv tn L c»m y amt — p Q dv dd Mh av am v My (av) = V av am k av ^mh My Ly Mh av am h — V av am h Lk My Ln y Mk am y L_ Lh Mk — l mk Mh 2 c is constant, ijr is constant. av (dW) = pv pv am k dv am y L ^m y Mh huh My ^mk Mh — Inih Mk p V av amv, Irriy Mk — l rnk My (dQ) = 0 TABLE I—FIRST DERIVATIVES 251 6, m b , e, n, £ constant. < and tjr are constant. d, m b , c, n, i constant. Indeterminate. 6, m b , e, Cj X constant, n and tlr are constant. 6, m b , £, tjr constant, n and £ are constant. m b , t, x, t|r constant, n and C are constant. 6, m b , n, £, x constant. Indeterminate. m b , n, t|r constant. £ and x are constant. 6, m b , n, x, 4 constant. £ and < are constant. 6, m b , (, ife constant. £ and n are constant. Group 119 Same as Group 118. Group 120 Group 121 Same as Group 118. Group 122 Same as Group 118. Group 123 Same as Group 118. Group 124- Group 125 Same as Group 118. Group 126 Same as Group 118. Group 127 Same as Group 118. 252 THERMODYNAMICS Group 128 p, m b , v, e, n constant, dv (M) = dm k dv am y dv Lh Py Ph l m y dv dm h _ ^mk Py Iniy Pk Imh Pk ^TOk Ph (^®y) (Pk ^TOh Ph Imk) (lUl "j - * ' ’ H - Uln) £p dv L" k to h Ph dv dm k _ dv (dlllh) \ a (Py Imk Pk ^ra y ) (HI] ~f“ * * * “J~ m n J Cp O tf dv L My aiYk dv Pk dm yJ dv (dlll k ) qq (.Py Imh Ph ^m y ) "1“ (l^l “f" + m n ) c % dv _ My dm h Ph 00 = " dv am y . dv dm k dv Ph Iniy Lh Py — n dv dm h _ lm y Pk Imk Py n 6m y L Lk Ph Imh Pk K is constant. = n dv dm k n dv dm y (dW) = 0 (dQ) = 0 Ph lm y l-mh Py Imk Ph ~ l mh Pk — n dv 6m h ^Ttty Pk ~ l mk Py TABLE I—FIRST DERIVATIVES 253 p, mb,v, e, ( constant. dv dm y l?7ly Mh Mh Imh My dv [l 7 Wy Mk tmk My ^mh Mk \ dv , 7 (dniy) — (/i.h l? dv U — Mk ^mh) (mi + • • • + m n ) c p dv Mh t— dm k Mk as;] + n (dlllh) (Mk dv W - My U (Mk + Lk) — ^ mk + W L k ) - (mi + • • • + m n ) C. dv Mk - . dm y My dm k + n (M> + Ly) — (Mk + L k ) <9v 6v (dm k ) = ~ a (Mh Imy My ^mh) H - (nil ~1“ * * * ~b m n ) C p ^h U U _ C/Hly .dm^ (>ly + Ly) ~dm y ( ' IJ ' h + lmh) _ lm y dv My r)m h , a \ dv (an) = n toT t dv n dm y X is constant. t|r is constant. (dW) = 0 <«»-•£ dv n ^mh L y T My _ T inh _ _ Lk T ’ Mk-y Mh Mh dv fL k — T My “T Mk n dm h ” dm y Imh My ^mh Mk l?n y Mh Lk Mh dv n dm h ^mk My ^m y Mk Group 130 p, m b , v, t, x constant. Indeterminate. 254 THERMODYNAMICS Group 181 p, m b , v, t, ijr constant. < and x are constant. Same as Group 129. Group 132 p, m b , v, n, ( constant, dv (M) = dm k dv Ln y Mh Lh My dV dm h LyMk LkMy dm y L_ ^ntk Mh — l mh Mk x dV (am y ) = gg Mh - Mk l mh (dmh) = Mk dv dd My dv dm h _ — n dv + (mi + • • • + m„) c p l - vm.h „ v r , dv Mh T- _ dm k ‘mi “wk dm k dmh Mk L» y My ^ dv mk + (nil + • • • + m n ) c p Mk dv dm v dm k _ n dv 7 _ dv , tmk 7 U (am k ) = - g My dv Mh Ln y My ^ dv mb — (mi + • • • + m n ) Cj Mh dv dm v (de) = ti dm h _ dv + n i_dm y i -Ji.i l mh ~ V/ tl (djc) = n dm k _ dv dm y dv dm k dv n dm y L Lz y Mh Ch My Lk Mh Lh Mk Ly Mh Lh My Lk Mh Lh Mk dirn/ my . dv n dm h _ Ly Mk “ l mk My — tl dv dm h Ly Mk ~ l mk My tjr is constant. (dW) = 0 (dQ) = 0 TABLE I—FIRST DERIVATIVES 255 Group 133 p, m b , v, n, x constant, c is constant. Same as Group 128. Group 134 p, m b , v, n, i|r constant. ( is constant. Same as Group 132. Group 135 p, m b , v, £ constant. £ and ^ are constant. Same as Group 129. Group 136 p , m b , v, i|r constant. Indeterminate. Group 137 p, m b , v, 2C> tfe constant. £ and ( are constant. Same as Group 129. Group 138 p, m b , e, n, ( constant. dv (69) = V V dm k _ dv ^mh My lm y Mh V dv dm h _ ^wk My lm y Mk dm ■y L Ivih Mk — l mk Mh \ dv Mk ^mh Mh ^ mk Mh (dm h ) = P dv dm k _ dv p (mi + • • • + m n ) c p dv \ + " ^mk yMh P dm / ^ mh Mk dv dv dm h V dd My Lk Mk 1 Mk dv dim - p (mi H-+ m n ) c v + n l my ^Mk — v ^My — P dm k /_ dv My dm k dv V dm. 256 THERMODYNAMICS Group 138 (Con.) (dm k ) = dv ~ p Te My Lh Mh l- my + p (mi + • • • + m n ) Cj dv L My dEL dv — Mh dm. — n lm,[nh—p “• ) —L„(p y — p (dv) = n dv dm k dv Imh My lm y Mh n <3m h dv dm h _ Cik My ^niy Mk n 5m, Lh Mk — l mk Mh 02 C) = P n pn 04) = Pn pn (dW) = p n pn (dQ) = 0 dv dm k _ dV dm y dv dm k dv dm y dv dm k _ dv Ch My ^Wly Mh Imh Mk — Z mk Mh p n dv dm h Lk My ^m y Mk dm •y L l?n y Mh — l mh My LkMh Lh Mk ^m y Mh ^mh My LkMh LhMk — p n dv dm h _ ^m y Mk Lk My — p n dv dm h L y Mk ^mk My Group 139 p, mb, e, n, ^ constant, v is constant. Same as Group 128. Group llf.0 p, m b , t, n, tjr constant. 6 is constant. Same as Group 98. Group 141 p, m b , e, C, x constant, v and tjr are constant. Same as Group 129. TABLE I—FIRST DERIVATIVES 257 Group 142 p, mb, c, 4 constant, v and 2c are constant. Same as Group 129. Group 143 p, m b , e, 2C> tfe constant, v and £ are constant. Same as Group 129. Group 144 p, m b , n, % constant, d is constant. Same as Group 102. Group 145 p, m b , n, tjr constant, v is constant. Same as Group 132. Group 146 p, m b , n, X) constant. dv m = v V Imh lm y dm k IT My " T Mh . - p dv dm h v m k 1 My dv dm y L_ I'mh Imk yM.-y Mh (dm y ) = ( n + p dv SO. Mh y “ Mh y 6 Mk + p (mi + • • • + Cp m ") ~e (dmh) = ( n + P Mh dv dd, dv dm k — Mk My ‘'Wk Q dv dm h _ l Mk % Q + p (mi + • • • + \ Cp m n )~ Mk dv dm v My dv dm k (drn k ) = - ( n + v \ Cp m ») 7 Mh dv dd dv dm. Lh ^m y My T ~ Mh T p (m l + • • • + My dv dm h _ 258 THERMODYNAMICS Group I 46 (Con.) (dv) = n dV lm y h e^ie fth ~T My . n dv dm h Imy Imk y Mk - y My dv n (dt) = dm pn y dv ^771k Lh T Mh ~ T Mk Iflth Lily dm k U My - T M \ — pn dv dm h pn 00 pn (dW) = pn dv dm y L ^mh ^mk y Mk ~ y Mh dv dm k dv dm y dv dm k dv l>n y Imh y Mh - y My link link yMh-y Mh .mh h'H y My - y Mh pn dv dm h _ ^TOy yMh ^mk Q My pn dv dm h L 0 l m k TOy My-Mk dm y L Ch ^mk T Mk - y Mh pn (dQ) = 0 Group 147 p, m b , C, x, 4 constant, v and e are constant. Same as Group 129. Group I 48 m b , v, c, n, < constant, dv (d0) = v dm k dv ^m y Mh ^mh My — V dV dm h lm y Mk “ ^mk My dm y L Ck Mh Lh Mk (dp) = n dv dm k _ dv Mh Ly My ^ mh n dv dm h Mk My l mk n dm y l_ Mh ^mk Mk ^ Wh TABLE I—FIRST DERIVATIVES 259 (dm y ) = dV , dV v ae + n apJL Group I 4.8 (Con.) Mh l m k Mk Lh . dv en Te~ v (mi + • ■ • + m n ) Cp dv dv Mk — Mh dm h dm k _ (dm h ) = dv . dv V— + d6 dp Mk ^m y My l m k . av * n dd" v (mi + • • • + m n ) c p My dv dv _ L dm k — Mk dm ■yJ (dm k ) = dv . dv v de + "apj Mh lm y My ^ mh + (djc) = v (mi + • • • + m n ) Cp dv My dv - dv dv _ L dm h Mh dm yJ v n V n dm k L r ii ' my dv dm y L Mh l m y My Lh Mh l mk Mk Cth — V n dv dm h Mk Irriy My l m k (dtjr) = V n Vn dv dm k dv My Imh - Mh 1 m y — vn dv dm h Lk My ^m y Mk dm y L. Mk ^mh Mh ^ mk (dW) (dQ) = 0 = 0 Group 149 m b , v, e, n, 2c constant, p is constant. Same as Group 128 . Group 150 mb, v, e, n, tjr constant. 6 is constant. Same as Group 108. 260 THERMODYNAMICS Group 151 m b , v, t, C, 2C constant. p and t|r are constant. Same as Group 129. Group 152 m b , v, e, tjr constant. p and 2C are constant. Same as Group 129. Group 153 m b , v, e, ifc constant. p and < are constant. Same as Group 129 Group 154 m b , v, n, x constant. 9 is constant. Same as Group 112. Group 155 m b , v, n, t{f constant. p is constant. Same as Group 132. Group 156 m b> v, n, constant. dv (69) = v dm k dv dm y L_ (dp) = n dv dm k _ Ly Ch Mh T" - My T Imk ^mh Mh 9 ~ Mk 7. ^771 h lm y My T" " Mh ¥ — v dv dm h ^m y My v m k d + n dv dm h Ly L»k TABLE I—FIRST DERIVATIVES 261 Group 156 (Con.) (dm y ) = dV S V (mh V — n — Uh —-n k — do dp] L e e . v (nil “I - • • • -f* sCp. dv m n ) -f + n — ' 0 dd dv _ Mh dm k Pk dv dm h _ + nv dv l mk _dm h 6 dv l mh dm k 8 _ (dm h ) = dv Yq n dv dp_ lm y lmh Mk T~^T v (mi + • • • + \ c p . dv m ^J + n de Pk dV <9m v Py dV dm k _ + n V dV l my _dm k 6 dv l m k dm v Q ■y v _J (dm k ) = - dv V dd n dv dp_ vm y Ph Y ~ Vy L mh 6 + v (mi + • • • + v Cp . dv m ^J + n ye Ph dv dm v Py dv dm h _ n V dv l my _<9m h 6 dv l mh <9m v 6 ■y w _i (dt) = V n dv d m k lm y lmh — V n dv dm h Ly lmh Mk T ~ My T vn dv (9m ■y L. v mh Mhy- lmh Pk T (dQ = vn dv Py t'mh dm k L 8 Ph Vm y d — vn dv lmh lm y dm h L 1Uy T ~ Mk T. Vn dv (9m y L I'm h Pk —Q - Ph "m k 6 (dW) = 0 (dQ) = 0 262 THERMODYNAMICS Grouv 157 m b> v, C, x, ifc constant. p and e are constant. Same as Group 129. mb, e, n, £, x constant. 6 and t|r are constant. mb, £, n, £, t}r constant. 6 and jc are constant. mb, £, n, 2C> ^ constant. 6 and £ are constant. nib, e, 2C> ^ constant. Indeterminate. m b, n, £, 2C, ^ constant. 9 and e are constant. Group 158 Same as Group 118. Group 159 Same as Group 118. Group 160 Same as Group 118. Group 161 Group 162 Same as Group 118. Second Derivatives f Group 1 * -+^> p c n P O u a a rH a •s > |<3> fD |co CO CO a > i ^ CO | co a a ~ m a o £ o w S TABLE II—SECOND DERIVATIVES 267 denotes all the component masses; mi all except mt. 268 THERMODYNAMICS «Si > I <£> <■© I CO * o Sx += a a m £ O o 6 a a 3 {£ + n I a I JS * So a © > I c > I e «® a t© > i s <© <© <© C <©> <© M a a <© I<© > a > <© !<© <© <© 1 ^ l<© <© <© <© 1 a ~i a fC scT B N i S. © a <© <© a denotes all the component masses; m; all except mk. 270 THERMODYNAMICS TABLE II-SECOND DERIVATIVES 271 denotes all the component masses; mi all except mk. Group 5* (Con.) 272 THERMODYNAMICS oi a o *- + > ft 1 I ns 1 1_ ns _1 ns a ft I a «t> > I L —I a 1 « * + a s£ ft 1_ +3 fl c3 -*-=> 72 £ O o W a os L 0) I > I ^ ft) I ft) ft) I a I I CO | co 15 <"0 + cS a O, co iT + a * denotes all the component masses j nij all except mhj m; all except m lc . 276 THERMODYNAMICS * s o ft Cb -+* a & -t-= m O o B <3s a i c 6 > ICO CO I ^ co ft «T > CO CO ft <35 B «o -C a CO + a > CO fO ft A 3. a a 1 s > CO a a co a a ccT CO a 1 1 a CO 0 a a > CO <■0 N I <31 “I a <31 a <■0 co a w CO -a a CO I N a <31 a <0 a CO I ^ I co >1 s a a 31 CO A a CO N I <30 r CO SX, CO A 3 . CO > co A 3 . a a <31 ^3 a CO a a <30* a co N I <31 I- 1 M 3 . CO CO denotes all the component masses; mj all except m*,; mi all except mk. j, x constant. 278 THERMODYNAMICS t * Si, S o i- B a. fX a S .M a. L Si j, i|r constant. TABLE II—SECOND DERIVATIVES 279 denotes all the component masses; mj all except mi,; mi all except mt. Group 15* (Con.) 280 THERMODYNAMICS 8 a 6 qT > I a to I to 54 . a + a a & a oT Ob’ “1 C3 M js a =t B ft to to + > to a ft <3b a to * to to l a oT > I a to I to to I I to Ito - 4 ^ a co (3 o o > a ♦n a > I to to I to to L to to > I A os' > I a to Ito to L l to |to TABLE II—SECOND DERIVATIVES 281 » 6 6 £ a a at a a I-1 a a < 5 >“ £ a ~1 a" a QS a denotes all the component masses; mj all except mb! m; all except m^. i, e constant. THERMODYNAMICS 282 * B co i 3 . a a <3S 1 1 c3 a __ a cO> *a CO a 1 CO 1 > I ^ I a a “I a L cJ a a a Sh J3 + JS . 3 . a co i_ £ CO 1 ^ CO \co ft. a a <3i 4) 6 a 1 M I a. co I <35 CO * 00 ft. a A A < 0 > > I ft, CO I J> I ft, <0 I r© L denotes all the component masses; mj all except mhj mi all except mk. Group 18 * (Con.) 284 THERMODYNAMICS dd \dm J 9t p , nuj p, mj , z \dm k /e, P . mj Pi ma Ldm h W*. p, mi TABLE II—SECOND DERIVATIVES 285 -» I ^ C I 3. a a I — oT — 1 £ J3 a. a © a 05 a l 1 <3* I - 1 -a a A c <© S 1 1 1 1 + + a L a J + N II N a a a a 1 ^ s <©> 1 M r-O =L 1 fO J L J L * O s © a a o a a a I Sh ro | <© oa a a n a a ■35 L m a a L <© I <© a * denotes all the component masses; mj all except mhj mi all except nik. Group 20* (Con.) 286 THERMODYNAMICS a. a CO + A s ft «r A a co * o ft > CO ft, «r A a «r> a JS a. > CO & ft a a A a. S £ ■a 1 1 ft / /— 5 a -< R <•£> a I i + ■X s os II X a ft 1 i s CO 1_ _1 a + A =t a ft «r A a I •M I S has x 6 ft a. L «£> CO -A> a o3 m fl O o *5^ • ► V > 0) ► 43 B co £< CO CO CO a <*T a. B 43 a CO a> > -*C3 a 03 -t-3 m G O o w fO ft 0) ► > CO CO a a ft) CO a £ > CO CO a. «T a g. CO 43 a CO > CO CO V > £ G< co a a, «T 44 a CO 43 a CO _1 ft) CO 0) > a sj a fo G, a o a co 44 a CO a 4) > J L CO G, CO TABLE II—SECOND DERIVATIVES 289 u ► denotes all the component masses; mj all except m h ; mi all except m k . Group 23* 290 THERMODYNAMICS -+j d d m a O o C •v > • • a « > a a a <*> <■0 i_ + d > a i d > a > i ^ ro | | a ro | ro <0 I I £■< \ro I d > I a «r> -d a ro + d > ro + ct> ro a ro ro a ro ro d ► ro I I —I a ro I TABLE II—SECOND DERIVATIVES 291 denotes all the component masses; mj all except mhj mi all except mk. t& dnoif) 292 THERMODYNAMICS * N > £ a a I_ .£) B «t> i + + a - 1-5 d 4-3 co a o o > *N • ^ a i_i N > £ 6 pC «> ft ft ft N ► f n cS a oi* > | ^ ft \ £ I ^ I ft N ► £ a ?c N > a a 1A I + N > co «t> a ft a. a «t> a «t + AJ I § h^> + a si N > £ —i ft i TABLE II—SECOND DERIVATIVES 293 denotes all the component masses; mj all except m h ; mj all except mt. 294 THERMODYNAMICS je B co X > a a H > * *•0 A 3 o 5 -. CO <0 > CO CO oS a a “1 a a «T B CO I <=£> \co + a a CO x > a a CO a <35 n a a <*r ©. J X cS > l H a a, J3 co co a + X > CO CO CO > CO CO a oT a I |r© > CO CO a «r a CO -C a CO CO CO -M 0 0 -t-a co 0 O o •s > a > | I co > CO CO x > a i a <35 - ( a co A ‘"O x > i a o a ^ leg- X > CO a CO TABLE II—SECOND DEKIVATIVES 295 S si denotes all the component masses; mj all except mt; mi all except mk. 296 THERMODYNAMICS >> > ►» > <© <© a <© <© a a . Oj s> l 3 1 s -' a 33 H <© <© a <© «© * © S-. > <© <© a «T a <© i ^ l<© + -a a «© ►» > •» t «© a <© R. © a <© a <© i <© i> ef <© l«© > «© <© a pi «r -M a <© & a <© «© -m a o3 - 1-3 CO fl O O -&* > a p. > i 5 © «© i«© >> > <© ►> > a © a ^Icg* > <© a «© TABLE II—SECOND DERIVATIVES 297 >> > a = 1 . l —I a > | ro J3 denotes all the component masses; mj all except mhj mi all except mk. 298 THERMODYNAMICS d 0) d « d 4> J3 B <0> d a> CO * a S3 © s- C5 CO CO CO + d of rfl 6 <•© B & <© a s£ <& B CO CO CO CO A B CO -+ft a -+^> CO P o o e •N u d o* d «© <© TABLE II—SECOND DERIVATIVES 299 denotes all the component masses; mj all except m h ; mi all except mk. Group 28* 300 THERMODYNAMICS N oT £ a a <*r cd 1_ + N «r CO CD Si, | Q=> N 4 ? a I Pi, _ 1 03 03 £ XI a. Pi, N oT a T3 a c 3 6 * denotes all the component masses; mj all except m h ; mi all except m k . Group 29 * 302 THERMODYNAMICS x «' X oT X © J3 B co X iT P, CO + X ©' a p co co rta a CO CO P. «T AS a Ph X ©* I n > <■0 p. QS AS a <■0 a CO > CO CO P, x « I Si. ft. «r J3 a + X oT > a ft —I f4. _I CD ft fO ft. <*> xi a X oT 73 d o 3 ft a XS a fD x a fO a denotes all the component masses; mj all except mh; mj all except mt. Group 30 * 304 THERMODYNAMICS ji a CO © s a, <■0 . ©‘ a a + ©* a > | ^ . « a n a «T © I a fC -a a CO I I \<0 t>> ©‘ £ a p. + >» ©~ CO I \co CO a CO J + a“ pi, > I <0 \ CO CO I_ cc> II © a > «T >> of =1 L A <"0 P, a I | ro > | fO | r© + <"0 xt a <■£> >. « a a denotes all the component masses; mj all except m h ; mi all except mk. 306 THERMODYNAMICS N a «t ft) ft) ft) ft ft * <30 3 o a o 3 -t-s CO 0 O o •s C ft ^ I ^ ft | co «t N a f “i a ?£ ccT a ft A a ft > «t ft a I ft a a “I a a ft N a* £ N -x' W' M C 1 l«t I - 1 <© A a X 0 * Pi, <© a a " ft 0 a <© A a <© <©> * <30 Ph © ft <3 -1-3 PJ c 3 -1-3 02 fl O o ><: sf *^» a i<© x a <© l«> I'"© > <©> <©> ft «r A a <©> A <© X 0 I A a <© > <©> <©> X o~ Pi. <© p. ftT a a <3S A a <© a <©> <©> X o' a a. 0 a <© pa, <© ft <0 a <©> 34 a <©> 1 <■© X o* <©> I_ a <© TABLE II—SECOND DERIVATIVES 309 c £ X <0 I —I sa. CO I > 1 ^ + a ft) \co X a. a X a. 6 * denotes all the component masses; mj all except mhj mi all except mk. 310 THERMODYNAMICS a 6 ft >, a S 54 , ft> ft A «t a a «t >> a I Ph ft a A ft ft A «T A a c 54 , ft A «T «t) A O a ft a «t> i 8 <£> 4 ^ a c 3 -+ 4 > m 0 O o > ft ft> A oT A a «t> «t> > ft a I I a A «T A a ft 54 , ft > ft ft A «cT A a ft .d a ft i e A a ft 54 , ft ft> >» e > a I + >> a 3 . I >> e 6 —I I > I ^ co A denotes all the component masses; mj all except mi,; mi all except m.k. 312 THERMODYNAMICS x N X N pC 6 CO x N I CO X N + X N a «T I- 1 a o B CO «r> * Vf- Go a s o a. CO CO S£> \fO CO cc> I^D JS B CO > CO CO X N A a CO X N I ~\ B a «r a CO CO CO x N a a «r> Pi, > N -C a co N S 5 , CO a a a CO CO a CO -d a >> N * 1 *-0 <30 Is 1 ^ a CO 1 co S 3 o S» +3 a & -+d> « o o •5* <"0 >> N CO >> N CO > CO CO >. N I "1 a a a <^o a CO JS a > CO CO >> N CO a a < 3 S a a «T a CO ■d a CO CO + N i-1 a o '~os a co | SX S is >. N CO CO f TABLE II—SECOND DERIVATIVES 315 denotes all the component masses; mj all except mhj mi all except mk. Group 86 316 THERMODYNAMICS s» X* >> X A B <0 x" £ <0 * ■+3 a c 3 ~ 4 —> m a o o hi >> x" <^D Q=> 1<"D >. x* ft. n > CO > x' ^3 a <0 >> M Si, ft «tT > <^> B ft «T ft i a >> x' ■ ft a Si, <0' ft a a fD g h^> <"0 >> x’ £ -14 J? CO Si, cr> TABLE II—SECOND DERIVATIVES 317 * i i ^ > H I- A 3 . a —i a 1 o O) a a > a I a 3 . «iT o3 a A £ G > 1 ^ > “1 a pi «r A a 0 Ad dYH2 / 0, p 7Yl\ or lim m 2 —> 0 m2 (am,),, = AS mi (2) (2') ( 3 ) (30 where 0 < A < A being a constant for each substance or component. 1 If m 2 can have both positive and negative values (see section 21), then A = O. 319 320 APPENDIX Now since m 2 = 1 — mi, we have Therefore mi = m 2 is continuous at and in the neighborhood of mi = 1 and hence we can choose a small positive quantity 8 such that when m 2 is less than 8, we have Integrating (5) at constant 0 and p, we have A0 LF Mi —-b K; mi and integrating (5') at constant 0 and p, we have (5) (50 ( 6 ) mi = - A0— + k: m, Now when m 2 = 0, then K = mi + A0 = £i 4-A0 and K' = pi = ft. Thus K' = K — A 0, and (6) and (60 become n (0, V) - Mi (0, V, mi) = A 0 —-— = A 0 — (7) mi mi and f i P) - Mi (0, P, mi) = A 0 ^ (70 For the dilute component Integrating (8) at constant temperature and pressure, we have A ni B m 2 M 2 = A 0 log- (9) mi . Bm 2 1 then Ah —» 0, and n 2 (0, p, Ah) becomes negatively infinite. Now transforming from gram masses to mole fractions we have d/X2 \ / m 2l ij±r I = lVlAr H^v , p,mi \ui\ 2 /0, p Thus in the limit as Ah —> 0, Ah —> 1 and hence lim dju 2 _ ,, d/L6 2 _ R0 m 2 or Ah —> 0 m2 dm 2 2 dAh M 2 We therefore find that we can identify the dilute solution laws with the definition of an ideal solution. We see that as Ah departs appreciably from unity we have the inequality m 2 d/x 2 dm 2 ^ Ah djX2 dN 2 but the dilute solution laws are only applicable for systems in which N 2 is very small, that is, Ah does not depart appreciably from unity. Hence if we wished we could set down the equation N 2 d/X2 \ dAh/0, p R e m 2 ’ 0 ^ N 2 ^ 5 as a general equation where <5 is some positive value equal to or less than one, this value depending on the particular system chosen. If the solution is an ideal solution then 8 = 1, if not then 8 is less than one and in general (dilute solutions) we know that it is usually very small. APPENDIX 323 FUGACITY AND ACTIVITY The fugacity is defined as a function of temperature and pressure in a simple system by the equation . Mf(»,p) -MUe.Po)* f(.9,p)=f(e,p.)e r# or M f (0, p) - M f (0, Vo) = R 0 log f(o, v) 'f(0, Vo) where e denotes the base of the Naperian logarithms, R = 83.147 X 10 6 ergs per degree per mole, M denotes the gram formula or molar weight of the substance, f (0, p) and £ (0, p 0 ) the zeta per unit mass at the states (0 , p) and (0> Vo) respectively, and / (0, v) an d / (0, Vo) the fugacities at the states (0, p) and (0, p Q ) respectively. To complete the definition of fugacity we must assign a value to/(9, Po). , Now assuming as a physical hypothesis that we are dealing only with systems such that, if we first choose a small positive quantity K to represent the maximum allowable error of our result we can pick a value 8 > 0 so that the characteristic equation (equation of state) of the system for values of p less than 8 and greater than zero v M "R0~ - V < will differ from p = by less than the value K, i.e. r R0 J ’ K where 0 < p < 8. Then where 0 < p < 8 we have m(^) = M» = — \dp/ e p Integrating this expression at constant temperature we have p m r (a, p) - m r (g, pq) — = e Ro f(d,p) = o 0 = K' Lewis completes the definition of fugacity by defining / (0, p 0 ) as p 0 where 0 < p < 5, f (0, p 0 ) = p 0 , 0 < p < 5. If / (0, p 0 ) were chosen as zero when p = 0 then f would become negatively infinite, giving us an indeterminate form for / (0, p) which therefore could not be evaluated. Now when the system is such that M p v = R 0 then f (0, p) = p, and hence a tabulation of fugacities for gases will give the deviation of these gases from the perfect gas law. For a liquid the fugacity of the liquid would be the fugacity of the vapor in equilibrium with it and thus, if this vapor obeyed the perfect gas law, the fugacity, / (0, p), of the liquid would be equal to its vapor pressure at the temperature 0. The fugacity of, let us say, component one in a binary solution of unit mass is defined as Mi m (9, p, mi) — Mi m (9, po, m )o ) fi (0, V, m) = fi (0, Po, m u ) e-o- To complete the definition we must assign a value to/i (0, p 0 , mj. The activity, a, is defined as a function of 0 and p for a simple substance by the equation m r (9, y) - m r (a, po a (0, p) = e r o or f (0, V) ~ f (0, Vi) = jjj 0 log a (0, p). To complete the definition of activity we must assign a value to Pi at each temperature. APPENDIX 325 The activity of, say, component one of a homogeneous binary solution of unit mass is defined as a functipn of 6, p, and mi by the equation Mi /ii (0, p, mi) — Mi pi (9, p i, mi 0 ) d\ ( 6 , p, mi) = e r 9 To complete the definition we must assign a value to p\ and mi 0 at each temperature. The standard states have been defined as follows. (1). For a gas the activity is made equal to the fugacity, a (0, p) — f V )• I n other words pi is so chosen that f ( 6 , pi) = 1 or M f ( 9 , p 0 ) M ( 9 , pi) R d P r 9 — y_ / (0,p.) (2). For a liquid or solid which may act as a solvent the activity is made equal to unity when the concentration of the substance is unity. In other words pi = p when mi 0 = 1, or m r (9, y) - m r ( 9 , pi) a (d, p) = e R0 — e° = 1 Thus a ( 6 , p, mi) = e Mi jui {9, P, mi) — Mi fi ( 9 , p) R 0 Now from the dilute solution laws or ideal solution definition we have Ni =e Mi mi (A, P, mi) — Mi fi (0, p) R 0 Hence for the series of states, i.e. in the regions, in which a homo¬ geneous binary system satisfies the ideal solution definition, ai (0, p, mi) = JVi where Ni > N 2 (3). For a solute. We perceive that in the region or series of states in which the ideal solution definition is satisfied the activity of the solute will be proportional to the mole fraction of the solute. 326 APPENDIX Furthermore, the limiting value of the activity, as m 2 approaches zero, will be zero, n a * ( e ’ V, ™i) = 0. m 2 —> 0 The standard state is so chosen that in the region in which the ideal solution definition is satisfied a 2 ( 0 , p, mi) = N 2 Thus if the solution acted as an ideal solution over the whole range of concentrations we would then have INDEX Activity, 325-6. Adiabatic changes of state, 121. Anisotropy, 127. Binary systems: experimental deter¬ mination of, 156; unit mass, 44, 45-7, 49, 53; variable mass, 43, 48, 52. Body forces, 139. Bulk modulus, anisotropic body, 132; isotropic body, 126. Cells, reversible, 154, 159-61. Characteristic equation, see equa¬ tion of state. Chi: definition, 4%, 70; differential, 73; diffl. for stressed systems, 117; second derivatives of, 73. Coefficients of elasticity, see elastic constants. Component: actual, 36; of strain, 106; of traction, 101, 105; posi¬ tive and negative values, 36-8; possible, 36. Compressibility, 126, 148, 149. Compression: cubical, 125; modulus of, see bulk modulus. Concepts, x, 1. Crystal systems, 129. Deformation, see strain. Dilatation: I48, 149; ellipsoid of, 84; linear at a point, 84. Dilute solution laws, 319; relation to ideal solutions, 322. Dissociation, 38. Elastic constants, 122, 129; moduli, see moduli. E. M. F. measurements, 159. Energy: see first law; kinetic, 134, 144. Enthalpy, see chi. Entropy, see second law. Equation of state: binary system, 44; n-component, 57; one-com¬ ponent, 14. Equilibrium: of rigid body, 97; state, 13; stress equations of, 99, 103. Extension (strain), 78. Extensive property, 13. Ferric chloride hexahydrate, 36. First law for binary system (UNIT MASS): derivatives, 49; inte¬ gral, 49; transformation of, 50; (VARIABLE MASS), 48. for irreversible process, 138. for n-component system (UNIT MASS): derivatives, 62, 64; dif¬ ferential, 68; integral, 61; trans¬ formation of, 63; (VARIABLE MASS): differential, 69; inte¬ gral, 60; transformation of, 62. for one-component system (UNIT MASS): derivatives, 33; differ¬ ential, 28; geometric interpre¬ tation of integral, 22; incom¬ plete statements, 28; independ¬ ence of path, conditions for, 23; integral, 22. (VARIABLE MASS): integral, 40. for strained systems: adiabatic changes, 121; derivatives, 111; differential, 116; integral, 111; isothermal changes, 120; trans¬ formation of integral, 112. zero energy, definition, 143. Force: body, 97, 101, 103; concept of, 8, 11; interior and exterior, rigid body, 97. Fugacity, 323. Fundamental equation: binary sys¬ tem, 159; definition of, 155. Gas constant, 321. Gibbs equations: 12, 69, 116; 93, 53, 65; 97, 74; 215, 320. Gravitational: field, 139; forces, 96. Green’s theorem, 104. Heat concept of, 137. for binary system (UNIT MASS): derivatives, 45, 4?; in¬ tegral. 45; transformation of, t 46j (VARIABLE MASS), 41 for irreversible process, 136. for n-comp. system: integral, 58; transformation of, 59. 327 328 INDEX Heat —Continued for one-comp, system (UNIT MASS): derivatives, 20 ; differ¬ ential, IS, 35; integral, 16; transformation of, 17. (VARI¬ ABLE MASS): 39. for strained system: derivatives, 109; integral, 108; transforma¬ tion of, 110 . Heat capacity at constant pressure: 20 , 45; de¬ rivatives of, 33, 34, 50, 53, 66; determination of, 152. at constant volume: 21, 47, 51; derivatives of, 34, 51, 67. at constant strain: 110, 113; de¬ rivatives of, 113, 115. at constant stress: 109; deriva¬ tives of, 111 . differences of, 35. Heterogeneous system, definition, 13. Homogeneous: strain, 85; system, 13. Homogeneous vs. heterogeneous, 139. Hooke’s law, 120 . Ideal solution laws, 321. Intensive property, 13. Irreversible process, 134- Isothermal changes of state, 120 . Kinetic energy, 134, 144- Latent heat of change of concentration, con¬ stant temp, and press.: 45, 50, 52, 55; derivatives of, 53, 66, 153; determination of, 154, 157. of change of concn., constant temp, and vol.: 47, 51; deriva¬ tives of, 51, 55, 67. of change of mass, see change of concn. of change of pressure: 20, 35, 45; derivatives of, 34, 35, 50, 53, 66. of change of strain: 111, 113; de¬ rivatives of, 113, 115. of change of stress: 109; deriva¬ tives of, 112 . of change of volume: 21, 35, 46, 51; derivatives of, 34, 35, 51, 55, 67. Length concept, 1. Line integral: 15, 18; conditions for independence of path, 23; geo¬ metric interpretation of, 22 Mass: concept, 6 ; fraction, 45, 57; variable to unit mass, 151. Moduli of elasticity: anisotropic body, 132; isotropic body, 125, 128. Modulus: of compression, see bulk modidus; of rigidity, see rigid¬ ity. Mole fraction, 321. Moments of forces, 99. Mu: binary system, 43, 56, 153; de¬ rivatives of, 50, 54, 55, 66, 68; determination of, 153, 157, 159; determination of in dilute solu¬ tions, 320; in ideal solutions, 321; one-comp, system, Non-equilibrium process, see irre¬ versible process. Non-homogeneous, relation to ho¬ mogeneous, 139, 143. Notation: of tables, 149; see nomen¬ clature, xv. Phase, 39. Poisson’s ratio: anisotropic body, 133; isotropic body, 127. Potassium chloride-water: 156; dis¬ sociation, 38. Potential, chemical, see mu. Pressure: 96; differential equation, 75. Property of system, 12. Psi: definition, 4%, 70; differential, 74; stressed systems, 118. Reversible process, 16, 143. Rigidity, modulus of: anisotropic body, 132; isotropic body, 127. Second derivatives of table II, 161. Second law at zero degrees absolute, 31. for binary system (UNIT MASS): derivatives, 53; in¬ tegral, 35; (VARIABLE MASS): 52. for irreversible process, 138. for n-comp. system (UNIT MASS): derivatives, 66 ; in¬ tegral. 65; transformation of, 67. (VARIABLE MASS): in¬ tegral, 64 ; transformation of, 66 . for one-comp, system (UNIT MASS): derivatives, 34; differ¬ ential. 32; integral, 31; (VARI¬ ABLE MASS): 41. for strained system: derivatives, 114; integral, 113; transforma¬ tion of, 114 ■ INDEX 329 Second law —Continued zero entropy, definition, 143. Solution laws, 319. Specific volume, 14, 4$, 57. State, 13. Strain: change of angle by strain, 82; conditions of compatibility, 92; definition of, 76; homoge¬ neous strain, 85, 87, 107; physi¬ cal interpretation of, 81, 83; pure, 87; small deformation, 91; strain components, 81, 106; tangent at a point, 90; trans¬ formations, 78. Strain—energy functions: adiabatic changes, 121; isothermal changes, 121; transformation of, 130. Stress: concept, 95; components, 124; specification of at a point, 101 . Stress-strain relations: isotropic body, 123; static vs. dynamic methods, 122. Temperature: absolute thermody¬ namic, 33; concept of, 9, 10. Tension, 96. Time, 4- Traction: 96, 101, 105; components of, 105. Volume: concept, 4‘, see specific vol. Work for binary system (UNIT MASS): derivatives, 45, 47; in¬ tegral, 44 • (VARIABLE MASS): 44. for irreversible process, 136. for n-comp. system: integral, 58. for one-comp, system (UNIT MASS): derivatives, 20; dif¬ ferential, 20 ; integral, 15; (VA¬ RIABLE MASS): 39. for strained systems: adiabatic process, 121; derivatives, 109; integral, 107; isothermal proc¬ ess, 120; transformation of, 110. Young’s modulus, 127, 133. Zeta: definition, 7$, 70; differential, 72; second derivatives, 72; stressed systems, 116, 117. Tire IlDPAtlY OF lLc lltAttUi Ul >1*- 00T i 7 1330 ■ r-'Z/.S'IY OF - ■ -