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Cornell University Library 
 TJ 265.P98 
 
 Thermodynamics of reversible cycles in g 
 
 3 1924 004 657 304 
 
8. la Cornell University 
 
 WJ Librar y 
 
 The original of this book is in 
 the Cornell University Library. 
 
 There are no known copyright restrictions in 
 the United States on the use of the text. 
 
 http://www.archive.org/details/cu31924004657304 
 

DEPARTMENT OF 
 
 HEAT-POWER 
 
 ENGINEERING 
 
 5 i^^tryJ tO+^UAAsfc 
 
 m-n4* 
 
THERMODYNAMICS 
 
 EEVEESIBLE CYCLES IN GASES 
 AND SATUEATED VAPOES. 
 
 FULL SYNOPSIS OF A TEN WEEKS' UNDERGRADUATE 
 COURSE OF LECTURES DELIVERED BY 
 
 M. I. PUPIN, Ph.D. 
 
 ARRANGED AND EDITED BY 
 
 MAX OSTERBERG, 
 
 Student in Electrical Engineering, Columbia College. 
 
 NEW YOEK: 
 
 JOHN WILEY & SONS. 
 
 London : CHAPMAN & HALL, Limited. 
 
 1898. 
 
Copyright, 1894, 
 
 BY 
 
 Max Osterberg. 
 
 ROBERT DRUMMOND, ELECTROTYPER AND PRINTER, NEW YORK. 
 
PREFACE. 
 
 My dear Mk. Osterberg: 
 
 It was quite a pleasure to examine your carefully worked 
 out notes on my lectures. I agree with you in the opinion that 
 the publication of these notes will be of much service to your 
 classmates and probably to others who may be interested in 
 the elementary features of thermodynamics. 
 
 Very sincerely yours, 
 
 M. I. Pupin, 
 
 Columbia College, New York, 
 December 20, 1893. 
 
 ill 
 
INTRODUCTION. 
 
 In this course on Theoretical Thermodynamics we shall 
 limit our discussion to those features of the science which 
 have a direct bearing upon the science of Caloric En- 
 gineering. The course forms, therefore, a theoretical in- 
 troduction to the practical course on Heat Engines. It 
 seems desirable, however, to mould our discussion in such a 
 way that it will serve at the same time the very important 
 purpose of forming an introduction to the study of some of 
 the best and most complete works on the subject. The 
 work of E. Olausius {Die mechanische Warmetheorie) is and 
 very probably will always remain the classical treatise on this 
 very important branch of exact sciences. We shall, therefore, 
 adopt the mathematical notation and follow as closely as prac- 
 ticable the method of discussion which is given in this great 
 work of Olausius, who, as you will presently see, is oneof the 
 principal founders of the beautiful science of thermody- 
 namics. 
 
 v 
 
THERMODYNAMICS 
 
 OP 
 
 REVERSIBLE CYCLES IN GASES AND 
 SATURATED VAPORS. 
 
 NATURE OF HEAT AND THE MATHEMATICAL STATE- 
 MENT OF THE FIRST LAW OF THERMODYNAMICS. 
 
 The Science of Thermodynamics investigates the formal 
 or quantitative laws which underlie all physical processes by 
 which the caloric state of material bodies is changed. 
 
 By physical process we simply mean any one of the almost 
 infinite variety of ways by which nature transforms one form 
 of energy into another, or one form of matter into another. 
 Heat is developed in almost every physical process, owing 
 principally to the presence of passive or frictional resistances, 
 unavoidable in material systems. 
 
 The first question to be considered is : Wliat is heat ? It 
 certainly must be one of the two fundamental physical 
 quantities, that is, it must be either a form of matter or a 
 
2 THERMODYNAMICS OF 
 
 form of energy. In the first case it will be subject to the 
 principle of conservation of matter , and in the second case it 
 will be subject to the principle of conservation of energy. 
 
 According to the old hypothesis heat is a substance ; but 
 since the weight of a body is independent of its temperature, 
 the heat substance, the so-called Caloric, had to lie supposed to 
 he imponderable. 
 
 The experiments of Sir Humphry Davy (1799*) and of 
 Rumford (1798 f) showed, however, that, 1st, the latent heat 
 contained in a given body could be increased, and, 2d, the 
 temperature of a body could be changed without a change in 
 the quantity of heat in the surrounding bodies. 
 
 The first was shown by Davy's experiment of melting two 
 pieces of ice by rubbing them one against the other ; the sec- 
 ond was shown by Bumford's experiment, in which heat was 
 developed by friction, which accompanies the boring of 
 metals. These experiments gave the death-blow to the old or 
 substance theory of heat. 
 
 They showed not only that heat cannot be a substance, but 
 that on the contrary it has all the physical characteristics of 
 the other fundamental physical quantity, which we call energy. 
 As such it must be subject to the princirjle of conservation of 
 energy. Joule ivas the first to shotv that heat and mechanical 
 work are not only convertible into each other, but that the 
 
 * Essay on Heat, Light, and the Combinations of Light, with a new 
 Theory of Respiration. Experiments II. and III. 
 
 f Count Rumford 's essay on " An Inquiry concerning the Source of 
 Heat which is excited by Friction," read before the Royal Society, 
 January 25th, 1798. See Memoir of Sir Benjamin Thompson, Count 
 Rumford, with notices of his Daughter, by George Edward Ellis. 
 
REVERSIBLE CYCLES IN OASES AND VAPORS. 3 
 
 ratio of conversion is constant (1839-1844). Later on he ex- 
 tended this relation to other forms of energy, like chemical 
 energy, etc. By these experiments Joule not only proved 
 that heat is a form of energy convertible into mechanical or 
 any other form of energy at a constant ratio, but he also fur- 
 nished the first exact experimental basis for the general prin- 
 ciple of conservation of energy, first enunciated by Julius 
 Robert Mayer about the same time (1842). It was, however, 
 clearly stated in its most general form and mathematically 
 elucidated by Helmholtz in 1847. 
 
 The principle of Conservation of Energy can be stated 
 as follows : Energy cannot be created nor can it be annihi- 
 lated by any physical processes which our mind can con- 
 ceive. The energy contained in an isolated material system 
 must, therefore, remain constant, although all sorts of 
 changes may be taking place within the system. These 
 changes, however, will consist in a transformation of one 
 form of energy into another or one form of matter into 
 another without loss. 
 
 When the energy of a body or system of bodies is changed 
 in amount, then the change is due to the transference of 
 energy from external bodies to the system considered, or vice 
 versa. In the first case, work is done upon the system; in the 
 second, the system does work upon external bodies. 
 
 If we heat a body, one part of the heat energy put into it 
 will appear as heat, another part as work done against ex- 
 ternal forces, like surface pressure, and another part as 
 potential energy overcoming the internal forces, like forces of 
 cohesion, chemical affinity, etc. The common unit of meas- 
 ure of all these forms of energy is the unit of mechanical 
 
4 THERMODYNAMICS OF 
 
 work, that is, the kilogramme-meter, The unit of heat is one 
 Mgr. calorie. 
 
 Def. — One Mgr. calorie is the heat necessary to raise the 
 temperature of a Mgr. of a standard body one degree. The 
 standard body being pure distilled water at 4° C, and at nor- 
 mal pressure (760 mm.). 
 
 The mechanical equivalent of heat is the number of Mgr. 
 meters which must be spent to generate that much heat. 
 
 It was found by experiment that one klg. calorie = 424 
 klg. m. (very nearly). The most distinguished experimental- 
 ists in this line being Robert Mayer, Joule, Hirn,* Rowland. 
 
 To express, say, 890 klgr. meters in calories, divide 890 by 
 one /("Joule") or 424 {J being the symbol of the mechani- 
 cal equivalent of heat). 
 
 Statement of the First Law of Theemodynamics foe 
 a Particular Class of Physical Processes. 
 
 We will now limit ourselves to that class of bodies and 
 processes in which the external work done is mechanical work, 
 and in that class of processes we shall select those processes 
 where the mechanical work is done in overcoming a certain 
 amount of pressure on the surface of a body to which we 
 apply the heat, and consider that the pressure is uniform and 
 normal at all points of the surface. 
 
 *Hirn, Recherches sur l'equivalent mecanique de la chaleur, p. 20; 
 Also, Theorie mecanique de la chaleur, 2d edition, part 1st, p. 85. 
 
REVERSIBLE CYCLES IN CASES AND VAPORS. 5 
 
 Expression for External Work due to Infinitesimal Expansion, 
 
 Consider an infinitesimal area of the surface du square 
 meters ; if the pressure per sq. m. is p kg., the total pressure 
 is pdu. Now suppose the body to expand, overcoming the 
 surface pressure: the work done by the act of displacing du 
 is equal to the normal displacement dn meters times the 
 pressure, that is pdudn kg. meters. 
 
 To find the total work done over the whole surface, we 
 must integrate (remembering that p is uniform all over the 
 surface). 
 
 / pdudn = p I dudn — pdv. 
 
 That is to say, for infinitely small increments of volume, dur- 
 ing which the surface pressure may be considered constant, 
 the external work is equal to the surface pressure multiplied 
 by the increment in volume. 
 
 Intrinsic Energy and Co-ordinates of a Body. 
 
 Let us now communicate an infinitely small quantity of 
 heat dQ to a body. Experience tells us that the following 
 changes in the body may be looked for : change in tempera- 
 ture, interna] aggregation, chemical constitution, electrical 
 and magnetic state, volume, etc., etc. If we now call the 
 energy which a body contains in consequence of its tempera- 
 ture, internal aggregation, chemical constitution, electrical and 
 magnetic state, etc., etc., the intrinsic energy of the body, and 
 denote it symbolically by the letter U, and if we denote by W 
 
6 THERMODYNAMICS OF 
 
 the external work which a body does against the surface 
 pressure in expanding from a given volume to some other 
 volume, then it is evident that the above changes in tempera- 
 ture, aggregation, chemical constitution, electrical and mag- 
 netic state, volume, etc., etc., imply a certain small change 
 dU in the intrinsic energy and a certain small external work 
 dW. According to the Principle of Conservation of Energy 
 the mechanical equivalent of dQ is equal to the mechanical 
 equivalent of dU plus the mechanical equivalent of dW. 
 Hence 
 
 dQ = dU+dW. 
 
 Tliis is the most general form of the first lata of thermody- 
 namics. It is a formal statement of the Principle of Conser- 
 vation of Energy for processes which involve a transformation 
 of a certain amount of heat. 
 
 Experimental Physics and Experimental Chemistry teach 
 us how to measure the physical quantities by means of which 
 we describe the various states of physical bodies, these quan- 
 tities being temperature, forces of cohesion, chemical affinity, 
 quantities of various chemical elements contained in a body, 
 magnetic and electrical intensity in the various parts of the 
 body, volume, pressure, etc., etc. The energy contained in v. 
 body depends on its state and not on the processes by means 
 of which the body has been brought into that state. Hence 
 the physical quantities just mentioned which describe the 
 state of the body will also determine completely the value of 
 the intrinsic energy TJ which the body has in that state. Tliis 
 is what we mean when we say that U is an analytical function 
 of the above physical quantities. 
 
REVERSIBLE CYCLES IN OASES AND VAPORS. 7 
 
 The external work which a body does against surface press- 
 ure while expanding from a given volume to some other 
 volume may or may not be independent of the process by 
 means of which this expansion is effected. To be general, we 
 shall suppose that it is not independent, that is to say, we shall 
 suppose that W is such a function of the physical quantities 
 which define the state of the body, that in passing from one 
 set of values of these quantities to another set the change in 
 the value of PFwill depend on the physical processes by means 
 of which we passed from the first to the second set. 
 
 This will be illustrated further on by actual examples. The 
 physical quantities, like temperature, magnetic and electrical 
 intensity, pressure, volume, chemical affinity, quantity of mat- 
 ter, etc., which define the state of a body are called the co- 
 ordinates of a body. "We say then that 6 T and T^are functions 
 of the co-ordinates of the body. 
 
 It is evident that the more complex the state of a body the 
 more co-ordinates shall we need to describe its state. For 
 the present we shall limit ourselves to those physical bodies 
 and physical processes in which temperature, volume, and 
 pressure suffice to describe completely the state of the body, 
 and the processes by means of which that state is varied. 
 
 U will therefore be a function of the pressure p, the 
 volume v, and the temperature t ; the same is true of IT'. 
 
 We shall measure p in kilogrammes per square meter, v 
 in cubic meters, and t in degrees centigrade. 
 
 It must be observed, however, that these three co-ordi- 
 nates of the body are not independent of each other. Thus 
 the surface pressure and the temperature of a body being 
 given, its volume is also determined. Varying the pressure 
 
8 THERMODYNAMICS OF 
 
 ■without varying the temperature, we can pass from any volume 
 to any other volume (within certain definite limits). 
 
 We can therefore express TJ in terms of two of these. In 
 what follows TJ will be considered a function of t and v unless 
 the contrary is stated. 
 
 We shall therefore have in the case of these bodies and 
 these physical processes 
 
 dQ=^Ldt+^dv + dW. 
 at ov. 
 
 Since the pressure is normal and uniform and constant for 
 infinitely small variations of volume and temperature, 
 
 dW — pdv; 
 
 Hence the first law of thermodynamics in the case of bodies 
 and processes, which we are considering, says : If we com- 
 municate an infinitely small amount of heat to a body 
 under pressure, a part of the heat is used up iu increasing 
 the internal energy of the body, part of which is potential 
 and part heat energy, and the rest appears as mechanical work 
 done against the surface pressure. 
 
 If a finite quantity of heat Q is communicated to a body, 
 causing the same to pass from a given state, in which its tem- 
 perature is t 1 , surface pressure p t , and total volume v t , to an- 
 other state, in which these quantities are t t , p a , v^ , then 
 
 Q=U,-U l +f*pdv. 
 
REVERSIBLE CYCLES IN GASES AJfi) VAPORS. 9 
 
 That is, the total quantity of heat is equal to the total in- 
 crement of internal energy plus the total external work. The 
 increment in internal energy is due to two changes: 1st, 
 change of temperature; 2d, change in the arrangement of 
 matter in the body. The first change produces a variation 
 in the thermometric heat; the second, in the latent heat of the 
 body. 
 
 If pressure is constant, then external work = p(v i — v t ). 
 But since this is not always the case, we must, in order to 
 find the numerical value of the integral fpdv, consider first 
 the law of variation of p with v. 
 
 For infinitely small variations of volume the pressure can 
 be considered constant. 
 
 II. 
 
 APPLICATION OP THE FIRST LAW OP THERMO- 
 DYNAMICS TO PERFECT GASES. 
 
 We pass now to the application of the First Law of Ther- 
 modynamics to the study of the simplest thermodynamic 
 processes that can be performed upon the simplest class of 
 physical bodies, that is, we pass to the application of the First 
 Liw of Thermodynamics to the study of the process of ex- 
 pansion and compression of perfect gases. 
 
 These bodies may justly be considered the simplest bodies, 
 because experimental research tells lis that their co-ordinates 
 describe the state of these bodies by the simplest formal rela- 
 tions that we know of, namely, 
 
 Eelatiou (1) contains what is known as Boyle's law, viz. : 
 
 p 1 v 1 =p 2 v, (1) 
 
 That is to say, if a given quantity of a perfect gas, say a kilo- 
 
10 THERMODYNAMICS OF 
 
 gramme of dry air at a given temperature, occupies a volume 
 v t when it is under a pressure p x , then the same quantity of 
 the gas at the same temperature when it is under pressure p, 
 will occupy the volume v a as given in above equation. 
 
 This relation is true for a large number of gases within a 
 large interval of pressure and temperature. 
 
 Eelation (2) contains what is known as the Mariotte-Gav- 
 Lussac law, viz. : 
 
 pv=p a v,(l + af) (2) 
 
 Here p is pressure in kg. per square meter and v is volume 
 in cubic meters of a unit weight (that is a kilogramme) of 
 a perfect gas when it is under the pressure p and when its 
 temperature is t deg. centigrade above the standard freezing- 
 point. p and i\ denote the corresponding quantities of 
 the same weight of the gas at the freezing-point, a is 
 called the temperature coefficient of expansion. It is found 
 to be very nearly constant for all temperatures within a large 
 interval, and to be = ^-fj (very nearly) for all perfect gases. 
 
 The Mariotte-Gay-Lussac law can also be written 
 
 pv = |g°(273 +t) = ^T= RT. 
 
 Here T is called the absolute temperature of the gas and Ji 
 is a constant which depends on the nature of the gas. It can 
 be calculated for every gas from experimental data, as will be 
 shown presently. 
 
 Choosing T and v as the co-ordinates of the gas, we have 
 
 dQ = %-„dT-\- |w?e + pdv. 
 
REVERSIBLE CYCLES IN OASES AND VAPORS. 11 
 
 On the Eates of Variation of the Intrinsic Energy 
 of a Perfect Gas. 
 
 When a small quantity of heat dQ is communicated to a 
 body, then a part of it will appear as an increment in the in- 
 trinsic energy of the body, and the other part will appear as 
 external work. 
 
 r)U 
 The first part consists of two terms, namely, r™ dT and 
 
 r\ TT r)TT 
 
 ■—dv. "Now— y^dT is the increment of intrinsic energy due 
 
 OV O-i 
 
 to the increase of the body's temperature by dT. The factor 
 
 r) TT 
 
 — 7=- is the rate at which internal energy varies with the tem- 
 
 O-l 
 
 perature, the volume being constant. If that rate were con- 
 
 r\ TT 
 
 stant, then the factor — = would mean the increment of 
 
 O-L 
 
 internal energy due to a rise of temperature of 1°. This 
 
 p,TT 
 
 part —j^dT appears as sensible heat of the body, i.e., heat 
 
 which can be measured by a thermometer. The other part 
 
 7)U 
 
 - — dv is the increment of intrinsic energy due to the change 
 
 of the body's volume, the temperature remaining constant. 
 
 7) TJ 
 
 The factor is the rate at which this increment takes 
 
 dv 
 
 place; if this rate remained constant for finite variations 
 
 r) TT 
 
 of volume, then would mean the increment of intrinsic 
 
 dv 
 
 energy for a unit increment in volume, the temperature re- 
 maining constant. 
 
12 THERMODYNAMICS OF 
 
 We have every evidence in favor of the hypothesis that 
 
 heat is a mode of motion of the molecules of a body. Its 
 
 temperature is due to the kinetic energy of this motion; 
 
 7)17 
 hence ^-^dT denotes the increment of the kinetic part of the 
 a J- 
 
 intrinsic energy. The other part of the increment of the internal 
 
 energy, namely, -=—dv, will be work done against the forces 
 ov 
 
 between these molecules. These forces will depend on the 
 
 distances of the molecules from each other, and therefore on 
 
 the volume of the body. The work done against these forces, 
 
 (when the temperature, and consequently the kinetic energy of 
 
 the body, remains constant), will therefore appear as potential 
 
 energy. Whenever there are no internal forces between the 
 
 ?)T7 
 molecules, the term -^—dv will be zero. That is, the intrin- 
 
 dv 
 
 sic energy of the body will be independent of the volume. 
 
 If such a body expands without doing work against external 
 pressure, its temperature will remain constant, if no heat is 
 communicated from without. For, from the equation 
 
 dQ = ^dT+pdv, 
 
 we see that when dQ = 0, there being no heat communicated 
 from without, and pdv — 0, there being no external work 
 done, then 
 
 ^-dT = 0. That is, dT - 0, since ~ cannot = 0. 
 3i 9 1 
 
REVERSIBLE CYCLES IN OASES AND VAPORS. 13 
 
 Vice versd, if when the body expands without doing external 
 work, and without receiving heat from without, its tempera- 
 ture remains constant, then 
 
 r^r-dv = 0. Hence — — = 0. 
 [dv dv 
 
 That is to say, the intrinsic energy of such a body will be in- 
 dependent of its volume. 
 
 Gay-Lussac's experiments disclosd such a behavior on the 
 part of perfect gases. These experiments were verified later 
 by Joule and Kegnault. We conclude, therefore, that for 
 perfect gases the first law of thermodynamics will have the 
 following form : 
 
 dQ = ^dT + pdv. 
 
 A perfect gas may be described as a body which, when expand- 
 ing without doing external work, will neither cool nor heat. 
 
 The Various Mathematical Forms of Statement of 
 the First Law for Perfect Gases. 
 
 r) IT 
 
 The rate 7-= has a definite name in thermodynamics. Sup- 
 pose that the quantity of gas under consideration is one unit 
 
 p. TT 
 
 weight, i.e., one kilogramme; --^=- means then the quantity of 
 
 heat, measured in mechanical units, which we would have to 
 communicate to one unit weight of the gas at constant volume, 
 
14 THERMODYNAMICS OF 
 
 in order to raise its temperature by one degree, if the rate at 
 which heat is supplied during a rise of temperature were 
 constant. This quantity of heat is called the specific heat of 
 the gas at constant volume, and is denoted by C„. Now 
 
 D/7T p. 7T 
 
 since p — and — = = C v> we can write down the first 
 
 v O-L 
 
 law of thermodynamics for perfect gases as follows : 
 
 T>rp 
 
 dQ = C v dT+—dv (1) 
 
 From equation (1) we can eliminate v or T by means of 
 the relation pv = RT, and obtain 
 
 dQ = (C v + B)dT-—dp .... (2) 
 
 and 
 
 dQ = ^vdp+^±^pdv (3) 
 
 Equations (1), (2), (3) represent three ways of stating mathe- 
 matically the first law of thermodynamics in the case of a 
 perfect gas. They differ from each other only in the selection 
 of the co-ordinates. 
 
 These expressions contain two physical constants of the gas, 
 viz., C v = specific heat at constant volume, and E. Both 
 these constants must, of course, be determined experimentally. 
 The quantity C v being difficult to determine experimentally, 
 it is preferred to put these formulse into a different form, 
 containing O p — specific heat at constant pressure. 
 
REVERSIBLE CYCLES IN OASES AND VAPORS. 15 
 
 To bring out the meaning of this physical constant, and 
 show its relation to C v , suppose that we have a unit weight 
 of air enclosed in a cylinder by means of a piston, which can 
 glide up and down the cylinder without friction. Its tem- 
 perature, volume, and pressure being T, v, and p, we must 
 have pv = RT. Clamp the piston, and apply heat to the gas 
 until its temperature rises by dT. Let dQ l be the mechani- 
 cal measure of this heat communicated to the gas during this 
 operation; then 
 
 dQ 1 = d ^dT=C v dT. 
 
 Now loosen the clamp, the volume will increase, since the 
 temperature is now T -\- dT. During this second operation 
 communicate heat to the expanding gas so as to keep its tem- 
 perature constant. 
 
 The volume will expand from v to v-\-^-^dT, overcoming 
 
 pressure^?. Observe that — 7=, represents the rate of variation of 
 
 volume due to variation of temperature, the pressure being 
 constant. Hence from pv = RT we obtain 
 
 dT~ p- 
 
 Let dQ, be the heat communicated during this second 
 operation; then 
 
 dQ,=p^dT=RdT. 
 *Q t + *G, = (C + X)dT. 
 
16 THERMODYNAMICS OF 
 
 At the end of these two operations we have a change in tem- 
 perature — dT, a change in volume = ^TpdT, and external 
 
 work equal to p^^dT. 
 
 Consider now another process, by means of which the same 
 change in temperature, the same change in volume, and the 
 same external work is produced. Let us call the rate at 
 which heat must be applied to the unit weight of the gas, 
 when it expands at constant pressure, C p . Apply heat until 
 the same change in volume is produced as before; the final 
 temperature will be T -\- dT, as before. The pressure in the 
 second case being the same as in the first, we shall have the 
 same external work clone during the expansion, and the in- 
 crement in internal energy will also be the same, since the 
 change of temperature is the same; hence the quantity of 
 heat communicated will also be the same. Call it dQ ; then 
 
 dQ = C p dT. 
 
 But since dQ = dQ t + dQ* , we must also have 
 C P =C V + E or C v = C p - R. 
 Substituting in equations (1), (2), (3), we obtain 
 
 TIT 
 
 dQ = (C p -R)dT+^dv>, . . . (i») 
 
 TIT 
 dQ=C p dT-^dp; ...... (2«) 
 
 dQ = ^—vdp+^pdv (3") 
 
REVERSIBLE CYCLES IN OASES AND VAPORS. 17 
 
 The Eatio of Specific Heats. 
 
 
 We will now introduce a new constant, namely, —f=k. 
 
 This physical constant of a gas was first introduced by Reg- 
 nault, and is of vital importance, for the reason that k can be 
 determined experimentally with great accuracy by determin- 
 ing the velocity of sound through the gas. 
 
 Regnault found k for air = 1.41. 
 
 C p we can also determine by experiment; hence we can cal- 
 culate R. 
 
 Vice versd, if R and C^are determined experimentally, then 
 both C„ and k can be calculated. 
 
 The relation P = R — O v , as it stands, applies to values of 
 these physical constants measured in mechanical units. Divid- 
 ing by J — the mechanical equivalent of heat — we obtain 
 
 C p R O v 
 
 ~J~ J ~J' 
 
 or 
 
 „ R 
 Op — j c v> 
 
 where c p and c v are measured in kg. calories. 
 
 Calculation of the Mechanical Equivalent of Heat. 
 
 An interesting relation is obtained from the last equation. 
 
 Remembering, namely, that -=f = k, we obtain /= T - r- . 
 
 U (« — l)'-p 
 
18 THERMODYNAMICS OF 
 
 This is the relation first pointed out by Robert Mayer, and 
 actually employed by him to calculate J. 
 
 Before we can apply it to any gas, we must find the value 
 of B. Let us find the same for air. 
 
 U ~ 273 * 
 
 Regnault found that at the barometric pressure of 760 mm. 
 the value for v t of one unit weight of air is .7733 cubic 
 meters. But since our units are kilogrammes and meters, we 
 have to change the above value of p accordingly. 
 
 The specific weight of mercury used by Regnault is 13.596; 
 we get therefore 
 
 p — lOO" 3 *" X 7.6*" x 13.596 = 10333 kg. per square meter. 
 
 Hence, since v a = .7733 cubic meter, 
 
 i2 = 10333_X^733 = 29 ^ 
 
 The values of h and c p for air were determined by Reg- 
 nault. They are k = 1.41, c p = .2375. Hence we obtain 
 
 T 29.27 X 1.41 .„.. 
 
 .41 X .2375 = kg.-meters. 
 
 The value of J obtained by direct experiments is very nearly 
 424. So much for the physical constants of a perfect gas. 
 
REVERSIBLE CYCLES IN GASES AND VAPORS. 19 
 
 A table of c v and c p for various gases will be given pres- 
 ently. 
 
 To sum up: The mathematical statement of the first law 
 of thermodynamics in the case of perfect gases expresses a 
 quantitative relation between the increments in volume, 
 pressure, and temperature of a gas when a small quantity of 
 heat d Q is added to or taken away from it. This relation 
 contains the physical constants R, G p , C v , k, which must be 
 determined by experiment. Experiment tells us that C p , just 
 like R, is independent of pressure and temperature. Hence 
 G v and Tc are also independent of these. This makes the 
 thermodynamic study of the behavior of perfect gases com- 
 paratively easy. 
 
 It is well now to consider how we can apply the first law of 
 thermodynamics to the solution of a few simple problems. 
 
 Problems. 
 
 We shall consider two problems in particular : 
 
 (1) We shall determine the amount of heat absorbed by a 
 gas when it expands under constant pressure. 
 
 (2) We shall determine the amount of heat absorbed by 
 a gas when it expands at constant temperature. 
 
 To solve these equations we must observe that a proper 
 selection among the three mathematical statements of the 
 first law of thermodynamics will afford us considerable mathe- 
 matical advantage. 
 
 For the first problem we select one of the formulae, con- 
 taining v and p as independent variables, and since p in this 
 
20 THERMODYNAMICS OF 
 
 case is to be constant, the term containing dp becomes zero. 
 Hence 
 
 dQ = — £-5 — vdp -\- -£pdv 
 
 becomes 
 
 dQ = HjLpdv. 
 
 Integrating from v = v t to v = v, , we get 
 
 § = fdQ = %pf v ° % av = & P {v t - »,). 
 
 Numerical Example. — Consider a kilogramme of air at 
 barometric pressure of 760 mm. ; that is, let the initial press- 
 ure be 10333 kilog. per square meter, let the initial volume 
 be .7733 cubic meter, and let i> 2 = 2v r 
 
 Since c p =.2375 kg. calories, and C p = .2375 X 424, 
 therefore the amount of heat absorbed by the gas while 
 expanding under constant pressure from v l to 2v 1 will be 
 
 Q = 10333 X. 8375 X 484 x ^ = ^ ^ ^ 
 
 27490 , 
 = - 1 g r kg.cal. 
 
 In the second problem, where we wish to determine the amount 
 of heat that must be supplied to the gas in order to expand 
 it under constant temperature, we take the formula (l a ) 
 
 dQ=(C p -B)dT+ — dv, 
 
REVERSIBLE CYCLES IN GASES AND VAPORS. 21 
 because there <1T being zero, when T is constant, we obtain 
 
 dQ=^-dv, and hence Q = RT J *— = RT log \ 
 
 which is the total amount of heat measured in kg. meters 
 which is absorbed by the gas while expanding isothermally 
 from volume v x to volume v v To determine the total amount 
 of work done, we put 
 
 W= / pdv = RT — = RTlog -\ 
 
 This result, compared with the preceding one, gives an 
 identity. It tells tts that the entire amount of heat put into 
 the gas was utilized to do external work ; for since the temper- 
 ature remains constant, none of the heat supplied goes to in- 
 crease the intrinsic energy of the gas. 
 
 Isothermal and Adiahatic Curves. 
 
 If we make the supposition that during the expansion of 
 the gas the temperature remains constant, we have from Ma- 
 riotte-Gay-Lussac's law pv = (J. 
 
 Represent this result graphically, taking p for ordinates and 
 v for abscissae. The resulting curve is an equilateral hyper- 
 bola refer-red to its asymptotes. Such a curve is called an iso- 
 thermal curve. (Fig. 1.) 
 
 There is evidently an isothermal curve for every tempera- 
 ture; hence an infinite number of them, but every one of 
 them is an equilateral hyperbola in the case of a perfect gas. 
 
22 THERMODYNAMICS OF 
 
 Tlie isothermal curves of a perfect gas form a system of 
 curves which never intersect. For suppose there is a point 
 where two curves do intersect, this would have to he a point 
 where at different temperatures the pressure is the same for 
 the same volume, — which is absurd. 
 
 Fig. 1. Fig. 2. 
 
 Let us suppose again that we diminish the pressure without 
 introducing additional heat. The piston will go up, hence 
 the temperature will diminish; and as soon as a point is 
 reached where pj\ — RT 1 the expansion will cease. 
 
 If we plot a curve expressing a relation between p and v at 
 any moment during this expansion, we get what is called AN 
 
 ADIABATKJ Or ISENTROPIC CURVE. (Fig. 2.) 
 
 The equation of this curve will be deduced presently. 
 
 Equation of the Adiabatic Curve of a Perfect Gas. 
 Since no heat is communicated during an adiabatic expan- 
 sion, the following relation must exist at any moment between 
 the physical constants of the gas : 
 
 RT 
 
 {C p -R)dT+*-^-dv. 
 
 (R - C p ) log |f = 22 log V f, 
 
REVERSIBLE CYCLES W OASES AND VAPORS. 23 
 
 or 
 
 Let us illustrate this by an example. 
 
 Suppose the initial temperature is 0° 0. above the freezing- 
 point, and a certain initial volume of a kg. of air is com- 
 pressed to £ the volume ; calculate the rise in temperature. 
 
 T x = 273, ^ = 2. 
 
 .-. |L = (2)-* 1 = 1.329, or T, = 363° C. 
 
 Hence t,= T,— 273 = 90° C. above the freezing-point. 
 Compressing the initial volume to £, we find 
 
 t - 209° C. 
 
 Compressing the initial volume to -fa, we find 
 
 t = 429° C. 
 
 We shall noiv calculate the external work done, when the gas 
 expands adiabatically. 
 From pv — ETwe get 
 
 
 £« = =£» -*. But we found ^ = (M*~\ 
 
 ^ ) , or, more generally, p = ^-. 
 
24 THERMODYNAMICS OF 
 
 The last equation is the equation of an adiabatic of a 
 perfect gas. It is evidently of the form 
 
 const. 
 9 = ~7S— 
 
 To calculate the external work done during an adiabatic 
 expansion from volume v 1 to v 2 , we have 
 
 = R .{T — T). 
 
 A family of adiabatic curves never intersect each other, but 
 every adiabatic curve intersects every isothermal curve. 
 
 The axes of p and v are asymptotes to the adiabatics of a 
 perfect gas, but the adiabatic passing through a given point 
 is more, steeply inclined to the p axis than the isothermal 
 passing through the same point. 
 
 Summary. 
 
 It has been shown so far that heat is a form of energy, and 
 that therefore it obeys the Principle of Conservation of Energy. 
 The constant ratio at which it is convertible into mechanical 
 energy is 424; that is to say, if the quantity of heat which must 
 be supplied to the unit mass of the standard substance (one 
 cubic decimeter of pure distilled water at 4° C. and 760 mm. 
 barometric pressure) in order to raise its temperature 1° C. be 
 called a unit of heat, then this unit of heat is equivalent to 
 424 mechanical units of work, the weight of the unit mass at 
 
REVERSIBLE CYCLES IN OASES AND VAPORS. 25 
 
 a definite place, that is, the kilogramme weight, being taken 
 as the unit of force and the meter as the unit of length. 
 Briefly stated, the mechanical equivalent of a kilogramme 
 calorie is 424 kilogramme-meters. After that it was shown 
 that if we add a quantity of heat to a body that heat will 
 appear partly as an increment of the intrinsic energy of the 
 body and partly as external work. For very small quantities 
 of heat this can be stated as follows : 
 
 dQ-dU. + dU^ + dW, 
 
 which is the most general form of the First Law of Thermo- 
 dynamics. In this equation dU 1 is the increment of the in- 
 trinsic energy due to increase of the sensible heat of the body, 
 and d Z7 2 is the increment of the intrinsic energy due to the 
 increment of internal potential energy of the body. The ex- 
 ternal work dWwa,s shown then to he pdv, as long as we limit 
 ourselves to a particular kind of external work, that is, work 
 done by the body in expanding against a uniform, normal 
 surface pressure. 
 
 Limiting ourselves to such physical processes in which the 
 state of the body, which is the seat of these processes, can be 
 described completely at any moment by temperature, vol- 
 ume, and pressure, it was shown that the first law of thermo- 
 dynamics can be stated as follows : 
 
 «e- 8 !* +(£+*)* 
 
 The application of the first law of thermodynamics to the 
 study of such physical processes in various classes of physical 
 
26 THERMODYNAMICS OF 
 
 bodies was then in order. We commenced with the simplest 
 class of physical bodies, that is, with perfect gases. Starting 
 with the Boyle and the Mariotte-Gay-Lussac Laws, and re- 
 membering that the intrinsic energy of a perfect gas is inde- 
 pendent of its volume, we showed that the First Law of Ther- 
 modynamics for perfect gases can be stated as follows : 
 
 dQ= C v dT + pdv, 
 
 when G v is the specific heat of the gas at constant volume, 
 measured in mechanical units. By the relations G p = C v + R 
 and pv = RT we then gave the various forms of statement of 
 the above equation, these various statements differing from 
 each other in the selection of the independent variables p, v, 
 and T, and the physical constants C p , C v , R, and k. An 
 important observation was then made, and-that was, that ac- 
 cording to Regnault's experiments, O p , and therefore C v , R, 
 and k, are the same for all temperatures, volumes, and press- 
 ures at which the gases retain their characteristic properties 
 of a perfect gas. 
 
 The application of these various forms of statement of the 
 First Law of Thermodynamics for perfect gases was then made, 
 and the isothermal and adiabatic changes of such gases dis- 
 cussed. We ended with the discussion of isothermal and adia- 
 batic or isentropic curves of a perfect gas. 
 
 We could, following the same method of discussion, apply 
 now the first law of Thermodynamics to the study of physical 
 processes of above description in the case of any other class of 
 physical bodies. It is evident, however, that since the fun- 
 damental relation between pressure, volume, and temperature 
 
REVERSIBLE CYCLES W GASES AND VAPORS. 27 
 
 in the case of physical bodies in general is far from being as 
 simple as in the case of perfect gases, — it is evident, we say, 
 that our discussion would lead to very complicated equations, 
 involving physical constants which, in very many cases, have 
 not as yet been determined experimentally. We shall there- 
 fore abstain from a general discussion, but pass on to the 
 application of the first law of thermodynamics, to the study 
 of reversible processes in vapors. This class of bodies is ex- 
 tremely important, particularly from an engineering stand- 
 point, because it is the action of heat upon these bodies that 
 is generally employed as a means of transforming heat energy 
 into mechanical energy, and vice versd. 
 
 Before making this next step, it is advisable to deduce 
 another general law, which, like the first law of thermody- 
 namics, underlies all reversible heat processes in nature. It 
 is called the Second Law of Thermodynamics. 
 
 We close now this part of our course with a discussion of 
 the various methods of expressing the specific heats of gases, 
 and the relations which exist between them. 
 
 On the Vaeious Wats of Expressing the Specific 
 Heats of Perfect Gases. 
 
 Consider the relation c p = c v -\ — j . 
 
 This relation says : Given the spec, heat of a gas at con- 
 stant pressure, the spec, heat at constant volume can be 
 calculated, if the constant R of the gas, and the mechanical 
 equivalent of heat, are known. 
 
 We mentioned previously that we could determine the 
 
28 THERMODYNAMICS OF 
 
 ratio of the spec, heats by means of the velocity of sound. 
 This gives another method of calculating c„ when c p is 
 known. But it is very difficult in some gases to determine (k) 
 by velocity measurements. In such cases we must take our 
 recourse to the above formula. 
 R can be calculated from the formula 
 
 K ~ 273' 
 
 It is possible, nowever, that the gas cannot oe well observed 
 at the freezing temperature; hence we must look for another 
 method of calculating R. 
 We can write in every case 
 
 7?-P v 
 
 At the same temperature and pressure we shall have for the 
 same weight of a standard gas, say well-dried air, 
 
 R' '=■££. Hence R = R' 4-. 
 1 v 
 
 V 
 
 The fraction -7- is the reciprocal value of the spec, weight 
 
 of the gas, that is of its density, the density of air being taken 
 is equal to unity. 
 
 Calling this spec, weight, that is, the density, d, we have 
 
 d 
 
REVERSIBLE CYCLES IN GASES AND VAPORS. 29 
 By substituting this value of R iu the first equation, we get 
 
 R' 1 
 
 R' in our case is 29.27. 
 Example. Calculate c v for air. 
 
 d=l, hence c„ = .2375 - IMi = .1684. 
 
 424 
 
 Z - * - J68T - L4L 
 
 That is to say, to raise the temperature one degree, a kilog. 
 of air requires only about -§- of the heat that a kilog. of stand- 
 ard water does. 
 
 The spec, heats denoted by c p and c v refer to a unit weight 
 of the gas, and their unit is that quantity of heat which is 
 necessary to raise the temperature of a unit mass of standard 
 water under standard conditions one degree; that is, in other 
 words, the gas is compared calorically by weight, to water. 
 While, as a matter of fact, it is sometimes more convenient 
 to compare in this respect gases with air, by considering 
 equal volumes, i.e., to determine the spec, heat in such a 
 manner, that the amount of heat necessary to raise the unit 
 volume of a gas one degree, is compared with the quantity 
 of heat necessary to raise a unit volume of air through one 
 degree (at the same temperature and pressure). 
 
 We can employ that method of comparison for both specific 
 
30 THERMODYNAMICS OF 
 
 heats, considering in one case that the gas and the air are 
 heated at constant pressure, and in the other that the gas and 
 air are heated at constant volume. 
 
 Let the volume of a unit weight of gas be v. 
 
 The amount of heat which a unit volume must get to raise 
 its temperature one degree at constant pressure must there- 
 fore be 
 
 v 
 
 Let the volume of a unit weight of air = w, , then the 
 amount of heat necessary to raise a unit volume through one 
 degree under the same conditions will be 
 
 -% = / 
 v 
 
 The ratio of these quantities evidently gives us y p ; that is, 
 the quantity of heat necessary to raise the temperature of the 
 unit volume of a gas at constant pressure measured in terms 
 of the heat necessary to raise the temperature of the unit 
 volume of air under the same conditions. Hence 
 
 'p — *,' — -' ■>, ~ „> 
 
 Similarly, 
 
 y c p v c- p 
 
 
REVERSIBLE CYCLES IN OASES AND VAPORS. 31 
 TABLE OF SPECIFIC HEATS OF GASES. 
 
 Air 
 
 Oxygen 
 
 Nitrogen 
 
 Hydrogen 
 
 Chlorine 
 
 Carb. monoxide. . 
 Hydroch. acid gas. 
 
 Carbonic acid 
 
 Steam 
 
 Ammonia 
 
 Alcohol 
 
 Ether 
 
 2 
 
 N a 
 
 H 2 
 
 Clj 
 
 CO 
 
 HC1 
 
 C0 2 
 
 H 2 
 
 NH 3 
 
 CnH 6 
 
 C 4 H 10 O 
 
 Density . 
 
 1. 
 
 1.1056 
 
 0.9713 
 
 0.0692 
 
 2.4502 
 
 0.9673 
 
 1.2596 
 
 1.5201 
 
 0.6219 
 
 0.5894 
 
 1.5890 
 
 2.5573 
 
 Specific Heat at 
 Const. Pressure. 
 
 0.2375 
 
 0.21751 
 
 0.24380 
 
 3.40900 
 
 0.12099 
 
 0.2450 
 
 0.1852 
 
 0.2169 
 
 0.4805 
 
 0.5084 
 
 0.4534 
 
 0.4797 
 
 1. 
 
 1.013 
 
 0.997 
 
 0.993 
 
 1.248 
 
 0.998 
 
 0.982 
 
 1.39 
 
 1.26 
 
 1.26 
 
 3.03 
 
 5.16 
 
 Specific Heat at 
 Const. Volume. 
 
 0.1684 
 
 0.1551 
 
 0.1727 
 
 2.411 
 
 0.0928 
 
 0.1736 
 
 0.1304 
 
 0.172 
 
 370 
 
 0.391 
 
 0.410 
 
 0.453 
 
 1. 
 
 1.018 
 
 0.996 
 
 0.990 
 
 1.350 
 
 0.997 
 
 0.975 
 
 1 55 
 
 1.36 
 
 1.37 
 
 3.87 
 
 6.87 
 
 III. 
 
 SECOND LAW OF THERMODYNAMICS. 
 
 Oaknot's Kbvbrsible Engine and Carnot's Cycle. 
 
 Let ns start with a process of expansion and compression 
 of a perfect gas enclosed in a cylinder which is impermeable 
 to heat. The piston which shuts the gas in, can glide up and 
 down without friction. The piston being under a given pres- 
 sure, the gas will at a given temperature occupy a definite 
 volume. Heat can be communicated to the gas through plug 
 p. Let the length oe on the abscissa denote the original 
 volume v of a gas, and the ordinate ea the initial pressure 
 p . Evidently these two conditions determine the tempera- 
 ture of the gas at that moment. Let the gas expand isother- 
 
32 
 
 THERMODYNAMICS OF 
 
 mally. To accomplish this imagine it connected to a large 
 reservoir A (Fig. 4) of temperature T 1 , the same as that of the 
 gas. By gradually diminishing the pressure and expanding 
 
 H3 
 
 =c 
 
 
 
 Fig. 3. 
 
 Fig. 4. 
 
 very slowly, the temperature will remain constant and equal 
 to T . Plotting the curve expressing the relation between p 
 and v, we get a part of an isothermal curve. Point b indicates 
 the volume and pressure at the end of the isothermal expan- 
 sion. Now cut off the heat reservoir A, and let the gas ex- 
 pand slowly still further (by diminishing the pressure), but 
 of course adiabatically, that is to say, the gas expands and 
 decreases in temperature on account of loss of heat due to 
 external work done. The curve of expansion, be, will be a 
 part of an adiabadic. Call the temperature, when volume v 2 
 has been reached, T r From now on compress the gas, but 
 having first connected the cylinder to a reservoir B of tem- 
 perature T 2 and of so large a capacity that it will take up 
 all the heat from the compressed' gas without changing its 
 temperature perceptibly. The heat generated in the gas by 
 compression will be given off to B, hence the compression 
 will be isothermal. The curve of compression will again be 
 
REVERSIBLE CYCLES IN GASES AND VAPORS. 33 
 
 a part of an isothermal. At this point we impose a. particular 
 condition, namely, that the gas be compressed until a point 
 is reached where the isothermal cuts that adiabatic curve 
 winch goes through the starting point a. 
 
 Arrived at that particular point, we disconnect the reservoir 
 B and continue the compression. From there on the gas is 
 compressed adiabatically, the temperature rises, and when the 
 original temperature T t has been reached the gas will evi- 
 dently have the same volume and will be under the same 
 pressure as at the beginning. These four operations con- 
 stitute what is called a Garnot Cycle. 
 
 Now we shall consider the work done by the gas and upon 
 the gas during the various operations in the cycle. The work 
 done during the first expansion can be represented by the 
 area abfe, that during the second by b cgf. 
 
 During the compressions work has been spent upon the gas 
 equal to the areas cdhg and hdae respectively. The difference 
 between the work done and the work expended is the work 
 gained. This gain must evidently be at the expense of heat 
 given up to the gas by the reservoir A. 
 
 Heat was taken up from A by the gas during the first 
 operation; let us call that Q i; and during the isothermal 
 compression heat was given out from the gas to the reservoir 
 B; let us call that Q t . Since on the whole a certain quantity 
 Q was transformed into work represented by the area abed, it 
 is evident that 
 
 Q, = & + Q- 
 
 Let us illustrate that method by an example. 
 
34 TEESMODTXAinCS OF 
 
 Say that we hare .5 kilcgr. of air, and an initial pressure of 
 10 atmospheres, that is equal to 7600 mm. barometric pressure 
 or 103,330 kilogr. per square meter. 
 
 Let the initial volume be ^ cubic m. 
 
 i\ = .1 cm. 
 
 p i = 103,330 klg. per sq. m. 
 Weight = .5 kg. 
 2/w, = RT^ 
 
 . „ 103330 X .1 X 2 
 
 • ■ T i = on oy = ,06 
 
 Suppose now fhat we work between the reservoirs A and B 
 at 706° and 293° respectively. Hence 
 
 T ± = 706°, y s = 293°. 
 
 We will proceed by allowing the gas to expand isothermally. 
 until the volume is equal to ;•, , and suppose for convenience 
 of calculation 
 
 v 
 
 It=e = 2.72. 
 
 Work done by the gas during this expansion = / pdv= J*'. 
 
 J v x 
 
RBVERSIBLB GTOLBS IN OASES AND VAPORS. 35 
 But in isothermal expansion pv = RT l = constant. 
 
 .-. W, = RT> -- = RT X log ^ = 29.27 X 706 kg. meters. 
 
 Jv, v V l 
 
 Heat taken up from A will be = — '^—- — kg. calories. 
 
 424 ° 
 
 Passing to the second operation we expand the gas from 
 now on adiabatically, until its temperature is 293°. The 
 final volume will be v 3 . 
 
 Let the work done during this operation be denoted by W t , 
 then 
 
 W m 
 
 K — I v& v ' 
 
 p V 
 But during this operation p — ~- 3 . 
 
 = 29,480 kg. meters. 
 
 Compress now isothermally until the point is reached where 
 the isothermal cuts the adiabatic which passes through the 
 starting point p l v 1 and denote this work of compression by 
 W,. Then if v t be the final volume, 
 
 W, = RTJog°f. 
 
36 THERMODYNAMICS OF 
 
 (v \* _1 T (v \ k_1 T . 
 
 But since I - 4 j = -=±, also I -M = -+, it follows that 
 
 ^^ov^=%.;lo g ^=-lo S ^;,.W 3 =-RT,hg\ 
 v l v a v s v, & v 3 & «/ s «\ 
 
 Similarly the work done during the last adiabatic compression 
 will be 
 
 W A 
 
 R 
 
 k 
 
 ri( y . - T >) = - w* 
 
 The total external work represented by the area abed is 
 W= W t + W t + W % + W t 
 
 = Rlog V -y i -T 1 ). 
 
 W 
 Efficiency = -=- 
 
 _ T x - 7, 
 
 This relation expresses the second law of thermodynamics. 
 but only in a very limited form ; that is to say, this relation 
 being simply a mathematical expression for the efficiency of a 
 particular machine, Carnot's engine, operating in a perfectly 
 definite manner, — that is, by simple reversible cycles, upon a 
 particular class of bodies, namely, perfect gases, — this relation 
 should not be considered to be anything more than it really is 
 
REVERSIBLE CTCLES IN OASES AND VAPORS. 37 
 
 namely, a law which holds true for a particular class of physi- 
 cal processes taking place in a particular class of physical 
 bodies. 
 
 It was in this form that the great French engineer, Sadi 
 Carnot, first discovered the law, seventy years ago, in 1824. 
 But he did more than that. He also pointed out that it is 
 only a particular form of a more general law, that is to say, a 
 law which holds true for a class of physical processes much 
 larger than the class we have just described. It was by a 
 train of reasoning first suggested by Carnot's great essay, 
 "Keflexions sur la puissance motrice du feu et sur les 
 machines propres a la developper," that subsequent inves- 
 tigators, and foremost among them Olausius and Eankine, 
 were able to extend the above relation into a general law, 
 called the second law of thermodynamics. 
 
 Before proceeding any further in our discussion towards the 
 generalization of the above efficiency relation into the second 
 law of thermodynamics let us first examine carefully the 
 foundation on which this relation rests. 
 
 Consider each one of the four operations which taken to- 
 gether constitute Carnot's cycle. Any one of them when 
 reversed will produce just the opposite effect. Take for in- 
 stance the second operation, that is, the first adiabatic expan- 
 sion. If at the completion of this operation we reverse the 
 operation, we can bring the gas back into the state which it 
 had at the beginning of this operation. The reversed opera- 
 tion will cost us mechanical work of compression equal to the 
 area bcgh. That work will appear as intrinsic energy in the 
 gas, that is, as heat, in consequence of which the temperature 
 of the gas is increased from T^ to T,. If the first operation be 
 
38 THERMODYNAMICS OF 
 
 called the direct and the second the reversed, then we can say 
 that the direct and the reversed operation just neutralize each 
 other. The same is true of every other operation of the cycle. 
 It follows, therefore, that the whole cycle can be reversed and 
 the reversed cycle will produce just the opposite effect of the 
 direct cycle, that is to say, it will deprive the reservoir B of 
 Q 2 units and add Q 1 units of heat to reservoir A, and it will 
 also transform Q 1 — Q^ = Q units of work into heat. In the 
 first cycle heat passes from a hot body to a cold body, but 
 some of this heat in its journey to the cold body is transformed 
 into mechanical work and therefore never reaches the cold 
 body. In the reversed cycle mechanical work is expended, in 
 order to transfer Q 2 units of heat from the cold body to the 
 hot, and in addition generate Q units of heat which are also 
 given up to the hot body. 
 
 Such a cycle is called a reversible cycle. The relation 
 
 q _ y, - t, 
 
 is evidently the same no matter whether the cycle be taken 
 in the direct or in the reversed sense. 
 
 Suppose now that during any one of the operations the gas 
 had changed chemically, but in such a way that its chemical 
 constitution could not be restored by reversing the operation. 
 Such an operation would not be reversible, nor would the 
 cycle of which this operation forms a component part be re- 
 versible. In such a cycle the above relation may or may not 
 be true. A special investigation is required to clear up this 
 point; but since we are not going to discuss it in our course 
 it is well that you should be told that it is not true. 
 
REVERSIBLE CYCLES IN GASES AND VAPORS. 39 
 
 It should also be observed that there are a great many 
 reversible cyclic processes in all branches of physical sciences 
 which do not apparently resemble Carnot's cycles of opera- 
 tions. A great many illustrations may be cited from chem- 
 istry, electricity and magnetism, etc. Whether in these 
 Carnot's law of efficiency or the general form of it which we 
 are about to deduce is applicable or not, must be decided by 
 investigations which are outside of the limits of our course. 
 What we propose to do is simply this : we are going to prove 
 the applicability of Carnot's law of efficiency, and therefore 
 of all its extensions, to all physical processes which consist of 
 operations any one of which can be performed reversibly by 
 a Carnot engine. 
 
 The first step in our work will be to prove that the efficiency 
 of a Carnot cycle depends on the temperatures of the reser- 
 voirs A and B as expressed above, no matter what the sub- 
 stance may be upon which we operate. To do this we shall 
 start from an axiom first stated by Clausius, and therefore 
 called the 
 
 Axiom of Clausius. 
 Heat cannot pass from a cold body to a hot body of itself. 
 
 This statement includes, that by conduction of heat, or by 
 concentration, reflection, etc., of radiant heat, no heat can be 
 transferred to a body at the expense of a colder one, unless 
 the process of transference involves the expenditure of some 
 sort of energy. 
 
 In the preceding discussion we saw that heat passed from a 
 cold to a hot reservoir, but not without expenditure of work. 
 
40 THERMODYNAMICS OF 
 
 So we might say, instead of " of itself," " without work or 
 compensation." 
 
 The truth of this axiom is proved in the same way as the 
 truth of Newton's axioms of motion, that is, by an appeal to 
 experience. 
 
 We shall now prove that the ratio between the heat trans- 
 formed into work by a simple reversible cycle and the heat 
 transferred from a hot body to a colder body during the same 
 cycle is independent of the material which we employ in the 
 cylinder of our ideal engine, and that it can depend only on 
 the temperatures of the reservoirs A and B. 
 
 For this purpose we employ two equal cylinders A and B 
 containing two different substances, upon which we propose 
 to perform simple reversible cycles. Each cycle is performed 
 between the same tivo reservoirs, so that the temperatures be- 
 tween which we operate are the same for each cycle. Perform 
 now cycle I with the cylinder A, and suppose that a quantity 
 of heat equal to Q is transformed into mechanical work, and 
 that a quantity of heat equal to Q^ is transferred to the colder 
 reservoir. Pass now to the other cylinder and perform a 
 similar cycle, but in such a way that again a quantity Q of 
 heat is transformed into mechanical work. A certain quantity 
 of heat, say Q/, will be transferred to the colder reservoir. 
 We wish to prove now that Q/ = Q^. For if it is not, sup- 
 pose that Q/ > Q a . After performing cycle I in the direct 
 order, by which a quantity of heat Q is transformed into 
 mechanical work and Q^ transferred to the colder reservoir, 
 employ cylinder B to perform cycle II in the reverse order, 
 so that the mechanical work obtained by cycle I is retrans- 
 formed into heat, and a quantity of heat equal to Q ' is trans- 
 
REVERSIBLE CYCLES IN GASES AND VAPORS. 41 
 
 f erred from the cold to the hot reservoir. At the end of the 
 two cycles the only change that still remains is the transfer 
 of a quantity of heat equal to Q/ — Q^ from the cold to the 
 hot reservoir, that is, a transference of heat from a body of 
 lower to a body of higher temperature without any compensa- 
 tion. This is contrary to the axiom of Clausius, hence the 
 supposition that Q/ > Q, must be dismissed. 
 
 Next suppose that Q/ < Q^. We can prove that this sup- 
 position also violates our axiom by going through cycle II in 
 the direct and cycle I iu the reverse order. 
 
 Hence Clausius' Axiom leads to the conclusion that Q/ = 
 Q r That is to say, ivhenever mechanical vwrk is obtained at 
 the expense of heat by a reversible cyclic operation, heat is 
 transferred from a hot to a cold body and the ratio of the ivork 
 obtained to the quantity of heat transferred from the hot to the 
 cold body is independent of the substance which is operated 
 upoti. 
 
 That ratio will therefore be the same for a perfect gas as 
 for any other substance. We proceed to calculate it for a 
 perfect gas. Let the temperatures of the two reservoirs be 
 7', and T t and suppose that 7", > T t . 
 
 We found previously that in the case of a perfect gas 
 
 Q ._ y, - t % 
 
 Q,~ t, ' 
 
 or 
 
 ft - Q+Q. -t | C,_ t x 
 Q Q "*" Q t,-t; 
 
 9* - T > _ 1 - r « 
 q r.-r, t,-t; 
 
42 THERMODYNAMICS OF 
 
 We see therefore that this ratio in the case of a perfect gas 
 and therefore also in the case of any other substance depends 
 on the two temperatures between which the cycle is performed. 
 
 A slight modification of this relation will give us another 
 form of stating the second law of thermodynamics applicable 
 to simple reversible cycles. Prom this relation we obtain 
 
 rp rp i rp • 
 
 J 2 J ] ■ L t 
 
 We obtained previously 
 
 G, 
 
 _ T x . 
 
 Q 
 
 T,-T,' 
 
 e, 
 
 Q 
 
 T, 
 
 t, - t; 
 
 hence 
 
 Combining this with the above we obtain 
 
 & = & or £_ft- or «.+^e,_ 
 
 — Q 2 is the heat given off to the cold reservoir by the sub- 
 stance operated upon, hence — (— Q t ) or + Q^ can be said to 
 be the heat received by the substance from the cold reservoir. 
 With that mental reservation we can write down 
 
 T, ^ T. 
 
REVERSIBLE CYCLES IN OASES AND VAPORS. 43 
 
 This is the more compact and at the same time more flexible 
 form of stating the second law of thermodynamics applicable 
 to simple reversible cycles. The English translation of this 
 mathematical statement is : If in a simple reversible cycle the 
 heats received by the substance operated upon be divided by the 
 temperatures at which these heats were received and the quo- 
 tients thus obtained be added together, then will the sum be 
 equal to zero. 
 
 This statement of the second law of a simple reversible 
 cycle is certainly not nearly as clear nor physically as in- 
 telligible as the other statement, which involved the efficiency 
 of an ideal reversible engine, but, as will be presently seen, it 
 enables us to extend in a very simple manner the applicability 
 of this law to complex reversible cycles and thus give the 
 second law of thermodynamics a much more general form. 
 
 General Form of the Second Law of Thermo- 
 dynamics. 
 
 Complex Cycles. 
 
 So far we have considered cycles between two temperatures 
 only. We now proceed to extend this to a so-called complex 
 cycle, that is, a cycle of operations between several reservoirs 
 of different temperatures. We commence with three reser- 
 voirs of temperature T lt T^, and T 3 . 
 
 Let ab (Fig. 5) denote an isothermal expansion at the tem- 
 perature T, , be an adiabatic expansion down to the tempera- 
 ture T a , cd an isothermal expansion at the temperature T^ , 
 de an adiabatic expansion down to the temperature T a , ef an 
 isothermal compression at the temperature T t , the isothermal 
 
44 
 
 THERMODYNAMICS OF 
 
 compression to continue until the isothermal curve ef cuts 
 the adiabatic through a, and lastly fa an adiabatic compres- 
 sion to the starting point. 
 
 During the expansions ab and cd the substance operated 
 upon takes up the quantities of heat <2, , Q 2 from the reser- 
 voirs A and B at temperatures T^ and T 2 , and during the 
 
 Fig. 5. 
 
 isothermal compression ef it takes up the negative quantity 
 of heat — Q 2 from the reservoir at temperature T % . The 
 external work done during the cycle is represented by the 
 area abcdcfa. 
 
 AVe proceed to show now that a similar relation holds true 
 in this case as in the simple cycle. 
 
 To do that, let us produce the adiabatic be until it cuts the 
 isothermal fe at g ; then will the whole complex cycle be 
 divided into two simple cycles abgfa and cdegc. 
 
 The negative quantity Q 3 consists of q, and g/ , given off to 
 the coldest reservoir during the isothermal compressions gf 
 and eg respectively. AVe can now write down the symbolical 
 
MEVBRSIBLE CYCLES IN GASES Aflh VAPORS. 45 
 
 statement- of the second law for each one of these two simple 
 cycles, viz. : 
 
 Mr + -fr = f or cycle abcgfa ; 
 
 ■*■ 1 ■* 3 
 
 Adding these two we obtain 
 
 
 or, since q, + y,' = Q„ 
 
 t, "*" y. "*" r. 
 
 In the same manner we can prove this relation for a com- 
 plex cycle divisible into any number of simple cycles and 
 obtain 
 
 -^ + -| + -|+...+-^ = 0, or ||=0. 
 
 If the cycle be of infinite complexity, that is to say, if it be 
 representable by an infinite number of infinitely small elements 
 of adiabatic and isothermal curves, it is evident that the 
 
 relation ~2~f, —- will still hold true. Of course each one of 
 
46 THERMODYNAMICS OF 
 
 the Q's will be infinitely small and the sum will -be an in- 
 finite sum. We can therefore express this relation in the 
 notation of the infinitesimal calculus, thus : 
 
 f 
 
 f = o. 
 
 A reversible cycle of this complexity will evidently in the 
 limit be representable by a continuous closed curve. The 
 area bounded by this curve will represent the external work 
 done during the cycle if the cycle be performed in the direct 
 sense, otherwise it will represent the external work spent in 
 order to transfer a certain amount of heat to the hot reservoir, 
 a part of which comes from the cold reservoir. 
 
 The statement of the second law of thermodynamics (appli- 
 cable to reversible cycles) as given by the last equation, though 
 very general, gives us but little information about the various 
 elements of which the cycle is composed. It gives us no in- 
 formation about the progress of the various physical processes 
 which taken together constitute the cycle. It has the defect 
 which is common to all integral laws. For such laws give us 
 the total effect of a series of physical processes without giving 
 us the separate effect of each process of the series. Take for 
 instance the integral laws of Keppler which describe the 
 motion of the planets around the sun. They tell us all about 
 the areas traced out by radii vectores of the planets during a 
 finite time, about the periodic times, and about the orbits of 
 the planets, but they do not tell us anything about the action 
 which is going on at any and every moment between the sun 
 
REVERSIBLE CYCLES IN GASES AND VAPORS. 47 
 
 and the planets to produce these integral effects. It took no 
 less a genius than that of a Newton to infer these differential 
 effects from the integral effects discovered by Keppler. The 
 inference was the law of gravitation. To draw that inference 
 Newton had to invent the infinitesimal calculus. We may 
 well assert that he had to invent it ; for a simple consideration 
 will convince you that the process of passing from an integral 
 physical law to the differential law — that is, the law which de- 
 scribes a physical process as it progresses from point to point 
 through infinitely small intervals of space and time — is exactly 
 the same as the mathematical process of passing from an in- 
 tegral to the differential of that integral. 
 
 We are now ready to discuss the differential of the integral 
 
 / 
 
 dQ 
 T 
 
 = 0. 
 
 Consider any one of the infinitely small terms -—^ in the 
 
 Pig. 6. 
 
 infinite series / -^. The subscripts 12 (Fig. 6) aenote that 
 dQ„ is the heat taken up during the infinitely small element 
 
48 THERMODYNAMICS OF 
 
 12 of the cycle and that T lt was the mean temperature of the 
 substance operated upon at that particular moment. It is evi- 
 dent that dQ^ depends on the instantaneous state of' the sub- 
 stance operated upon at that moment, and also upon the 
 nature of the operation represented by the element 12. In 
 other words, dQ„ depends on the volume, pressure, and tem- 
 perature, and the method of their variation during the opera- 
 tion represented by the element 12. Now all these things are 
 completely defined by the co-ordinates of the extremities of the 
 element, hence dQ^ is a function of these co-ordinates. The 
 
 same is true of the infinitely small quantity -pfr 2 - 
 
 12 
 
 Consider now a quantity <p, and suppose it to be a finite, 
 continuous, and singly-valued function of the pressure, 
 volume, and temperature of the substance operated upon. 
 will therefore have definite values at every point of any 
 cyclic diagram. 
 
 Denote by <p t , 2 , <p 3 , . . . <p n the values of at the 
 various points 1, 2, 3, . . . n all around the cyclic diagram 
 and consider the differences. 
 
 0, - 0, = 40; 
 
 03 - <£, = AJ>; 
 
 04 - 0= = 40; 
 
 0» — 0n-l = 4-i0; 
 
 0, — 0n = 40- 
 
 It is evident that for every cyclic diagram 
 
 A x <f> + 40 + • • • + 40 = 0. 
 
REVERSIBLE CYCLES IN OASES AND VAPORS. 49 
 
 When the points 1, 2, 3, ... n are infinitely near each 
 other, then A x <p, Ajf>, etc., are infinitely small quantities or 
 what we call in calculus the complete differentials of <p at the 
 various points of the curve. We express that by saying that 
 in the limit we shall have the differences djp, djp, d 3 0, etc., 
 but an infinite number of them and their sum, which will 
 still be zero for every cycle, can be expressed in the ordinary 
 way, thus: 
 
 fd<t> = 0. 
 
 This relation will hold true for every function of pressure, 
 volume, and temperature which has definite values at every 
 point of the cyclic diagram and varies continuously from 
 point to point of the diagram. There are evidently an in- 
 finite number of such functions. 
 
 If that integral is extended not all around the cyclic dia- 
 gram, but only between two points, say points 1 and a, then 
 we shall have the integral evidently equal to a — <t> x ; that 
 is to say, 
 
 /: 
 
 d<p= (pa — 0,- 
 
 The value of the integral will depend therefore on the 
 position of the initial and final point of integration and not 
 on the shape of the cyclic diagram between the two points. 
 
 This relation throws much light upon our integral jd<p — 0. 
 
50 
 
 THERMODYNAMICS OF 
 
 For, consider two integrals / d(p along a portion AB of a 
 cyclic diagram (Fig. 7) ; one integral from A to B along one 
 
 Pig. 7. 
 
 part of the diagram in the clockwise direction, and the other 
 along the other part of the diagram in anti-clockwise direc- 
 tion. Denote the first by subscript (AB) l , the second by sub- 
 script (AB),. Then 
 
 / d<p = <p B — <f> A and / d<p = <p B — 
 
 1/ (-4S), J(AS), 
 
 <Pa 
 
 Denote now the integral from B to A along the second part 
 of the cycle by subscript (BA); then evidently 
 
 I d(p= — dcf>= — (<p B — 0a)' 
 
 J(.BA) J(AB\ 
 
i REVERSIBLE CYCLES IN GASES AND VAPORS. 51 
 
 It follows therefore that 
 
 J d<f>+ I dtp= I d</>- M0=(0i,-0j)-(0 fl -0j=O, 
 
 «/(ii)l i/(BA) i/(iB), W (ABh 
 
 as it should be, since 
 
 / d<p -\- I d<p = integral all around the cycle. 
 
 We can therefore say that the integral all around the cyclic 
 diagram vanishes because the value of the integral between 
 any two points of the diagram depends on the position of 
 the two points and not on the shape of the diagram between 
 these two points. The only other way in which that integral 
 could vanish would be that dcp is zero for every element of 
 every cycle, in which case <p would be a constant. This is 
 contrary to our supposition, for we supposed that it is a 
 variable function of volume, pressure, and temperature. 
 A careful inspection of the series which in the limit gave 
 
 the integral / d<p will convince you that this integral be- 
 tween any two points of any cyclic diagram depends on 
 the position of these two points and on nothing else, because 
 the quantity under the integral sign is a perfect differen- 
 tial of a function <p which is finite, continuous, and singly 
 valued at every point of any cyclic diagram. We conclude, 
 
 therefore, since the integral j -^ taken around a diagram 
 
52 THERMODYNAMICS OF 
 
 representing any reversible cyclic process vanishes, that -^ 
 
 is the complete differential of a function S which is a finite, 
 continuous, and singly-valued function of the volume, press- 
 ure, and temperature of the substance upon which we oper- 
 ate. We express this by writing 
 
 f = « 
 
 This is the most general form of the second law of thermo- 
 dy?mmics. This statement of the second law bears about the 
 same relation to the other statement, 
 
 / 
 
 *« •= 0, 
 
 as the Newtonian law of universal gravitation bears to the 
 three laws of Keppler which describe the motion of planets 
 around the sun. It tells us all about the progress of the 
 infinite number of infinitely small processes which, taken 
 together, constitute the whole reversible cyclic process of 
 which the cyclic diagram is a geometrical representation. 
 We shall illustrate this statement by reference to a par- 
 ticular physical process when we come to the discussion of 
 the application of the two laws of thermodynamics to the 
 study of saturated vapors, which we are now ready to take up. 
 This statement of the second law of thermodynamics of 
 reversible processes was first given by Clausius. The func- 
 
REVERSIBLE CYCLES IN OASES AND VAPORS. 53 
 
 tion S was named by him the Entropy of the system. It 
 is impossible to give a general and yet complete definition 
 of the physical meaning of the entropy function, because 
 this meaning will vary considerably with the nature of the 
 process that we may be considering. Hence it can be dis- 
 cussed only in connection with the discussion of each par- 
 ticular process under consideration. It is desirable, however, 
 to illustrate its meaning by considering a few simple pro- 
 cesses. 
 
 1st. Suppose a body expands reversibly at constant tem- 
 perature from volume v 1 and pressure p i to volume v^ and 
 pressure p,. Let the heat that must be supplied to the 
 body be Q, then 
 
 that is, the heat is equal to the absolute temperature multi- 
 plied by the increase of the entropy. In the case of a 
 perfect gas that heat is also equal to the external work 
 
 done by the gas during the expansion, and this work we 
 
 v 
 found to be equal to RT log — . 
 
 Hence for isothermal expansions of perfect gases we have 
 
 5, -5, = 5 log J. 
 
 2d. If the gas expands adiabatically, then, since no heat is 
 supplied during such expansion, we shall have 
 
 = TdS, hence dS = 0. 
 
54 THERMODYNAMICS OF 
 
 That is to say, the entropy remains constant. It is on 
 this account that adiabatic expansion is called sometimes an 
 isentropic expansion, the word isentropic meaning that the 
 entropy remains constant. 
 
 IV. 
 
 APPLICATION OF THE TWO LAWS OF THERMODY- 
 NAMICS TO SATURATED VAPORS. 
 
 We are now ready to take up the discussion of the rever- 
 sible processes in the next simplest class of physical bodies, 
 that is, saturated vapors. Our discussion will be framed 
 after the model of our discourse on the reversible cyclic pro- 
 cesses in the case of perfect gases. We shall therefore com- 
 mence with an appeal to experience for information about 
 the physical properties of saturated vapors. We did the 
 same thing in the case of perfect gases by taking account of 
 the laws of Boyle and Mariotte^Gay-Lussac, and of the 
 experimental researches of Eegnault on the specific heats of 
 perfect gases. We shall then put the two laws of thermo- 
 dynamics into a suitable form by taking proper account of 
 this information about the physical properties of saturated 
 vapors. This will give us two mathematical statements of 
 these laws involving certain physical constants of saturated 
 vapors. Some of these constants are known from experi- 
 mental researches just as in the case of perfect gases; others 
 we shall have to calculate with the assistance of the two laws. 
 Hence after modelling properly our two fundamental laws, 
 
REVERSIBLE CYCLES IN GASES AND VAPORS. 55 
 
 we shall proceed, just as we did in the case of perfect gases, 
 to calculate and tabulate the various physical constants of 
 saturated vapors. Having done that, we shall then be ready 
 to pass on to the discussion of isothermal and adiabatic 
 expansions of saturated vapors, calculating in each case the 
 external work done and the heat supplied to the expanding 
 vapor. Since water-vapor is the substance most generally 
 employed in our vapor-engines, it is advisable to concentrate 
 our attention as much as possible upon that vapor. The 
 discussion will nevertheless lose very little of its generality, 
 since in dealing with saturated water-vapor we have to 
 consider the same questions that would have to be con- 
 sidered in any other vapor. 
 
 Physical Properties oe Saturated Vapors. 
 
 Consider a cylinder A (Pig. 8) in which a piston B can sMde 
 np and down without friction. Let the volume enclosed by 
 
 ''"W'WM 
 
 .'AWA WA' " ' ' . .~T 
 
 I 
 
 Fig. 8. 
 
 the piston be denoted by v, and let a part a of this volume 
 contain a volatile liquid. The other part b will be vapor of 
 
56 THERMODYNAMICS OF 
 
 that liquid. Let the temperature be uniform throughout 
 and equal to T. Experience tells us that this vapor, as long 
 as it is in contact with its own liquid, has a definite tension 
 which depends on the nature of the liquid and on its tempera- 
 ture and on nothing else. Let the pressure on the piston be 
 just such as to be in equilibrium with the tension of the 
 vapor at the temperature T. Suppose now that the piston is 
 pushed down through a very small distance ; the volume of 
 the vapor is diminished; experience tells us that the vapor 
 will condense. Vice versd, if the piston is slightly raised, 
 keeping the temperature constant, then some more liquid 
 will evaporate so as to keep the density in the vapor volume 
 the same as before. We infer, therefore, that vapor when in 
 contact with its own liquid has the maximum density for that 
 temperature ; in other words, it is saturated. 
 
 Both the tension and the density of the saturated vapor 
 increase with temperature. 
 
 Saturated vapors do not obey the Mariotte-Gay-Lussac law, 
 not even approximately, except for low temperatures. So, for 
 instance, this law cannot be applied to saturated water-va- 
 pors at any temperature which is more than only a very few 
 degrees centigrade above the freezing-point. 
 
 When a liquid is brought into a space which does not con- 
 tain the vapor of this liquid, then the liquid will evaporate 
 until its vapor has the maximum density in every part of the 
 surrounding space. The process of evaporation will be slow 
 if the space contains a gas at considerable pressure, but 
 rapid if the space is a vacuum. This is due to the slowness 
 with which the vapor diffuses through a gas of considerable 
 tension. Experience tells us that evaporation is accompanied 
 
REVERSIBLE CYCLES IN GASES AND VAPORS. 57 
 
 by a cooling off of the liquid, and the cooling is the more rapid 
 the quicker the process of evaporation. You are acquainted 
 with the well-known experiment of freezing mercury by 
 bringing it in contact with a volatile liquid which is made to 
 evaporate very rapidly by the action of a pump. If we wish 
 to maintain the temperature of an evaporating liquid constant, 
 then heat must be supplied to it. Conversely, if the saturated 
 vapor be compressed in contact with its own liquid, heat will 
 be developed, and if we wish to keep the temperature constant, 
 then this heat must be taken away. The quantity of heat 
 which is developed by compressing a kilogramme of saturated 
 vapor to liquid at the same temperature and pressure is called 
 the latent heat of saturated vapor at that temperature. This 
 is a very important physical constant and has been deter- 
 mined experimentally for a large number of vapors. 
 
 When the temperature of a liquid has reached that point at 
 which the tension of its vapor is the same as the pressure of 
 the surrounding space, then the vaporization will be rapid, 
 because it will take place not only on the surface of the liquid, 
 but also in the interior parts, especially if the liquid, as is 
 usually the case, contains small bubbles of a gas, like air, dis- 
 solved in it. When this takes place we say that the liquid is 
 boiling. The boiling point of pure distilled water at 760 mm. 
 barometric pressure is 100° C, and it rises with the rise of 
 barometric pressure. The boiling point of water is raised by 
 the solution of foreign substances in it. Sea-water, for in- 
 stance, has a considerably higher boiling point than pure dis- 
 tilled water at the same barometric pressure. As long as the 
 boiling process continues, the temperature of the liquid re- 
 
58 THERMODYNAMICS OF 
 
 mail-is constant; the heat supplied to the liquid is all utilized 
 to transform the liquid into steam. 
 
 If, however, the liquid does not contain any dissolved gases, 
 then the process of boiling is considerably retarded. There is 
 superheating and the boiling is accompanied by the well- 
 known " bumping " or concussive boiling. 
 
 When the contact of a saturated vapor with its own liquid 
 is cut off and the vapor allowed to expand slowly, it will cool 
 off, of course, and its density will diminish. The ratio of the 
 diminution of the density to the diminution of temperature 
 is different for different vapors. In some vapors the tem- 
 perature diminishes more rapidly than the density. Such 
 vapors will condense during adiabatic expansion. If, how- 
 ever, the temperature diminishes more slowly than the dens- 
 ity, then the vapor will be superheated, that is to say, it will 
 not remain saturated during adiabatic expansion. In the 
 first case we have superheating and in the second we have 
 condensation during adiabatic compression. 
 
 On the Physical Constants of Saturated Vapors. 
 
 Let the total weight of the liquid and its saturated vapor 
 in the above-mentioned cylinder be M. Denote by T the 
 absolute temperature of the whole mass, and by v the volume. 
 
 Let m = weight of the vapor in kilogrammes ; 
 
 p = tension of the saturated vapor in kgr. per sq. m. ; 
 a = specific volume of the liquid; 
 s = " " " " vapor. 
 
REVERSIBLE CYCLES IN OASES AND VAPORS. 59 
 We shall have then 
 
 ms -+- (M — m)cr = v, 
 or 
 
 m{s — c) + Ma — v. 
 
 In what follows we shall write u for s — a, hence 
 
 raw -f- Mcr — v. 
 
 It must be observed, however, that (in the case of water es- 
 pecially) s is large in comparison to cr, so that in most cases 
 we may consider u practically equal to s. 
 
 When the temperature T and the weight m of the saturated 
 vapor are given, then the state of the whole mass in the cylin- 
 der is completely known. For we know then the pressure 
 and the volume (since the density is completely determined 
 by T), and that is all that we require to know. We can there- 
 fore select T and m as our variables. 
 
 In order to bring out clearly the meaning of some other 
 physical constants of saturated vapors, we shall consider now 
 two distinct physical processes which we shall call definitional 
 processes. 
 
 Definitional Process (A). 
 
 If a small quantity of heat clQ is communicated to the mass 
 without changing the weight m of the saturated vapor, we shall 
 have the following changes produced: 
 
60 THERMODYNAMICS OF 
 
 First. The temperature of the liquid will increase by a small 
 quantity dT. Hence if C denote the specific heat of the 
 liquid at the temperature T (measured in kg. meters), then 
 (M — m)CdT will be equal to that part of clQ which is used 
 up in raising the temperature of the liquid by dT. 
 
 Secondly. The temperature of the vapor is increased 
 by the same amount dT. Hence its density will increase; 
 and if we did not diminish the volume slightly by 
 pushing down the piston, the weight m of the vapor 
 part of our whole mass would increase. Supposing now 
 that by a slight depression we keep m constant, there will 
 be no vaporization and the rest of dQ will go to increase 
 simply the temperature of the m kg. of the saturated vapor 
 by dT degrees. 
 
 This point needs a little fuller discussion on account of its 
 importance in the theory of the steam-engine. Suppose that 
 we have 1 kilogramme of vapor in saturated state but not in 
 contact with its own liquid, and that we wish to increase its 
 temperature by dT degrees centigrade and still keep it 
 saturated. We can proceed in the following way : 
 
 We communicate to it a quantity of heat dQ^ which is just 
 sufficient to raise its temperature by dT degrees. The vapor 
 is then superheated, that is to say, its density is smaller than 
 the density corresponding to its saturated state at that 
 temperature T -\- dT. In order to increase the density so 
 as to make it equal to the density of saturation at the 
 temperature T + dT, we now compress it isothermally 
 until the proper density is reached. Let the heat developed 
 by this compression be dQ, , then dQ t — dQ q = the heat 
 which must be supplied to 1 kg. of the saturated vapor in 
 
REVERSIBLE CYCLES IN OASES AND VAPORS. 61 
 
 order to raise its temperature by dT degrees and still keep 
 the vapor saturated at this increased temperature. 
 
 ' ' is denoted by H and called the specific heat (in 
 
 kg.-m.) of saturated vapor. It is evidently equal to the heat 
 which would have to be supplied to one kilogramme of satu- 
 rated vapor in order to raise its temperature by one degree and 
 maintain it in saturated condition, if the ratio of the supply 
 of heat to the rise of temperature remained constant. 
 
 Hence in our problem above the part of dQ which is ex- 
 pended in raising the temperature of the m kg. of saturated 
 vapor without superheating and without increasing the quan- 
 tity of m will be mHdT. 
 
 The total heat contained in the mass will be increased be- 
 cause the temperature of the mass is increased. The rate at 
 which the heat of the mass increases with the increase of 
 temperature, everything else remaining the same, is expressed 
 
 in the well-known notation — %. Hence the increment in the 
 
 ol 
 
 heat due to the increment dT of the temperature is ^LdT. 
 
 al 
 
 This is evidently equal to the heats mentioned under 1 and 2. 
 We can therefore write 
 
 f£-tf T =(M-m) CdT + mHdT, 
 
 ol 
 
 or 
 
 ^- = m(H-C) + MC. 
 
62 THERMODYNAMICS OF 
 
 Definitional Process (B). 
 
 Let the piston rise so as to allow a small quantity 
 dm of the liquid to evaporate. In consequence of thin 
 evaporation the temperature of the mass begins to sink ; 
 we therefore supply heat in order to maintain it con- 
 stant. Let dQ be equal to the heat supplied to the total 
 mass during the isothermal evaporation of the quantity 
 
 dm. The ratio -j^ is denoted by p and called the heat of 
 
 evaporation. It evidently means the number of units of heat 
 which must be supplied, in order to evaporate isothermally 
 one kg. of water at T deg. centig., and the pressure of satu- 
 rated vapor at that temperature into saturated vapor. Let 
 
 7\0 
 
 — denote the rate at which the heat of the whole mass in- 
 
 dm 
 
 creases when the water is evaporated isothermally, then 
 
 —dm is evidently the total increment of heat when dm unit 
 
 dm J 
 
 weights of the liquid pass isothermally into that many units 
 of saturated vapor. But we have just seen that this is also 
 equal to pdm. Hence 
 
 ^dm = pdm, 
 dm 
 
 or 
 
 dQ 
 
 dm 
 
REVERSIBLE CYCLES IN GASES AND VAPORS. 63 
 
 The two equalities obtained by the definitional processes 
 (A) and (B) enable us now to deduce a very elegant state- 
 ment of the two laws of thermodynamics for saturated vapors. 
 
 Forms of Statement of the Two Laws of Thermo- 
 dynamics eor Saturated Vapors. 
 
 We start with the general statements of these two laws : 
 
 dQ — dU + pdv 
 and 
 
 dQ = TdS . . 
 
 By carrying out the indicated differentiation, remember- 
 ing that T and m are our variables — that is to say, remember- 
 ing that Q, U, 8, and v are in our discussion of saturated 
 vapors functions of T and m and of nothing else — we obtain 
 the following relations: 
 
 We commence with the first law. 
 
 lQjT+ dQ dm-(^+v^\dT+(^4-v^\dm 
 
 This is an identical equation in dm and dT; hence we must 
 have, according to well-known rules in algebra, 
 
 dQ_dU, 9» 
 dT~dT +P dT' 
 
 dm dm p dm 
 
64 THERMODYNAMICS OF 
 
 Comparing these two relations with the expressions for 
 -^ and — - obtained by the definitional processes (A) and (B), 
 we infer that 
 
 ^, + p^ = m(H-C,+MG, 
 
 
 It must be observed that these two equalities are nothing 
 more nor less than a slightly different mathematical statement 
 of the first law. Nor will any other mathematical relation 
 which we can derive from these two equalities by any mathe- 
 matical operations whatever contain a single grain of truth 
 which is not already contained in the original statement of 
 that law. "What do we gain, then, by going through all 
 these tedious mathematical operations ? Do we arrive at a 
 simpler statement of this law ? No, we do not. Mathemati- 
 cally speaking, the expression 
 
 dQ = dU+pdv 
 
 is the simplest form in which that law can be stated. Before 
 answering the above question let us first perform the last 
 mathematical operation and obtain the final form of state- 
 ment of the first law. Differentiate the first of the above 
 equations with respect to m and the second with respect to T, 
 and subtract the first from the second so as to eliminate U. 
 
REVERSIBLE CYCLES IN OASES AND VAPORS. 65 
 
 Kemember, however, that p, H, M, and G are independent 
 of m. We obtain 
 
 §£ _ (IT- n\- A ( d l-\ _ A ( ®E\ - <?l <?! 
 QT K* v >-QT\ P dm) dmVdTJ ~ dT dm' 
 
 Remembering now that v = um -\- Mcr, hence — = u, we 
 
 obtain finally 
 
 dm 
 
 ff-(#-e) = *f§, (I) 
 
 This is the statement of the first law of thermodynamics 
 for saturated vapors at which we were aiming continually 
 during our tedious mathematical transformations. Now 
 in what respect is this statement superior to the general 
 statement dQ = dU + pdv? Evidently in this: it con- 
 tains nothing but physical constants of the liquid aud its 
 saturated vapor, most of which, as we shall presently see, are 
 capable of exact experimental measurement. A very famous 
 scientist said once that a department of human knowledge 
 becomes an exact science when it can express its laws in 
 terms of things which are capable of exact experimental 
 measurement. The intrinsic energy U, to be sure, is a thing 
 which we can and did express in terms of a definite unit, the 
 kilogramme-meter; we can also give a perfectly satisfactory 
 definition for it. We can evidently define it in two ways, 
 
66 THERMODYNAMICS OF 
 
 which we may call the absolute and the relative method of 
 definition. The absolute definition describes U as the energy 
 which we can obtain from the body by a series of processes 
 which would reduce the body to a state in which its intrinsic 
 energy is zero. The relative definition describes U as the 
 energy which we can obtain from a body by a series of pro- 
 cesses which will reduce the body to a state which we call 
 the normal state; and we can select any state for our normal 
 state. For instance, we can define the intrinsic energy of a 
 kilogramme of saturated water-vapor at any temperature as 
 the energy which we can obtain from it by reducing it to a 
 kilogramme of water at 0° C. and a pressure equal to the 
 tension of its saturated vapor at that temperature. Still, 
 when it comes to exact experimental measurement it is not 
 TJ that we measure but the physical constants p, H, 0, and 
 u, and from the measurements of these the relative value of 
 TJ can then in a very limited number of cases be obtained by 
 calculation. Hence the desirability of stating the first law of 
 thermodynamics of saturated vapors in terms of these physi- 
 cal constants which are capable of exact experimental meas- 
 urement instead of in terms of TJ, a quantity which is not 
 capable of direct experimental measurement. It is clear 
 now that the ultimate object of all our long mathematical 
 operations was to rid ourselves by a process of elimination 
 of the quantity TJ. 
 
 We proceed now to perform the same process of elimina- 
 tion on the general statement of the second law. Carrying 
 out the differentiations indicated in 
 
 dQ = TdS 
 
REVERSIBLE CYCLES IN OASES AND VAPORS. 67 
 
 and remembering that T and m are our independent variables, 
 we obtain 
 
 &*+*&'=*&"+*&'■ 
 
 This is an identical equation in cZTand dm, and therefore 
 the following relations will hold true : 
 
 8© _ T dS_ 
 ~dT dT' 
 
 dm dm' 
 
 Comparing these two relations with those obtained by the 
 definitional processes (A) and (B) we obtain 
 
 r|^ = m(H - G) + MO, 
 
 T— - o 
 
 Differentiating now the first equation with respect to m 
 and the second with respect to T and then subtracting the 
 first from the second in order to eliminate 8, we obtain 
 
 dS dp ,tt p\ 
 dm-dT~ {H ~ C) - 
 
68 THERMODYNAMICS OF 
 
 But since 
 
 dm dm ' ' 
 
 we obtain 
 
 |=|£-(fl-0 (II) 
 
 This is our final statement of the second law of thermo- 
 dynamics for reversible processes in saturated vapors. It 
 contains, just like the statement (I) of the first law, a mathe- 
 matical relation between the experimentally measurable 
 physical constants of the liquid and its saturated vapor. 
 
 By combining (I) and (II) we obtain another very impor- 
 tant relation between these constants, viz., 
 
 <=Tu^ (Ill) 
 
 Observation. — It is well to call your attention here to a 
 remark which was made in connection with our deduction of 
 the differential statement of the second law, 
 
 dQ = TdS, 
 
 from the integral statement 
 
 S" 
 
REVERSIBLE CYCLES IF GASES AND VAPORS. by 
 
 The remark was to the effect that the integral statement 
 described the total effect of a series of processes which taken 
 together constitute a reversible cycle, without telling us 
 anything about the separate effects of each particular process 
 of that series. The differential statement, on the other 
 hand, describes completely every minute step in the progress 
 of the reversible cycle. The relations (II) and (III) illus- 
 trate the truth of this remark very forcibly; for they tell us 
 that during every interval of time, no matter how small, the 
 reversible operations will progress in such a way as to main- 
 tain those relations between the physical constants and the 
 variable co-ordinates of the liquid and its vapor which are 
 given by (II) and (III). But do not forget that relation (II) 
 and to same extent relation (III) also is only another way of 
 stating the differential form of the second law, 
 
 dQ = TdS. 
 
 More than this. The integral statement limits us to the 
 study of those cycles which are composed of reversible opera- 
 tions only. The differential statement enables us to break 
 through these limits and study cycles consisting of operations 
 some of which are not reversible. For it is evident that we 
 can apply (II) and (III) to every reversible operation of the 
 cycle, and the irreversible operations we can attack by 
 the first law or in any other way that we may consider cor- 
 rect and convenient. This has a very important technical 
 significance, for we shall presently see that the cyclic opera- 
 tions in those types of steam and other caloric engines which 
 prevail to-day are by no means reversible in all their parts. 
 
70 THERMODYNAMICS OF 
 
 Since numerical values of the physical constants p, H, and 
 C have been tabulated by various experimentalists in terms 
 of the kg. calorie and not the kg. meter as the unit, it is 
 advisable to substitute in our equations (I), (II), and (III) 
 the numerical values of p, H, and C'in terms of the kg. calorie 
 as unit. Let 
 
 r = the numerical value of p when the unit is a kg. calorie, 
 c — " " 
 h= " " 
 
 then 
 
 P_ 0_ H_ 
 J~ r ' J -°' J - n > 
 
 hence we obtain from (I), (II), and (III) by these substitu- 
 tions 
 
 dT + c l- JdT , . . . . (I) 
 
 tt 
 
 " 
 
 a 
 
 a 
 
 tt 
 
 a tt a 
 
 tt 
 
 " H 
 
 it 
 
 tt 
 
 tt 
 
 a u tt 
 
 g r + c-h=J r , (II-) 
 
 These three equations are our fundamental equations in the 
 thermodynamics of reversible processes in saturated vapors. 
 We now proceed to the next part of our programme, and 
 
REVERSIBLE CYCLES IN OASES AND VAPORS. 71 
 
 that is the numerical calculation of the physical constants of 
 saturated vapors. 
 
 Numeral Calculation op the Physical Constants of 
 Saturatfed Vapors. 
 
 Our three fundamental equations (P), (IP), and (IIP) repre- 
 sent two independent relations [since (III a ) has been deduced 
 from (P), that is the first law, and (IP), that is the second 
 law of thermodynamics] between five unknown quantities, 
 
 dp 
 viz., r, h, u, c, and — £-. Hence they enable us to calculate 
 
 ol 
 
 any two of these quantities when the remaining three are 
 known. That is to say, three of these physical constants of 
 saturated vapors must be determined experimentally and the 
 other two can then be calculated by means of the fundamental 
 laws of thermodynamics. What is meant by experimental 
 determination is simply this: It must be found by experi- 
 mental investigation in what way these physical constants 
 depend on the variable co-ordinates of the body. In our dis- 
 cussion of the general physical properties of saturated vapors 
 we saw that all the physical constants of such a vapor depend 
 on the temperature only, hence the object of an experimental 
 investigation will be to express these physical constants as 
 functions of the temperature T. Three of the above con- 
 stants having been experimentally determined, the remaining 
 two can then be also expressed as functions of the temperature 
 by simple calculation, starting with our fundamental equa- 
 tions (I 3 ), (IP), and (IIP). 
 In most cases r, c, and p have been determined experiment- 
 
72 THERMODYNAMICS OF 
 
 ally, so that it is generally h and u, that is, the specific heat 
 and the specific volume of saturated vapors, that we have to 
 calculate. 
 
 We now proceed to carry out these calculations for a par- 
 ticular vapor, and select water vapor for reasons given above. 
 
 Regnault in his classical researches (Relation des experiences, 
 Mem. de l'Acad. t. xxi, 1847, etc.) made the quantities c, r, 
 and p, that is, the specific heat of water, the heat of evapora- 
 tion, and the tension of saturated vapor of water, the subject 
 of very careful experimental investigations. His results have 
 been verified by all investigators in that field of physical re- 
 search. We shall, therefore, use the values obtained from 
 the results of his investigations. 
 
 The specific heat of water is expressed, according to 
 Regnault's experimental data, by the following formula : 
 
 C-1+ .00004* + .0000009*% 
 
 where t is the temperature in degrees centigrade above the 
 freezing point. 
 
 For the heat of evaporation experimental data furnish the 
 following relation : 
 
 r = 606.5 + .305* - / cdt. 
 
 Substituting the value of c and carrying out the integration, 
 we obtain 
 
 r = 606.5 - .695* - .00002* 2 - .0000003* 3 . 
 
REVERSIBLE CYCLES IN GASES AND VAPORS. 73 
 
 For this long formula, Olausius (Mechanische Warmetheorie, 
 vol. i. p. 137) substitutes the following: 
 
 r = 607 - .708*. 
 
 To show the agreement between the values of r obtained 
 from these two formulae the following table is given for com- 
 parison : 
 
 t 
 
 0° 
 
 50° 
 
 100° 
 
 150° 
 
 200° 
 
 
 606.5 
 607 
 
 571.6 
 571.6 
 
 536.5 
 536.2 
 
 500.7 
 500.8 
 
 464.3 
 465.4 
 
 The agreement is excellent. We shall, therefore, employ 
 Clausius' simpler expression. 
 
 Taking the experimental data of Regnault's researches, 
 Clausius* constructed the following table for the tension^ 
 of saturated water- vapor at various temperatures : 
 
 t in C. ° by 
 
 p in mm. 
 
 t in C. ° by 
 
 p in mm. 
 
 Air Therm. 
 
 of Hg. 
 
 Air Therm. 
 
 of Hg. 
 
 -20 
 
 .91 
 
 110 
 
 1073.7 
 
 -10 
 
 2.08 
 
 120 
 
 1489 
 
 
 
 4.60 
 
 130 
 
 2029 
 
 10 
 
 9.16 
 
 140 
 
 2713 
 
 20 
 
 17.39 
 
 150 
 
 3572 
 
 30 
 
 31.55 
 
 160 
 
 4647 
 
 40 
 
 54.91 
 
 170 
 
 5960 
 
 50 
 
 91.98 
 
 180 
 
 7545 
 
 60 
 
 148.79 
 
 190 
 
 9428 
 
 70 
 
 233.09 
 
 200 
 
 11660 
 
 80 
 
 354.64 
 
 210 
 
 14308 
 
 90 
 
 525 45 
 
 220 
 
 17390 
 
 100 
 
 760 
 
 230 
 
 20915 
 
 * Mechanische Warmetheorie, vol. i. p. 149. 
 
74 THERMODYNAMICS OF 
 
 Observation 1. To transform p from pressure in mm. 
 Hg to pressure in kg. per sq. m. we simply remember that 
 760 mm. barometric pressure is equivalent to 10,333 kg. per 
 square meter. If we now wish to know how many kg. per 
 sq. m. correspond to, say, 11,660 mm. barometric pressure, we 
 simply put down the proportion 
 
 p: 10,333 :: 11,660:760, 
 
 where p is the pressure in kg. per sq. m., corresponding to 
 11,660 mm. barometric pressure. 
 
 10333X1166 1KQ „ nl 
 p = — = 158,530 kg. per sq. m. 
 
 (about 229 pounds per square inch). 
 
 This, then, is the pressure of saturated water-vapor at 200° C. 
 Observation 2. If we refer the tension p and the tem- 
 perature t of the above table to a set of rectangular axes, 
 measuring off t as the abscissae and the corresponding p's as 
 the ordinates, we obtain the curve as in diagram on opposite 
 page. 
 
 The value of the differential coefficient -—■ (which is evi- 
 
 dt \ 
 
 dently the same as -p^\ at any temperature represents the 
 
 tangent of the angle which the tangent line to the curve at 
 the point corresponding to that temperature makes with the 
 axis of t. An inspection of the curve shows that this angle 
 remains practically constant for considerable intervals of 
 temperature, certainly for intervals of 10°. 
 
REVERSIBLE CYCLES IN GASES AND VAPORS. 75 
 
 800 p. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 4 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 > 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 , 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 I 
 
 
 
 500 - 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 J 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 . 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ""i 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 V 
 
 .JL. 
 
 %- 
 
 
 
 - 
 
 
 
 
 
 
 
 
 
 - 
 
 
 
 
 
 
 15 30 45 60 75 90 
 
 Consider now any interval of 10° and let 
 dp _ 
 tit 
 
 105 
 
 a for that interval. 
 
 ldp d ,, . a . ,, 
 .'• y21~di ' %P> = v same lnterval - 
 
76 THERMODYNAMICS OF 
 
 This relation enables us to calculate in a very simple manner 
 
 the quantity — — for any temperature. This is a quantity 
 
 P (it 
 
 which we shall need presently. Suppose we wish to calculate 
 
 (— -jt) , that is, the value of --=(■ for the temperature of 
 \p at Jus p dt r 
 
 t = 25°. We select, therefore, that 10° interval the mean 
 
 temperature of which is 25°, that is, the interval between 20° 
 
 and 30°. We obtain 
 
 logp so - logp 10 = a j d lj 
 
 The mean value of p within the limits of integration is very 
 nearly the value of p at 25°, that is,^ 26 . 
 
 ••• l°g.Pso - log P™ = £- ( dt = ~. 
 
 But 
 
 " = (!b 
 
 L( C !P) - ] °SP™ - Io £A. _ Log P, - Log,,, 
 " p n \dt)» 10 M10 "' 
 
 where Log stands for Briggs' logarithm, and M is the modulus 
 of that system. 
 
 We are ready now to take up the calculation of the re- 
 maining physical constants, that is, to find the values of the 
 
REVERSIBLE CYCLES IN GASES AND VAPORS. 77 
 
 specific heat aud the specific volume of saturated water-vapor 
 for various temperatures within those limits within which the 
 other physical constants were studied experimentally. 
 
 1. Calculation of h. 
 From the second law 
 
 we obtain 
 
 dr j, _ v 
 
 , dr , r 
 
 h = ^ + c -: 
 
 dt ' rf + 273 
 
 Regnault's experiments gave us 
 
 r = 606.5 + .305£ - / cdt. 
 
 S' 
 
 5 = .305 -& 
 
 dt 
 
 Therefore 
 
 dr 
 
 ^ + , = .305. 
 
78 
 
 THERMODYNAMICS OF 
 
 In the last term of the expression for h we shall substitute 
 for r the simple expression given by Clausius and obtain 
 
 h = .305 - 
 
 607 -.7 08* 
 
 273 + * " 
 
 The following table contains the calculated values of h for 
 several temperatures : 
 
 t 
 
 0° 
 
 50° 
 
 100° 
 
 150° 
 
 200° 
 
 h 
 
 - 1.916 
 
 - 1.465 
 
 - 1.133 
 
 -.879 
 
 -.676 
 
 The physical meaning of this negative specific heat will be 
 considered further on in connection with adiabatic expansion 
 of saturated water-vapor. 
 
 We pass on to the calculation of the specific volume of 
 saturated water-vapor. 
 
 2. Calculation of the Specific Volume of Saturated 
 Water-vapor. 
 
 The first question to consider is, whether in its saturated 
 state water- vapor follows the law of Mariotte-Gay-Lussac, 
 and if it does not, to what extent and in what way deviations 
 occur. 
 
 Making use of the fundamental relation (III"'), we find, 
 after replacing (s — a) for u, 
 
 r _ T(s-a )dp 
 J di 
 
REVERSIBLE CYCLES IN OASES AND VAPORS. 79 
 
 or 
 
 273r_ _ 273p(s - a) 
 ,ldp 
 p dt 
 
 r ldp /(273 + 0" 
 
 If the Mariotte-Gay-Lussac law were applicable to satu- 
 rated water - vapor, then from ^ — const, we should have 
 
 ps X 273 
 
 jp3+T) = const ' 
 
 Since s differs but little from (s — <r), we should have 
 
 P(* ~ °-) 273 _ const 
 
 Comparing this to the equation just obtained from the funda- 
 mental relation (IIP), we see that if the Mariotte-Gay-Lussac 
 law were applicable to saturated water-vapor we should have 
 
 273r _ p(s - <r) 273 _ 
 
 -^ldp-J(2W+W~ 
 pdt 
 
 Having calculated - ■— by the method described above, and 
 r from the formula r = 607 — .708*5, and substituted their 
 
80 THERMODYNAMICS OF 
 
 values in the last equation, Clausius obtained the following 
 table : 
 
 
 »v jrv »> 2 73 + f 
 
 ( in C,° by Air 
 Thermometer. 
 
 Calculated from 
 Regnault's data. 
 
 Calculated from 
 Clausius' formula. 
 
 5 
 
 30.93 
 
 30.46 
 
 15 
 
 30.60 
 
 30.38 
 
 25 
 
 30.40 
 
 30.30 
 
 35 
 
 30.23 
 
 30.20 
 
 45 
 
 30.10 
 
 30.10 
 
 55 
 
 29.98 
 
 30.00 
 
 G5 
 
 29.88 
 
 29.88 
 
 75 
 
 29.76 
 
 29.76 
 
 85 
 
 29.65 
 
 29.63 
 
 95 
 
 29.49 
 
 29.48 
 
 105 
 
 29.47 
 
 29.33 
 
 115 
 
 29.16 
 
 29.17 
 
 125 
 
 28.89 
 
 28.99 
 
 135 
 
 28.88 
 
 28.80 
 
 145 
 
 28.65 
 
 28.60 
 
 155 
 
 28.16 
 
 28.38 
 
 165 
 
 28.02 
 
 28.14 
 
 175 
 
 27.84 
 
 27.89 
 
 185 
 
 27.76 
 
 27.62 
 
 195 
 
 27.45 
 
 27.33 
 
 205 
 
 26.89 
 
 27.02 
 
 215 
 
 26.56 
 
 26.68 
 
 225 
 
 26.64 
 
 26.32 
 
REVERSIBLE CYCLES IN CASES AND VAPORS. 81 
 
 The figures given in the column headed " Calculated from 
 
 Clausius' formula" will be explained presently. 
 
 We can easily see from this table that the Mariotte-Gay- 
 
 1 273 
 
 Lussac law does not hold true; for -jp(s — u) dimin- 
 
 ishes steadily with the increase of temperatures; evidently 
 because the specific volume of saturated steam diminishes 
 more rapidly than the pressure increases with increase of 
 temperature. Clausius found that this formula could be 
 reduced to the more simple one, viz., 
 
 1 , . 273 , 
 
 where m = 31.549, 
 n= 1.0486, 
 a = .007138. 
 
 By means of this formula Clausius calculated the figures 
 given in the column headed " Calculated from Clausius' for- 
 mula." 
 
 After this brief digression we return to the calculation of 
 the 
 
 Specific Volume of Saturated Steam. 
 
 We had 
 
 1 . , 273 , 
 
82 THERMODYNAMICS OF 
 
 Since we commit an error of a small fraction only of one per 
 cent by neglecting cr in comparison with s, we may write 
 
 J, ,,273 + 1 
 
 Let v' = volume of a kilogr. of air at a pressure p and 
 temperature t, then will 
 
 ^ = ^'(273 + 0; 
 
 For a given temperature t, say the freezing temperature, 
 this ratio can be determined experimentally, as will be ex- 
 plained presently. 
 
 Denoting the ratio -, at the freezing temperature by (-) 
 
 we shall have 
 
 (?) = F< m " W >2T3 ; 
 
 s _fs\m — ne at 
 
 ' ' v' \W m — n 
 
 The value of f — ,] can be obtained easily by the following 
 
 method ; 
 
REVERSIBLE CYCLES IN GASES AND VAPORS. 83 
 
 Experiment tells us that saturated water-vapor at low tem- 
 peratures obeys the Mariotte-Gay-Lussac law so nearly that 
 we may use the law in practical calculations without commit- 
 ting an appreciable error. On that hypothesis the density of 
 steam at 0° C. is easily obtained. 
 
 2 cub. m. H at 0° 0. at a given pressure weigh 2 X .06926 
 as much as 1 cub. m. of air at the same temperature and 
 pressure; 
 
 1 cub. m. at 0° C. at a given pressure weighs 1 X 1.1056 
 as much as 1 cub. m. of air at the same temperature and 
 pressure; 
 
 At low temperature, 2 cub. m. H and 1 cub. m. O unite into 
 2 cub. m. saturated water-vapor (very nearly). 
 
 .•. 2 cub. m. H„0 at 0° C. at a given pressure weigh 
 2 X .06926 + 1.1056 as much as 1 cub. m. of air at the 
 same temperature and pressure ; 
 
 1 cub. m. of H Q at 0° C. at a given pressure weighs 
 
 2 X .06926 + 1.1056 . 
 - as much as 1 cub. m. of air at the 
 
 & 
 
 same temperature and pressure. 
 
 - (*).= 
 
 1 1 
 
 i(2 X .06926 + 1.1056) _ .622' 
 
 Therefore 
 
 s i. m — ne 
 
 at 
 
 v' .622 m — n' 
 
84 ThaRMODTNAMICS OF 
 
 which can also be expressed by a more convenient formula 
 given by Clausius, thus : 
 
 -, = M- NBK 
 v 
 
 In this formula 
 
 M= 1.663, 
 W- .05527, 
 /3 = 1.007164. 
 
 This formula was used by Clausius to calculate the specific 
 volume of saturated water-vapor at various temperatures. 
 The results are given in the table on page 85. Parallel with 
 these figures are given the figures obtained experimentally by 
 two English engineers, Fairbairn and Tate.* 
 
 It must be observed that for every temperature we must 
 'mow first the pressure of saturated vapor at that tempera- 
 ture, and, inserting that pressure in pv' = RT, calculate v'. 
 
 Observe the close agreement between experiment and 
 theory as worked out by Clausius ; observe also the enormous 
 increase of the density of steam with rising temperature. 
 
 There is a temperature at which the density of steam is 
 the same as that of water at the same temperature. This is 
 called the critical temperature of water. Above the critical 
 temperature the vapor cannot be liquefied by pressure alone. 
 
 It is highly probable that all bodies have a critical tem- 
 perature. In the case of the so-called perfect gases the criti- 
 
 * Transactions of the Royal Society of London, 1860, vol. 150, p. 185, 
 
REVERSIBLE CYCLES IN OASES AND VAPORS. 85 
 Table of Specific Volume of Saturated Water-vapors. 
 
 t in C.° 
 
 From 
 
 Faiibairn and Tate's 
 
 Clausius' Form. 
 
 Experimental Table. 
 
 58.21 
 
 8.23 
 
 8.27 
 
 68.52 
 
 5.29 
 
 5.33 
 
 70.76 
 
 4.83 
 
 4.91 
 
 77.18 
 
 3.74 
 
 3.72 
 
 77.49 
 
 3.69 
 
 3.71 
 
 79.40 
 
 3.43 
 
 3.43 
 
 83.50 
 
 2.94 
 
 3.05 
 
 86.83 
 
 2.60 
 
 2.62 
 
 92.66 
 
 2.11 
 
 2.15 
 
 117.17 
 
 0.947 
 
 0.941 
 
 118.23 
 
 0.917 
 
 0.906 
 
 118.46 
 
 0.911 
 
 0.891 
 
 124.17 
 
 0.769 
 
 0.758 
 
 128.41 
 
 0.681 
 
 0.648 
 
 130.67 
 
 0.639 
 
 0.634 
 
 131.78 
 
 0.619 
 
 0.604 
 
 134.87 
 
 0.569 
 
 0.583 
 
 137.46 
 
 0.530 
 
 0.514 
 
 139.21 
 
 0.505 
 
 0.496 
 
 141.81 
 
 0.472 
 
 0.457 
 
 142.36 
 
 0.465 
 
 0.448 
 
 144.74 
 
 0.437 
 
 0.432 
 
 cal temperature is very low; hence the necessity of cooling in 
 the liquefaction of gases.* 
 
 We now come to the last and most important part of our 
 discussion of the application of the two laws of thermo- 
 dynamics to the study of reversible processes in saturated 
 vapors in general and water-vapor in particular, that is, to 
 the discussion of adiabatic and isothermal expansion of satu- 
 rated vapors. 
 
 * See the researches of Avenarius in Poggendorf 's Annalen, 1874. 
 
86 THERMODYNAMICS OF 
 
 Abiabatic Expansion of Saturated Water-vapok. 
 
 We commence with the deduction from our fundamental 
 relations of a mathematical expression which simplifies con- 
 siderably our discussion of adiabatic expansion. 
 
 By the definitional processes (A) and (B) we obtained 
 
 ^ = p- 
 dm P > 
 
 .\dQ = ^dT+ p^dm = \m(H - 0) + MC}dT + pdm. 
 
 J. 0i7l 
 
 From the second law (Fundamental Belation II) we 
 obtain 
 
 7T-r- dp -P 
 
 Substituting this value of (H — 0) in the last equation, we 
 derive the relation 
 
 dQ = ] m(|§, - f) + MO } dT + pdm. 
 
 Considering now that p, the heat of evaporation, depends 
 on temperature only, it is evident that 
 
 ^{lT=dp; 
 
REVERSIBLE CYCLES IN GASES AND VAPORS. 87 
 
 .'• m— mdT-}- pdm — mdp -f pdm = dimp) ; 
 
 01 
 
 .'. dQ = d(mp) + (MO - n ^)dT, 
 
 or 
 
 dQ = Td(^\ + MCdT. 
 
 This is the expression which we started out to deduce. 
 Iu adiabatic expansion dQ — 0, hence 
 
 or 
 
 TdiEj(\ + MCdT = 0, 
 
 Td(^j + McdT = 0j 
 
 r fcdT 
 
 mr m l r l _ M / cdT 
 ~~T~~T~ ~ ~ '' 
 
 Calculation is simplified very much and the error com- 
 mitted is very small if we put c = constant, during the expan- 
 sion, provided that the expansion does not extend over a very 
 large interval of temperature. In practice this never occurs. 
 
 mr m,r, , ,_ , T 1 
 •'• ~ji - -jr- = +Mc log y 5 , 
 
 or 
 
 m 
 
 T ( m t r x , , , , T, 1 
 
88 THERMODYNAMICS OF 
 
 (a) Condensation during Adidbatic Expansion of Saturated 
 Water-vapor. 
 
 A very interesting but, from an economical standpoint, 
 a very objectionable phenomenon takes place when satu- 
 rated water-vapor expands adiabatically. Owing to the fact 
 that during such an expansion the temperature (in con- 
 sequence of external work done by the expansion) sinks more 
 rapidly than the density diminishes, the vapor condenses. 
 Hence in order to prevent this condensation it would be 
 necessary to supply heat during the expansion. This, then, 
 is the physical meaning of the negative specific heat, as found 
 above, of saturated water-vapor. During adiabatic compres- 
 sion the vapor would, of course, become superheated. The 
 last equation enables us to calculate the amount of this con- 
 deusation. For example, suppose that m, kilogrammes of 
 saturated water-vapor at initial temperature T x are enclosed 
 in a cylinder by a piston which can glide up and down with- 
 out friction. Let there be no liquid mass in the cylinder. 
 Hence M = m l5 and as the piston goes up slowly owing to the 
 gradual diminution of the pressure, we shall have at any mo- 
 ment 
 
 in 
 m. 
 
 = *&+'**%)' 
 
 Starting with initial temperature T l = 150° -f 273, Clausius* 
 calculated the ratio of the vapor weight in at various tem- 
 peratures T to the initial weight »?,. The following table 
 contains the interesting results of this calculation : 
 
 * JMechanisclie Warmetheorie, vol. i. p. 164. 
 
REVERSIBLE CYCLES IN OASES AND VAPORS. 89 
 
 t 
 
 150° 
 
 1S5« 
 
 100° 
 
 75° 
 
 50° 
 
 25° 
 
 m 
 
 1 
 
 .956 
 
 .911 
 
 .866 
 
 .821 
 
 .776 
 
 The amount of condensation is, therefore, very consider- 
 able. 
 
 This phenomenon contributes to what is known in steam- 
 engineering under the name of cylinder condensation, a pro- 
 cess which, as is easily seen, pulls down the output of the 
 steam-engine. Other and probably quite as serious causes 
 con-tribute to this objectionable process in the cylinder of a 
 steam-engine. Various devices have been suggested and 
 tried in order to overcome this evil, but the discussion of 
 these is outside of the limits of this course. 
 
 (/?) Calculation of the External Work done during an 
 Adiabatic Expansion. 
 
 W- 
 
 - I pdv. 
 
 dp 
 
 But v = mu -f- Ma. 
 
 .'. pdv =pd(mu) — d{mup) — mu^^dT. 
 From our fundamental equation (III) we have 
 
 Tu 
 
 
 mp 
 
 ,'. pdv = d(mup) qrdT. 
 
90 THERMODYNAMICS OF 
 
 Since v changes adiabatically, we can make use of the re- 
 lation deduced in the last paragraph when we considered the 
 process of condensation during adiabatic expansion. The re- 
 lation is 
 
 Td(^-)+MCdT=0, 
 
 or 
 
 ^-dT - d{mp) = MCdT. 
 
 .'. pdv = d{mup) — d(mp) — MCdT. 
 
 d{mup) - / d(mp) - MO j dT 
 
 = mup — m i u i p 1 — [mp — ?«,pj + MQ{T y — T). 
 
 Since the quantities p and C are given, by the experimental 
 formulae discussed above, in kg.-calories, it is preferable to ex- 
 press PFthus: 
 
 W = mup — m l u 1 p l — J{(mr — m^,) — Mc(T, — T)}. 
 
 This formula enables us to calculate the external work W 
 in kg.-meters when a certain initial quantity m 1 of saturated 
 water-vapor at temperature T 1 , being in contact witli a quan- 
 tity M — m l of its own liquid at the same temperature, ex- 
 pands adiabatically until its temperature has gone down to a 
 temperature T. Thus: 
 
REVERSIBLE CYCLES IN GASES AND VAPORS. 91 
 
 Since T t and T are given, p l and p can be found by a refer- 
 ence to Kegnault's table of vapor tensions. Since to, and M 
 are also given, JMc{T x — T) and Jm^r x can be easily calcu- 
 lated. There still remain mr and mu to be calculated. The 
 quantity mr we have already calculated in the preceding 
 paragraph and found 
 
 mr 
 
 = r$P-**f> 
 
 The quantity mu can be easily found thus : 
 From the fundamental relation (IIP) we have 
 
 T dp 
 
 mr = w T -—. 
 J dT 
 
 Substituting this value of mr in the last equation, we ob- 
 tain 
 
 J f m l r 1 
 dp\ T, 
 dT 
 
 ■mu - , {-,:',' ■ Mc log -pY 
 
 The quantity -^, is the tangent of the angle which the 
 
 tangent line to the curve of vapor tension, given above, at 
 the point corresponding to temperature T = 273 + t, makes 
 with the axis of t, and can be easily found either graphically 
 or by calculation, thus : 
 
 Suppose we wish to calculate -?-=, for the temperature 
 
92 THERMODYNAMICS OF 
 
 t = 110°. On account of the gradual ascent of the curve of 
 vapor tension, we have 
 
 Eef erring now to Eegnault's table we find p lt0 in mm. = 
 1489 and j9 110 in mm. = 1073.7. Hence, according to the ex- 
 planation given before, in order. to transform these pressures 
 into kg. per square m. we have to divide by 760 and mul- 
 tiply by 10,333. We obtain 
 
 \dt)J 
 
 (1489 - 1073.7)^|F = 564 64 
 
 It is seen, therefore, that the two laws of thermodynamics 
 in connection with the experimental data on saturated water- 
 vapor enable us to calculate all the quantities which enter 
 into the expression for the external work done by the adia- 
 batic expansion of such a vapor. 
 
 It should be noticed that this expression is very much 
 different from the corresponding expression obtained for the 
 adiabatic expansion of a perfect gas. This difference would 
 not exist if saturated vapors obeyed the law of Mariotte-Gay- 
 Lussac. 
 
 The following table is taken from Clansius * ; its verifica- 
 tion is recommended as a very useful exercise. It contains the 
 external work per kg. of saturated vapor when the initial 
 temperature is t = 150° C. and the initial quantity of the 
 liquid part is zero, hence M = ??? v 
 
 * Mech. "Warmetheorie, vol. i. p. 167. 
 
REVERSIBLE CYCLES IN OASES AND VAPORS. 93 
 
 t 
 
 150° 
 
 125° 
 
 100° 
 
 75° 
 
 50° 
 
 35° 
 
 w 
 
 
 
 11,300 
 
 23,200 
 
 35,900 
 
 49,300 
 
 63,700 
 
 Thus the external work done by a kg. of saturated water- 
 vapor in expanding adiabatically from initial temperature of 
 t = 150° C. until its temperature has sunk down to 100° C. 
 is 23200 kg. meters. 
 
 Obs. 1. In evaporating a kg. of water at 150° to a kg. of 
 steam at 150° work must be done against the vapor pressure 
 corresponding to that temperature. This work can be easily 
 found to be 18,700 kg. meters. 
 
 Obs. 2. In the above calculation Clausius assumed J — 
 423.55. 
 
 (y) External Work done during Isothermal Expansion. 
 
 During isothermal expansion pressure remains constant, 
 since temperature remains constant. The heat supplied is 
 utilized to compensate two distinct processes which, taken 
 together, constitute the operation of isothermal expansion. 
 The two processes are: separation of the vapor from the 
 liquid mass, and overcoming of the external pressure. Ac- 
 cording to our definition of p, that is, the latent heat of 
 evaporation, the total heat necessary to evaporate m kilo- 
 grammes of vapor is mp. That part of this total heat which 
 does the external work is given by 
 
 W 
 
 — J pdv. 
 
94 THERMODYNAMICS OF 
 
 But v = mu + Ma, and in isothermal expansion 
 
 dv = —- dm — udm. 
 dm 
 
 Sim 
 
 ,\ W = I pudm = pum, 
 
 since p and u remain constant during this expansion 
 
 We are ready now to take up the discussion of a complete 
 cycle of operations upon water-vapor. 
 
 THE INDICATOK DIAGEAM OF A SIMPLE CYCLE. 
 
 "We shall arrange the cycle in such a way that it shall ap- 
 proach as nearly as possible the cycles which occur in actual 
 steam-engines, without introducing the difficult questions 
 concerning the various causes which produce considerable 
 differences between the results obtained by pure theory and 
 the results obtained from the actual operation of the various 
 types of steam-engines. The existence of these differences is 
 no proof of the weakness of the fundamental features of our 
 theory. On the contrary, these very differences prove its 
 strength, for they point out the important fact that some of 
 the changes or phenomena which take place during the cyclic 
 operations of actual steam-engines have not been taken into 
 account in our theoretical calculations. Careful experimental 
 investigations of these ignored phenomena and the study of 
 their bearing upon the performance of the steam-engine, not 
 only mean the gradual extension towards a complete theory 
 
REVERSIBLE CYCLES IN OASES AND VAPORS. 95 
 
 of the steam-engine, but they also, by widening out that 
 theory, bring us nearer and nearer to a perfect steam-engine. 
 Fig. 9 represents symbolically the essential parts of a 
 modern steam-engine. A is the boiler, maintained at con- 
 stant temperature T, by the action of the heat supplied by 
 fuel. C is the condenser maintained at constant tempera- 
 
 ture T . To simplify our discussion, we shall suppose that 
 the condenser is maintained at constant temperature by 
 taking away all the heat that may be supplied to it during 
 the cycle which we are considering, and not by injection of 
 cold water. B is the cylinder in which a piston can glide Up 
 and down without friction. We shall suppose that the cylin- 
 der is impermeable to heat. The action of the pump D will 
 be explained presently. We shall now suppose that by special 
 mechanical contrivances we can at any moment connect the 
 
96 THERMODYNAMICS OF 
 
 cylinder to or disconnect it from either the boiler A or the 
 condenser G. 
 
 Operation 1. 
 
 At the beginning of the cycle the piston is at the bottom of 
 the cylinder. Steam is now admitted from the boiler, and 
 when its quantity is m 1 the connection with the boiler is cut 
 off. Suppose that a quantity of the liquid equal to M — m 1 
 also enters with the steam. The piston goes up until the 
 volume «, in the cylinder is 
 
 i>, = JMjMj + Mcr. 
 
 Since the pressure during this operation is constant and 
 equal to p 1 , (the saturated-vapor tension at the initial tem- 
 perature T^), the external work W 1 done during this operation 
 is given by 
 
 W 1 = p 1 (m 1 u 1 + Ma). 
 
 The quantity of heat Q^ consumed during this isothermal 
 expansion is m^p^ 
 
 In order to represent the cycle graphically, refer the press- 
 ures on the piston and the volume in the cylinder to a set of 
 rectangular axes OA and OB, taking the pressures in kg. per 
 sq. m. for ordinates and the corresponding volumes in cub. m. 
 for abseissse (Fig. 10*). 
 
 *This diagram was taken from an actual steam-engine. Boiler 
 gauge, 165 ; vacuum gauge, 12.75 ; rev. per minute, 76 ; back pressure, 
 17 lbs. 
 
REVERSIBLE CYCLES IN OASES AND VAPORS. 97 
 
 The first operation will be represented by the isothermal 
 line ab. The length of ab = Ob' represents the numerical 
 
 * 
 
 Fig. 10. 
 
 value of v 1 =m 1 u i -\- Mcr. The area abb'O is numerically 
 equal to W v 
 
 Operation 2. 
 
 Disconnect the cylinder from the boiler, and by gradually 
 diminishing the pressure let the steam expand from this point 
 on adiabatically until its temperature has sunk to T„ the 
 temperature of the condenser. During this operation the 
 steam, as was explained before, not only remains saturated 
 but even condenses rapidly. The adiabatic curve will be 
 somewhat like the curve be. Its exact form can be traced very 
 accurately by theoretical calculation, but since this is a some- 
 what tedious process we shall omit it from our discussion. 
 The external work done during this operation is represented 
 by the area lec'b'. Denoting it by W^ we shall have, as has 
 been shown before, 
 
 W,=m^p % — m l u l p l — J{(m,r, — m/,) — Mc{T l — T 2 ). }. 
 
96 THEBMODYNAMICS OF 
 
 Operation 3. 
 
 Connect the cylinder with the condenser and compress. 
 The initial volume in the cylinder is 
 
 v, = (171^ + M a). 
 
 Let the connection with the condenser be maintained until 
 the piston reaches its initial position again. Since the 
 pressure during this operation remains constant and equal 
 to j? a , the work done during this operation will be, denoting it 
 by W t , 
 
 w* = — P* v , ~ -~A( JM » M » + Ma )' 
 
 The curve of compression cd is an isothermal ; it is evidently 
 a straight line parallel to OB. The area CO' Od represents 
 numerically the value of W t . 
 
 This operation completes the cycle as far as the process in 
 the cylinder is concerned. The diagram abed is called the 
 indicator diagram of the cycle. Its area is numerically equal 
 to the external work done during the cyclic process. Let 
 this be W, then 
 
 W= W,+ W,+ W % 
 
 - Mc(T, - T,)}] + [-.?,(«,«, + Ma)] 
 = Ma( Pl - Pt ) + JMc{T x - y.) - J{m t r t - wi.r,). 
 
REVERSIBLE CYCLES IN GASES AND VAPORS. 99 
 
 Corrections for the Work expended upon tlie Pump. 
 
 The cycle given above is not complete because the state of 
 the boiler and condenser is not the same at the end of the 
 cycle as in the beginning of it. In the first place, the quantity 
 of the water in the condenser is increased and that in the 
 boiler diminished by M kilogrammes. To make the cycle 
 complete it is necessary now to transfer this quantity M from 
 the condenser- to the boiler by the action of the pump. The 
 work of the pump subtracted from W gives us then the avail- 
 able external work of the complete cycle. 
 
 The work of the pump is easily calculated thus : 
 In order to transfer the quantity M from the condenser to 
 the pump the piston of the pump must go up until it has in- 
 creased the available volume in the cylinder of the pump by 
 Mcr. During the upward stroke of the latter's piston a valve 
 connects the pump to the condenser, and the condenser press- 
 ure p a forces the quantity M into the cylinder. The work of 
 the pressure p t will evidently be 
 
 W t = p t M<r. 
 
 The valve connecting the pump to the condenser is then 
 turned off, and another valve is turned on which connects the 
 pump to the boiler. The piston of the pump is now pressed 
 down and the quantity M of the fluid is forced into the boiler 
 against the pressure p t . This work is evidently 
 
 W t = —pjMcr. 
 
100 THERMODYNAMICS OF 
 
 Adding W t + W & to W will give us W, that is, the available 
 external work done during the complete cycle. Hence 
 
 W = JMc(T t - TJ - J(m,r, - m.r,). 
 
 The quantity of heat expended so far to gain this amount of 
 external work is m^r^ But this is not all. For in order to 
 establish the same state of the boiler and condenser as at the 
 beginning of the cycle it is necessary to heat the quantity M 
 of the liquid, after it enters the boiler again, from the tem- 
 perature T^ to the temperature T, , for which an additional 
 quantity of heat, MC\T 1 — TJ, is required. We have there- 
 fore on the whole a gain of W units of available external 
 work at the expense of (very nearly) J{m 1 r 1 + Mc{T^ — TJ] 
 units of heat. 
 
 W 
 
 efficiency = E, 
 
 Jlm^+McW-TJ 
 
 or 
 
 Me{T, -r,)+ OTi r, -i»,r, _ „ 
 Mc(T, - T t ) + wi.r, - J °- 
 
 Calculation of the Efficiency in Terms of the Initial and 
 Final Temperatures. 
 
 In order to obtain a more suggestive comparison of this 
 result to the result which we obtained in the case of a simple 
 reversible cycle with a perfect gas, it is desirable to express E 
 in terms of initial and final temperature. The second opera- 
 
REVERSIBLE CYCLES IN OASES AND VAPORS. 101 
 
 tion being an adiabatic expansion, we have the following 
 relation : 
 
 Substituting this value of mj\ in the expression for E, we 
 obtain 
 
 / j-\ T — T 
 
 Mc[T, -T t +T t log J ) + in.r, -^-^ 
 
 E ~ Mc{T, - T,) + m,r r ' * 
 
 To simplify this equation somewhat, suppose that at the be- 
 ginning of the cycle no part of the mass is liquid. Hence 
 
 m 1 — M, 
 and 
 
 c(T, - T,) + r, T, 
 
 / y y\ 
 
 It can be easily shown that 1-= — ! - 7 =-log-^j is always 
 
 greater than unity as long as y, < T t . Hence the efficiency 
 of our cycle is smaller than that of a simple reversible cycle 
 performed upon a perfect gas. Now the efficiency of a 
 
102 THERMODYNAMICS OF 
 
 simple reversible cycle is independent of the substance 
 operated ujion, hence we are forced to admit that the cycle 
 described above is not a simple reversible cycle. A moment's 
 consideration will convince you that it evidently cannot be 
 considered as such. For in such a cycle heat should be 
 transmitted between bodies of equal or very nearly equal 
 temperatures; in other words, there should be at no moment 
 during the cycle a contact between bodies of widely different 
 temperatures. In our cycle this condition was not fulfilled, 
 for the temperature of the feed-water forced into the boiler by 
 the pump was not supposed to be the same as that of the 
 water in the boiler. The operation of the steam-engine which 
 we discussed is as near an approach to perfect reversibility as 
 the action of any steam-engine in operation to-day. And yet 
 its efficiency is below that of a Carnot reversible engine. We 
 traced the cause of this to the fact that the machine does not 
 operate by perfectly reversible cycles, and since no types of 
 steam-engines which now prevail operate by perfectly reversi- 
 ble cycles, it follows that the efficiency of the now prevailing 
 types of steam-engines would even under ideal conditions be 
 lower than that of Oarnot's reversible engine. 
 
 This remarkable fact is but a single illustration of a general 
 principle first announced by a young French engineer, Kicolas 
 Leonhard Sadi Carnot, in his immortal essay, " Reflexions 
 sur la, puissance mot rice du feu et sur les machines propres 
 a la developper" (Paris, 1S24). This principle may be stated 
 as follows: 
 
BEVEBSIBLE CYCLES IN GASES AND VAP0B8. 103 
 
 Garnot's Principle. 
 
 Of all engines working between the same source of heat and 
 refrigeration a reversible engine gives the maximum efficiency. 
 
 This principle contains Oarnot's solution of the problem : 
 " What are the conditions under which the maximum work 
 can be obtained from a given quantity of heat?" 
 
 Add to this principle the hypothesis that the efficiency of 
 a reversible engine is independent of the material operated 
 upon, or, what amounts to the same thing, that the efficiency 
 of all reversible engines is the same, and the second law of 
 thermodynamics in all its generality follows at once. But 
 this very hypothesis may be said to be the fundamental 
 hypothesis of Carnot. There is no doubt, therefore, that to 
 this great Frenchman belongs the glory of having discovered 
 one of the most important laws in the whole range of physi- 
 cal sciences. 
 
 If the first law of thermodynamics be defined as the prin- 
 ciple of equivalence betzveen heat and mechanical work and 
 not as a particular case of the Principle of Conservation of 
 Energy, then, according to historic evidences which we 
 possess to-day, Carnot is certainly the discoverer of the first 
 law as well. Observe here that from this principle of 
 equivalence to the general principle of conservation of 
 energy there are but few and easy steps.* 
 
 It is an established historical fact that Carnot had made 
 the necessary preparations to verify experimentally his ideas 
 on the two fundamental laws of the theory of heat when a 
 
 * See notes following the edition of 1878 of his great work, Re- 
 flexions, etc. (Gauthier-Villars, Paris). 
 
104 THERMODYNAMICS OF 
 
 premature death deprived the world of one of its most brill- 
 iant intellects. Carnot died in 1832, when only 36 years of 
 age. 
 
 Reversible Steam-engine. 
 
 We shall now close our discussion of thermodynamics of 
 saturated vapors by describing a simple reversible process 
 performed with saturated water-vapor. This will give us a 
 pretty faithful picture of the possible type of a perfectly 
 reversible steam-engine; it will also afford us an oppor- 
 tunity of showing by actual calculation that a Carnot engine 
 when operating upon a body which is not a perfect gas has 
 the same efficiency as when it operates upon such a gas. 
 
 Place a quantity of m, kilogrammes of water of temper- 
 ature T x into the cylinder of a Carnot engine. Let there be 
 two large reservoirs A and B of temperatures T t and T 3 re- 
 spectively. Let the piston be in contact with the liquid at 
 the beginning of the operation. 
 
 Connect the cylinder to A and allow the piston to go up 
 slowly. There will be evaporation, but the temperature and 
 consequently the pressure also will remain constant, because 
 A supplies the heat of evaporation. Let the evaporation 
 continue until all the water is evaporated. Denote the heat 
 supplied by A during this operation by Q x and the external 
 work done by W t , then 
 
 Q, = w.P, and W 1 = p^m,. 
 
 Disconnect the reservoir and let the steam expand adiabat- 
 ically until the temperature 1\ of the reservoir B is reached 
 
REVERSIBLE G7GLES IN GASES AND VAPORS. 105 
 
 The steam will remain saturated on account of condensation. 
 The external work W t done during this operation will be 
 
 W t = p l ii,m l — p l M l m 1 + m,p, - m,p, + m^T, - T t ). 
 
 Connect now to reservoir B and compress until the weight 
 of steam is reduced to m 3 . Temperature and therefore pres- 
 sure remain constant during this isothermal operation. The 
 external work W 3 done during this operation will be 
 
 W, = I pdv. 
 
 Bui. since v = mu 4-m,a and dv = —-dm = udm, 
 
 dm 
 
 /t»i 
 .'. W,= l pudm = p % ujrn 3 — m 3 ). 
 
 The quantity m 3 must be such that the next adiabatic opera- 
 tion shall be capable of reducing the whole quantity to liquid 
 at T L degrees. 
 
 By disconnecting the cylinder from the reservoir B we 
 compress adiabatically. The steam will condense and the 
 temperature will rise. But since the steam is continually in 
 contact with its own liquid it will remain saturated. At the 
 end of this operation we shall have again to, kg. of water at 
 T t degrees. The work done during this operation, denoting 
 it by W t , can be calculated from the relation which we de- 
 duced before, namely, 
 
 W - pum - p,u t m t + v, - nip + m 1 C(T % - T t ) ; 
 
106 THERMODYNAMICS OF 
 
 but we must remember that the weight m of steam at the end 
 of our operation is zero; hence 
 
 K = -jw». + m >P, + m^T, - TJ. 
 
 Denote the total external work during this cycle by W ' 
 then 
 
 W'= W x + W % + W s + W t 
 
 + [P> u A m , - »».)] + [~ ftV, + tn 3 p, + m, C{T a - T,)]. 
 
 The expressions in the brackets stand for the various W's. 
 This expression reduced, gives 
 
 W - (m, - m t )p, + W.P,. 
 
 We can express W in terms of the initial and final tem- 
 perature by the relation deduced in our discussion of the 
 condensation during adiabatic expansion, viz., 
 
 T 
 mj) t = - 7>, log jj?; 
 
 T 
 
 .: (m, - m t )p t = - -^m^. 
 -* i 
 
 This substituted in the expression for W gives 
 
 T — T 
 JP = », lPl -!--_\ 
 
 i 
 
REVERSIBLE CYCLES IN GASES AND VAPORS. 107 
 The efficiency of the cycle is, according to definition, 
 
 T.-T. 
 
 E=~ = 
 
 jp m iPi rp rp 
 
 The efficiency, therefore, is just the same as that obtained 
 by operating in a reversible manner upon a perfect gas. It 
 is, therefore, higher than the efficiency of the cycles of any 
 other type of steam-engine. 
 
 But why, then, should the now prevailing types of steam- 
 engines be so widely different from Carnot's type if a devia- 
 tion from this type brings with it a loss of efficiency ? The 
 complete answer to this question will be given in the course 
 on mechanical engineering. Let two suggestions suffice here: 
 first, efficiency is not the only guiding consideration in the 
 construction of steam-engines, or for that matter in the con- 
 struction of any other kind of engines. The question of out- 
 put is a very serious matter too, and in order to increase the 
 output we must very often make some sacrifice in the 
 efficiency. Secondly, the difference between the efficiency 
 of the now prevailing types of steam-engines and that of the 
 ideal type just described is due far more to other things than 
 to the fact that the operations which we discussed above 
 are not perfectly reversible. Consider the losses due to 
 radiation, the action of the conductingc ylinder-walls on 
 the expanding steam, etc., etc. So that the completeness of 
 the reversibility of the operations through which the steam 
 passes in these engines is not quite as important a matter as 
 it might appear at first sight. 
 
108 THERMODYNAMICS OF 
 
 SUMMAET. 
 
 1. Method of Reasoning which led to the General Form of the 
 Second Law of Thermodynamics. 
 
 After discussing the application of the first law of thermo- 
 dynamics to the study of reversible operations performed 
 upon perfect gases we passed on to a similar discussion for 
 the next simplest class of bodies, that is, saturated vapors. 
 We suggested, however, that it would be advisable to deduce 
 first another general law which, together with the first law, 
 forms the foundation of the science of thermodynamics : we 
 mean the second law. In doing that we followed practically 
 in the steps of the great Carnot. For he had to discover 
 the second law before he was able to pass from the study of 
 the reversible gas-engine to that of the steam-engine. 
 
 The first and simplest form of the second law was obtained 
 by considering the efficiency of a Carnot engine operating 
 reversibly upon a perfect gas by simple cycles. This form 
 consisted in a mathematical statement which expressed that 
 the efficiency of such cycles depends on the extreme tempera- 
 tures of the cycles, thus : 
 
 Q _ T X -T % 
 
 We then introduced the axiom of Clausius and by means of 
 this axiom we showed that the efficiency of a simple reversible 
 cycle is independent of the substance operated upon. Hence 
 it is always 
 
 Q _ y, - r, 
 
REVERSIBLE CYCLES IN GASES AND VAPORS. 109 
 "We then transformed this relation into the following : 
 
 qi ~T~ rp V* 
 
 It was stated that this transformed expression is preferable 
 because it lends itself more readily to generalization. In 
 order to effect this generalization we passed on to the con- 
 sideration of a complex reversible cycle and obtained the rela- 
 tion 
 
 ~7p ~r rp T/T7 T • • • T ~fn~ — «» 
 
 or 
 
 ■■"^ rp v * 
 
 We finally considered a reversible cycle of infinite com- 
 plexity and obtained 
 
 / 
 
 d 4=o, 
 
 where the integral is to be extended all around the closed 
 curve which represents graphically the reversible cycle of 
 infinite complexity, that is to say, all around the indicator 
 diagram. 
 
 This is the integral form of the second law for reversible 
 cycles. But inasmuch as the integral laws in general describe 
 resultant effects only of a series of physical processes and 
 throw but very little light upon each process of that series, we 
 
110 THERMODYNAMICS OF 
 
 proceeded to deduce from the integral law / -=- = a dif- 
 ferential law, that is to say a relation which will hold true 
 during every interval, no matter how small, of the entire time 
 during which the cyclic process is completed. We obtained 
 
 That is to say, the heat absorbed by a body during any one 
 of the infinite number of infinitely small operations which 
 taken together form the complete reversible cycle is equal to 
 the absolute temperature of the body multiplied by the incre- 
 ment of the entropy of the body. The definition of the 
 entropy being : 1st. Mathematical and complete. It is a finite, 
 continuous, and singly-valued function of the co-ordinates 
 which define the state of the body, that is, of pressure, volume, 
 and temperature, which satisfies the above differential equa- 
 tion for any reversible operation. 2d. Physical meaning. Its 
 variation during an isothermal operation is equal to the heat 
 absorbed or given off, divided by the absolute temperature at 
 which the operation takes place. Its variation is zero during 
 an adiabatic operation. 
 
 2. Forms of the Tivo Laws in the case of Reversible Opera- 
 tions on Saturated Vapors. 
 
 Starting, then, with our two laws of thermodynamics, 
 
 dQ — dU -\-pdv, 
 
 dQ = TdS, 
 
REVERSIBLE CYCLES IN GASES AND VAPORS. Ill 
 
 we took up the discussion of saturated vapors, and in this 
 discussion we followed up as much as possible the line of 
 reasoning which we employed in the discussion of reversible 
 processes in perfect gases. 
 
 1st. We discussed the most essential physical properties of 
 saturated vapors by consulting carefully the records of physi- 
 cal research. 
 
 2d. "We then performed two simple processes for the purpose 
 of defining the physical constants of a saturated vapor. These 
 two processes we called the definitional processes (A) and (B). 
 They gave us the following two definitional relations : 
 
 ^ = m(H-C) + MG, 
 
 dm ' 
 
 These two definitional relations enabled us then to express the 
 two laws of thermodynamics in the case of saturated vapors 
 in a very convenient form, viz., 
 
 ^L+C-E = u^, (I) 
 
 |£-+(7-tf=|, (II) 
 
 and by combining these two we obtained a third very con- 
 venient relation: 
 
 fi=Tu^- (Ill) 
 
112 THERMODYNAMICS OF 
 
 By introducing the kg. calorie as the unit of heat (as is gener- 
 ally done in experimental researches) we obtained 
 
 QT + C - h -jQT' ' • • • (I) 
 
 §r + c - h = i (IP) 
 
 _ Tu dp 
 r -~JdT" • • • (III) 
 
 3. Application of the Two Laws to the Study of the Physical 
 
 Constants of Saturated Vapor. 
 
 We then proceeded to apply these fundamental thermo- 
 dynamical relations to the study of saturated vapors, and in 
 particular to the study of saturated water-vapor. We divided 
 this discussion into three parts. One part related to the dis- 
 cussion of the constants c, li, r, u, and p, especially to the 
 methods of calculating two of them, h and u, when the other 
 three are given by experimental data. The negative value of 
 the specific heat, the increase of density with the temperature, 
 and the deviation of the behavior of saturated water-vapor 
 from the law of Mariotte-Gay-Lussac were discussed with 
 particular emphasis. 
 
 4. Application of the Two Laws to the Study of Isothermal 
 and Adiaoatic Expansion of Saturated Water-vapor. 
 
 Following the method of discussion which we employed in 
 the case of perfect gases we then took up the very important 
 
REVERSIBLE CYCLES IN CASES AND VAPORS. 113 
 
 part of our work in thermodynamics, that is, adiabatic and 
 isothermal expansion of saturated water-vapor, the condensa- 
 tion which takes place during adiabatic expansion and the 
 superheating which takes place during adiabatic compression 
 were particularly dwelt upon. The external work done dur- 
 ing adiabatic and isothermal operations received our most 
 careful attention. 
 
 5. Non-reversible and Reversible Cycles of Operation upon 
 Saturated Water-vapor. 
 
 We were ready then to take up the question of the theo- 
 retical efficiency of the non-reversible and the reversible types 
 of steam-engines. This question is, from an engineering 
 standpoint, the most important of all questions in thermo- 
 dynamics. It is also exceedingly important from a historical 
 point of view, for the study of this question led Carnot to the 
 discovery of the two laws of thermodynamics. 
 
 The external work done during each component operation 
 of these cycles was calculated and expressed by f ormulse which 
 admit of being reduced to actual figures, so that the action of 
 each component operation could be studied by finding the 
 numerical value of this action in terms of kg. meters. The 
 theoretical efficiency of non-reversible machines was then 
 shown to be smaller than that of the reversible type, and the 
 efficiency of this was then demonstrated by actual calculation 
 to be the same as that of a Carnot reversible engine operating 
 by simple reversible cycles upon a perfect gas. 
 
 There remains still another important application of the 
 two laws of thermodynamics which is within the limits of 
 
114 THERMODYNAMICS. 
 
 this purely theoretical course. It is the application to the 
 study of the flow of gases and vapors. In this part of our 
 discussion we propose to follow closely the beautifully worked 
 out Chapter IX in Peabody's " Thermodynamics of the Steam- 
 engine." 
 
 These are the most essential elements of theoretical thermo- 
 dynamics, upon which as a strong foundation the practical 
 engineer must raise the vast structure of the beautiful science 
 of Caloric Engineering. 
 
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