" 
 
 2 7 1997
 
 HARPER'S SCIENTIFIC MEMOIRS 
 
 EDITED BY 
 
 J. S. AMES, PH.D. 
 
 PROFESSOR OF PHYSICS IN JOHNS HOPKINS UNIVERSITY 
 
 VI. 
 THE SECOND LAW OF THERMODYNAMICS 
 
 SCIENCE & ENGINEERING 
 LIBRARY 
 
 AUG 2 7 1997 
 
 PHYSICS COLLECTION 
 UCLA
 
 THE SECOND LAW OF 
 THERMODYNAMICS 
 
 MEMOIRS BY CARNOT, CLAUSIUS 
 AND THOMSON 
 
 TRANSLATED AND EDITED 
 
 BY W. F. MAGIE, FH D. 
 
 PROFESSOR OF PHYSICS IN PRINCKTON UNIVERSITY 
 
 NEW YORK AND LONDON 
 
 HARPER & BROTHERS PUBLISHERS 
 1899
 
 HARPER'S SCIENTIFIC MEMOIRS. 
 
 XDITKO BT 
 
 J. S. AMES, PH.D., 
 i-Boruwok op riiTsioa IN JOHNS HOPKINS VNIVKEBITY. 
 
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 Qc 
 
 PREFACE 
 
 AFTER the invention of the steam-engine in its present form 
 by James Watt, the attention of engineers and of scientific 
 men was directed to the problem of its further improvement. 
 With this end in view, the young Sadi Carnot, in 1824, pub- 
 lished the Reflexions sur la Puissance Motrice du Feu, of which 
 this translation is given in this volume. In* this really great 
 memoir, Carnot examined the relations between heat and the 
 work done by heat used in an ideal engine, and by reducing 
 the problem to its simplest form and avoiding all special ques- 
 tions relating to details, he succeeded in establishing the condi- 
 tions upon which the economical working of all heat-engines 
 depends. It is not necessary here to animadvert upon the use 
 made by Carnot of the substantial theory of heat, and the con- 
 sequent failure of the proof of his main proposition when the 
 true nature of heat was appreciated. It is sufficient to say 
 that though the proof was invalid, the proposition remained 
 true, and carried with it the truth of such of Carnot's deduc- 
 tions as were based solely upon it. 
 
 Carnot's memoir remained for a long time unappreciated, 
 and it was not until use was made of it by William Thomson 
 (now Lord Kelvin), in 1848, to establish an absolute scale of 
 temperature, that the merits of the method proposed in it were 
 recognized. In his first paper on this subject Thomson re- 
 tained the substantial theory of heat, but the evidence in favor 
 of the mechanical theory became so strong that he soon after 
 adopted the new view. Applying it to the questions treated 
 by Carnot, he found that Carnot's proposition could no longer 
 be proved by denying the possibility of " the perpetual motion," 
 and was led to lay down a second fundamental principle to serve 
 in the demonstration. This principle is now called the Second 
 Law of Thermodynamics. A part of the memoir in which this
 
 PREFACE 
 
 principle is stated :ml many of its consequences developed is 
 given in this volume. It was publislu-d in March. is.M. 
 
 In the previous year Clansius published a discussion of the 
 same question as that treated by Thomson, in which he lays 
 down a principle for use in the demonstration of Cairnot's propo- 
 sitton. which, while not the same in form as Thomson's, is the 
 same in content, and ranks as another statement of the Second 
 Law of Thermodynamics. \\\^ paper is also given in this vol- 
 ume. While not so powerful or so inclusive as Thomson's, it 
 deserves attention for the clearness and simplicity of its form. 
 Clausing followed up this paper by others, and subsequently 
 published a book in which the subject of Thermodynami--- 
 L r iven a systematic treatment, and in which he introduced and 
 developed the important function called by him the entropy. 
 
 The science of Thermodynamics, founded by the labors of 
 these three illustrious men, has led to the most important de- 
 velopments in all departments of physical science. It lia- 
 pointed out relations among the properties of bodies which 
 could scarcely have been anticipated in any other way : it ha- 
 laid the foundation for the Science of Chemical Physics ; and. 
 taken in connection with the kinetic theory of gases, as devel- 
 oped by Maxwell and Boltzmann, it has furnished a funeral 
 view of the operations of the universe which is far in advance 
 of any that could have been reached by purely dynamical rea- 
 soning.
 
 GENERAL CONTENTS 
 
 PAGE 
 
 Preface v 
 
 Reflections on the Motive Power of Heat. By Sadi Carnot 3 
 
 Biographical Sketch of Carnot 60 
 
 Ou the Motive Power of Heat, and on the Laws which can be De- 
 duced from it for the Theory of Heat. By R. Clausius 65 
 
 Biographical Sketch of Clausius 107 
 
 The Dynamical Theory of Heat. (Selected Portions.) By William 
 
 Thomson (Lord Kelvin) Ill 
 
 Biographical Sketch of Lord Kelvin 147 
 
 Bibliography 149 
 
 INDEX.. .. 151
 
 REFLECTIONS ON 
 THE MOTIVE POWER OF HEAT 
 
 BY 
 
 SADI CAENOT 
 
 Paris, 1824
 
 CONTENTS 
 
 PAG I 
 
 Ueat-enginet 8 
 
 JbB of Temperature 7 
 
 Reversible Processes 11 
 
 "Carnot's Cycle " 10 
 
 Efficiency a Function of Limiting Temperatures -2(\ 
 
 Specific Heats of Gate* 21 
 
 Motive Ptoeer of Air, Steam, Alcohol Vapor 40 
 
 Greatest Efficiency 49 
 
 I ;,rinvt Types of Machines 62 
 
 Advantages and Disadvantages of Steam 55
 
 REFLECTIONS ON 
 
 THE MOTIVE POWER OF HEAT AND 
 
 ON ENGINES SUITABLE FOR 
 
 DEVELOPING THIS POWER 
 
 BY 
 
 SADI CARNOT 
 
 IT is well known that heat may be nsed as a cause of motion, 
 and that the motive power which may be obtained from it is 
 very great. The steam-engine, now in such general use, is a 
 manifest proof of this fact. 
 
 To the agency of heat may be ascribed those vast disturb- 
 ances which we see occurring everywhere on the earth ; the 
 movements of the atmosphere, the rising of mists, the fall of 
 rain and other meteors,* the streams of water which channel 
 the surface of the earth, of which man has succeeded in util- 
 izing only a small part. To heat are due also volcanic erup- 
 tions and earthquakes. From this great source we draw the 
 moving force necessary for our use. Nature, by supplying 
 combustible material everywhere, has afforded us the means of 
 generating heat and the motive power which is given by it, at 
 all times and in all places, and the steam-engine has made it 
 possible to develop and use this power. 
 
 The study of the steam-engine is of the highest interest, 
 owing to its importance, its constantly increasing use, and the 
 great changes it is destined to make in the civilized world. It 
 has already developed mines, propelled ships, and dredged 
 rivers and harbors. It forges iron, saws wood, grinds grain, 
 spins and weaves stuffs, and transports the heaviest loads. In 
 the future it will most probably be the universal motor, and 
 
 * [Any atmospJieric phenomenon was formerly called a meteor.]
 
 MKMuIKS ON 
 
 will furnish the power now obtained from animals, from water- 
 falls, and from air-currents. Over the first of these motors it 
 has the advantage of economy, and over the other two the in- 
 calculable advantage that it can be used everywhere and always. 
 and that its work need never be interrupted. If in the future 
 the steam-engine is so perfected as to render it less costly to 
 construct it and to supply it with fuel, it will unite all desir- 
 able qualities and will promote the development of the indus- 
 trial arts to an extent which it is difficult to foresee. It is, in- 
 deed, not only a powerful and convenient motor, which can be 
 set up or transported anywhere, and substituted for other 
 motors already in use, but it lends to the rapid extension of 
 those arts in which it is used, and it can even create arts hith- 
 erto unknown. 
 
 The most signal service which has been rendered to England 
 by the steam-engine is that of having revived the working of 
 her coal-mines, which had languished and was threatened with 
 extinction on account of the increasing ditlirulty of excavation 
 and extraction of the coal.* We may place in the second rank 
 the services rendered in the manufacture of iron, as much by 
 furnishing an abundant supply of coal, which took the place of 
 wood as the wood began to be exhan-te.l. a- l>y the powerful 
 machines of all kinds tho use of which it either facilitated or 
 made possible. 
 
 Iron and fire, as every one knows, are the mainstays of the 
 mechanical arts. Perhaps there is not in all England a single 
 industry whose existence is not dependent <>n these agents, and 
 which does not use them extensively. If England were to-day 
 to lose its steam-engines it would lose also its coal and iron, 
 and this loss would dry up all its sources of wealth and destroy 
 its prosperity ; it would annihilate this colossal power. The 
 destruction of its navy, which it considers its strongest support, 
 would be, perhaps, less fatal. 
 
 The safe and rapid navigation by means of steamships is an 
 
 One may wifely say that the mining of coal has increased tenfold since 
 
 the Jnv.-nti. f tho steam engine. Tin* mining of copper, <>f tin. mid <>f 
 
 iron has increased almost as much. The effect produced half n century 
 ago in the mines of Knd.ind i> imw IH-III^ r< \ -uted in the irnld and -liver 
 mines of the New W<H|,|, the working of which was steadily declining. 
 principally on account of the insulli. ! -ncy of the motors used for the ex- 
 cavation and extraction of the minerals. 
 4
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 entirely new art due to the steam-engine. This art has already 
 made possible the establishment of prompt and regular com- 
 munication on the arms of the sea, and on the great rivers of 
 the old and new continents. By means of the steam-engine 
 regions still savage have been traversed which but a short time 
 ago could hardly have been penetrated. The products of civ- 
 ilization have been taken to all parts of the earth, which they 
 would otherwise not have reached for many years. The navi- 
 gation due to the steam-engine has in a measure drawn to- 
 gether the most distant nations. It tends to unite the peoples 
 of the earth as if they all lived in the same country. In fact, 
 to diminish the duration, the fatigue, the uncertainty and dan- 
 ger of voyages is to lessen their length.* 
 
 The discovery of the steam-engine, like most human inven- 
 tions, owes its birth to crude attempts which have been attrib- 
 uted to various persons and of which the real author is not 
 known. The principal discovery consists indeed less in these 
 first trials than in the successive improvements which have 
 brought it to its present perfection. There is almost as great 
 a difference between the first structures where expansive force 
 was developed and the actual steam-engine as there is between 
 the first raft ever constructed and a man-of-war. 
 
 If the honor of a discovery belongs to the nation where it 
 acquired all its development and improvement, this honor can- 
 not in this case be withheld from England : Savery, Newcomen, 
 Smeaton, the celebrated Watt, Woolf, Trevithick, and other 
 English engineers, are the real inventors of the steam-engine. 
 At their hands it received each successive improvement. It is 
 natural that an invention should be made, improved, and per- 
 fected where the need of it is most strongly felt. 
 
 In spite of labor of all sorts expended on the steam-engine, 
 and in spite of the perfection to which it has been brought, 
 its theory is very little advanced, and the attempts to bet- 
 ter this state of affairs have thus far been directed almost at 
 random. 
 
 The question has often been raised whether the motive power 
 
 * We speak of diminishing the danger of voyages ; in fact, though the 
 use of the steam-engine in ships is attended with some dangers, these are 
 al ways exaggerated and are compensated for by the ability of ships to keep 
 a definite course, and to resist winds which would otherwise drive the ves- 
 sel on the coast, or on shoals or reefs. 
 5
 
 MEMOIRS OX 
 
 of heat is limited or not ;* whether there is a limit to the pos- 
 sible improvements of the steam-engine which, in the nature of 
 the case, cannot be passed by any means ; or if, on the other 
 hand, these improvements are capable of indefinite extension. 
 Inventors have tried for a long time, and are still trying, to 
 find whether there is not a more efficient agent than water Ky 
 which to develop the motive power of heat; whether, for e\- 
 ample, atmospheric air does not offer great advantages in this 
 respect. We propose to submit these questions to a critical 
 examination. 
 
 The phenomenon of the production of motion by heat hus 
 not been considered in a sufficiently general way. It has leen 
 treated only in connection with machines whoso nature ami 
 mode of action do not admit of a full investigation of it. In 
 such machines the phenomenon is, in a measure, imperfect and 
 incomplete ; it thus becomes difficult to recognize its principles 
 and study its laws. To examine the principle of t lie production 
 of motion by heat in all its generality, it must be conceive.! in- 
 dependently of any mechanism or of any particular agent ; it 
 is necessary to establish proofs applicable not only to steam- 
 engines f but to all other heat-engines, irrespective of the work- 
 ing substance and the manner in which it acts. 
 
 The machines which are not worked by heat for instance, 
 those worked by men or animals, by water-falls, or by air cur- 
 rents can be studied to their last details by the principles of 
 mechanics. All possible cases may be anticipated, all imagi- 
 nable actions are subject to general principles already well es- 
 tahlished ;md applicable in all circumstances. The theory of 
 snch machines is complete. Such a theory is evidently lacking 
 for heat-engines. We shall never possess it until the laws of 
 physics are so extended and generalixed as to make known in 
 advance all the effects of heat acting in a definite \\-.\\ on any 
 body whatsoever. 
 
 * The expression motive power here signifies the useful cfTcci that a 
 motor in capable of producing. This effect may always he nnnsiin.1 in 
 terms of Ibe elevation of a weight through a certain distance ; it is u>. .1- 
 UP'. I. as it well known, by the product of the weight ami the In ijiii to 
 which it is raised. 
 
 f We distinguish hen- between the stcnm-cnginc and the heat-engine in 
 general. which can be worked by nny agent, and uot by water vapor only, 
 to realize the motive power of heat. 
 6
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 We shall take for granted in what follows a knowledge, at 
 least a superficial one, of the various parts which compose an 
 ordinary steam-engine. "We think it unnecessary to describe 
 the fire-box, the boiler, the steam -chest, the piston, the con- 
 denser, etc. 
 
 The production of motion in the steam-engine is always ac- 
 companied by a circumstance which we should particularly no- 
 tice. This circumstance is the re-establishment of equilibrium 
 in the caloric* that is, its passage from one body where the 
 temperature is more or less elevated to another where it is 
 lower. What happens, in fact, in a steam-engine at work? 
 The caloric developed in the fire-box as an effect of combus- 
 tion passes through the wall of the boiler and produces steam, 
 incorporating itself with the steam in some way. This steam, 
 carrying the caloric with it, transports it first into the cylinder, 
 where it fulfils some function, and thence into the condenser, 
 where the steam is precipitated by coming in contact with cold 
 water. As a last result the cold water in the condenser receives 
 the caloric developed by combustion. It is warmed by means 
 of the steam, as if it had been placed directly on the fire-box. 
 The steam is here only a means of transporting caloric ; it thus 
 fulfils the same office as in the heating of baths by steam, 
 with the exception that in the case in hand its motion is ren- 
 dered useful. 
 
 We can easily perceive, in the operation which we have just 
 described, the re- establishment of equilibrium in the caloric 
 and its passage from a hotter to a colder body. The first of 
 these bodies is the heated air of the fire-box ; the second, the 
 water of condensation. The re-establishment of equilibrium of 
 the caloric is accomplished between them if not complete- 
 ly, at least in part ; for, on the one hand, the heated air after 
 having done its work escapes through the smoke-stack at a 
 much lower temperature than that which it had acquired by 
 the combustion ; and, on the other hand, the water of the con- 
 denser, after having precipitated the steam, leaves the engine 
 with a higher temperature than that which it had when it 
 entered. 
 
 The production of motive power in the steam-engine is 
 
 * [Caloric u lieat considered as an indestructible substance. T/ie word is 
 used by Carnot interchangeably with fen, fire, or heat.] 
 
 7
 
 MKMOIRS ON 
 
 therefore not due to a real consumption of the caloric, but t //> 
 transfer from a hotter /<> a n>1tli>r fault/ that is to say, to tin- iv- 
 establishment of its equilibrium, which is assumed to have been 
 destroyed by a chemical action such as combustion, or by some 
 other cause. We shall soon see that this principle is applica- 
 ble to all engines operated by heat. 
 
 According to this principle, to obtain motive power it is not 
 enough to produce heat ; it is also necessary to provide cold, 
 without which the heat would be useless. For if there exist- 
 ed only bodies as warm as our furnaces, how would the con- 
 densation of steam be possible, and where could it be sent if it 
 were once produced? It cannot be replied that it could be 
 ejected into the atmosphere, as is done with certain engines,* 
 since the atmosphere would not receive it. In the actual state 
 of things the atmosphere acts as a vast condenser for the steam. 
 because it is at a lower temperature ; otherwise it would soon 
 be saturated, or, rather, would be saturated in advance.f 
 
 Everywhere where there is a difference of temperature, and 
 where the re-establishment of equilibrium of the caloric can lie 
 effected, the production of motive power is possible. Water 
 vapor is one agent for obtaining this power, but it is not the 
 only one ; all natural bodies can be applied to this purpose, for 
 they are all susceptible to changes of volume, to successive 
 contractions and dilatations effected by alternations of heat and 
 cold ; they are all capable, by this change of volume, of over- 
 coming resistances and thus of developing motive power. A 
 
 * Some high -pressure engines eject vapor into the atmosphere 
 of condensing it. They are used mostly in places where it is difficult to 
 procure a current of cold water sutn.-i.-ni t ctTcct condensation. 
 
 t The existence of water in a liquid state, which is here necessarily as- 
 sumed, since without it the steam engine could not be supplied, presup- 
 poses the existence of a pressure capable of preventing it from evaporating, 
 and consequently of a pressure equal to or greater than the tension of the 
 vapor at the temperature of the water. If such a pressure were not ex- 
 erted by the atmosphere a quantity of water vapor would instantly be pro- 
 duced sufficient to exert this pressure on itself, and this pressure must 
 always be overcome in ejecting the steam of the engine into the new at- 
 mosphere. This is evidently equivalent to overcoming the ton-inn whi.-ii 
 is exerted by the vapor after it has been condensed by the ordinary means. 
 
 If a very high temperature were to prevail at the surface of the earth, as 
 it almost certainly does in its interior, all the water of the oceans would 
 exist in the form of vapor in the atmosphere, and there would be no water 
 in a liquid state. 
 
 8
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 solid body, such as a metallic bar, when alternately heated and 
 cooled, increases and diminishes in length and can move bod- 
 ies fixed at its extremities. A liquid, alternately heated and 
 cooled, increases and diminishes in volume and can overcome 
 obstacles more or less great opposed to its expansion. An 
 aeriform fluid undergoes considerable changes of volume with 
 changes of temperature ; if it is enclosed in an envelope capa- 
 ble of enlargement, such as a cylinder furnished with a piston, it 
 will produce movements of great extent. The vapors of all bod- 
 ies which are capable of evaporation, such as alcohol, mercury, 
 sulphur, etc., can perform the same function as water vapor. 
 This, when alternately heated and cooled, will produce motive 
 power in the same way as permanent gases, without returning to 
 the liquid state. Most of these means have been proposed, several 
 have been even tried, though, thus far, without much success. 
 
 We have explained that the motive power in the steam-engine 
 is due to a re-establishment of equilibrium in the caloric ; this 
 statement holds not only for steam-engines but also for all heat- 
 engines that is to say, for all engines in which caloric is the 
 motor. Heat evidently can be a cause of motion only through 
 the changes of volume or of form to which it subjects the body ; 
 those changes cannot occur at a constant temperature, but are 
 due to alternations of heat and cold ; thus to heat any sub- 
 stance it is necessary to have a body warmer than it, and to 
 cool it, one cooler than it. We must take caloric from the 
 first of these bodies and transfer it to the second by means of 
 the intermediate body, which transfer re-establishes, or, at least, 
 tends to re-establish, equilibrium of the caloric. 
 
 At this point we naturally raise an interesting and important 
 question : Is the motive power of heat invariable in quantity, 
 or does it vary with the agent which one uses to obtain it 
 that is, with the intermediate body chosen as the subject of the 
 action of heat ? 
 
 It is clear that the question thus raised supposes given a cer- 
 tain quantity of caloric* and a certain difference of temperature. 
 
 * It is unnecessary to explain here what is meant by a quantity of ca- 
 loric or of heat (for we use the two expressions interchangeably), or to de- 
 scribe how these quantities are measured by the calorimeter ; nor shall we 
 explain the terms latent heat, degree of temperature, specific heat, etc. 
 The reader should be familiar with these expressions from his study of the 
 elementary treatises of physics or chemistry. 
 9
 
 M HMO IRS OX 
 
 For example, we suppose that we have at onr disposal a body. . I . 
 maintained at the temperature 100 degrees, and another Im.ly. 
 B, at degrees, and inquire what quantity of motive power 
 will be produced by the transfer of a given quantity of caloric 
 for example, of so much as is necessary to melt a kilogram of 
 ice from the first of these bodies to the second ; we inquire if 
 this quantity of motive power is necessarily limited ; if it varies 
 with the substance used to obtain it; if water vapor offers in 
 this respect more or less advantage than vapor of alcohol or of 
 mercury, than a permanent gas or than any other substance. 
 We shall try to answer these questions in the light of the con- 
 siderations already advanced. 
 
 We have previously called attention to the fact, which is self- 
 evident, or at least becomes so if we take into consideration the 
 changes of volume occasioned by heat, that wherrrrr ///>/> {.* a 
 difference of temperature tkt production <i/nifin- jtmn'r is />ns*i/i/,: 
 Conversely, wherever this power can be employed, it is possible 
 to produce a difference of temperature or to destroy the equili- 
 brium of the caloric. Percussion and friction of bodies are 
 means of raising their temperature spontaneously* to a higher 
 degree than that of surrounding bodies, and consequently of 
 destroying that equilibrium in the caloric which had previously 
 existed. It is an experimental fact that the temperature of 
 gaseous fluids is raised by compression and lowered by expan- 
 sion. This is a sure method of changing the temperature of 
 bodies, and thus of destroying the equilibrium of the caloric in 
 the same substance, as often as we please. Steam, when used 
 in a reverse way from that in which it is used in the steam- 
 engine, can thus be considered as a means of destroying the 
 equilibrium of the caloric. To be convinced of this, it is only 
 necessary to notice attentively the way in which motive power 
 is developed by the action of heat on water vapor. Let us 
 consider two bodies, A and B, each maintained ut a constant 
 temperature, that of A being higher than that of />' : these two 
 bodies, which can either give up or receive heat without a 
 change of temperature, perform the funetions of two indefi- 
 nitely great reservoirs of calorie. We will call the first body 
 the source and the second the refrigerator. 
 
 If we desire to produce motive power by the transfer of a 
 
 * [ That it, without the communication of heat.] 
 10
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 certain quantity of heat from the body A to the body B we 
 may proceed in the following way : 
 
 1. We take from the body A a quantity of caloric to make 
 steam that is, we cause A to serve as the fire-pot, or rather 
 as the metal of the boiler in an ordinary engine ; we assume 
 the steam produced to be at the same temperature as the 
 body A. 
 
 2. The steam is received into an envelope capable of enlarge- 
 ment, such as a cylinder furnished with a piston. We then in- 
 crease the volume of this envelope, and consequently also the 
 volume of the steam. The temperature of the steam falls when 
 it is thus rarefied, as is the case with all elastic fluids ; let us 
 assume that the rarefaction is carried to the point where the 
 temperature becomes precisely that of the body B. 
 
 3. We condense the steam by bringing it in contact with B 
 and exerting on it at the same time a constant pressure until it 
 becomes entirely condensed. The body B here performs the 
 function of the injected water in an ordinary engine, with the 
 difference that it condenses the steam without mixing with it 
 and without changing its own temperature.* The operations 
 which we have just described could have been performed in a 
 reverse sense and order. There is nothing to prevent the for- 
 
 * It will perhaps excite surprise that B, being at the same temperature 
 as the steam, can condense it. Without doubt this is not rigorously possi- 
 ble, but the smallest difference in temperature will determine condensa- 
 tion. This remark is sufficient to establish the propriety of our reasoning. 
 In the same way, in the differential calculus, to obtain an exact result it is 
 sufficient to be able to conceive of the quantities neglected as capable of 
 being indefinitely diminished relative to the quantities retained in the equa- 
 tion. 
 
 The body B condenses the steam without changing its own temperature. 
 We have assumed that this body is maintained at a constant temperature. 
 The caloric is therefore taken from it as fast as it is given up to it by the 
 steam. An example of such a body is furnished by the metallic walls of 
 the condenser when the vapor is condensed in it by means of cold water 
 applied to the outside, as is done in some engines. In the same way the 
 water of a reservoir can be maintained at a constant level, if the liquid runs 
 out at one side as fast as it comes in at the other. 
 
 One could even conceive the bodies A and B such that they would re- 
 main of themselves at a constant temperature though losing or gaining 
 quantities of heat. If, for example, the body A were a mass of vapor 
 ready to condense and the body B a mass of ice ready to melt, these bodies, 
 as is well known, could give out or receive caloric without changing their 
 temperature. 
 
 11
 
 MEMOIRS ON 
 
 mat ion of vapor by means of the caloric of the body B, and its 
 compression from the temperature of 11, in such a way that it 
 acquires the temperature of the body A, and then its condensa- 
 tion in contact with A, under a pressure which is maintained 
 constant until it is completely liquefied. 
 
 In the first series of operations there is at the same time a 
 production of motive power and a transfer of caloric from the 
 body A to the body R ; in the reverse series there is at the 
 same time an expenditure of motive power and a return of the 
 caloric from B to A. Hut if in each case the sumo quantity of 
 vapor has been used, if there is no loss of motive power or of 
 caloric, the quantity of motive power produced in the first 
 case will equal the quantity expended in the second, and the 
 quantity of caloric which in the first case passed from A to B 
 will equal the quantity which in the second case returns from 
 B to A, so that an indefinite number of such alternating oper- 
 ations can be effected without the production of motive power 
 or the transfer of caloric from one body to the other. Now if 
 there were any method of using heat preferable to that whieh 
 we have employed, that is to say, if it were possible that the 
 caloric should produce, by any process whatever, a larger quan- 
 tity of motive power than that produced in our first series of 
 operations, it would be possible, by diverting a portion of this 
 power, to effect a return of caloric, by the method just indi- 
 cated, from the body B to the body A that is, from the refrig- 
 erator to the source and thus to re-establish things in their 
 original state, and to put them in position to recommence an 
 operation exactly similar to the first one, and so on : there 
 would thus result not only the perpetual motion, but an indef- 
 inite creation of motive power without consumption of caloric 
 or of any other agent whatsoever. Such a creation is entirely 
 contrary to the ideas now accepted, to the laws of mechanics 
 and of sound physics ; it is inadmissible.* We may hence con- 
 
 * The objection will perhaps here be made that perpetual motion has 
 only been demonstrated lo be impossible in the case of mechanical actions, 
 and that it may not be so when we employ the agency of lira: 
 iricity ; but can we conceive of the phenomena of heat ami <.f !< triciiy 
 a* due lo any other cause than some motion of bodies, nml, as such. slnmM 
 they not be subject to the general laws of mechanics? 15- -i ! -. <l . \\,- not 
 know a potttriori that all the attempts made to produce perpetual motion 
 by any means whatever have been fruitless ; that no truly perpetual inutiou
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 elude that the maximum motive power resulting from the use of 
 steam is also the maximum motive power which can be obtained 
 by any other means. We shall soon give a second and more 
 rigorous demonstration of this law. What has been given 
 should only be regarded as a sketch (see page 15). 
 
 It may properly be asked, in connection with the proposition 
 just stated, what is the meaning of the word maximum 9 How 
 can we know that this maximum is reached and that the 
 steam is used in the most advantageous way possible to produce 
 motive power ? 
 
 Since any re-establishment of equilibrium in the caloric can 
 be used to produce motive power, any re-establishment of equi- 
 librium which is effected without producing motive power 
 should be considered as a veritable loss: now, with little re- 
 flection, we can see that any change of temperature which is not 
 due to a change of volume of the body can be only a useless re- 
 establishment of equilibrium in the caloric.* The necessary 
 condition of the maximum is, then, that in bodies used to obtain 
 the motive power of heat, no change of temperature occurs which is 
 
 has ever been produced, meaning by that, a motion which continues in- 
 definitely without change in the body used as an agent ? 
 
 The electromotive apparatus (Volta's pile) has sometimes been considered 
 capable of producing perpetual motion; the attempt has been made to re- 
 alize it by the construction of the dry pile, which is claimed to be unal- 
 terable ; but, in spite of all that has been done, the apparatus always dete- 
 riorates perceptibly when its action is sustained for some time with any 
 energy. 
 
 The general and philosophical acceptation of the words perpetual motion 
 should comprehend not only a motion capable of indefinite continuance 
 after it has been started, but also the action of an apparatus, of a set of 
 bodies, capable of creating motive power in an unlimited quantity, and of 
 setting in motion successively all the bodies of nature, if they are originally 
 at rest, and of destroying in them the principle of inertia, and finally capa- 
 ble of furnishing in itself all the forces necessary to move the entire uni- 
 verse, to prolong and to constantly accelerate its motion. Such would be 
 a real creation of motive power. If this were possible, it would be useless 
 to search for motive power in combustibles, in currents of water and 
 air. We should have at our disposal an inexhaustible source from which 
 we could draw at will. 
 
 * We do not here take into consideration any chemical action between the 
 bodies used to obtain the motive power of heat. The chemical action which 
 occurs in the source is in a sense preliminary, an fiction not designed to im- 
 mediately create motive power, but to destroy equilibrium in the caloric, to 
 produce a difference in temperature which shall finally result in motion. 
 13
 
 MEMOIRS ON 
 
 not due to a change of volume. Conversely, every time that this 
 oon.lition is fulfilled, the maximum is attained. 
 
 This principle should not be lost sight of in the construction 
 of heat-engines. It is the foundation upon which they rest. If 
 it cannot be rigorously observed, it should at least be departed 
 from as little as possible. 
 
 Any change of temperature which is not due to a change of 
 volume or to chemical action (which we provisionally assume not 
 to occur in this case) is necessarily due to the direct transfer of 
 caloric from a hotter to a colder body. This transfer takes 
 place principally at the points of contact of bodies at different 
 temperatures; thus such contacts should be avoided as much 
 as possible. They doubtless cannot be avoided entirely, but at 
 least care should be taken that the bodies brought in contact 
 should differ but little in temperature. 
 
 When we assumed in the previous demonstration that the ca- 
 loric of the body A was used to produce steam, we supposed the 
 steam to be produced at the same temperature as that of the 
 body A; thus the only contact was between two bodies of equal 
 temperature ; the change of temperature which the steam after- 
 wards experienced was due to expansion and consequently to a 
 change of volume; finally condensation was effected without 
 contact of bodies of different temperatures. It was effected by 
 the exercise of a constant pressure on the steam brought in con- 
 tact with the body B, at the same temperature as that of the 
 body It. The condition of the maximum was thus fulfilled. In 
 reality things would not occur exactly as we have supposed. I n 
 order to effect a transfer of the caloric from one body to the 
 other, the first must have the higher temperature ; but this dif- 
 ference may be supposed to be as small an we please ; wo may, 
 in theory, consider it zero without invalidating the arirumt -nt. 
 
 A more valid objection may be made to our demonstration, 
 namely: 
 
 When we produce steam by taking caloric from the body .1. 
 Mini when this steam is afterward condensed by contact with 
 the body h, the water used to form it, which was a>snmed to In-. 
 at the beginning, at the temperature of the \\\ . I. i-. at the 
 end of the operation, at the temperature of the hody IS that is. 
 it is colder. If we wish to recommence an operation similar to 
 the first, to develop a new quantity of motive power with the 
 same instrument and the same steam, we must first re-establish 
 14
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 the original state of things and bring the water to the temper- 
 ature which it had at first. This can no doubt be done by 
 placing it immediately in contact with the body A ; but in that 
 case there is contact between bodies of different temperatures 
 and loss of motive power.* It would become impossible to 
 perform the reverse operation that is, to cause the caloric 
 used in raising the temperature of the liquid to return to the 
 body A. 
 
 This difficulty can be removed by supposing the difference of 
 temperature between the body A and the body B infinitely 
 small ; the quantity of heat needed to bring the liquid back to 
 its original temperature is also infinitely small and negligible 
 relatively to that finite quantity which is needed to produce the 
 steam. 
 
 The proposition being thus demonstrated for the case in which 
 the difference of temperature of the two bodies is infinitely 
 small may easily be extended to cover the general case. In fact, 
 if we desire to produce motive power by the transfer of caloric 
 from the body A to the body Z, the temperature of the latter 
 body being very different from that of the former, we may 
 imagine a series of bodies B, C, D . . .at temperatures interme- 
 diate between those of the bodies A and Z, and chosen in such 
 a manner that the differences between A and B, B and C . . . 
 shall be always infinitely small. The caloric which proceeds 
 from A arrives at Z only after having passed through the bodies 
 B, C, D . . . and after having developed in each of these trans- 
 fers the maximum of motive power. The reverse operations are 
 here all possible, and the reasoning on page 11 becomes rigor- 
 ously applicable. 
 
 According to the views now established we may with pro- 
 
 *This kind of loss is always met with in steam-engines. In fact,, the 
 water which supplies the boiler is always colder than that which it already 
 contains, and hence a useless re-establishment of equilibrium in the ca- 
 loric takes place between them. It is easy to see a posteriori that this re- 
 estjiblishment of equilibrium entails a loss of motive power if we reflect 
 that it would be possible to heat the water supply before injecting it by 
 using it as water of condensation in a small accessory engine, in which 
 stenm taken from the large boiler could be used and in which condensation 
 would occur at a temperature intermediate between that of the boiler and 
 that of the principal condenser. The force produced by the small engine 
 would entail no expenditure of heat, since all that it would use would re- 
 cuter the boiler with the water of condensation. 
 15
 
 MEMOIRS ON 
 
 priety compare the motive power of heat with that of a water- 
 fall; both have a maximum which cannot be surpassed, whatever 
 may be, on the one hand, the machine used to receive the action 
 of the water and whatever, on the other hand, the substance 
 used to receive the action of the heat. The motive power of 
 fulling water depends on the quantity of water and on the 
 height of its fall; the motive power of heat depends also on the 
 quantity of caloric employed and on that which might be named, 
 which we, in fact, will call, its descent* that is to say. on t he dif- 
 ference of temperature of the bodies between which t he exchange 
 of caloric is effected. In the fall of water the motive power is 
 strictly proportional to the difference of level between the higher 
 and lower reservoirs. In the fall of caloric the motive power 
 doubtless increases with the difference of temperature between 
 the hotter and colder bodies, but we do not know whether it is 
 proportional to this difference. We do not know, for example, 
 whether the fall of the caloric from 100 to 50 degrees furnishes 
 more or less motive power than the fall of the same caloric from 
 50 degrees to zero. This is a question which we propose to ex- 
 amine later. 
 
 We shall give here a second demonstration of the funda- 
 mental proposition stated on page 13 and present this propo- 
 sition in a more general form than we have before. 
 
 When a gaseous fluid is rapidly compressed its temperature 
 rises, and when it is rapidly expanded its temperature falls. 
 This is one of the best established facts of experience ; we shall 
 take it as the basis of our demonstration, f When tin tem- 
 perature of a gas is raised and we wish to bring it back to its 
 
 The matter here treated being entirely new. we are obliged to employ 
 expressions hitherto unused and which are not perhaps as clear as could be 
 desired. 
 
 f The facts of experience which best prove the change of temperature 
 of a gas by compression or expansion are the following : 
 
 1. The fall of temperature indicated by a thermometer placed under tin- 
 receiver of an air pump in which a \.n mini is produced. This is very p< t 
 rrptililr with a Bregucl thermometer ; it may amount to upwards of 40 or 
 50 degrees. The cloud, which is formed in this operation seems to in tin to 
 ii,c cii<lriisalion of water vapor caii-cil hy the coolintr of the air. 
 
 2. The ignition of tinder in the so-called fire-syringe (pneumatic tinder- 
 box), which is, as is well known, a small pump in which air may !>< rapidly 
 compressed. 
 
 8. The fall of temperature indicated by a thermometer placed in a re- 
 16
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 original temperature without again changing its volume, it is 
 necessary to remove caloric from it. This caloric may also be 
 removed as the compression is effected, so that the temperature 
 of the gas remains constant. In the same way, if the gas is 
 rarefied, we can prevent its temperature from falling, by fur- 
 nishing it with a certain quantity of caloric. "We shall call the 
 caloric used in such cases, when it occasions no change of tem- 
 perature, caloric due to a change of volume. This name does not 
 indicate that the caloric belongs to the volume; it does not be- 
 long to it any more than it does to the pressure, and it might 
 equally well be called caloric due to a change of pressure. We are 
 ignorant of what laws it obeys in respect to changes of volume : 
 it is possible that its quantity changes with the nature of the 
 
 ceptacle in which air has been compressed, and from which it is allowed to 
 escape by opening a stopcock. 
 
 4. The results of experiments on the velocity of sound. M. de Laplace 
 has shown that to harmonize these results with theory and calculation we 
 must assume that air is heated by a sudden compression. 
 
 The only fact which can be opposed to these is an experiment of MM. 
 Gay-Lussac and Welter, described in the Annales de Chimie et de Physique. 
 If a small opening is made in a large reservoir of compressed air, and the 
 bulb of a thermometer is placed in the current of air escaping through this 
 opening, no perceptible fall of temperature is indicated by the thermometer. 
 
 We may explain this fact in two ways : 
 
 1. The friction of the air against the walls of the opening through which 
 it escapes may perhaps develop enough heat to be noticed ; 2. The air 
 which impinges immediately upon the bulb of the thermometer may re- 
 cover by its shock against the bulb, or rather by the detour which it is 
 forced to make by the encounter, a density equal to that which it had in 
 the receiver, somewhat in the same way as a current of water rises above 
 its level when it meets a fixed obstacle. 
 
 The change of temperature in gases occasioned by a change of volume 
 may be considered one of the most important facts in physics, because of 
 the innumerable consequences which it entails, and at the same time as one 
 of the most difficult to elucidate and to measure by conclusive experi- 
 ments. It presents singular anomalies in several cases. 
 
 Must we not attribute the coldness of the air in high regions of the at- 
 mosphere to the lowering of its temperature by expansion ? The reasons 
 hitherto given to explain this coldness are entirely insufficient ; it has been 
 said that the air in high regions, receiving but a small amount of heat 
 reflected by the earth, and itself radiating into celestial space, would lose 
 caloric and thus become colder ; but this explanation is overthrown when 
 we consider that at equal elevations the cold is as great or even greater on 
 elevated plains than on the tops of mountains or in parts of the atmosphere 
 distant from the earth. 
 
 B 17
 
 MEMOIRS ON 
 
 gas, or with its density or with its temperature. Experiment 
 has tanght us nothing on this subject; it has taught us only 
 that this caloric is developed in greater or less quantity by the 
 compression of elastic fluids. 
 
 This preliminary idea having been stated, let us imagine an 
 elastic fluid atmospheric air, for example enclosed in a cylin- 
 drical vessel abed (Fig. 1) furnished with a movable diaphragm 
 or piston cd; let us assume also the 
 two bodies A, B both at constant tem- 
 peratures, that of A being higher than 
 that of B, and let us consider the series 
 of operations which follow : 
 
 1. Contact of the body A with the 
 air contained in the vessel //// <>r with 
 the wall of this vessel, which wall is 
 supposed to be a good conductor of 
 caloric. By means of this contact the 
 air attains the same temperature as the 
 body .4; cd is the position of the pis- 
 ton. 
 
 2. The piston rises gradually until 
 it takes the position ef. Contact is al- 
 ways maintained between the air ami 
 
 flKrr^ ^ the body A, and the temperature thus 
 1 remains constant during the rarefac- 
 
 tion. The body A furnishes the ca- 
 loric necessary to maintain a constant 
 Fig. i temperature. 
 
 3. The body A is removed and the 
 
 air is no longer in contact with any body capable of supply- 
 ing it with caloric; the piston, however, continues to move 
 and passes from the position ef to the position gh. The air is 
 rarefied without receiving caloric and its temperature falls. 
 Let us suppose that it falls until it becomes equal to that of 
 the body />'; at this instant the piston ceases to move and 
 occupies the position ////. 
 
 4. The air is brought in contact with the body /.' : it is com- 
 pressed by the piston as it returns from the position //// to the 
 position nl. The air, however, remains at a constant tempera- 
 ture on account of its contact with the body B, to which it gives 
 up its caloric. 
 
 18
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 5. The body B is removed and the compression of the air con- 
 tinued. The temperature of the air, which is now isolated, 
 rises. The compression is continued until the air acquires the 
 temperature of the body A. The piston during this time passes 
 from the position cd to the position ik. 
 
 6. The air is again brought in contact with the body A\ the 
 piston returns from the position ik to the position ef, and the 
 temperature remains constant. 
 
 7. The operation described in No. 3 is repeated, and then the 
 operations 4, 5, 6, 3, 4, 5, 6, 3, 4, 5, and so on, successively. 
 
 In these various operations a pressure is exerted upon the 
 piston by the air contained in the cylinder ; the elastic force 
 of this air varies with the changes of volume as well as with 
 the changes of temperature ; but we should notice that at equal 
 volumes that is, for similar positions of the piston the tem- 
 perature is higher during the expansions than during the com- 
 pressions. During the former, therefore, the elastic force of 
 the air is greater, and consequently the quantity of motive 
 power produced by the expansions is greater than that which is 
 consumed in effecting the compressions. Thus there remains 
 an excess of motive power, which we can dispose of for any 
 purpose whatsoever. The air has therefore served as a heat-en- 
 gine ; and it has been used in the most advantageous way pos- 
 sible, for there has been no useless re-establishment of equilib- 
 rium in the caloric. 
 
 All the operations described above can be carried out in a 
 direct and in a reverse order. Let us suppose that after the 
 sixth step, when the piston is at ef, it is brought back to the 
 position ik, and that, at the same time, the air is kept in con- 
 tact with the body A ; the caloric furnished by this body dur- 
 ing the sixth operation returns to its source that is, to the body 
 A and the condition of things is the same as at the end of the 
 fifth operation. If now we remove the body A and move the 
 piston from ef to cd, the temperature of the air will fall as 
 many degrees as it rose during the fifth operation and will equal 
 that of the body B. A series of reverse operations to those 
 above described could evidently be carried out ; it is only neces- 
 sary to bring the system into the same initial state and in each 
 operation to carry out an expansion instead of a compression, 
 and conversely. 
 
 The result of the first operation was the production of a cer- 
 19
 
 MKMOIRS ON 
 
 tain quantity of motive power and the transfer of the caloric 
 from the body -1 to the body B ; the result of the reverse opera- 
 tion would be the consumption of the motive power product- 1 
 and the return of the caloric from the body B to the body .1 : 
 so that the two series of operations in a sense annul or neutralize 
 each other. 
 
 The impossibility of making the caloric produce a larger 
 quantity of motive power than that which we obtained in our 
 first series of operations is now easy to prove. It may be de- 
 monstrated by an argument similar to that used on pa ire 11. 
 The argument will have even a greater degree of rigor : the air 
 which serves to develop the motive power is brought back, 
 at the end of each cycle of operations, to its original condi- 
 tion, which was, as we noticed, not quite the case with the 
 steam.* 
 
 We have chosen atmospheric air as the agency employed t.. 
 develop the motive power of heat; but it is evident that the 
 same reasoning would hold for any other gaseous substance, and 
 even for all other bodies susceptible of changes of temperature 
 by successive contractions and expansions that is, for all 
 bodies in Nature, at least, all those which are capable of develop- 
 ing the motive power of heat. Thus we are led to establish this 
 general proposition : 
 
 The motive power \ of /ie<tt i* imli'i>i->nlrnt of the ay> 
 
 We implicitly assume-, in our demonstration. tli.-U if a body experi- 
 cnces any changes, ami returns exactly to its origin tl stair. vt. t a certain 
 numtterof transformations thnl is to say, to its original Mate determined by 
 its density, its temperature, and its mode of aegrcpitioii : \v. a-Mime. I say, 
 that the body contains the same quantity of hc:it as it contained nl first, or. 
 in other words, thatthcqtinniitiesof heat absorbed and te|.-as< d in ii- 
 transformations exactly compensate one another. Thi fact has never IN-CM 
 called in question ; it was ftt first admitted wiihoul consider iti-m and after 
 wanis verified in many cases by exporimcnts with the ralorim- -IIT. To 
 deny it would he to overthrow UK* entire theory of heat, of wliich it is die 
 i foundation. It may IK; rcmarkc.l, in passinp. that tin- fiin<l:inicntal prin 
 i-ipli-soii which the theory of heat rests should l jfiven the nvM cueful 
 i \ itnination. Severnl experimental fat is si-<-m to Ix; almost inexplicable in 
 Hi- actual Htaic of that theory. [The doubts here expressed as to the 
 validity of the assumptions made with respect to the nature of li 
 vclo|M*d in (-arnot's mind into an actual rejection of those assumption*, an. I 
 led him to suspect the true nature of heat. See Life </ ' ' ' KM. ] 
 
 f [// may be well to notife a<j,iiit t/i.it Curnnt utet " motir, 
 lyminymout ifilh the more modern term " work."\ 
 
 m
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 ployed to develop it ; its quantity is determined solely by the tem- 
 peratures of the bodies between which, in the final result,* the 
 transfer of the caloric occurs. 
 
 It is understood in this statement that the method used for 
 developing motive power, whatever it may be, attains the highest 
 perfection of which it is capable. This condition will be ful- 
 filled, as we remarked above, if there is no change of tempera- 
 ture in the bodies which is not due to a change of volume or, 
 which amounts to the same thing differently expressed, if the 
 temperatures of the bodies which come in contact with each 
 other are never perceptibly different. 
 
 Various methods of developing motive power may be adopted, 
 either by the use of different substances or of the same sub- 
 stance in different states; for example, by the use of a gas at 
 two different densities. 
 
 This remark leads us naturally to the interesting study of aeri- 
 form fluids, a study which will conduct us to new results con- 
 cerning the motive power of heat, and will give us the means 
 of verifying in some particular cases the fundamental proposi- 
 tion stated above, f 
 
 It can easily be seen that our demonstration will be simplified 
 if we suppose the temperatures of the bodies A and B to be very 
 slightly different. Then the movements of the piston will 
 be very small during operations 3 and 5, and these operations 
 may be suppressed without perceptible influence on the de- 
 velopment of motive power. That is, a very small change of 
 volume ought to be sufficient to produce a very small change 
 of temperature, and this change of volume is negligible com- 
 pared with that of operations 4 and 6, which are unrestricted 
 in extent. 
 
 If we suppress operations 3 and 5 in the series above de- 
 scribed, it is reduced to the following: 
 
 1. Contact of the gas contained in abed (Fig. 2) with the body 
 A, and passage of the piston from cd to ef; 
 
 2. Removal of the body A, contact of the gas enclosed in abcf 
 with the body B, and return of the piston from ef to cd ; 
 
 3. Removal of the body B, contact of the gas with the body 
 
 * [That is, upon tlie completion of a cycle of operations.] 
 \ We shall suppose in what follows that the reader is familiar with the 
 latest progress of modern physics in the departments of heat and gases. 
 21
 
 1 KMOIRS ON 
 
 A, and passage of the piston from cd to ef that is to say, 
 tition of the first operation, and so on. 
 
 The motive power resulting from the operations 1, 2, 3, taken 
 together, will evidently be the difference between that which 
 is produced by the expansion of the gas while its temperature 
 equals that of the body A and that which is consumed to com- 
 press the gas while its temperature equals that of the body B. 
 
 Let us suppose that the operations 1 and 'I are performed 
 with two gases which are chemically different, but which are 
 subjected to the same pressure for example, that of the atiuos- 
 
 Pig. 
 
 phere ; these gases behave in the same circumstances in exactly 
 the same way that is to say, their expansive forces, originally 
 equal, remain so irrespective of changes of volume ami temper- 
 ature, provided that these changes are the same in both. This 
 is an evident consequence of the laws of Mariotte ami <>f M M. 
 <ia v-Lussiic and Dalton, which laws are eommoii t<> all elastic 
 fluids, and in virtue of which the same relation- exist in all 
 these fluids between the volume, expansive force, ami temper- 
 ature. Since two different gases, taken at the same tempera- 
 ture and under the same pressure, should liehave alike under 
 the same circumstances, they should produce eijual quantities 
 of motive power when subjected to the operations above de- 
 scribed. Now this implies, according to the fundamental 
 proposition which we have established, that two equal quanti- 
 ties of caloric are employed in these operations that is, that 
 the quantity of caloric transferred from the body A to the body
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 B is the same whichever of the two gases is used in the opera- 
 tions. The quantity of caloric transferred from the body A to 
 the body B is evidently that which is absorbed by the gas in 
 the increase of its volume, or that which it afterwards emits 
 during compression. We are thus led to lay down the follow- 
 ing proposition : 
 
 When a gas passes without change of temperature from one defi- 
 nite volume and pressure to another, the quantity of caloric ab- 
 sorbed or emitted is alivays the same, irrespective of the nature of 
 the gas chosen as the subject of the experiment. 
 
 For example, consider 1 litre of air at the temperature of 
 100 degrees and under the pressure of 1 atmosphere*, if the 
 volume of this air is doubled, a certain quantity of heat must 
 be supplied to it in order to maintain it at the temperature of 
 100 degrees. This quantity will be exactly the same if, instead 
 of performing the operation with air, we use carbonic acid gas, 
 nitrogen, hydrogen, vapor of water, or of alcohol that is, if we 
 double the volume of 1 litre of any one of these gases at the 
 temperature of 100 degrees and under atmospheric pressure. 
 
 The same thing would be true, in the reverse sense, if the 
 volume of the gas, instead of being doubled, were reduced one- 
 half by compression. 
 
 The quantity of heat absorbed or set free by elastic fluids 
 during their changes of volume has never been measured by 
 direct experiment. Such an experiment would doubtless pre- 
 sent great difficulties, but we have one result which for our pur- 
 poses is nearly equivalent to it ; this result has been furnished 
 by the theory of sound, and may be received with confidence be- 
 cause of the rigor of the demonstration by which it has been 
 established. It may be described as follows : 
 
 Atmospheric air will rise in temperature 1 degree centigrade 
 when its volume is reduced by y|^ by sudden compression.* 
 
 The experiments on the velocity of sound were made in air 
 under a pressure of 760 millimetres of mercury and at the tem- 
 perature of G degrees ; and it is only in these circumstances that 
 Poisson's statement is applicable. We shall, however, for the 
 
 * M. Poisson, to whom we owe this statement, has shown that it agrees 
 very well with the results of an experiment by MM. Clement and Desormes 
 on the behavior of air entering into a vacuum or rather into slightly rare- 
 fied air. It agrees also, very nearly, with a result obtained by MM. Gay- 
 Lussac and Welter. (See note, p. 32.)
 
 MEMOIRS ON 
 
 sake of convenience, consider it to hold at a tcmperatnre of 
 degrees, which is only slightly different. 
 
 Air compressed by -pj-y and so raised in temperature 1 degree 
 differs from air heated directly by the same amount only in its 
 density. If we call the original volume V, the compression 1>\ 
 Y^-y reduces it to V 'j\j V. Direct heating under constant 
 pressure, according to the law of M. Gay-Lussuc, should in- 
 crease the volume of the air by ^ of that which it would have 
 at degrees; thus the volume of the air is in one process re- 
 duced to V -rfj T", and in the other increased to V+ 5^-7 T. 
 The difference between the quantities of heat present in tin- 
 air in the two cases is evidently the quantity used to raise its 
 temperature directly by 1 degree ; thus the quantity of heat ab- 
 sorbed by the air in passing from the volume I' , ],., V to the 
 volume r+^Tis equal to that which is necessary to raise 
 its temperature 1 degree. 
 
 Let us now suppose that, instead of heating the air while sub- 
 jected to a constant pressure and able to expand freely, we en- 
 close it in an envelope not capable of expansion, and then raise 
 its temperature 1 degree. The air thus heated 1 degree dif- 
 fers from air compressed by yfy, by having its volume larger 
 by y^. Thus, then, the quantity of heat which the air gives up 
 by a reduction of its volume by jfa is equal to that which is re- 
 quired to raise its temperature 1 degree at constant volume. 
 As the differences, V T fyJ r , F, and F+ ? $ T I . are .-mall in 
 comparison with the volumes themselves, we may consider the 
 quantities of heat absorbed by the air in passing from the first 
 of these volumes to the second, and from the first to the third. 
 M sensibly proportional to the changes of volume. \\ e thus 
 obtain the following relation : 
 
 The quantity of heat required to raise the temperature of air 
 under constant pressure 1 degree is to the quantity required to 
 raise it 1 degree at constant volume in the ratio of the numbers 
 
 rir + jfr to rfy, 
 or. multiplying both terms by 11G.2G7, in the ratio of the nnin- 
 
 ;7+116 to 267. 
 
 This is the ratio between the capacity for heat of air under 
 constant pressure and its capacity at constant volume. If tin- 
 first of these two capacities is expressed by unity the other will 
 
 be expressed by the number - - " < . approximately, 0.700. 
 
 Ji
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 Their difference 10.700 or 0.300 will evidently express the 
 quantity of heat which will occasion the increase of volume of 
 the air when its temperature is raised 1 degree under constant 
 pressure. 
 
 From the law of MM. Gay-Lussac and Daltou this increase 
 of volume will be the same for all other gases ; from the the- 
 orem demonstrated on page 23 the heat absorbed by equal in- 
 crements of volume is the same for all elastic fluids; we are 
 thus led to establish the following proposition : 
 
 The difference between the specific heat under constant pressure 
 and the specific heat at constant volume is the same for all gases. 
 
 It must be noticed here that all the gases are assumed to be 
 taken at the same pressure for example, the pressure of the 
 atmosphere and also that the specific heats are measured in 
 terms of the volumes. 
 
 Nothing is now easier than to construct a table of the specific 
 heats of gases at constant volume with the aid of our knowl- 
 edge of their specific heats under constant pressure. We 
 present this table, the first column of which contains the re- 
 sults of direct experiments by MM. Delaroche and Berard on 
 the specific heat of gases under atmospheric pressure. The 
 second column contains the numbers in the first diminished by 
 0.300. 
 
 TABLE OF THE SPECIFIC HEAT OF GASES 
 
 GASES 
 
 SPECIFIC HEAT UN- 
 DER CONSTANT 
 PRESSURE 
 
 SPECIFIC HEAT 
 AT CONSTANT 
 VOLUME 
 
 Atmospheric air 
 
 1 000 
 
 700 
 
 Hydrogen 
 
 903 
 
 603 
 
 Carbonic acid . 
 
 1 258 
 
 958 
 
 Oxygen . . 
 
 0.976 
 
 676 
 
 Nitrogen . . . . 
 
 1 000 
 
 700 
 
 Nitrons oxide 
 
 1 350 
 
 1 050 
 
 Olefiant gas 
 
 1.553 
 
 1 253 
 
 Carbonic oxide . . 
 
 1.034 
 
 0.734 
 
 The numbers in the two columns are referred to the same 
 unit, to the specific heat of atmospheric air under constant 
 pressure. 
 
 The difference between the corresponding numbers in the 
 two columns being constant, the ratio between them should be 
 25
 
 MEMOIRS ON 
 
 variable ; thus the ratio between the specific heats of gases un- 
 der constant pressure and at constant volume varies for the 
 different gases. 
 
 \\ f have seen that the temperature of the air when it under- 
 goes a sudden compression of t f ff of its volume rises 1 degree. 
 That of other gases should also rise when they are similarly 
 roiii|>ivss<_-<l. The temperature should rise, not equally for all, 
 but in the inverse ratio of their specific heats at constant 
 volume. In fact, the reduction of volume being, by hypothesis, 
 always the .-aim-, the quantity of heat due to this reduction 
 should also be always the same, and consequently should cans.' 
 a rise of temperature depending only on the specific heat of the 
 gas after its compression, and evidently in an inverse ratio to 
 that specific heat. It is therefore easy to construct the table 
 of elevations of temperature of the different gases for a com- 
 pression 
 
 TABLE OF THE ELEVATION OF Till: TKM I'KKATl UK OF &A8E8 
 1H K TO COMPRESSION 
 
 ELEVATION OK TKMi'i I: \ 
 
 GASES TI-KK Kon A in in . noa 
 
 OF VOLfMK OK ,\ f 
 
 o 
 
 Atmospheric air ........................ 1.000 
 
 Hydrogen ............................ 1.160 
 
 Carbonic acid .......................... 0.730 
 
 Oxygen ................................ 1.035 
 
 Nitrogen .............................. 1.000 
 
 Nitrous oxide .......................... 0.< 
 
 Olefiant gas ............................ 0.558 
 
 Carbonic oxide ......................... 0.955 
 
 A second compression of T fy of the new volume would, as we 
 shall soon see, again raise the temperature of these gases In an 
 amount nearly equal to the first ; but this would not be the 
 case for a third, a fourth, or a hundredth compression of tin- 
 same sort. The capacity of gases for heat changes with their 
 volume; it is quite possible that it changes also with their 
 temperature.* 
 
 [It ,u found ty ReynauH (M.-in. !.- I'Acadfimle, asnrf., p. 58) tl. 
 tprtifit heat of the "permanent" gate* it independent of prtmure and tern- 
 i re.} 
 
 ''!
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 We shall now deduce from the general proposition presented 
 on page 20 a second theorem which will be the complement of 
 that which has just been demonstrated. 
 
 Let us suppose that the gas contained in the cylinder abed 
 (Fig. 2) is transferred to the receptacle a'b'c'd' (Fig. 3), which is 
 of equal height, but which has a different and larger base ; the 
 gas will increase in volume and diminish in density and elas- 
 tic force in the inverse ratio of the two volumes abed, a'b'c'd'. 
 The total pressure exerted on each piston, cd, c'd', will be the 
 same, for the surfaces of these pistons are in the direct ratio of 
 the volume. 
 
 Let us suppose that the operations described on page 21 as 
 performed on the gas contained in abed are performed on the 
 gas in a'b'c'd' that is, let us suppose that the piston c'd' is given 
 displacements equal in amplitude to those given the piston cd, 
 and that it occupies successively the positions c'd' correspond- 
 ing to cd, and e'f corresponding to ef. At the same time let us 
 subject the gas, by means of the two bodies A, B, to the same 
 variations of temperature as those to which it was subjected 
 when enclosed in abed ; the total force exerted on the piston 
 will be the same in both cases at corresponding instants. This 
 results immediately from Mariotte's law* ; in fact, the densities 
 of the two gases are always in the same ratio for similar posi- 
 tions of the pistons, and, their temperatures being always equal, 
 the total pressures exerted on the pistons are always in the same 
 ratio. If this ratio is at any time that of equality, the pressures 
 will bo always equal. 
 
 Further, as the movements of the two pistons have equal am- 
 plitudes, the motive power they both produce will evidently be 
 the same, from which we may conclude, from the proposition 
 
 * Mariotte's law, upon which our demonstration is based, is one of thebest- 
 establishcd physical laws. It has served as a foundation for several theo- 
 ries verified by experiment, and which verify in their turn the laws on 
 which they rest. We may also cite, as an important verification of 
 Mariotte's law and also of the law of MM. Gay-Lussac and Dalton for a 
 large range of temperature, the experiments of MM. Dnlong and Petit. 
 (See Annales de Chimie et de Physique, Feb., 1818, vol. vii., p. 122.) We 
 mav also cite the still more recent experiments of Davy and Faraday. 
 
 The theorems here deduced would perhaps not be exact if applied out- 
 side of certain limits either of density or of temperature. They should 
 only be taken as true within the limits within which the laws of Mariotte, 
 Gay-Lussac, and Dalton are themselves established. 
 27
 
 MEMOIRS ON 
 
 on page 20, that the quantities of heat used by each are equal 
 that is to say, that the same quantity of heat passes from .1 t<> 
 // in each case. 
 
 The heat taken from the body A and given to the body /.' i> 
 nothing other than the heat absorbed by the expansion of the 
 gas and afterwards set free by < onipn-iuii. We are thus led 
 to establish the following theorem : 
 
 II lit'ii (lie I'olnitli' of an flitxtir jlinil rlminiis. without r}/n)irjt< nf 
 
 tim/ierature, from I' to I". "//// tin- /<<//////< ,,/',/ i/nantitii of tin' 
 same gas, equal in weight and at tin' >v////- fn,ij r,tf >//. clntn<i>x 
 from IT 1 toV', the quantities of heat ali.^,rl,,;l or .^7 f !<> from nn-li 
 will be equal when the ratio of U' to I /> /mtl to tint! of U 
 toV. 
 
 This theorem may be stated in another form, as follows: 
 
 When a gas changes in volume without change of fi-m/H-ratiin- 
 the quantities of heat which it absorbs or gives KI> an- in arith- 
 metical progression when the increments or reductions of volume 
 are in geometrical /;/-o///v /'///. 
 
 When we compress one litre of air maintained at a tempera- 
 ture of 10 degrees and reduce its volume to J a litre, it gives 
 out a certain quantity of heat. This quantity will bo al\\a\- 
 the same if we further reduce the volume from J to {, from J 
 to 1 . and so on. 
 
 If, instead of compressing the air, we allow it to expand to 2 
 litres, 4 litres,8 litres, etc., successively, we must supply it \\iih 
 equal quantities of heat in order to keep its tomporuturc c,nst ant. 
 
 This easily explains why the temperature of air risrs \\-hrn it 
 is suddenly compressed. We know that this temperature is 
 sufficient to ignite tinder and even to cause the air to become 
 luminous. If we assume for the time lieing the specific heat 
 of air as constant, in spite of changes of volume and tempera- 
 ture, the temperature will increase in arithmetical pm^ressinn 
 as the volume is diminished in geometrical progression. Start- 
 ing with this as given, and admitting that an elevation of inn 
 perature of 1 degree corresponds to a compression of ^{ r . it i> 
 easy to conclude that when air is reduced to fa of its original 
 volume its temperature should rise about 300 degrees, which 
 is enough to ignite tinder.* 
 
 When the volume l mlur<i l,y ,}, that In. when it becomes \\\ of 
 that which it wa* at first the temperature risen 1 <i<
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 The elevation of temperature would evidently be still greater 
 if the capacity of the air for heat were to become less as its 
 volume diminishes ; now this is probable, and seems to be con- 
 firmed by the results of the experiments of MM. Delaroche and 
 Berard on the specific heat of air taken at different densities. 
 (See the Memoir published in the Annalesde Chimie et de Phy- 
 sique, vol. Ixxxv., pp. 72, 224.) 
 
 The two theorems given on pages 23 and 28 are sufficient for 
 the comparison of the quantities of heat absorbed or released in 
 the changes of volume of elastic fluids, whatever may be the den- 
 sity and chemical nature of these fluids, provided always that 
 they are taken and maintained at a certain invariable tempera- 
 ture ; but these theorems do not give us any means of compar- 
 ing quantities of heat absorbed or released by elastic fluids whose 
 volumes are changed at different temperatures. Thus we do 
 not know the relation between the heat released by 1 litre of air 
 reduced in volume one-half when its temperature is kept at zero 
 and the heat released by the same litre of air reduced in volume 
 one -half when its temperature is kept at 100 degrees. The 
 knowledge of this relation is connected with the knowledge of 
 the specific heat of the gases at different degrees of tempera- 
 ture, and on other data which Physics, in its present state, can- 
 not furnish. 
 
 The second of our theorems affords a means of knowing by 
 what law the specific heat of gases varies with their density. 
 
 Let us suppose that the operations described on page 21, in- 
 stead of being performed with two bodies, A, B, whose temper- 
 
 A now reduction of T | ff brings the volume to (Ul) 4 . and tlie tempera- 
 ture should rise another degree. 
 
 After x such reductions the volume is (Hf )*. aQ d the temperature should 
 be higher by x degrees. 
 
 If we set (Hi^T 1 ?. and tuke tue logarithms of both sides, we find 
 x = 300 about. 
 
 If we set (Hf)* = i we find that x = 80, which shows that the tem- 
 perature of air compressed to one half of its original volume rises 80 de- 
 
 All this is dependent on the hypothesis that the specific heat of air does 
 not change when the volume diminishes ; but if, for the reasons given on 
 pages 31 and 32, we consider the specific heat of air compressed to one-half 
 its volume as reduced in the ratio of 700 to 616, we must multiply 80 de- 
 grees by J^, which brings it to 90 degrees. 
 29
 
 MEMOIRS ON 
 
 atures differ by an infinitely small quantity, are performed with 
 two bodies whose temperatures differ by a finite quant ity, say 
 byl. 
 
 In a complete cycle of operations the body A furnishes to the 
 elastic fluid a certain quantity of heat which may be dividt <! 
 into two portions : 1, the quantity required to keep the tem- 
 perature of the fluid constant during expansion ; 2, that re- 
 quired to change the temperature of the fluid from that of the 
 body B to that of the body A, after the fluid has been restored 
 to its original volume and is put in contact with the body J. 
 Let us call the first of these quantities a and the second b. 
 The total caloric furnished by the body A will be expressed by 
 a + b. 
 
 The caloric transmitted by the fluid to the body B may also 
 be divided into two parts ; one of which, b', is due to the cooling 
 of the gas by the body B, the other, ', is that released by the 
 gas during the reduction of its volume. The sum of these two 
 quantities is a'-f b' ', this should be equal to a -f- b, for after a 
 complete cycle of operations the gas returns exactly to its 
 original state.* It must have given up all the caloric with 
 which it had at first been supplied. We then have 
 
 a+b=a'+b', 
 or, a a' = b b'. 
 
 Now, from the theorem given on page 28, the quantities a and 
 a' are independent of the density of the gas, always provided 
 that the quantity of the gas by weight remains the -aim- ami 
 that the variations of volume are proportional to the original 
 volume. The difference a a' should satisfy the same con- 
 ditions, and consequently also the difference b' b, which is 
 equal to it. But b' is the caloric necessary to raise the temper- 
 ature of the gas contained in nhnl one degree (Fig. 2); b' is the 
 caloric released by the gaa when it is enclosed in //. and its 
 temperature falls one degree. These quantities can serve as a 
 measure of the specific heats. We an- thus led to establish the 
 following proposition : 
 
 [The M* here made of the caloric theory titiata the demonttration and 
 Uttdt to erroneout eondunont.} 
 
 M
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 The change made in the specific heat of a gas in consequence of 
 a change of volume depends only upon the relation between the 
 original volume and that which results from the change that is 
 to say, the difference between the specific heats does not depend 
 on the absolute magnitudes of the volumes but on their ratio. 
 
 This proposition may be stated in another way, namely : 
 
 When the volume of a gas increases in geometrical progression 
 its specific heat increases in arithmetical progression. 
 
 Thus, if a is the specific heat of air taken at a given den- 
 sity, and a + h its specific heat when its density is one-half this, 
 its specific heat will be a+'2h when its density is one-quarter 
 this, a+'3h when its density is one-eighth this, and so on. 
 
 The specific heats are here referred to weight. They are 
 supposed to be taken at constant volume ; but, as we shall see, 
 they would follow the same law if they were taken under con- 
 stant pressure. 
 
 In fact, to what cause is due the difference between the 
 specific heats taken at constant volume and under constant 
 pressure ? It is due to the caloric required in the latter case 
 to produce the increase of volume. Now, by Mariotte's law, 
 the increase of volume of a gas, for a given change of tempera- 
 ture, should be a definite fraction of the original volume, which 
 fraction is independent of the pressure. From the theorem 
 given on page 28,if the ratio between the original volume and the 
 changed volume is given, the heat required to produce the in- 
 crease of volume is determined thereby. It depends only on this 
 ratio and on the quantity of the gas by weight. "We must then 
 conclude that : The difference between the specific heat under con- 
 stant pressure and that at constant volume is always the same, 
 whatever the density of the gas, provided that the quantity of the 
 gas by weight remains the same. These specific heats both in- 
 crease as the density of the gas diminishes, but their difference 
 does not change.* Since the difference between the two capaci- 
 
 * MM. Gay-Lussac and Welter have found by direct experiments, cited 
 in the Mecanique Celeste and in the Annales de Chimie et de Physique, July, 
 1822, page 267, that the ratio between the specific heat under constant press- 
 ure and that at constant volume varies very little with the density of the 
 gas. From what we have just seen, it is the difference and not the ratio 
 that should remain constant. However, as the specific heats of gases, for a 
 given weight, vary very little with their density, it is clear that the ratio 
 also will experience only very small changes. 
 31
 
 M KM Ml US ON 
 
 ties for heat is constant, when one increases in arithmetical 
 progression the other will increase in a similar progression ; thus 
 our law applies to specific heats taken under constant pressure. 
 
 We have tacitly supposed that the specific heat inn. 
 with the volume. This increase is shown in the experiments 
 of MM. Delaroche and Hi' rani; these physicists have found 
 that the specific heat of air under the pressure of 1 meter of 
 mercury is 0.967 (see the memoir already referred to), taking as 
 the unit the specific heat of the same weight of air under tin- 
 pressure of 0.7GO meter. From the law followed hy the specific 
 heats with respect to pressure, observations made of tln-m in 
 two particular cases permit us to calculate them in all pos- 
 sible cases ; thus, by using the result of the experiment of M M. 
 Delaroche and Berard, which has just been cited, we have con- 
 structed the following table of the specific heats of air under 
 various pressures : 
 
 
 -PM'IFIC HEAT, 
 
 
 SPECIFIC HEAT. 
 
 PRESSURE 
 
 THAT OK AIR UNDER 
 
 PKKSsritK 
 
 THAT OF AIK 1 MU.Il 
 
 IN 
 
 ATMOSPHERIC 
 
 IN 
 
 ATMOSIMIKICK 
 
 ATMOSPHERES 
 
 PRESSURE 
 
 \TMOH-IIKKI-.S 
 
 PRESSURE 
 
 
 BEING 1. 
 
 
 aura i 
 
 TiiW 
 
 1.840 
 
 1 
 
 l.(MM) 
 
 X 
 
 1.756 
 
 2 
 
 0.916 
 
 i 
 
 L.678 
 
 ,688 
 
 4 
 
 8 
 
 0.748 
 
 Vr 
 
 ..~><>1 
 
 16 
 
 
 A 
 
 .I'M 
 
 .' ! '.' 
 
 0.580 
 
 
 ,886 
 
 64 
 
 0.496 
 
 i 
 
 .'.'">'.' 
 
 US 
 
 0.411 
 
 I 
 
 '[.;. : , 
 
 
 
 1 
 
 .084 
 
 51SJ 
 
 ii > | j 
 
 1 
 
 .000 
 
 M'.'l 
 
 o!l60 
 
 From the experiments of MM. Gay-Lussac and WclU-r. tlie ratio of the 
 specific heat of atmospheric air under constant pressure to tint ai rn 
 stant volume is 1.8748, a uumtx-r \\lii< h is nearly constant for all pr. 
 and for all temperatures. In the previous iliHriivsion we have l>c<>n lid, 
 
 by other considerations, to the number ~~na-j =1.44, \\ hieli dilTers fnun 
 
 this by fo, and we have used this number to construct a table of thesperiiic 
 hents of gases at constant volume. Neither this taM.- nr (lie tai.l.- uivrn 
 on page 88 should lie considered as accurate. Tln-y an- intcmlr.1 mainly 
 to set forth the laws followed by the specific heats of aeriform fluids.
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 The numbers in the first column are in geometrical progres- 
 sion, while those in the second are in arithmetical progression. 
 We have carried the table out to extreme compressions and 
 rarefactions. It is to be supposed that air, before attaining a 
 density 1024 times its ordinary density that is, before becom- 
 ing more dense than water would be liquefied. The specific 
 heats vanish, and even become negative if we prolong the 
 table beyond the last number given. It seems probable that the 
 numbers in the second column decrease too rapidly. The ex- 
 periments on which we based our calculation were made within 
 too narrow limits to enable us to expect great exactness in the 
 numbers obtained, especially in the extreme values. 
 
 Since, on the one hand, we know the law by which heat is 
 evolved by the compression of a gas, and, on the other, the law 
 by which the specific heat varies with the volume, it will be 
 easy for us to calculate the increase of temperature of a gas 
 compressed without loss of caloric.* In fact, the compression 
 can be considered as consisting of two successive operations : 
 1, compression at a constant temperature, and, 2, restoration 
 of the caloric emitted. In the second operation the tempera- 
 ture will rise in the inverse ratio to the specific heat which 
 the gas acquires by the reduction of its volume. "We can de- 
 termine the specific heat by means of the law above demon- 
 strated. From the theorem on page 28 the heat set free by 
 compression should be represented by an expression of the form 
 
 s = A + B log v, 
 
 s being the heat, v the volume of the gas after compression, 
 A and B arbitrary constants dependent on the original volume 
 of the gas, on its pressure, and on the units which are chosen. 
 
 The specific heat, which varies with the volume in accordance 
 with the law just demonstrated, should be represented by an 
 expression of the form, 
 
 A' and B' being arbitrary constants different from A and B. 
 The increase of temperature which the gas receives by com- 
 
 pression is proportional to the ratio -, or to the ratio -p " . 
 
 * [This demonstration is erroneous in that it assumes tJie, materiality of 
 heat, and also tfie change of specific heat with volume. The conclusions are 
 
 invalid.]
 
 MEMOIRS ON 
 
 It may be represented by this ratio itself; tlins, if we represent 
 it by /, we shall have A + li log r 
 
 ~ A ~ /' io 
 
 If the original volume of the gas is 1 and the original u-nipt'ni- 
 ture zero, we shall have at the same time / = 0, log y = 0, ami 
 hence A ; t will then express not only the increase of tem- 
 perature, but the temperature itself above the thermometric zero. 
 We must not think that we can apply the formula just given 
 to very large changes in the volume of the gas. We have taken 
 the rise of temperature to be in the inverse ratio to the specific 
 heat, which implies that the specific heat is constant at all tem- 
 peratures. Large changes of volume in the gas occasion large 
 changes of temperature, and there is no evidence that the specific 
 heat is constant at different temperatures, especially when these 
 temperatures are widely separated from each other. This con- 
 stancy of specific heat is only an hypothesis assumed in the case 
 of gases from analogy, and verified fairly well for solids and li(j aids 
 within a certain range of the thermometric scale, but which the 
 experiments of MM. Pulongaml Petit have shown to IK- inexact 
 when extended to temperatures much above 100 degrees.* 
 
 We sec no reason to assume a priori the constancy of the specific 
 heat of bodies at various temperatures that is to say, to assume that 
 equal quantities of heat will produce equal increments in tin- temperature 
 of a body, even when neither the state in>r tlic density of the body U 
 clinnired ; as. for example, in the case of an elastic fluid enclosed in n riirul 
 envelope. Direct experiments on solid and liquid bodies have proved that 
 between zero and 100 degrees equal im -r. im-nt* of heat produce nearly 
 equal increments of temperature ; but the more recent experiments of 
 MM Dulong and Petit (see AnnaU* de Cliimi, ,t <! l'/i?/*iqnf, February. 
 MUM -li. and April, 1818) Imve shown that this relation does not hold for 
 temperatures much over 100 degrees, whether they are measured by the 
 mercury thermometer or by the air thermometer. 
 
 Not only do the specific heats not u-nwin the same at different t. in 
 peratures. but the ratios between them do not remain the same ; so Unit 
 DO thermometric scale can establish the constancy of all specific heats at 
 the same time. It would be interesting to examine whether the same 
 irregularities would obtain in gaseous substances, but the necessary experi- 
 ments present almost insurmountable ditllculiL-r 
 
 It teems probable that the irregularities of the specific beats of solid 
 bodies may be attributed to latent heat, employed in producing a commence- 
 ment of fusion, a softening which in many cases becomes perceptible in 
 these bodies long before complete fusion occurs. We can support this 
 opinion by the following observation: From the experiments of MM. Dulong 
 and Petit, the increase of specific hent with the temperature is more r.ij.i.i 
 84
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 According to a law due to MM. Clement and Desormes,* 
 established by direct experiment, water vapor, under whatever 
 pressure it is formed, always contains the same quantity of heat 
 in equal weights ; this amounts to saying that the vapor when 
 compressed or expanded without loss of heat is always in such 
 a condition as to saturate the space which it occupies, if it is 
 originally in this condition. Water vapor in this condition can 
 thus be considered a permanent gas, and should follow all the 
 laws of gases. Consequently, the formula 
 
 A + B\ogv 
 ~ A'+U'\ogv 
 
 should be applicable and should agree with the table of tensions 
 constructed by M. Dalton from his direct experiments. 
 
 We can, in fact, satisfy ourselves that our formula, with a 
 suitable determination of the arbitrary constants, represents 
 very approximately the results of experiment. The unimpor- 
 tant discrepancies which we find in it are no more than may 
 reasonably be attributed to errors of observation.! 
 
 with solids than with liquids, though the latter have a larger dilatability. 
 If the cause which we have proposed to account for this irregularity is the 
 real one, it would disappear entirely with gases. 
 
 * [/it has been shown by Rankine and Clausim that this law does not hold.] 
 t To determine the arbitrary constants A, B, A, B', from data taken 
 from M. Dalton's table, we must begin by calculating the volume of the 
 vapor by means of its pressure and temperature, the quantity of the vapor 
 by weight being always constant. This is made easy by the laws of Ma- 
 riotte and Gay-Lussac. The volume will be given by the equation 
 
 in which v is the volume, t the temperature, p the pressure, and c a constant 
 quantity which depends on the weight of the vapor and the units chosen. 
 The following is the table of the volumes occupied by a gram of vapor 
 formed at various temperatures and consequently under various pressures : 
 
 t 
 
 p 
 
 OR THE TENSION OP THE VAPOR 
 
 
 
 OR THE VOLUME OP A GRAM 
 
 OR CEOTIGRADE DEGREES 
 
 KXI'KKSSKl) IN MILLIMETRES 
 
 OP VAPOR EXPRESSED 
 
 
 OP MERCURY 
 
 IS LITRES 
 
 Deg. 
 
 Mm. 
 
 Lit. 
 
 
 
 5.060 
 
 185.0 
 
 20 
 
 17.32 
 
 58.2 
 
 40 
 
 53.00 
 
 20.4 
 
 60 
 
 144.6 
 
 7.96 
 
 80 
 
 352.1 
 
 8.47 
 
 100 
 
 760.0 
 
 1.70 
 
 The first two columns in this table are taken from the Traite de Physique 
 85
 
 MKMOIRS ON 
 
 We shall now retnrn to our principal subject, the motive 
 power of heat, from which we have already digressed too far. 
 
 We have shown that the quantity of motive power developed 
 by the transfer of caloric from one body to another depends 
 essentially on the temperatures of the two bodies, but we have 
 not discussed the relation between these temperatures and the 
 quantities of motive power produced. It would seem at tir.-t 
 natural enough to suppose that for equal differences of tem- 
 perature the quantities of motive power produced are equal 
 that is, for example, that a given quantity of caloric passing 
 from a body, A, kept at 100 degrees, to a body, B, kept at 50 de- 
 grees would develop a quantity of motive power equal to that 
 which would be developed by the transfer of the same caloric 
 from a body, #, kept at 50 degrees to a body, C, kept at zero. 
 Such a law would indeed be a very remarkable one, hut we. do 
 not see sufficient reason to admit it a priori. We shall examine 
 this question by a rigorous method. 
 
 Let us suppose that the operations described on page 21 are 
 performed successively on two quantities of atmospheric air 
 equal in weight and volume but taken at different temperat :in >. 
 and let us suppose also that the differences of temperature l>e- 
 
 of M. Biot (vol. i., pp. 272 mid 531). The third is calculated by menus of 
 the above formula, mid from tin- expuriincntul fact that tin- vapor of water 
 under atmospheric pressure occupies a volume 1700 times us gn -at a> ili.it 
 which it occupies when in the liquid state. 
 
 By using three numbers from the first column and the corresponding 
 numbers from the third, we can easily determine the constants of our 
 equation 
 
 t _ A + n lotr 
 
 logc 
 
 We shall not enter into the dotnils of the calculation necessary to de- 
 termine tin-si- quantities; it will be enough for us to say that the following 
 
 B = - 1000. & = 8.80. 
 satisfy sufficiently well the prescrilted conditions, so that the equation 
 
 = 19.04 
 
 expresses very approximately the relation existing between the volume of 
 the vnpor and its temperature. 
 
 It is to be noticed that the quantity IT is positive and very small. \\ hirli 
 tends to confirm the proposition tint the specific heat of an elastic fluid in- 
 with the volume, but at a wry slow rate. 
 80
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 tween the bodies A and B are the same in both cases ; thus, 
 for example, the temperatures of these bodies will be in one 
 case 100 and 100 h (h being infinitely small), and in the 
 other, 1 and 1 h. The quantity of motive power produced 
 is in each case the difference between that which the gas fur- 
 nishes by its expansion and that which must be used to restore 
 it to its original volume. Now this difference is here the same 
 in both cases, as >ve may satisfy ourselves by a simple argument, 
 which we do not think it necessary to give in full ; so that the 
 motive power produced is the same. Let us now compare the 
 quantities of heat used in the two cases. In the first case the 
 quantity used is that which the body A imparts to the air in 
 order to keep it at a temperature of 100 degrees during its ex- 
 pansion ; in the second, it is that which the same body imparts 
 to it to maintain its temperature at 1 degree during an exactly 
 similar change of volume. If these two quantities were equal 
 it is evident that the law which we have assumed would follow. 
 But there is nothing to prove that it is so ; we proceed to prove 
 that these quantities of heat are unequal. 
 
 The air which we first supposed to occupy the space abed 
 (Fig. 2) and to be at a temperature of 1 degree, may be made to 
 occupy the space abef, and to acquire the temperature of 100 
 degrees by two different methods : 
 
 TL. It may first be heated without change of volume, and then 
 expanded while its temperature is kept constant. 
 
 2. It may first be expanded while its temperature is kept 
 constant, and then heated when it has acquired its new vol- 
 ume. 
 
 Let a and b be the quantities of heat used successively in the 
 first of the two operations, and b' and ' the quantities used in 
 the second ; as the final result of these two operations is the 
 same, the quantities of heat used in each should be equal ; we 
 then obtain 
 
 from which we have 
 
 a'a=b b'. 
 
 We represent by a' the quantity of heat necessary to raise the 
 temperature of the gas from 1 to 100 degrees when it occupies 
 the volume abef, and by a the quantity of heat necessary to 
 raise the temperature of the gas from 1 to 100 degrees when it 
 occupies the volume abed. 
 
 37
 
 MEMOIRS ON 
 
 The density of the air is less in the first case than in the sec- 
 ond, and from the experiments of MM. Delaroche and Beranl. 
 already cited on page 32, its capacity for heut should be a little 
 greater. 
 
 As the quantity a' is greater than the quantity a, b should be 
 greater than b', consequently, stating the proposition generally, 
 we may say that : 
 
 The quantity of hrat <lnc t<> the clntmjr of volume of a gas be- 
 comes greater as the teiiifn nttiirc /"> rai*nl. 
 
 Thus, for example, more caloric is required to maintain at 
 100 degrees the temperature of a certain quantity of air whose 
 volume is doubled than to maintain at 1 degree t)ie tempera- 
 ture of the same quantity of air during a similar expansion. 
 
 These unequal quantities of heat will, however, as we have 
 seen, produce equal quantities of motive power for equal de- 
 scents of caloric occurring at different heights on the thermo- 
 metric scale; from which we may draw the following conclu- 
 sion : 
 
 The descent of caloric prmlnn's more motive power at lower de- 
 ffftU of temjH'raturr limn tit higher.* 
 
 Thus a given quantity of heat will develop more motive 
 power in passing from a body whose temperature is kept at 1 
 degree to another whose temperature is kept at zero than if 
 the temperatures of these two bodies had been lul and loo 
 respectively. It must be said that the difference should lie very 
 small ; it would be zero if the capacity of air for heat remained 
 constant in spite of changes of density. According to the ex- 
 periments of MM. Delaroche and Beranl. this capacity \arie> 
 very little, so little, indeed, that the differences notiee<l miu'ht 
 strictly be attributed to errors of observation or to s..me eir- 
 cumstances which were not taken into account. 
 
 It would be out of the question for us, with the experimental 
 data at our command, to determine rigorously the law l.y which 
 
 * [ The preff fling drmon*tration if erroMout in contequenfe of the. attump- 
 Hun of the materiality of heat. The ennclwtion it inform correct, but only 
 becaute of the erroneout ute of a variable tpeciftc heat of air. If tftit be con- 
 tutored eonttant. at Carnot point* out. the efficiency ihould br. <m ///,//. 
 the tame at all temperaturet. The ratio of thr n,> ../.,.../ t-> th, 
 
 heat vted hould be equal to the difference of tempernt'tr, i,,<itti)>lied by a eon- 
 tfiint. I/if ' C>i run (' fnitftiini." At M note kiunr, (hit / / Con- 
 
 t.ii.t. but it the reciprocal of the abtolute temperature of the touree of htat. ]
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 the motive power of heat varies at different degrees of the 
 thermometric scale. It is connected with the law of the varia- 
 tions of the specific heat of gases at different temperatures, 
 which has not been determined with sufficient exactness.* We 
 
 * If we admit that the specific heat of a gas is constant when its volume 
 docs not change, but only its temperature varies, analysis would lead us to 
 a relation between the motive power and the therraometric degree. We 
 shall now examine the way in which this may be done; it will also give us 
 an opportunity of showing how some of the propositions formerly estab- 
 lished should be stated in algebraic form. 
 
 Let / be the quantity of motive power produced by the expansion of a 
 given quantity of air changing from the volume 1 litre to the volume v 
 litres at constant temperature. If v increases by the infinitely small quan- 
 tity dv, r will increase by the quantity dr, which, from the nature of mo- 
 tive power, will be equal to the increase of volume do multiplied by the 
 expansive force which the elastic fluid then has. If p represents the ex- 
 pansive force, we shall have the equation 
 
 (1) dr=pdv. 
 
 Let us suppose the constant temperature at which the expansion occurs to 
 be equal to t degrees centigrade. Representing by q the elastic force of 
 the air at the same temperature, t, occupying the volume of 1 litre, we 
 shall have from Mariotte's law 
 
 = , from which p = -- 
 
 lp v 
 
 Now if Pis the elastic force of the same air always occupying the volume 
 1, but at the temperature zero, we shall have from M. Gay-Lussac's law 
 
 T> 
 
 If, for the sake of brevity, we represent by N the quantity ~, the equa- 
 tion will become 
 
 by using which we have, from equation (1), 
 
 v 
 Considering t constant, and taking the integrals of the two terms, we ob- 
 
 If we suppose that r=Q when c=l, we shall have (7=0, from which 
 
 (2) r = ^V(< + 267)logc. 
 
 This is the motive power produced by the expansion of the air at the tem- 
 perature t, whose volume has changed from 1 to . If instead of working
 
 MHMOIRS OX 
 
 shall now endeavor to determine definitively the motive power 
 of heat, and in order to verify our fundamental proposition 
 
 at the temperature t we work in exactly the same way at the temperature 
 t+dt, the power developed will be 
 
 r + / = y (t + (It + 2G7 1 log r. 
 Subtracting equation (2) we obtain 
 
 (8) Sr = N log oft. 
 
 Let be the quantity of heat used to keep the temperature of the gns 
 constant during its expansion. From the discussion on page 21 ir will be 
 the power developed by the descent of the quantity of heat < from the degree 
 t + dt to the degree t. Let represent the motive power developed by the 
 descent of a unit of heat from t degrees to zero; since from the gi-m ml 
 principle established ou page 21 this quantity n should depend only on t, 
 it may be represented by the function J-'t, fn.ni which u = Ft. 
 
 When t increases and becomes i-\-<it. n becomes v + du, from which 
 
 Subtracting the preceding equation we have 
 
 du=F(t+dl)-Fl = Ftdt. 
 
 This is evidently the quantity of motive power produced by the descent of 
 a unit quantity of heal from the degree t + dt to the degree /. 
 
 If the quantity of heat, instead of being a unit, had been < . the motive 
 power produced would have been 
 
 (4) edu = eF'tdt. 
 
 But edu is the same as (r. both being the power developed by the descent of 
 the quantity of heat < from the degree t + dt to the degree t ; consequently, 
 
 edu = Si; 
 and, from equations (3) and (4), 
 
 eF'tdt = JVlog// ; 
 
 or, dividing by Ftdt, and representing by T the fraction ^- . which is a 
 function of t only, we have 
 
 The equation e = T log c 
 
 is the analytical expression of the law stated on page 28 ; it is the same for 
 
 all gases, since the laws we have used are common i- all 
 
 If we represent by * the quantity of heal required to chant'' 1 the volume 
 of the ait with which we arc working from 1 to r. and the temper-it ure from 
 zero to t, the difference between and r will IK; the quantity of heat required 
 to change the temperature of the air, while its volume remains 1. from /< m 
 to t. This quantity depends on / only. It will be some function of /, and 
 we shall have, if we call it U, 
 
 If wo differentiate this equation with rc*pe<-t to t only anil represent l.\ /' 
 and I' the differential coefficients of T and U, it will become 
 
 (5) ^ 
 
 40
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 that is, to show that the quantity of motive power produced 
 is really independent of the agent used we shall choose sev- 
 
 -37 is nothing other than the specific heat of the gas at constant volume, 
 and our equation (5) is the analytical expression of the law stated on page 
 31. 
 
 If we suppose the specific heat to be constant at all temperatures an 
 
 hypothesis which was discussed on page 34 the quantity -r will be inde- 
 
 pendent of t, and, to satisfy equation (5) for two particular values of v, 
 T and U' must also be independent of t ; we shall then have T' = C, a con- 
 stunt quantity. Multiplying T and C by dtaud integrating both sides we 
 find 
 
 T=Ct + C l ; 
 
 but as T=jj^ we have 
 
 N 
 
 T 
 
 Multiplying both sides by dt and integrating we obtain 
 
 or, changing the arbitrary constants, and remembering that Ft is zero 
 when t = 0, we have 
 
 (6) Pt = Alog(l + ^. 
 
 The nature of the function Ft is thus determined, and may serve us as a 
 means of calculating the motive power developed by any descent of heat. 
 But this last conclusion is based on the hypothesis of the constancy of the 
 specific heat of a gas whose volume does not change an hypothesis which 
 experiment has not yet sufficiently verified. Until there are further proofs 
 of its validity equation (6) can only be admitted for a small part of the 
 thermometric scale. 
 
 The first term in equation (5) represents, as we have said, the specific 
 heat of the air occupying the volume . Experiment has taught us that 
 this specific heat varies only slightly in spite of considerable changes of 
 volume, so 'that the coefficient T' of log v must be a very small quantity. 
 If we assume that it is zero and multiply the equation T' = Q by dt and 
 then integrate, we have 
 
 T= C, a constant quantity. 
 But 
 
 from which 
 
 f-M^. 
 
 from which we may conclude by a second integration that 
 Ft = At + B. 
 41
 
 MEMOIRS ON 
 
 eral such agents atmospheric air, water vapor, and alcohol 
 vapor. 
 
 Let us take first atmospheric air. The operation is effected* 
 according to the method indicated on page 21. We make the 
 following hypotheses : 
 
 The air is taken under atmospheric pressure ; the tempera- 
 ture of the body A is j^Vr f a degree above zero and that of the 
 body B is zero. We see that the difference is, as it should be, 
 very small. The increase of. the volume of the air in our operu- 
 tion will be ^fa+yfa of the original volume ; this is a very 
 small increase considered absolutely, but large relatively to the 
 difference of temperature between A and B. 
 
 The motive power developed by the two operations described 
 on page 21 taken together will be very nearly proportional to the 
 increase of volume and to the difference bet ween the two press- 
 ures exerted by the air when its temperature is 0.001 and zero. 
 
 According to the law of M. Gay-Lussac, this difference is 
 i g 7*0 OF of the elastic force of the gas, or very nearly T jrVoT f 
 the atmospheric pressure. 
 
 The pressure of the atmosphere is equal to that of a column 
 of water 10 T <jfr meters high ; , , ,i of this pressure is equal to 
 that of a water column 8> ^ oirg x 10.40 meters in height. 
 
 As for the increase of volume, it is, by hypothesis, 1 1 + jii 
 of the original volume that is, of the volume occupied by 1 
 kilogram of air at zero, which is equal to 0.77 cubic meters, 
 if we take into account the specific gravity of air; thus the 
 product, ( Tl + lW O.T7 II * 1 l<UO 
 
 expresses the motive power developed. This power is here 
 estimated in cubic meters of water raised to the height of 1 
 meter. 
 
 If we carry out the multiplications indicated, we find for the 
 product 0.000000372. 
 
 Let us now try to determine the quantity of heat used to ob- 
 tain this resultthat is, the quantity transferred from the body 
 A to the body B. The body A furnishes : 
 
 when t = 0, D is zero ; thus 
 
 Ft 
 
 tlint is to nay. the motive power produced is exactly proportional to the 
 descent of the caloric. This is the analytical expression of the statement 
 made on page 88. 
 
 42
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 1. The heat required to raise the temperature of 1 kilogram 
 of air from zero to 0.001. 
 
 2. The quantity required to maintain the temperature of the 
 air at 0.001 when it undergoes an expansion of 
 
 The first of these quantities of heat may be neglected, as it is 
 very small in comparison with the second, which is, from the 
 discussion on page 24, equal to that required to raise the tem- 
 perature of 1 kilogram of air under atmospheric pressure 1 
 degree. 
 
 The specific heat of air by weight is 0.267 that of water, from 
 the experiments of MM. Delaroche and Berard on the specific 
 heat of gases. If, then, we take for the unit of heat the quantity 
 required to raise 1 kilogram of water 1 degree, the quantity re- 
 quired to raise 1 kilogram of air 1 degree will be 0.267. Thus 
 the quantity of heat furnished by the body A is 
 
 0.267 unit. 
 
 This quantity of heat is capable of producing 0.000000372 unit 
 of motive power by its descent from 0.001 to zero. 
 
 For a descent one thousand times as great, or of one degree, 
 the motive power will be very nearly one thousand times as 
 great as this, or 
 
 0.000372. 
 
 Now if, instead of using 0.267 unit of heat, we use 1000 units, 
 the motive power produced will be given by the proportion 
 ^Z^^-IOJLQ, from which -r = ff = 1.395 units. 
 
 Thus if 1000 units of heat pass from a body 
 whose temperature is kept at 1 degree to another 
 at zero, they will produce by their action on air 
 1.395 units of motive power. 
 
 We shall compare this result with that which 
 is obtained from the action of heat on water 
 vapor. 
 
 Let us suppose that 1 kilogram of water is 
 contained in the cylinder abed (Fig. 4) between 
 the base ab and the piston cd, and let us assume 
 also the existence of two bodies, A, B, each 
 maintained at a constant temperature, that of 
 A being higher than that of B by a very small 
 quantity. We shall now imagine the following 
 operations : 
 
 43 
 
 Fig
 
 MEMOIRS ON 
 
 1. Contact of the water with the body A, change of the 
 position of the piston from cd to ef, formation of vapor at the 
 temperature of the body A to fill the vacuum made 1>\ tilt- 
 increase of the volume. We shall assume the volume atn-t 
 to be large enough to contain all the water in a state of 
 vapor ; 
 
 x'. Removal of the body A, contact of the vapor with the 
 body B, precipitation of a part of this vapor, decrease of its 
 elastic force, return of the piston from efto ab, and liquefaction 
 of the rest of the vapor by the effect of the pressure combined 
 with the contact of the body B; 
 
 3. Removal of the body B, new contact of the water with 
 the body A, return of the water to the temperature of this 
 body, a repetition of the first operation, and so on. 
 
 The quantity of motive power developed in a complete cy- 
 cle of operations is measured by the product of the volume 
 of the vapor multiplied by the difference between its tensions 
 at the temperatures of the body A and of the body B respec- 
 tively. 
 
 The heat used that is, that transferred from the body A to 
 the body B is evidently the quantity which is required to 
 transform the water into vapor, always neglecting the small 
 quantity necessary to restore the water from the tempt -rature 
 of the body li to that of the body A. 
 
 Let us suppose that the temperature of the body A is 100 
 degrees and that of the body B 99 degrees. From 1C. Dahoifs 
 table the difference of these tensions will be 26 millimetres of 
 mercury or 0.36 meter of water. The volume occupied by tin- 
 vapor is 1700 that of the water, so that, if we use l kilo-ram, 
 it will be 170Q litres or 1.700 cubic meters. Thus the motive 
 power developed is 
 
 1.700x0.36 = 0.611 unit 
 of the sort which we used before. 
 
 The quantity of heat used is the quantity required to trans- 
 form the water into vapor, the water beini: already at a ti-m- 
 |H-rature of 100 degrees. This quantity has !><. u d< t< rmiind 
 I >y experiment ; it has been found equal to A.*>< decrees, or, 
 -leaking with greater precision, to 550 of our units of 
 heat 
 
 Thus it. c,i i unit of motive power result from the use of 550 
 units of heat. 
 
 44
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 The quantity of motive power produced by 1000 units of heat 
 will be given by the proportion 
 
 550 1000 . , . , 611 
 
 = , from which x = - = 1.112. 
 
 0.611 x ' 550 
 
 Thus 1000 units of heat transferred from a body maintained 
 at 100 degrees to one maintained at 99 degrees will produce 
 1.112 units of motive power when acting on the water vapor. 
 The number 1.112 differs by nearly ^ from 1.395, which was the 
 number previously found for the motive power developed by 
 1000 units of heat acting on air ; but we must remember that 
 in that case the temperature of the bodies A and B were 1 
 degree and zero, while in this case they are 100 and 99 degrees 
 respectively. The difference, is indeed the same, but the tem- 
 peratures on the thermometric scale are not the same. In order 
 to obtain an exact pomparison it would be necessary to calculate 
 the motive power developed by the vapor formed at 1 degree 
 and condensed at zero, and also to determine the quantity of 
 heat contained in the vapor formed at 1 degree. The law of 
 MM. Clement and Desormes, to which we referred on page 35, 
 gives us this information. The heat used in turning water 
 into vapor (chaleur constituante) is always the same at whatever 
 temperature the vaporization occurs. Therefore, since 550 
 degrees of heat are required to vaporize the water at the tem- 
 perature of 100 degrees, we must have 550 4-100, or 650 degrees, 
 to vaporize the same weight of water at zero. 
 
 By using the data thus obtained, and reasoning in other 
 respects quite in the same way as we did when the water was 
 at 100 degrees, we readily see that 1.290 is the motive power 
 developed by 1000 units of heat acting on water vapor between 
 the temperatures of 1 degree and zero. 
 
 This number approaches 1.395 more nearly than the other. 
 
 It only differs by y 1 ^, which is not outside the limits of prob- 
 able error, considering the large number of data of different 
 sorts which we have found it necessary to use in making this 
 comparison. Thus our fundamental law is verified in a par- 
 ticular case.* 
 
 * In a memoir of M. Petit (Annales de Chimie et de Physique, July, 1818, 
 page 294) there is a calculation of the motive power of heat applied to air 
 and to water vapor. The results of this calculation are much to the advan- 
 tage of atmospheric air ; but this is owing to a very inadequate way of 
 considering the action of heat. 
 
 45
 
 MEMOIRS ON 
 
 We shall now examine the case of heat acting on alcohol vapor. 
 
 The method used in this case is exactly the same as in the 
 case of water vapor, but the data are different. Pure alcohol 
 boils under ordinary pressure at 78.7 centigrade. According 
 to MM. Delaroche ami P.t'-rard, 1 kilogram of this substance 
 absorbs 207 units of heat when transformed into vapor at this 
 same temperature, 78.7. 
 
 The tension of alcohol vapor at 1 degree below its boiling- 
 point is diminished by .,'.,. und is J. less than atmospho it- 
 pressure (this is at least the result of the experiments of M. 
 Hi-tancour, an account of which was given in the second part 
 of M. IVony's An-h i/tff /< lliidrnnUque, pages 180, !!">).* 
 
 We find, by use of these data, that the motive ]u>\vrr de- 
 veloped, in acting on 1 kilogram of alcohol at the temperatures 
 77.7 and 78.7, would be 0.251 unit. 
 
 This results from the use of 207 units of heat. For 1000 
 units we must set the proportion 
 
 207 1000 . 
 - = , from which Z= 1.230. 
 
 ''. J ' 1 *C 
 
 This number is a little greater than 1.112, resulting from the use 
 of water vapor at 100 and 99 degrees ; but if \vi- assume the water 
 vapor to be employed at 78 and 77 degrees, we find, by the law 
 of MM. Cli'ment and Desormes, 1.212 for the motive power 
 produced by 1000 units of heat. As we see, this number ap- 
 proaches 1.230 very nearly ; it only differs from it by -j^. 
 
 * M. Dalton thought that he hiul discovered thnt the vapors of different, 
 liquids exhibited equal tensions tit temperatures on tlie tin -rm.mn -trie scale 
 equally distant from their boiling-points ; this law Is, however, not rigor- 
 ously, but only approximately, correct. The same is true of tlie law of 
 the ratio of the latent heat of vapors to their densities (see ex 
 from a memoir of M. C. Despreiz. Annale* de Chimit et de Ptiyrique, vol. 
 xvi.. p. 105. and vol. xxiv.. p. 828). Questions of this kind are closely con- 
 nected with those relating to the motive power of heat Davy and Km .<>., \ 
 recently tried to recognize the changes of tension of liquefied gases for 
 small changes of temperature, after having made excellent experiments 
 on the liquefaction of gnses by the effect of a considerable pressure, 
 had In view the use of new liquids in the production of motive p>\\, t 
 (see Annale* de Chimie et fa Physiqnt, January. 1*24. p. 80). From the 
 theory given above we can predict that the use of these liquids presents no 
 advantage for the economical use of heat. The advantage could only be 
 realized at the low temperature at which it would be possible to work, and 
 by the use of sources from which, for this reason, it would become pos- 
 sible to extract caloric. 
 
 46
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 We should have liked to have made other comparisons of this 
 kind for example, to have calculated the motive power de- 
 veloped by the action of heat on solids and liquids, by the freez- 
 ing of water, etc. ; but in the present state of Physics we are 
 not able to obtain the necessary data.* The fundamental law 
 which we wish to confirm seems, however, to need additional 
 verifications to be put beyond donbt; it is based upon the the- 
 ory of heat as it is at present established, and, it must be con- 
 fessed, this does not appear to us to be a very firm foundation. 
 New experiments alone can decide this question; in the mean 
 time we shall occupy ourselves with the application of the the- 
 oretical ideas above stated, and shall consider them as correct 
 in the examination of the various means proposed at the pres- 
 ent time to realize the motive power of heat. 
 
 It has been proposed to develop motive power by the action 
 of heat on solid bodies. The mode of procedure which most 
 naturally presents itself to our minds is to firmly fix a solid 
 body a metallic bar, for example by one of its extremities, 
 and to attach the other extremity to a movable part of the 
 machine ; then by successive heating and cooling to cause the 
 length of the bar to vary, and thus produce some movement. 
 Let us endeavor to decide if this mode of developing motive 
 power can be advantageous. We have shown that the way to 
 get the best results in the production of motion by the use of 
 heat is to so arrange the operations that all the changes of tem- 
 perature which occur in the bodies are due to changes of vol- 
 ume. The more nearly this condition is fulfilled the better the 
 heat will be utilized. Now, by proceeding in the manner just 
 described, we are far from fulfilling this condition ; no change 
 of temperature is here due to a change of volume ; but the 
 changes are all due to the contact of bodies differently heated, 
 to the contact of the metallic bar either with the body which 
 furnishes the heat or with the body which absorbs it. 
 
 The only means of fulfilling the prescribed condition would 
 be to act on the solid body exactly as we did on the air in the 
 operations described on page 18, but for this we must be able to 
 produce considerable changes of temperature solely by the 
 change of volume of the solid body, if, at least, we desire to 
 
 * The data lacking are the expansive force acquired by solids and 
 liquids for u given increase of temperature, and the quantity of heat ab- 
 sorbed or emitted during changes in the volume of these bodies. 
 47
 
 MKNh'IRS ON 
 
 use considerable descents of caloric. Now this seems to be 
 impracticable, for several considerations lead ns to think that 
 the changes in the temperature of solids or liquids by compres- 
 sion and expansion are quite small. 
 
 1. We often observe in engines (in heat-engines particularly) 
 solid parts which are subjected to very considerable forces, some- 
 times in one sense and sometimes in another, and although those 
 forces are sometimes as great as the nature of the substances 
 employed will permit, the changes in temperature are scarcely 
 perceptible. 
 
 2. In the process of striking medals, of rolling plates, or of 
 drawing wires, metals undergo the greatest compressions to 
 which we can subject them by the use of the hardest and most 
 resisting materials. Notwithstanding this the rise in tempera- 
 ture is not great, for if it were, the steel tools which we u.<- in 
 these operations would soon lose their temper. 
 
 3. We know that it is necessary to exert a very great force 
 on solids and liquids to produce in them a reduction of volume 
 comparable to that which they undergo by cooling (for ex- 
 ample, by a cooling from 100 degrees to zero). Now, cooling 
 requires a greater suppression of caloric than would be n><|iiiiv<l 
 by a simple reduction of volume. If this reduction were pro- 
 duced by mechanical means the heat emitted could not change 
 the temperature of the body as many degrees as the cooling. 
 It would, however, require the use of a force which would cer- 
 tainly be very considerable. Since solid bodies are susceptible 
 to but small changes of temperature by changes of volume, and 
 since, moreover, the condition for the best use of heat, in the de- 
 velopment of motive power is that any change of temperature 
 should be due to a change of volume, solid bodies do not seem 
 to be well adapted to realize this power. 
 
 This is equally true in the case of liquids ; the same reasons 
 could be given for rejecting them.* 
 
 We shall not speak hero of the practical difficulties, which are 
 innumerable. The movements produced by the expansion and 
 compression of solids or liquids can only be very small. To 
 extend these movements we should be forced to use complicated 
 
 * The recent experiment* of M. Oersted on the compressibility of W.H.T 
 hnvc shown that for a pressure of 5 atmospheres the temperature of the 
 liquid undergoes no perceptible change. (See Annalet d Chimie et de 
 Phyrique. February, 1828, p. 192.) 
 
 48
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 mechanisms and also materials of the greatest strength to trans- 
 mit enormous pressures ; and, finally, the successive operations 
 conld only proceed very slowly compared with those of the 
 ordinary heat-engine, so that even large and expensive ma- 
 chines would produce only insignificant results. 
 
 Elastic fluids, gases, or vapors are the instruments peculiar- 
 ly fitted for the development of the motive power of heat ; they 
 unite all the conditions necessary for this service ; they may be 
 easily compressed, and possess the property of almost indefinite 
 expansion ; changes of volume occasion in them great changes 
 of temperature, and finally they are very mobile, can be easily 
 and quickly heated and cooled, and readily transported from 
 one place to another, so that they are able to produce rapidly 
 the effects expected of them. 
 
 We can easily conceive of many machines fitted for the de- 
 velopment of the motive power of heat by the use of elastic 
 fluids, but however they are constructed in other respects, the 
 following conditions must not be lost sight of : 
 
 1. The temperature of the fluid should first be raised to the 
 highest degree possible, in order to obtain a great descent of 
 caloric and consequently a great production of motive power. 
 
 2. For the same reason the temperature of the refrigerator 
 should be as low as possible. 
 
 3. The operations must be so conducted that the transfer of 
 the elastic fluid from the highest to the lowest temperature 
 should be due to an increase of volume that is, that the cool- 
 ing of the gas should occur spontaneously by the effect of ex- 
 pansion. 
 
 The limits to which the temperature of the fluid can be 
 raised in the first operation are determined only by the tem- 
 perature of combustion ; they are very much higher than ordi- 
 nary temperatures. The limits of cooling are reached in the 
 temperature of the coldest bodies which we can conveniently 
 use in large quantities ; the body most used for this purpose is 
 the water available at the place where the operation is car- 
 ried on. 
 
 As to the third condition, it introduces difficulties in the 
 realization of the motive power of heat, when the object is to 
 profit by great differences of temperature, that is to utilize 
 great descents of caloric. For in that case the gas must change 
 from a very high temperature to a very low one, by expansion, 
 D 49
 
 MEMOIRS ON 
 
 which requires a great change of volume and density. To 
 effect this the gas must at first be subjected to a very high 
 pressure, or it must acquire by expansion an enormous volume, 
 either of which conditions is difficult to realize. Tin- first 
 necessitates the use of very strong vessels to contain the gas 
 when it is at a high pressure and temperature; the second re- 
 quires the use of vessels of a very large size. 
 
 In fact, these are the principal obstacles in the way of profit- 
 ably using in steam-engines a large portion of the motive power 
 of heat. We are of necessity limited to the use of a small de- 
 scent of caloric, although the combustion of coal furnishes us 
 with the means of obtaining a very great one. In the use of 
 steam-engines the elastic fluid is rarely developed at a pressure 
 higher than atmospheres, which pressure corresponds to near- 
 ly lf0 degrees centigrade, and condensation is rarely effected at 
 a temperature much below 40 degrees; the descent of caloric 
 from 100 to 40 degrees is 120 degrees, while we can obtain by 
 combustion a descent of from 1000 to 2000 degrees. 
 
 To conceive of this better, we shall recall what we have pre- 
 viously called the descent of caloric : It is the transfer of heat 
 from a body, A, at a high temperature to a body, 11, whose tem- 
 perature is lower. We say that the descent of caloric is lun 
 degrees or 1000 degrees when the difference of temperature 
 between the bodies J ami />' is 100 or 1000 decrees. In a 
 steam-engine working under a pressure of 6 atmospheres the 
 temperature of the boiler is 100 degrees. This is the tempera- 
 ture of the body A ; it is maintained by contact with the fur- 
 nace at a constant temperature of lH" decree*, ami affords a 
 continual supply of the heat necessary to the formation of 
 steam. 
 
 The condenser is the body /?; it is maintained by means of a 
 current of cold water at an almost constant temperature of 4<> 
 degrees, and continually absorbs the caloric which is rarrie.l t<> 
 it by the steam from the body A. The difference of tempera- 
 ture between these two bodies is 16040, or 120 degrees ; it is 
 for this reason that we say that the descent of caloric is in this 
 case 120 doL' 
 
 1 il i< capable of producing by combustion a higher t -m- 
 
 peratlire than lIMM) decrees, ami the temperature, of the e.uld 
 
 water which we ordinarily use is about 1<> decrees, so that we can 
 
 easily obtain a descent of caloric of 1000 degrees, of which only 
 
 50
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 120 degrees are utilized by steam-engines, and even these 120 
 degrees are not all used to advantage ; there are always con- 
 siderable losses due to useless re-establishments of equilibrium 
 in the caloric. 
 
 It is now easy to perceive the advantage of those engines 
 which are called high-pressure engines over those in which the 
 pressure is lower : this advantage depends essentially upon the 
 power of utilizing a larger descent of caloric. The steam being 
 produced under greater pressure is also at a higher tempera- 
 ture, and as the temperature of condensation is always nearly 
 the same the descent of caloric is evidently greater. 
 
 But to obtain the most favorable results from high-pressure 
 engines the descent of caloric must be used to the greatest ad- 
 vantage. It is not enough that the steam should be produced 
 at a high temperature, but it is also necessary that it should 
 attain a sufficiently low temperature by its expansion alone. 
 It should thus be the characteristic of a good steam-engine not 
 only that it uses the steam under high pressure, but that it uses 
 it under successive. pressures which are very variable, very dif- 
 ferent from each other, and progressively decreasing.* 
 
 *This principle, which is the real basis of the theory of the steam-engine, 
 has been developed with great clearness by M. Clement in a memoir pre- 
 sented to the Academy of Sciences a few years ago. This memoir has 
 never been printed, and I owe my acquaintance with it to the kindness of 
 the author. In it not only is this principle established, but applied to va- 
 rious systems of engines actually in use ; the motive power of each is cal- 
 culated by the help of the law cited on p. 35 and compared with the results 
 of experiment. This principle is so little known or appreciated that Mr. 
 Perkins, the well-known London mechanician, recently constructed an 
 engine in which the steam, formed under a pressure of 35 atmospheres, a 
 pressure never before utilized, experienced almost HO expansion, as we 
 may easily be convinced by the slightest knowledge of the engine. It is 
 composed of a single cylinder, which is very small, and at each stroke is 
 entirely filled with steam formed under a pressure of 35 atmospheres. The 
 steam does no work by expansion, for there is no room for the expansion 
 to take place : it is condensed as soon as it passes out of the small cylinder. 
 It acts only under a pressure of 35 atmospheres, and not, as the best usage 
 would require, under progressively decreasing pressures. This engine of 
 Mr. Perkins does not realize the hopes which it at first excited. It was 
 claimed that the economy of coal in this machine was T 9 ff greater than in 
 the best machines of Watt, and that it also possessed other advantages 
 over them. (See Annales de Ghimie et de Physique, April, 1823, p. 429.) 
 These assertions have not been verified. Mr. Perkins's engine may never- 
 theless be considered a valuable invention in that it has proved it to be 
 51
 
 MEMOIRS ON 
 
 In order to show, to a certain extent, a /W^r/Wi the advan- 
 tage of high-pressure engines, let us assume that the strain 
 formed under atmospheric pressure is contained in H cylimln- 
 
 feasiblc to use steam under much higher pressures than ever before, and 
 because when properly modified it may lead us to really useful tesults. 
 
 Wait, to whom we <>\v.- almost nil the great improvements in the strum 
 engine, and who has brought these machines to a state of perfection which 
 can hardly IK? surpassed, was the first to use steam under pr.>i:i-e**ivrly 
 decreasing pressures. In many cases he checked the introduction of the 
 steam into the cylinder at one-half, one-third, or one-quarter of tin- stroke 
 of the piston, which was thus completed under a picssiirc which constant 
 ly diminished. The first engines working on this principle dale from 177s 
 Walt had conceived the idea in 1769, and took out a patent for it in 17^,' 
 
 A table annexed to Watt's patent is here presented. In it lie supposes 
 the vapor to enter the cylinder during the first quarter of the stroke of the 
 piston, and he then calculates the mean pressure by dividing the stroke in;. . 
 twenty parts: 
 
 l-ARTW Or TIIK 
 
 PATH FRO* TUB HBAD OF THK 
 CYLINDER 
 
 DKRKA8IKO PRKMfRK Or TlUt VAPOK, TH 
 TOTAL PMK881-HK RUM. 1 
 
 
 0.05 \ 
 0.101 Steam entering 
 
 i) i:> freely from 
 
 ,1.000} 
 
 \1 no,,/ 
 
 1. 000 V Total pressure. 
 
 
 o-jo 
 
 the boiler. 
 
 M. -00, 
 
 Quarter 
 
 .0.25 
 
 
 ; i.ooo ) 
 
 
 0.80 
 
 
 0.810 
 
 
 085 
 
 
 0.714 
 
 
 0.40 
 
 
 0625 
 
 
 0.45 
 
 
 o 560 
 
 Half. 
 
 .0.50 
 
 ' 
 
 0.500. .Half the original pressure. 
 
 
 
 ii V, 
 060 
 0.65 
 0.70 
 
 a. 78 
 
 The steam cut off. 
 and moving the 
 piston by expan- 
 sion aloue. 
 
 o :, 
 0417 
 0.885 
 0875 
 
 0.888.. One third. 
 
 
 OSO 
 
 
 0.811 
 
 
 0.85 
 
 
 o IM 
 
 
 0.90 
 
 
 
 Bottom of 
 
 095 
 
 
 
 cylinder 
 
 .1.00 
 
 
 0.250. .One-quarter. 
 
 
 Total 
 
 11 588 
 
 
 Mean pressure, 11 ' 888 = 
 
 0.579. 
 
 On this showing he remarks that the mean pressure is more thnn half of 
 the original pressure, so that a quantity of steam equal t ne quarter \\ dl 
 produce an effect greater than one-half [freely intrmlncctl fmm tin hnlrr 
 until the end of the ftrolce}. 
 
 Watt here assumes that the i \p:m-in <>f the steam is in accordance with 
 Mariotte's law. This assumption should not be considered correct, be-
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 cal vessel, abed (Fig. 5). under the piston cd, which at first 
 
 touches the base ab; the steam, after moving the piston from 
 
 ab to cd, will subsequently act in a manner with 
 
 which we need not occupy ourselves. Let us 
 
 suppose that after the piston has reached cd it is 
 
 forced down to ef without escape of steam, or 
 
 loss of any of its caloric. It will be compressed 
 
 into the space abef, and its density, elastic force, 
 
 and temperature will all increase together. 
 
 If the steam, instead of being formed under 
 atmospheric pressure, were produced exactly in 
 the state in which it is when compressed into a 
 abef, and if, after having moved the piston from #%. 5 
 
 ab to ef by its introduction into the cylinder, it 
 should move it from ef to cd solely by expansion,- the motive 
 power produced would be greater than in the first case. In 
 fact, an equal movement of the piston would take place under 
 the influence of a higher pressure, although this would be va- 
 riable and even progressively decreasing. 
 
 The steam would require for its formation a precisely equal 
 quantity of caloric, but this caloric would be used at a higher 
 temperature. 
 
 It is from considerations of this kind that engines with two 
 cylinders (compound engines) were introduced, which were in- 
 vented by Mr. Hornblower and improved by Mr. Woolf. With 
 
 cause, on the one hand the temperature of the elastic fluid is lowered by ex- 
 pansion, and on the other there is nothing to show that a part of this fluid 
 does not condense by expansion. Watt should also have taken into ac- 
 count the force necessary to expel the steam remaining after condensa- 
 tion, whose quantity is greater in proportion as the expansion has been 
 carried further. Dr. Robinson added to Watt's work a simple formula 
 to calculate the effect of the expansion of steam, but this formula is af- 
 fected by the same errors to which we have just called attention. It has, 
 however, been useful to constructors in furnishing them with a means of 
 calculation sufficiently exact to be of use in practice W T e have thought it 
 worth while to recall these facts because they are little known, especially 
 in France. Engines have been constructed there after the models of in- 
 ventors but without much appreciation of the principles on which these 
 models were made. The neglect of these principles has often led to grave 
 faults. Engines which were originally well conceived have deteriorated 
 in the hands of unskilful constructors, who, wishing to introduce unim- 
 portant improvements, have neglected fundamental considerations which 
 they did not know enough to appreciate. 
 
 53
 
 MEMOIRS ON 
 
 respect to the economy of fuel, they are considered the best en- 
 gines. They are composed of a small cylinder, which at each 
 stroke of the piston is more or less and often entirely tilled 
 with steam, and of a second cylinder, of a capacity usually 
 fonr times as great, which receives only the steam which has 
 already been used in the first one. Thus the volume of the 
 steam at the end of this operation is at least four times its 
 original volume. It is carried from the second cylinder direct- 
 ly into the condenser; but it is evident that it could he carried 
 into a third cylinder four times as large as the second, where its 
 volume would become sixteen times its original volume. The 
 chief obstacle to the use of a third cylinder of this kind is tin- 
 large space which it requires, and the size of the openings 
 which are necessary to allow the steam to escape.* 
 
 We shaU say nothing more on this subject, our object ii<>t 
 being to discuss the details of construction of heat-engines. 
 These should be treated in a separate work. No such work 
 exists at present, at least in France, f 
 
 * It is easy to perceive the advantages of having two cylinders, for when 
 there is only one the pressure on the piston will vary very much he! ween the 
 beginning and end of the stroke. Also, all the portions of the machine de 
 signed to transmit the action must be strong enough to re-ist tin- first im- 
 pulse, and filled together pcifectly so as to avoid sudden motions by \\liii-h 
 they might be damaged and which would soon wear them out. Tins \\onld 
 be specially true of the walking beam, the supports, tin- conm< -ting-rod, 
 the crank, and of the first cog-wheels. In these parts tin- irreirnlarity of 
 the impulse would be roost felt and would do tin- most damage The 
 steam-chest would also have to lie strong enough to resist tin- i 
 pressure employed, and large enough to contain the v:i|>or when its volume 
 is increased. If two cylinders are used the capacity of the first need not 
 be great, so that it is easy to give it the strength required, while the second 
 must be large but need not be very strong. 
 
 Engines with two cylinders have been planned on proper principles hut 
 have often fallen far short of yielding a good iwulis n.s miL'lit have been 
 expected of them. Thi- is the case principally IK-CM u*e the dimer>- 
 the different parts are difficult to arrange and are often not in proper pro- 
 portion to each other. There are no good models of the-e engines, while 
 there are excellent ones of those constructed after Watt's plan To this is 
 due the it regularity which we observe in the effects produced by the 
 former. whil those produced by the latter are almost uniform. 
 
 4 In the work entitled De la fifc*MM Vintral. by M. Heron de V ill, .fosse, 
 vol. iii , p. .V) itq., we find a good description of the steam ermine- M..W 
 used in mining. The subject Ir.s b. en treated wilh sufficient ful 
 England in the Kneyflnptnlia Brittinnica. Some of the data which we 
 have employed have been taken from the latter work. 
 54
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 "While the expansion of the steam is limited by the dimen- 
 sions of the vessels in which it dilates, the degree of conden- 
 sation at which it is possible to begin to use it is only limited 
 by the resistance of the vessels in which it is generated name- 
 ly, of the boilers. In this respect we are far from having 
 reached the possible limits. The character of the boilers in 
 general use is altogether bad; although the tension of the steam 
 is rarely carried in them beyond 4 to 6 atmospheres, they often 
 burst and have caused serious accidents. It is no doubt quite 
 possible to avoid such accidents and at the same time to make 
 the tension of the steam greater than that commonly employed. 
 
 Besides the high - pressure engines with two cylinders of 
 which we have been speaking, there are also high-pressure en- 
 gines with one cylinder. Most of these have been constructed 
 by two skilful English engineers, Messrs. Trevithick and Viv- 
 ian. They use the steam under a very high pressure, some- 
 times 8 or 10 atmospheres, but they have no condenser. The 
 steam, after its entrance into the cylinder, undergoes a certain 
 expansion, but its pressure is always greater than that of the 
 atmosphere. When it has done its work, it is ejected into the 
 atmosphere. It is evident that this mode of procedure is en- 
 tirely equivalent, with respect to the motive power produced, 
 to condensing the steam at 100 degrees, and that we lose a part 
 of the useful effect, but engines thus worked can dispense with 
 the condenser and air-pump. They are less expensive than the 
 others, and are not so complicated ; they take less room, and can 
 be used where it is not possible to obtain :i current of cold water 
 sufficient to effect condensation. In such places they possess 
 an incalculable advantage, since no others can be used. They 
 are used principally in England to draw wagons for the carriage 
 of coal on railroads, either in the interior of mines or on the 
 surface. 
 
 Some remarks may still be made on the use of permanent 
 gases and vapors other than water vapor in the development of 
 the motive power of heat. 
 
 Various attempts have been made to produce motive power 
 by the action of heat on atmospheric air. This gas, in com- 
 parison with water vapor, presents some advantages and some 
 disadvantages, which we shall now examine. 
 
 1. It has this notable advantage over water vapor, that 
 since for the same volume it has a much smaller capacity for 
 55
 
 MEMOIRS <>N 
 
 heat it cools more for an equal expansion, as is proved by what 
 we have previously said. We have seen the importance of effect- 
 ing the greatest possible changes of temperature by changes of 
 volume alone. 
 
 2. Water vapor can be formed only by the aid of a boiler, 
 while atmospheric air can be heated directly by combustion 
 within itself. Thus a considerable loss is avoided, not only in 
 the quantity of heat, but also in its thermometric degree. This 
 advantage belongs exclusively to atmospheric air ; the other 
 gases do not possess it; they would be even more difficult to 
 heat than water vapor. 
 
 3. In order to produce a great expansion of the air, and to 
 cause thereby a great change of temperature, it would be neces- 
 sary to subject it in the first place to rather a high pressure, to 
 compress it by an air-pump or by some other means before heat- 
 ing it. This operation would require a special apparatus which 
 does not form a part of the steam-engine. In it the water is 
 in a liquid state when it enters the boiler, and requires only a 
 small force-pump to introduce it. 
 
 4. The cooling of the vapor by the contact of the refrigerat- 
 ing body is more rapid and easy than the cooling of air could 
 be. It is true that we have the resource of eject inn it into the. 
 atmosphere. This procedure would have the further advan- 
 tage that we could then dispense with a refrigerator, which is 
 not always at our disposal, but in that case the air must nt 
 expand so far that its pressure is lower than that of the a; 
 phere. 
 
 5. One of the most serious drawbacks to the employment of 
 steam is that it cannot be used at high temperatures except 
 with vessels of extraordinary strength. This is not true <>f air. 
 for which there is no necessary relation between its tempera- 
 ture and elastic force. The air, then, seems bettor fitted than 
 steam to realize the motive power of the descent of caloric at 
 high temperatures; perhaps at low temperatures water vapor 
 would be preferable. We can even conceive of the possibility 
 of making the same heat act successively in air and in water 
 vapor. All that would be necessary would be to keep the tem- 
 perature of the air sufficiently high, after it had Wn nsi-d. and 
 instead of ejecting it immediately into the atmosphere, to sur- 
 round a steam-boiler with it, as if it came directly from the 
 fire-box. 
 
 N
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 The use of atmospheric air for the development of the mo- 
 tive power of heat presents very great practical difficulties which, 
 however, may not be insurmountable. These difficulties once 
 overcome, it will doubtless be far superior to water vapor.* 
 
 As for other permanent gases, they should be finally rejected ; 
 they have all the inconveniences of atmospheric air without 
 any of its advantages. 
 
 The same may be said of other vapors in comparison with 
 water vapor. 
 
 * Among the attempts made to develop the motive power of heat by the 
 use of atmospheric air, we should notice particularly those of MM. Niepce, 
 which were made in France several years ago by means of an apparatus, 
 called by the inventors pyreolophore. This instrument consists essentially 
 of a cylinder, furnished with a piston, and tilled with atmospheric air at 
 ordinary density. Into this is projected some combustible substance in a 
 highly attenuated form, which remains in suspension for a moment in the 
 air and is then ignited. The combustion produces nearly the same effect 
 as if the elastic fluid were a mixture of air and combustible gas of air 
 and carburetted hydrogen, for example a sort of explosion occurs and a 
 sudden expansion of the elastic fluid, which is made use of by causing it 
 to act altogether against the piston. This moves through a certain dis- 
 tance, and the motive power is thus realized. There is nothing to prevent 
 a renewal of the air and a repetition of the first operation. This very in- 
 genious engine, which is especially interesting on account of the novelty 
 of its principle, fails in an esseniial particular. The substance used for the 
 combustible (lycopodium powder, that which is used to produce flames on 
 the stage) is so expensive, that all other advantages are outweighed, and 
 unfortunately it is difficult to make use of a moderately cheap combustible, 
 for it requires a substance that is very finely pulverized, in which the igni- 
 tion is prompt, is propngnted rapidly, and which leaves little or no residue. 
 
 Instead of following MM. Niepce's operations it would seem to us better 
 to compress the air by air-pumps and to conduct it through a perfectly 
 sealed fire-box into which the combustible is introduced in small quan- 
 tities by some mechanism which is easy to conceive of; to allow it to de- 
 velop its action in a cylinder with a piston or in any other envelope capable 
 of enlargement ; to eject it finally into the atmosphere, or even to pass it 
 under a steam-boiler in order to utilize its remaining heat. 
 
 The chief difficulties which we should have to meet in this mode of 
 operation would be the enclosure of the fire-box in a sufficiently solid en- 
 velope, the suitable control of the combustion, the maintenance of a moder- 
 ate temperature in the several parts of the engine, and the prevention of O; 
 rapid deterioration of the cylinder and piston. We do not consider these 
 difficulties insurmountable. 
 
 It is said that successful attempts have been made in England to develop 
 motive power by the action of heat on atmospheric air. We do not know 
 what these are, if, indeed, they have really been made. 
 57
 
 MEMOIRS ON 
 
 It would no doubt be preferable if there were an abundant 
 supply of a liquid which evaporated at a higher temperature 
 than water, the specific heat of whose vapor was less for equal 
 volume, and which did not injure the metals used in the con- 
 struction of an engine ; but no such body exists in natim-. 
 
 The use of alcohol vapor has been suggested, and engines 
 have even been constructed in order to make it possible, in 
 \\hii-li the mixture of the vapor with the water of condensation 
 is avoided by applying the cold body externally instead of in- 
 troducing it into the engine. 
 
 It was thought that alcohol vapor possessed a notable advan- 
 tage on account of its having a greater tension than that of 
 water vapor at the same temperature. We see in this only an- 
 other difficulty to be overcome. The principal defect of water 
 vapor is its excessive tension at a high temperature, and this 
 defect is still more marked in alcohol vapor. As for the ad- 
 vantage which it was believed to have with respect to a larger 
 production of motive power, we know from the principles stated 
 above that they are imaginary. 
 
 Thus it is with the use of water vapor and atmospheric air 
 that the future attempts to improve the steam-engine shun 1.1 
 be made. AH efforts should be directed to utilize by means of 
 these agents the largest possible descents of caloric. 
 
 We shall conclude by showing how far we are from the reali- 
 zation, by means already known, of all the motive power of the 
 combustibles. 
 
 A kilogram of coal burned in the calorimeter furnishes a 
 quantity of heat capable of raising the tempi-rat uru of about 
 7000 kilograms of water 1 degree that is, from the definition 
 given (page 43) it furnishes 7000 units of heat. The largest 
 descent of caloric which can be realized is measured by the dif- 
 ference of the temperature produced by combustion and that 
 of the refrigerating body. It is difficult to see any limit to the 
 temperature of combustion other than that at which the com- 
 bination of the combustible with oxygen is effected. Let us 
 assume, however, that this limit is 1000 degrees, which is cer- 
 tainly within the bounds of truth. We shall assume the t< m 
 peratnre of the refrigerator to be degrees. 
 
 \\ have calculated approximately (page 4. r >) the quantity of 
 motive power developed by 1000 units of In at in passing from 
 the temperature 100 to the temperature 99, and have found
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 it to be 1.112 units, each equal to 1 meter of water raised 1 
 meter. 
 
 If the motive power were proportional to the descent of caloric, 
 if it were the same for each thermometric degree, nothing would 
 be easier than to estimate it from 1000 to degrees. Its value 
 would be 
 
 1.112 x 1000 = 1112. 
 
 But as this law is only approximate, and perhaps at high 
 temperatures departs a good deal from the truth, we can only 
 make a very rough estimate. Let us suppose the number 1112 
 to be reduced one-half that is, to 560. 
 
 Since one kilogram of coal produces 7000 units of heat, and 
 since the number 560 is referred to 1000 units, we must multi- 
 ply it by 7, which gives us 
 
 7 x 560 = 3920, 
 
 which is the motive power of one kilogram of coal. 
 
 In order to compare this theoretical result with the results 
 of experiment, we shall inquire how much motive power is actu- 
 ally developed by one kilogram of coal in the best heat-engines 
 known. 
 
 The engines which have thus far offered the most advanta- 
 geous results are the large engines with two cylinders used in 
 the pumping out of the tin and copper mines of Cornwall. The 
 best results which they have furnished are as follows : Sixty- 
 five million pounds of water have been raised one English foot 
 by the burning of one bushel of coal (the weight of a bushel is 
 88 Ibs.). This result is equivalent to raising 195 cubic meters 
 of water one meter by the use of one kilogram of coal, which 
 consequently produces 195 units of motive power.* 
 
 * The result given here was furnished by an engine whose large cylin- 
 der was 35 inches in diameter, with a 7-foot stroke ; it is used to pump 
 out one of the mines of Cornwall, called " Wheal Abraham." This result 
 should in a way be considered as an exception, for it only was accomplished 
 for a short time during one month. A product of 30 million Ibs. raised 
 one English foot by a bushel of coal weighing 88 Ibs. is generally consid- ( 
 ered to be an excellent result for a steam-engine. It is sometimes reached 
 by the engines made on Watt's system, but has rnrely been exceeded. 
 This result expressed in French units is equal to 104000 kilograms raised 
 one meter by the burning of one kilogram of coal. 
 
 By what we ordinarily understand as one horse- power in the calculation
 
 MEMOIRS ON 
 
 195 units are only one-twentieth of 3920, the theoretical max- 
 imum ; consequently only ^ of the motive power of the combus- 
 tible has been utilized. 
 
 We have, moreover, chosen our example from among the 
 best steam-engines known. Most of the others have been vrry 
 inferior. For example, G'haillot's engine raises '-.'<> cubic meters 
 of water 33 meters in consuming 30 kilograms of coal, which is 
 equivalent to 22 units of motive power to 1 kilogram, a result 
 nine times less than that cited above, and one hundred and 
 eighty times less than the theoretical maximum. 
 
 We should not expect ever to employ in practice all the mo- 
 tive power of the combustibles used. The efforts which one 
 would make to attain this result would be even more harmful 
 than useful if they led to the neglect of other important con- 
 siderations. The economy of fuel is only one of the conditions 
 which should be fulfilled by steam-engines ; in many cases it is 
 only a secondary consideration. It must often yield the prece- 
 dence to safety, to the solidity and durability of the engine, to 
 the space which it must occupy, to the cost of its construction, 
 etc. To be able to appreciate justly in each case the consider- 
 ations of convenience and economy which present themselves, 
 to be able to recognize the most important of those which are 
 only subordinate, to adjust them all suitably, and finally to 
 reach the best result by the easiest method such should be the 
 power of the man who is called on to direct and co-ordinate the 
 labors of his fellow-men, and to make them concur in attaining 
 a useful purpose. 
 
 BIOGRAPHICAL SKETCH 
 
 NICOLAS-LEON A RD-S ADI CARXOT was born in Paris on June 
 1, 17% ; the son of the illustrious engineer, soldier, and states- 
 man who played so prominent a part in the history of France 
 during the Revolution. He was educated at the Keolc I'i.l\ - 
 
 of the efficiency of steam-engines, a 10 horse-power engine should raise 
 10 x 75, or 750 kilograms 1 meter iu a second, or 750 x 8600 = 3700000 
 kilograms 1 meter in an hour. 
 
 If we suppose eacli kilogram of coal to raise 104000 kilograms, it is 
 necessary to divide 2700000 by 104000 to find the coal burned in one hur 
 by the 10 horse-power engine, which gives us YoY = 28 kilograms. Hut 
 it is very rare that a 10 horse-power engine consumes less than 26 kilograms 
 of coal an hour. 
 
 60
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 technique, and served for several years as an officer of engineers 
 and on the general staff. His inclinations towards the study of 
 science were so strong that they controlled the whole course of 
 his life. While still engaged in his profession he devoted such 
 time as he could spare to scientific investigations, and he at 
 last resigned from the army in order to obtain more leisure for 
 studious pursuits. He died of the cholera on August 24, 1832. 
 The memoir on the "Motive Power of Heat" is the only one 
 which he published. It shows that he possessed a mind able 
 to penetrate to the heart of a question, and to invent general 
 methods of reasoning. The extracts from his note-book, pub- 
 lished by his brother, indicate that he was also fertile in devis- 
 ing experiments. It is interesting to note that the doubt of 
 the validity of the substantial theory of heat, expressed by him 
 in his memoir, developed later into complete disbelief, and that 
 he not only adopted the mechanical theory of heat, but planned 
 experiments to test it similar to those of Joule, and calculated 
 that the mechanical equivalent of heat is equal to 370 kilogram- 
 meters.
 
 ON THE MOTIVE POWER OF HEAT, AND 
 
 ON THE LAWS WHICH CAN BE 
 
 DEDUCED FROM IT FOR THE 
 
 THEORY OF HEAT 
 
 BY 
 
 R. CLAUSIUS 
 
 (Poggendorff's Annalen, vol. Ixxix., pp. 376 and 500. 1850)
 
 CONTEXTS 
 
 MM 
 
 Work of Carnot and Clapeyron 65 
 
 Dynamical Theory of Heat 66 
 
 Equivalence of Heat and Work 67 
 
 Camot't Cycle 
 
 Application to Change of State 78 
 
 Second Late of Thermodynamics 88 
 
 Carnot't /*/<// 90 
 
 Application of Clapeyron'* Equation .):{ 
 
 Mechanical Equivalent, of Heat 105
 
 ON THE MOTIVE POWER OF HEAT, AND 
 
 ON THE LAWS WHICH CAN BE 
 
 DEDUCED FROM IT FOR THE 
 
 THEORY OF HEAT 
 
 BY 
 
 R. CLAUSIUS 
 
 SINCE heat was first used as a motive power in the steam- 
 engine, thereby suggesting from practice that a certain quantity 
 of work may be treated as equivalent to the heat needed to pro- 
 duce it, it was natural to assume also in theory a definite rela- 
 tion between a quantity of heat and the work which in any 
 possible way can be produced by it, and to use this relation in 
 drawing conclusions about the nature and the laws of heat it- 
 self. In fact, several fruitful investigations of this sort have 
 already been made ; yet I think that the subject is not yet ex- 
 hausted, but on the other hand deserves the earnest attention 
 of phvsicists, partly because serious objections can be raised to 
 the conclusions that have already been reached, partly because 
 other conclusions, which may readily be drawn and which will 
 essentially contribute to the establishment and completion of 
 the theory of heat, still remain entirely unnoticed or have not 
 yet been stated with sufficient definiteness. 
 
 The most important of the researches here referred to was 
 that of S. Carnot,* and the ideas of this author, were after- 
 wards given analytical form in a very skilful way by Clapey- 
 ron.f Carnot showed that whenever work is done by heat and 
 
 * Reflexions sur la puissance motrice du feu, et sur Us machines propres a 
 developper cette puissance, par 8. Carnot. Paris, 1824. I have not been 
 able to obtain a copy of this book, and am acquainted with it only through 
 the work of Clapeyron and Thomson, from the latter of whom are quoted 
 the extracts afterwards given. 
 
 f Journ. de I'ficole Poly technique, vol. xix. (1834), and Pogg. Ann., vol. lix. 
 E 65
 
 MKMoIKS ON 
 
 no permanent change occurs in the condition of the work in <r 
 body, a certain quantity of heat passes from a hotter to a c>|.|rr 
 body. In the steam-engine, for example, by means of the 
 steam which is developed in the boiU'r and precipitated in the 
 condenser, heat is transferred from the grate to the eonden> T. 
 This transfer he considered as the heat change, correspond in;: 
 to the work done, lie says expressly that no heat is lost in tin- 
 process, but that the quantity of heat remains unchanged, and 
 adds : " This fact is not doubted ; it was assumed at first with- 
 out investigation, and then established in many cases by calori- 
 metric measurements. To deny it would overthrow tin- whole 
 theory of heat, of which it is the foundation." I am nut a wan-. 
 however, that it has been sufficiently proved by experiment 
 that no loss of heat occurs when work is done ; it may, perhaps. 
 on the contrary, be asserted with more correctness that even if 
 such a loss has not been proved directly, it has yet been sh>wn 
 by other facts to be not only admissible, but even highly prob- 
 able. If it be assumed that heat, like a substance, cannot 
 diminish in quantity, it must also be assumed that it cannot 
 increase. It is, however, almost impossible to explain the heat 
 produced by friction except as an increase in the quantity of 
 heat. The careful investigations of Joule, in whieh heat is 
 produced in several different ways by the application of me- 
 chanical work, have almost certainly proved not only the pos- 
 sibility of increasing the quantity of heat in any ciivumst 
 but also the law that the quantity of heat developed i- propor- 
 tional to the work expended in the operation. To this it mu.-t 
 be added that other facts have lately become known \\hich 
 support the view, that heat is not a substance, but con>i.-i> in a 
 motion of the least parts of bodies. If this view is correct, it 
 is admissible to apply to heat the general mechanical principle 
 that a motion may be transformed into work, and in siu-h a 
 manner that the loss of //.-> rira is proportional to the work ac- 
 complished. 
 
 These facts, with which Carnot also was well acquainted, and 
 the importance of which he has expressly re -oirnixi-d. almost 
 compel us to accept the equivalence between heat and work, on 
 the modified hypothesis that the accomplishment of work re- 
 quires not merely a change in the >iistribution of heat, hut also 
 an actual consumption of heat, ami that, conversely, heat can 
 be developed again by the expenditure of work. 
 ' M
 
 THE SECOXD LAW OF THERMODYNAMICS 
 
 In a memoir recently published by Holtzmann,* it seems at 
 first as if the author intended to consider the matter from this 
 latter point of view. He says (p. 7) : " The action of the heat 
 supplied to the gas is either an elevation of temperature, in 
 conjunction with an increase in its elasticity, or mechanical 
 work, or a combination of both, and the mechanical work is 
 the equivalent of the elevation of temperature. The heat can 
 only be measured by its effects ; of the two effects mentioned 
 the mechanical work is the best adapted for this purpose, and 
 it will accordingly be so used in what follows. I call the unit 
 of heat the heat which by its entrance into a gas can do the me- 
 chanical work a that is, to use definite units, which can lift a 
 kilograms through 1 meter." Later (p. 12) he also calculates 
 the numerical value of the constant a in the same way as Mayer 
 had already done,f and obtains a number which corresponds 
 with the heat equivalent obtained by Joule in other entirely 
 different ways. In the further extension of his theory, how- 
 ever, in particular in the development of the equations from 
 which his conclusions are drawn, he proceeds exactly as Clapey- 
 ron did, so that in this part of his work he tacitly assumes that 
 the quantity of heat is constant. 
 
 The difference between the two methods of treatment has 
 been much more clearly grasped by W. Thomson, who has ex- 
 tended Carnot's discussion by the use of the recent observations 
 of Regnault on the tension and latent heat of water vapor. J He 
 speaks of the obstacles which lie in the way of the unrestricted 
 assumption of Carnot's theory, calling special attention to the 
 researches of Joule, and also raises a fundamental objection 
 which may be made against it. Though it may be true in any 
 case of the production of work, when the working body has re- 
 turned to the same condition as at first, that heat passes from a 
 warmer to a colder body, yet on the other hand it is not gener- 
 ally true that whenever heat is transferred work is done. Heat 
 can be transferred by simple conduction, and in all such cases, 
 if the mere transfer of heat were the true equivalent of work, 
 there would be a loss of working power in Nature, which is 
 hardly conceivable. Nevertheless, he concludes that in the 
 
 * Ueber die Warme und Elasticit&t der Oase und Dampfe, von C. Holtz- 
 mann, Mannheim, 1845 : also Pogg. Ann., vol. 72a. 
 
 f Ann. der Chem. und Pfiarm. of WOhler and Liebig, vol. xlii., p. 239. 
 J Transactions of Uie Royal Society of Edinburgh, vol. xvi. 
 67
 
 MEMOIRS ON 
 
 present state of the science the principle adopted by Carnot is 
 still to he taken as the most probable basis for an investigation 
 of the motive power of heat, saying : " If we abandon this prin- 
 ciple, we meet with innumerable other difficulties insuperable 
 without further experimental investigation, and an entire it-- 
 construction of the theory of heat from its foundation."* 
 
 I believe that we should not be daunted by these difficulties, 
 but rather should familiarize ourselves as much as possible 
 with the consequences of the idea that heat is a motion, since 
 it is only in this way that we can obtain the means \\here\\iih 
 to confirm or to disprove it. Then, too, I do not think the 
 difficulties are so serious as Thomson does, since even though 
 we must make some changes in the usual form of presentation, 
 yet I can find no contradiction with any proved facts. It is 
 not at all necessary to discard Caruot's theory entirely, a 
 step which we certainly would find it hard to take, since it 
 has to some extent been conspicuously verified by experience. 
 A careful examination shows that the new method does not 
 stand in contradiction to the essential principle of Carnot, but 
 only to the subsidiary statement that no hmt /> l^t, sim-e in 
 the production of work it may very well be the case that at the 
 same time a certain quantity of heut is consumeti and another 
 quantity transferred from a hotter to a colder body, and hoth 
 quantities of heat stand in a definite relation to the work that 
 is done. This will appear more plainly in the sequel, and it 
 will there be shown that the consequences drawn fmm the two 
 assumptions are not only consistent with one another, but are 
 even mutually confirmatory. 
 
 1. CONSEQUENCES OF THE PRINCIPLE OP THE EQl'IY \U.\- K 
 OF HEAT A Xli \\oKK 
 
 We shall not consider here the kind of motion which ran lie 
 conceived of as taking place within bodies, further than to as- 
 sume in general that the particle-; of l>odie>an- in motion, ami 
 that their heat is the measure of their ris viva, or rath, r still 
 more generally, we shall only lay down a principle condition.-.! 
 by that assumption as a fundamental prineiple, in the words: 
 In all cases in which work is produced by the agency of li.at. 
 a quantity of heat is consumed which is proportional to the 
 
 * Math, and Phyt. I\ijr, vol. I., p. 119, note. 
 68
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 work done ; and, conversely, by the expenditure of an equal 
 quantity of work an equal quantity of heat is produced. 
 
 Before we proceed to the mathematical treatment of this 
 principle, some immediate consequences may be premised 
 which affect our whole method of treatment, and which may 
 be understood without the more definite demonstration which 
 will be given them later by our calculations. 
 
 It is common to speak of the total heat of bodies, especially 
 of gases and vapors, by which term is understood the sum of 
 the free and latent heat, and to assume that this is a quantity 
 dependent only on the actual condition of the body considered, 
 so that, if all its other physical properties, its temperature, its 
 density, etc., are known, the total heat contained in it is com- 
 pletely determined. This assumption, however, is no longer 
 admissible if our principle is adopted. Suppose that we are 
 given a body in a definite state for example, a quantity of gas 
 with the temperature t and the volume t' and that we subject 
 it to various changes of temperature and volume, which are 
 such, however, as to bring it at last to its original state again. 
 According to the common assumption, its total heat will again 
 be the same as at first, from which it follows that if during 
 one part of its changes heat is communicated to it from with- 
 out, the same quantity of heat must be given up by it in the 
 other part of its changes. Now with every change of volume 
 a certain amount of work must be done by the gas or upon it, 
 since by its expansion it overcomes an external pressure, and 
 since its compression can be brought about only by an exertion 
 of external pressure. If, therefore, among the changes to which 
 it has been subjected there are changes of volume, work must 
 be done upon it and by it. It is not necessary, however, that 
 at the end of the operation, when it is again brought to its 
 original state, the work done by it shall on the whole equal 
 that done upon it, so that the two quantities of work shall 
 counterbalance each other. There may be an excess of one or 
 the other of these quantities of work, since the compression 
 may take.place at a higher or lower temperature than the ex- 
 pansion, as will be more definitely shown later on. To this 
 excess of work done by the gas or upon it there must corre- 
 spond, by our principle, a proportional excess of heat consumed 
 or produced, and the gas cannot give up to the surrounding 
 medium the same amount of heat as it receives.
 
 MKMnlRS OX 
 
 The same contradiction to the ordinary assumption about 
 the total hi-at may be presented in another way. If tin- gas at 
 / and i' is brought to the higher temperature /, and tin- larger 
 volume t>j, the quantity of heat which must be imparted to it 
 is, on that assumption, independent of the way in which the 
 change is brought about ; from our principle, however, it is dif- 
 ferent, according as the gas is first heated wink- its vohun 
 is constant, and then allowed to expand at the << instant tem- 
 peratnre /,, or is first expanded at the constant temperature/,,. 
 and then heated, or as the expansion and heating aiv inter- 
 changed in any other way or even occur together, since in all 
 these cases the work done by the gas is different. 
 
 In the same way, if a quantity of water at the temperature 
 / is changed into vapor at the temperature /, and of the 
 volume r,. it will make a difference in the amount of lu-at 
 needed if the water as such is first heated to /, and then 
 evaporated, or if it is evaporated at / and the vapor then 
 brought to the required volume and temperature. /-, and /,. or 
 finally if the evaporation occurs at any intermediate tempera- 
 ture. 
 
 From these considerations and from the immediate applica- 
 tion of the principle, it may easily be seen what conception 
 must he formed of luti-nt heat. I'sing again the example al- 
 ready employed, we distinguish in the quantity of heat which 
 must be imparted to the water during its changes the t'rtr and 
 the Inti-nt heat. Of these, however, we may consider only the 
 former as really present in the vapor that has l.ecii formed. 
 The latter is not merely, as its name implies. r//, >/,/,,/ from 
 our perception, hut it is n<nr}n /< jm:cnf ; it is nmsitinnl during 
 the changes in doing work. 
 
 In the heat consumed we must still introduce :i distinction 
 that is to say, the work done is of two kinds, l-'irst. there i> a 
 certain amount of work done in overcoming the mutual attrac- 
 tions of the particles of the water, and in M -parating them to 
 such a distance from one another. that they are in the Mate of 
 vapor. Secondly, the vapor during its evolution iyu>t push 
 back an external pressure in order to make room for itself. 
 The former work we shall call the i/i/rnml. the latter the t- 
 ln-H'il work, and shall partition the latent heat accordingly. 
 
 It can make no difference with respect to the //,/, / W work 
 whether the evaporation goes on at / or at /,, or at any intcr- 
 70
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 mediate temperature, since we must consider the attractive 
 force of the particles, which is to be overcome, as invariable.* 
 
 The external work, on the other hand, is regulated by the 
 pressure as dependent on the temperature. Of course the 
 same is true in general as in this special example, and there- 
 fore if it was said above that the quantity of heat which must 
 be imparted to a body, to bring it from one condition to an- 
 other, depended not merely on its initial and final conditions, 
 but also on the way in which the change takes place, this 
 statement refers only to that part of the latent heat which cor- 
 responds to the external work. The other part of the latent 
 heat, as also the free heat, are independent of the way in which 
 the changes take place. 
 
 If now the vapor at t l and v, is again transformed into water, 
 work will thereby be expended, since the particles again yield 
 to their attractions and approach each other, and the external 
 pressure again advances. Corresponding to this, heat must be 
 produced, and the so-called liberated heat which appears during 
 the operation does not merely come out of concealment but is 
 actually made new. The heat produced in this reversed opera- 
 tion need not be equal to that used in the direct one, but that 
 part which corresponds to the external work may be greater or 
 less according to circumstances. 
 
 We shall now turn to the mathematical discussion of the sub- 
 ject, in which we shall restrict ourselves to the consideration of 
 the permanent gases and of vapors at their maximum density, 
 since these cases, in consequence of the extensive knowledge 
 we have of them, are most easily submitted to calculation, and 
 besides that are the most interesting. 
 
 Let there be given a certain quantity, say a unit of weight, 
 of a permanent gas. To determine its present condition, three 
 magnitudes must be known : the pressure upon it, its volume, 
 
 ' * It cannot he raised, as an objection to this statement, that water at t, 
 has less cohesion than at. t t , and that therefore less work would be needed 
 to overcome it. For a certain amount of work is used in diminishing the 
 cohesion, which is done while the water as such is heated, and this must 
 be reckoned in with that done during the evaporation. It follows at once 
 that only a part of the heat, which the water takes up from without whilfe 
 it is being heated, is to be considered as free heat, while the remainder is 
 used in diminishing the cohesion. This view is also consistent with the 
 circumstance that water has so much greater a specific heat than ice, and 
 probably also than its vapor. 
 
 71
 
 MEMOIRS ON 
 
 and its temperature. These magnitudes are in a mutual re- 
 lationship, which is expressed by the combined laws of Mariotte 
 and Gay-Lussac*, and may be represented by the equation : 
 
 (I.) pv=R(a+t), 
 
 where p, v, and t represent the pressure, volume, and tem- 
 perature of the gas in its present condition, a is a constant. 
 the same for all gases, and /,' is also a constant, which in its 
 
 complete form is ~^i if jo *><> ant * 'o are tne corresponding 
 
 values of the three magnitudes already mentioned for any other 
 condition of the gas. This last constant is in so far different 
 for the different gases that it is inversely proportional to their 
 specific gravities. 
 
 It is true that Regnault has lately shown, by a very careful 
 investigation, that this law is not strictly accurate, yet the de- 
 partures from it are in the case of the permanent gases \erv 
 small, and only become of consequence in the case of those 
 gases which can be condensed into liquids. From this it seems 
 to follow that the law holds with greater accuracy the more 
 removed the gas is from its condensation point with respect to 
 pressure and temperature. We may therefore, while the ac- 
 curacy of the law for the permanent gases in their ordinary 
 condition is so great that it can be treated as complete in most 
 investigations, think of a limiting condition for each gas. in 
 which the accuracy of the law is actually complete. We shall. 
 in what follows, when we treat the permanent gases as sin-h. 
 assume this ideal condition. 
 
 According to the concordant investigations of Regnault and 
 
 Magnus, the value of - for atmospheric air is equal to 0.003G65, 
 
 if the temperature is reckoned in centigrade degrees from the 
 freezing-point. Since, however, as has been mentioned, the 
 gases do not follow the M. and G-. law exactly, the same value 
 
 of - will not always be obtained, if the measurements are 
 
 made in different circumstances. The number here given 
 holds for the case when air is taken at under the pressure of 
 one atmosphere, and heated to 100 at constant volume, ami the 
 
 * This law will hereaflcr. for brevity, be called tin- M. ami <} law, anil 
 Muriuiu-'a law will be called the M. law. 
 78
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 increase of its expansive force observed. If, on the other hand, 
 the pressure is kept constant, and the increase of its volume 
 observed, the somewhat greater number 0.003670 is obtained. 
 Further, the numbers increase if the experiment is tried 
 under a pressure higher than the atmospheric pressure, while 
 they diminish somewhat for lower pressures. It is not there- 
 fore possible to decide with certainty on the number which 
 should be adopted for the gas in the ideal condition in which 
 naturally all differences must disappear ; yet the number 
 0.003665 will surely not be far from the truth, especially since 
 this number very nearly obtains in the case of hydrogen, which 
 probably approaches the most nearly of all the gases the ideal 
 condition, and for which the changes are in the opposite sense 
 to those of the other gases. If we therefore adopt this value 
 
 of we obtain 
 
 In consequence of equation (I.) we can treat any one of the 
 three magnitudes p, v, and t for example, p as a function of 
 the two others, v and t. These latter then are the independent 
 variables by which the condition of the gas is fixed. We shall 
 now seek to determine how the magnitudes which relate to the 
 quantities of heat depend on these two variables. 
 
 If any body changes its volume, mechanical work will in general 
 be either produced or expended. It is, however, in most cases 
 impossible to determine this exactly, since besides the external 
 work there is generally an unknown amount of internal work done. 
 To avoid this difficulty, Carnot employed the ingenious method 
 already referred to of allowing the body to undergo its various 
 changes in succession, which are so arranged that it returns at 
 last exactly to its original condition. In this case, if internal 
 work is done in some of the changes, it is exactly compensated 
 for in the others, and we may be sure that the external work, 
 , which remains over after the changes are completed, is all the 
 work that has been done. Clapeyron has represented this proc- 
 ess graphically in a very clear way, arid we shall follow his pres- 
 entation now for the permanent gases, with a slight alteration 
 rendered necessary by our principle. 
 
 In the figure, let the abscissa oe represent the volume and 
 the ordinate ea the pressure on a unit weight of gas, in a con- 
 dition in which its temperature = t. We assume that the gas 
 73
 
 MEMOIRS ON 
 
 Fig. I 
 
 is contained in an extensible envelope, which, however, cannot 
 exchange heat with it. If, now. it is allowed to expand in this 
 envelope, its temperature would fall if no heat were imparted 
 to it. To avoid this, let it be put in contact, during its ex- 
 pansion, with a body, A, which is kept at the constant tempera- 
 ture t, and which imparts 
 just so much heat to the 
 gas that its temperature also 
 remains equal tot. During 
 this expansion at constant 
 temperature, its pressure 
 diminishes according to the 
 M. law, and may be repre- 
 sented by the ordinate of a 
 curve, ab, which is a por- 
 tion of an equilateral hy- 
 perbola. When the volume 
 of the gas has increased in this way from oe to of, tin- body .1 
 is removed, and the expansion is allowed to continue with- 
 out the introduction of more heat. The temperature will 
 then fall, and the pressure diminish more rapidly than before. 
 The law which is followed in this part of the process may In- 
 represented by the curve be. When the volume of the gas has 
 increased in this way from of to Of/, and its temperature has 
 fallen from t to T, we begin to compress it, in order to restore 
 it again to its original volume oe. If it were left to itself its 
 temperature would airain rise. This, however, we do not per- 
 mit, but bring it in contact with a body, li, at the constant tem- 
 perature r, to which it at once gives up the heat that is pro- 
 duced, so that it keeps the temperature T ; and while it is in 
 contact with this body we compress it so far (by the amount ////) 
 that the remaining compression he is exactly sufficient to raise 
 its temperature from T to /. if during this last coinpn -<-ion it 
 gives up no heat. During the former compression the piv.-smv 
 increases according to the M. law. and is represented l,\ tic 
 portion cd of an equilateral hyperbola. During the latter, on 
 the other hand, the increase is more rapid and is represented 
 by the curve da. This curve must end exactly at n. for since 
 at the end of the operation the volume and temperature have 
 again their original values, the same must be true of the 
 pressure also, which is a function of them both. The gas is 
 74
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 therefore in the same condition again as it was at the begin- 
 ning. 
 
 Now, to determine the work produced by these changes, for 
 the reasons already given, we need to direct our attention only 
 to the external work. During the expansion the gas does work, 
 which is determined by the integral of the product of the dif- 
 ferential of the volume into the corresponding pressure, and 
 is therefore represented geometrically by the quadrilaterals 
 eabf and fbcg. During the compression, on the other hand, 
 work is expended, which is represented similarly by the quad- 
 rilaterals (jcdk and hdae. The excess of the former quantity of 
 work over the latter is to be looked on as the whole work pro- 
 duced during the changes, and this is represented by the quad- 
 rilateral abed. 
 
 If the process above described is carried out in the reverse 
 order, the same magnitude, abed, is obtained as the excess of 
 the work expended over the work done. 
 
 In order to make an analytical application of the method 
 just described, we will assume that all the changes which the 
 gas undergoes are infinitely small. We may then treat the 
 curves obtained as straight lines, as they are represented in the 
 accompanying figure. \Ve may also, in determining the area 
 of the quadrilateral 
 abed, consider it a par- 
 allelogram, since the 
 error arising there- 
 from can only be a 
 quantity of the third 
 order, while the area 
 itself is a quantity of 
 the second order. On 
 this assumption, as 
 may easily be seen, 
 the area may be repre- 
 sented by the product 
 ef.bk, if k is the point in which the ordinate bf cuts the lower 
 side of the quadrilateral. The magnitude bk is the increase 
 of the pressure, while the gas at the constant volume of has 
 its temperature raised from r to t that is, by the differential 
 t T = dt. This magnitude may be at once expressed by the 
 aid of equation (I.) in terms of v and t, and is 
 75 
 
 Fig. 2
 
 MEMO IKS ON 
 W 
 
 * T 
 
 If, further, we denote the increase of volume ef by dv, we ob- 
 tain the area of the quadrilateral, and so, also, 
 
 (1) The work done = 
 
 We must now determine the heat consumed in these changes. 
 The quantity of heat which must be communicated to a gas, 
 while it is brought from any former condition in a definite way 
 to that condition in which its volume = v and its temperature 
 = t, may be called Q, and the changes of volume in the above 
 process, which must here be considered separately, may l>e rep- 
 resented as follows: efby dv, fig by d'v, eh by St>, and fgby 3'r. 
 During an expansion from the volume oe v to the volume 
 of v 4- dv at the constant temperature /, the gas must receive 
 the quantity of heat 
 
 di- 
 
 and correspondingly, during an expansion from oh v + 2t to 
 og = v + 2r + d'v at the temperature t dl, the quantity of 
 heat, 
 
 [8^-K-l* 
 
 In the case before us this latter quantity must be taken as 
 negative in the calculation, because the real process was a com- 
 pression instead of the expansion assumed. Pnrini: tin- expan- 
 sion from of to og and the compression from oh to oe, the gas 
 h:is neither gained nor lost heat, and hence the quantity of 
 heat which the gas has received in excess of that which it has 
 given up that is, the heat m, ,*,,,/ 
 
 The magnitudes Iv and d'v must be eliminate'! fn"n this ex- 
 pression. For this purpose we have first, immediately from the 
 inspection of the figure, the following equation : 
 
 l-'r<>in the condition that during the compression from // to 
 ml therefore also conversely during an expansion from <>< 
 to oh occurring under the same conditions, and similarly dur- 
 76
 
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 ing the expansion from of to og, both of which occasion a fall 
 of temperature by the amount dt, the gas neither receives nor 
 gives up heat, we obtain the equations 
 
 Eliminating from these three equations and equation (2) the 
 three magnitudes d'v, cv, and I'v, and also neglecting in the 
 development those terms which, in respect of the differentials, 
 are of a higher order than the second, we obtain 
 
 (3) The heat consumed = -L- ( ( - 1 4- ( ( -rr I dvdt. 
 L d t \ dv } dv \ a t } J 
 
 If we now return to our principle, that to produce a certain 
 amount of work the expenditure of a proportional quantity of 
 heat is necessary, we can establish the formula 
 The heat consumed _ . 
 
 The work done 
 
 where A is a constant, whicli denotes the heat equivalent for 
 the unit of work. The expressions (1) and (3) substituted in 
 this equation give 
 
 R.dvdt 
 
 v 
 or 
 
 (IL) dt \dv) dv \dt 
 
 We may consider this equation as the analytical expression 
 of our fundamental principle applied to the case of permanent 
 gases. It shows that Q cannot be a function of v and t, if 
 these variables are independent of each other. For if it were, 
 then by the well-known law of the differential calculus, that if 
 a function of two variables is differentiated with respect to botji 
 of them, the order of differentiation is indifferent, the right- 
 hand side of the equation should be equal to zero. 
 
 The equation may also be brought into the form of a complete 
 differential equation, 
 
 77
 
 MK MM IKS ON 
 
 in which r is an arbitrary function of v and /. This differen- 
 tial equation is naturally not integrable, but becomes so only 
 if a second relation is given between the varial>K-s. by which / 
 may be treated as a function of t'. The reason for this is 
 found in the last term, and this corresponds exactly to the 
 >.rlirnal work done during the change, since the dilTerential of 
 this work is ]><h\ from which we obtain 
 
 if we eliminate/) by means of (I.). 
 
 We have thus obtained from equation (Il.rt) what was in- 
 troduced before as an immediate consequence of onr principle. 
 that the total amount of heat received by the gas durin.ir a 
 change of volume and temperature can be separated into two 
 parts, one of which, T, which comprises the //>, heat that has 
 entered and the heat consumed in doing internal work, if any 
 such work has been done, has the properties which are com- 
 monly assigned to the total heat, of being a function of / and /. 
 and of being therefore fully determined by the initial and final 
 conditions of the gas, between which the transformation has 
 taken place; while the other part, which comprises the heat 
 consumed in doing external work, in dependent not only on the 
 terminal conditions, but on the whole course of the changes 
 between these conditions. 
 
 Before we undertake to prepare this equation for further 
 conclusions, we shall develop the analytical e\|iiv-.-ion of our 
 fundamental principle for the case of vapors at their maximum 
 density. 
 
 In this case we have no right to apply the M. and (i. law, and 
 so must restrict ourselves to the principle alone. In order to 
 obtain an equation from it, we again use the method given by 
 Carnot and graphically presented by Clapcyion. \\ith a slight 
 modification. Consider a liquid contained in a \c-el impen- 
 etrable by heat, of which, however, only a part is filled by 
 the liqni I. while the rest is left free for the vapor, \\hirh is at 
 the maximum density corresponding to its temperature./. Tho 
 total volume of both liquid and vapor is represented in the ac- 
 companying figure by the abscissa oe, and the pressure of the 
 78
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 vapor by the ordinate ea. Let the vessel now yield to the 
 pressure and enlarge in volume while the liquid and vapor are 
 in contact with a body, A, 
 at the constant temperature 
 /. As the volume increases, 
 more liquid evaporates, but 
 the heat which thus becomes 
 latent is supplied from the 
 body A, so that the temper- 
 ature, and so also the press- 
 ure, of the vapor remain 
 unchanged. If in this way 
 
 the total volume is increased 
 from oe to of, an amount of Fi ff- 3 
 
 external work is done which 
 
 is represented by the rectangle eabf. Now remove the body 
 A and let the vessel increase in volume still further, while heat 
 can neither enter nor leave it. In this process the vapor already 
 present will expand, and also new vapor will be produced, and 
 in consequence the temperature will fall and the pressure dimin- 
 ish. Let this process go on until the temperature has changed 
 from t to r, and the volume has become oy. If the fall of press- 
 ure during this expansion is represented by the curve be, the 
 external work done in the process =fbcg. 
 
 Now diminish the volume of the vessel, in order to bring the 
 liquid with its vapor back to its original total volume, oe; and 
 let this compression take place, in part, in contact with the body 
 B at the temperature r, into which body all the heat set free 
 by the condensation of the vapor will pass, so that the temper- 
 ature remains constant and = r, in part without this body, so 
 that the temperature rises. Let the operation be so managed 
 that the first part of the compression is carried out only so far 
 (to oh) that the volume he still remaining is exactly such that 
 compression through it will raise the temperature from r to t 
 again. During the former diminution of volume the pressure 
 remains invariable, = gc, and the external work employed is 
 equal to the rectangle gcdh. During the latter diminution of 
 volume the pressure increases and is represented by the curve 
 da, which must end exactly at the point , since the original 
 pressure, ea, must correspond to the original temperature, t. 
 The work employed in this last operation is = hdae. At the 
 
 79
 
 MEMOIRS ON 
 
 end of the operation the liquid and vapor are again in the same 
 condition as at the beginning, so that the excess of tin- fj-tt-nml 
 work done over that employed is also the total work done. It 
 is represented by the quadrilateral abed, and its area must also 
 be set equal to the heat consumed during the same time. 
 
 For our purposes we again assume that the changes just de- 
 scribed are infinitely small, and on this assumption represent 
 the whole process by the accompanying figure, in which the 
 curves ad and be which occur in 
 Fig. 3 have become straight lines. 
 So far as the area of the quadrilat- 
 eral abed is concerned, it may again 
 be considered a parallelogram, and 
 may be represented by the product 
 ef.bk. If, now, the pressure of tin; 
 vapor at the temperature t and at 
 
 / a ' its maximum tension is represented 
 Pig 4 by/?, and if the temperature ditTcr- 
 
 ence t T is represented by */-. 
 
 hftve =$*. 
 
 The line ef represents the increase of volume, which occurs in 
 consequence of the passage of a certain quantity of liquid, 
 which may be called dm, over into vapor. Representing now 
 the volume of a unit weight of the vapor at its maximum <!en- 
 sity at the temperature t by *, and the volume of the same 
 quantity of liquid at the temperature / by 9, we have evidently 
 
 ef-(s- )dm, 
 and consequently the area of the quadrilateral, or 
 
 (5) The work done = (* - ef^dmdt. 
 
 In order to represent the quantities of heat concerned, we 
 will introduce the following symbols. The quantity of heat 
 which becomes latent when a unit weight of the liquid evapo- 
 rates at the temperature / and under the corresponding press- 
 ure, is called r, and the specific heat of the liquid is called 
 
 Both of these quantities, as well as also *, a, and ^, are to be 
 
 considered functions of t. Finally, let us denote by //// tin- 
 quantity of heat which must be imparted to a unit weight of 
 80
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 the vapor if its temperature is raised from t to t -f- dt, while it is 
 so compressed that it is again at the maximum density for this 
 temperature without the precipitation of any part of it. The 
 quantity 7* is likewise a function of t. It will, for the pres- 
 ent, be left undetermined whether it has a positive or negative 
 value. 
 
 If we now denote by /i the mass of liquid originally present 
 in the vessel, and by tit the mass of vapor, and further by dm 
 the mass which evaporates during the expansion from oe to of, 
 and by d'm the mass which condenses during the compression 
 from og to oh, the heat which becomes latent in the first opera- 
 tion and is taken from the body A is 
 
 rdm, 
 
 and that which is set free in the second operation and is given 
 to the body B is j 
 
 In the other expansion and in the other compression heat is 
 neither gained nor lost, so that, at the end of the process, 
 
 (6) The heat consumed=rdm(r jrdt}d'm. 
 
 In this expression the differential d'm must be replaced by dm 
 and dt. For this purpose we make use of the conditions under 
 which the second expansion and the second compression oc- 
 curred. The mass of vapor, which condenses during the com- 
 pression from oh to oe, and which would be evolved by the cor- 
 responding expansion from oe to oh, may be represented by Sm, 
 and that which is evolved by the expansion from of to og by 
 I'm. We then have at once, since at the end of the process the 
 same mass of liquid p and the same mass of vapor m must be 
 present as at the beginning, the equation 
 
 dm + Km d'm -f Im. 
 
 Further, we obtain for the expansion from oe to oh, since in 
 it the temperature of the mass of liquid p and the mass of va- 
 por m must be lowered by dt without the emission of heat, the 
 equation 
 
 and similarly for the expansion from of to og, by substituting 
 fidm and m+ dm for /* and m, and Z'm for m, 
 
 81
 
 M KMOIRS ON 
 
 If from these three equations and (6) we eliminate the magni- 
 tudes d'm, 3m, and I'm, and reject terms of higher order than 
 the second, we have 
 
 (7) 
 
 The formulas (?) and (5) must now he connected in the same 
 way as that used in the case of the permanent gases, that is, 
 
 and we obtain as the analytical expression of the fundamental 
 principle in the case of vapors at their maximum density the 
 equation 
 
 (III.) + e -h = A(*-.)'%. 
 
 If, instead of using our principle, we adopt the assumption 
 that the quantity of heat is conx/anf. \ve must replace (III.), as 
 appears from (7), by 
 
 (8) 
 
 This equation has been used, if not exactly in the same form, 
 at least in its general sense, to obtain a value for tin- magni- 
 tude A. So long as Watt's law is considered true for water. 
 that the sum of the free and latent heats of a quantity of vapor 
 at its maximum density is equal for all temperatures, and that 
 therefore ^ r 
 
 rf/+'=; 
 
 it must be concluded that for this liquid 7<=0. This conclu- 
 sion has, in fact, often been stated us con-ret, in that it has 
 been said that if a quantity of vapor is at its maximum dmsit v. 
 and then compressed or expanded in a ves>i-l impermeable hy 
 heat, it remains at its maximum .im.-itv. \\\\\ sinee Ki-nault* 
 has corrected Watt's law by substituting for it the approximate 
 relation ,/.. 
 
 the equation (8) gives for 7* the value 0.305. It would there- 
 fore follow that the quantity of vapor formerly considered in 
 
 mm. de rAcad.. t. xxl., the 9tb and 10.1, M, moires.
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 the vessel impermeable by heat would be partly condensed by 
 compression, and on expansion would not remain at the maxi- 
 mum density, since its temperature would not fall in a way to 
 correspond to the diminution of pressure. 
 
 It is entirely different if we replace equation (8) by (III.). 
 The expression on the right-hand side is, from its nature, always 
 positive, and it therefore follows that h must be less than 0.305. 
 It will subsequently appear that the value of this expression is 
 so great that h is negative. We must therefore conclude that 
 the quantity of vapor before mentioned is partly condensed, 
 not by compression, but by expansion, and that by compression 
 its temperature rises at a greater rate than the density increases, 
 so that it does not remain at its maximum density. 
 
 It must be admitted that this result is exactly opposed to 
 the common view already referred to ; yet I do not believe that 
 it is contradicted by any experimental fact. Indeed, it is more 
 consistent than the former view with the behavior of steam 
 as observed by Pambour. Pambour* found that the steam 
 which issues from a locomotive after it has done its work 
 always has the temperature at which the tension, observed 
 at the same time, is a maximum. From this it follows 
 either that A=0, as it was once thought to be, because this 
 assumption agreed with Watt's law, accepted as probably true, 
 or that h is negative. For if h were positive, the temperature 
 of the vapor, when released, would be too high in comparison 
 with its tension, and that could not have escaped Pambour's 
 notice. If, on the other hand, h is negative, according to our 
 former statement, there can never arise from this cause too 
 low a temperature, but a part of the steam must become liquid, 
 so as to maintain the rest at the proper temperature. This 
 part need not be great, since a small quantity of vapor sets free 
 on condensation a relatively large quantity of heat, and the 
 water formed will probably be carried on mechanically by the 
 rest of the steam, and will in such researches pass unnoticed, 
 the more likely as it might be thought, if it were to be observed, 
 that it was water from the boiler carried out mechanically. 
 
 The results thus far obtained have been deduced from the 
 fundamental principle without any further hypothesis. The 
 equation (II. a) obtained for permanent gases may, however, be 
 
 * Traite des Locomotives, secoud edition, and Theorie des Machines d Vupeur, 
 second edition.
 
 MEMOIRS ON 
 
 made much more fruitful by the help of an obvious subsidiary 
 hypothesis. The gases show in their various relation 
 pecially in the relation expressed by the M. and (J. law be- 
 tween volume, pressure, and temperature, so great a regularity 
 of behavior that we are naturally led to take the view that the 
 mutual attraction of the particles, which acts within solid ami 
 liquid bodies, no longer acts in gases, so that while in the case 
 of other bodies the heat which produces expansion must over- 
 come not only the external pressure but the internal attraction 
 as well, in the case of gases it has to do only with the external 
 pressure. If this is the case, then during the expansion of a 
 gas only so much heat becomes latent as is used in doing ex- 
 ternal work. There is, further, no reason to think that a gas. 
 if it expands at constant temperature, contains more ir<-<- heat 
 than before. If this be admitted, we have the law : a per- 
 manent gas, when expanded at constant frt/t/irrti/nrr, tnkcs /</> 
 only xo much heat ns /.< mnsumed in tl<n'n</ r.rfrrntil imrk ilnrimi 
 the expansion. This law is probably true for any <:as with the 
 same degree of exactness as that attained by the M. and (J. law 
 applied to it. 
 
 From this it follows at once that 
 
 since, as already noticed, R - dv represents the external work 
 v 
 
 done during the expansion dv. It follows that the fimetion (' 
 which occurs in (II. a) does not contain /. and the equation 
 therefore takes the form 
 
 (ll.b) dQ=cdt+AR <i'\ 
 
 where c can be a function of / only. It is even probable that 
 this magnitude r, which represents the speciii.- heat of the gag 
 at constant volume, is a constant. 
 
 Now in order to apply this equation to special cases, we must 
 introduce the relation between the variable- (J. f, and /-. which 
 is obtained from the conditions of each separate ,-i-e. mi> the 
 equation, and so make it integrable. We shall here < .n.-ider 
 only a few simple examples of this sort, which are eiiher in- 
 teresting in themselves or become so by comparison with other 
 theorems already announced. 
 
 84
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 We may first obtain the specific heats of the gas at constant 
 volume and at constant pressure if in (II. b) we set #=const., 
 and j3 = const. In the former case, dv=Q, and (II. b) becomes 
 
 In the latter case, we obtain from the condition j9=const., by 
 the help of equation (I.), 
 
 , Rdt 
 
 *>= , 
 
 or 
 
 dv tit 
 
 v 
 and this, substituted in (II. b), gives 
 
 if we denote by c' the specific heat at constant pressure. 
 
 It appears, therefore, that the difference of the two specific 
 heats of any gas is a constant magnitude, AR. This magni- 
 tude also involves a simple relation among the different gases. 
 
 The complete expression for R is - ) , where p , v , and t 
 
 are any three corresponding values of p, v, and t for a unit 
 of weight of the gas considered, and it therefore follows, as 
 has already been mentioned in connection with the adoption 
 of equation (I.), that R is inversely proportional to the specific 
 gravity of the gas, and hence also that the same statement must 
 hold for the difference c' c=AR, since A is the same for all 
 gases. 
 
 If we reckon the specific heat of the gas, not with respect to 
 the unit of weight, but, as is more convenient, with respect to 
 the unit of volume, we need only divide c and c' by v , if the 
 volumes are taken at the temperature t and pressure j . Des- 
 ignating these quotients by / and y, we obtain 
 
 (11) y ' 
 
 I'o 
 
 In this last quantity nothing appears which is dependent on 
 the particular nature of the gas, and the difference of the specific 
 heats referred to the unit of volume is therefore the same for all 
 gases. 
 
 This law was deduced by Clapeyron from Carnot's theory, 
 85
 
 MEMOIRS ON 
 
 though the constancy of the difference c' c, which we have 
 deduced before, is not found in his work, where the expression 
 given for it still has the form of a function of the temperature. 
 If we divide equation (11) on both sides by y, we have 
 
 (12) *-l=-.-J^-, 
 
 in which k, for the sake of brevity, is used for the quotient , 
 
 or, what amounts to the same thing, for the quotient. This 
 
 quantity has acquired special importance in science from the 
 theoretical discussion by Laplace of the propagation of sound 
 in air. The excess of fin's quotient over unity is therefore. /<// t In- 
 different yases, inversely /n-ojtortiontil to tin- sped lie Ite/i/s of tin' 
 same at constant volume, ifike$tttn r>f< mil lo tin unit ,,f ruhnne. 
 This law has, in fact, been found by Dulong from experiment* 
 to be so nearly accurate that he has assumed it, in view of its 
 theoretical probability, to be strictly accurate, and has there- 
 fore employed it, conversely, to calculate the specific heats of 
 the different gases from the values of k determined by obser- 
 vation. It must, however, be remarked that the law is only 
 theoretically justified when the M. and G. law holds, whii-h is 
 not the case with sufficient exactness for all the gases employed 
 by Dulong. 
 
 If it is now assumed that the specific heat of gases at con- 
 stant volume f is constant, which has been stated al>ove to bo 
 very probable, the same follows for the specific heat at con- 
 stant pressure, and consequently the quotient of the tir 
 
 heats C k is a constant. This law, which Poisson has already 
 c 
 
 assumed as correct on the strength of the experiment! of (lay- 
 Lussac and Welter, and has made the basis of his investigations 
 on the tension and heat of gases, f is therefore in ^<><\ agree- 
 ment with our present theory, while, it would not bo possible 
 on Carnot's theory as hitherto developed. 
 
 If in equation (II. A) we set @=const., we obtain the follow- 
 ing equation between v and / : 
 
 Ann. df Chim. et de Phyt., xli.. and Po^g. Ann., xvi. 
 f Traiti de Mecanique, second edition, vol. ii., p. 646. 
 M
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 (13) 
 which gives, if c is considered constaiit, 
 
 v '.(a+t)=const., 
 
 AH c' 
 or, since from equation (10),^ = 1 = # 1, 
 
 C C 
 
 v*~ l (rt-M)=const. 
 
 Hence we have, if v , t , and j are three corresponding values 
 of M, and/,, a +j /, o 
 
 (14) a + t-\v 
 
 If we substitute in this relation the pressure p first for v and 
 then for t by means of equation (I.), we obtain 
 
 These are the relations which hold between volume, temper- 
 ature, and pressure, if a quantity of gas is compressed or ex- 
 panded within an envelope impermeable by heat. These equa- 
 tions agree precisely with those which have been developed by 
 Poisson for the same case,* which depends upon the fact that 
 he also treated k as a constant. 
 
 Finally, if we set t = const, in equation (Il.b), the first term 
 on the right drops out, and there remains 
 
 (17) dQ=AR f ^dv, 
 from which we have 
 
 Q = AR (a + t) log v + const., 
 
 or, if we denote by v , p , t , and Q the values of v, p, t, and 
 Q, which hold at the beginning of the change of volume, 
 
 (18) Q-Q =AR(a + t )\og?-. 
 
 From this follows the law also developed by Carnot : If a gas 
 changes its volume without changing Us temperature, the quanti- 
 ties of heat evolved or absorbed are in arithmetical progression, 
 while the volumes are in geometrical progression. 
 
 * Traite de Mecanique, vol. ii., p. 647. 
 87
 
 M KM <! KS OX 
 
 Further, if we substitute for R in (18) the complete expres- 
 sion " , we have 
 
 (19) Q-Q =A Po r \og^. 
 
 If now we apply this equation to the different gases, not by 
 using equal weights of them, but such quantities as have at tln> 
 outset equal volumes, t' , it becomes in all its parts indt-pen- 
 tlent of the special nature of the gas, and agrees with the 
 well-known law which Dulong proposed, guided by the ahuve- 
 nientioned simple relation of tin- magnitude k 1, that nil 
 gases, if equal volumes of tin in art' taken at tin' sunn' /////</////// n 
 and under flu- same fires* n re, ami if they art' then finii/n-'-ssnf <>r 
 expanded by an equal fraction <>f their minims, fill- 
 absorb an equal quantify <>f heat. Kipiation (19) is, however, 
 much more general. It states in addition, that the quantity of 
 heat is independent of tin- temperature at irhieh flu- ml nine oftkt 
 yas is altered, if only the quantity of tin- gas employed is always 
 determined so that the original volume r is always the same 
 at the different temperatures ; and it states further, that if fhf 
 original pressure is ilijt'in-nf in the <////'/</// <<>. //// quantities 
 <>f heat are proportional to it. 
 
 II. CONSEQUENCES OP CARNOT's PRIN'lIM.i: IX rnxxr.i THx 
 WITH THE ONE ALREADY IXTISiMH I l> 
 
 Carnot assumed, as has already ln-i-n nifiitiuncd, that tin- 
 equivalent of the work done by In-at is fnnn<l in tin' mn-f fr 
 of heat from a hotter to a colder body, while tin' quantity ;/ ' Itmt 
 remains undiminished. 
 
 The latter part of this assumption namely, that the quan- 
 tity of heat remains undimiiiislii'd contradicts our former prin- 
 ciple, and must therefore be rejected if we are to retain that 
 principle. On the other hand, the first part may still obtain in 
 all its essentials. For though we do not need a special equiva- 
 lent for the work done, since we have assumed as such an act nal 
 consumption of heat, it still may well be possible that such a 
 transfer of heat occurs at tin- sunn- ///// as the consumption of 
 heat, and als<. stands in a definite relation to the work done. 
 It becomes important, therefore, to consider whether this as- 
 sumption, besides the mere possibility, has also a sufficient 
 probability in its favor.
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 A transfer of heat from a hotter to a colder body always oc- 
 curs in those cases in which work is done by heat, and in which 
 also the condition is fulfilled that the working substance is in 
 the same state at the end as at the beginning of the operation. 
 For example, we have seen, in the processes described above 
 and represented in Figs. 1 and 3, that the gas and the evapo- 
 rating water took up heat from the body A as their volume in- 
 creased, and gave it up to the body B as their volume dimin- 
 ished : so that a certain quantity of heat was transferred from 
 A to B, and this was in fact much greater than that which we 
 assumed to be consumed, so that in the infinitely small changes, 
 which are represented in Figs. 2 and 4, the latter was an in- 
 finitesimal of the second order, while the former was one of 
 the first order. Yet, in order to establish a relation between 
 the heat transferred and the work done, a certain restric- 
 tion is necessary. For since a transfer of heat can take place 
 without mechanical effect if a hotter and a colder body are im- 
 mediately in contact and heat passes from one to the other by 
 conduction, the way in which the transfer of a certain quantity 
 of heat between two bodies at the temperatures t and r can be 
 made to do the maximum of work is to so carry out the proc- 
 ess, as was done in the above cases, that two bodies of different 
 temperatures never come in contact. 
 
 It is this maximum of work which must be compared with 
 the heat transferred. When this is done it appears that there 
 is in fact ground for asserting, with Carnot, that it depends 
 only on the quantity of the heat transferred and on the tempera- 
 tures t and T of the two bodies A and B, but not on the nature 
 of the substance by means of which the work is done. This 
 maximum has, namely, the property that by expending it as 
 great a quantity of heat can be transferred from the cold body 
 B to the hot body A as passes from A to B when it is produced. 
 This may easily be seen, if we think of the whole process for- 
 merly described as carried out in the reverse order, so that, for 
 example, in the first case the gas first expands by itself, until 
 its temperature falls from t tor, is then expanded in connection 
 with B, is then compressed by itself until its temperature is 
 again t, and finally is compressed in connection with A. In this 
 case more work will be employed during the compression than 
 is produced during the expansion, so that on the whole there 
 is a loss of work, which is exactly as great as the gain of work in
 
 MEMOIRS ON 
 
 the former process. Further, there will be just as much heat 
 taken from the body B as was before given to it, and just a> 
 much given to the body A as was before taken from it, whence it 
 follows not only that the same amount of heat is produced as was 
 formerly consumed, but also that the heat which in the former 
 process was transferred from A to B now passes from B to A. 
 
 If we now suppose that there are two substances of which the 
 one can produce more work than the other by the transfer of a 
 given amount of heat, or, what comes to the same thing, needs 
 to transfer less heat from A to B to produce a given <|ii;mtity 
 of work, we may use these two substances alternately by pro- 
 ducing work with one of them in the above process, and by e\- 
 pending work upon the other in the reverse process. At the 
 end of the operations both bodies are in their original condi- 
 tion ; further, the work produced will haVe exactly count er- 
 balanced the work done, and therefore, by our former principle, 
 the quantity of heat can have neither increased nor diminished. 
 The only change will occur in the ttixtrilmtion of the heat, since 
 more heat will be transferred from B to A than from A to B, 
 and so on the whole heat will be transferred from B to A. By 
 repeating these two processes alternately it would be possible, 
 without any expenditure of force or any other change, to trans- 
 fer as much heat as we please from a cold to a hot body, and this 
 is not in accord with the other relations of heat, since it always 
 shows a tendency to equalize temperature differences and 
 therefore to pass from hotter to colder bodies. 
 
 It seems, therefore, to be theoretically admissible to retain 
 the first and the really essential part of Carnot's assumptions, 
 and to apply it as a second principle in conjunction with the 
 first ; and the correctness of this method is, as we shall soon 
 see, established already in many cases by its consequences. ' 
 
 On this assumption we may express the maximum of work 
 which can be produced by the transfer of a unit of heat from t lie 
 body A at the temperature t to the body B at the temperature 
 r, as a function of / and r. The value of this function must 
 naturally be smaller as the difference tr is smaller, and when 
 this is infinitely small (=dt) it must go over into the pni.l- 
 uct of dt and a function of t only. For this latter case, with 
 which we will concern ourselves for the present, the work may 
 
 be expressed by the form ( ,-d(, where C is a function of / only. 
 90
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 In order to apply this result to the permanent gases, we re- 
 turn to the process represented in Fig. 2. In that case the 
 quantity of heat, 
 
 passed during the first expansion from A to the gas, and by the 
 first compression the part of it expressed by 
 
 or by 
 
 (dQ\ [d/(lQ\ (I S<lQ\-] 7 1f 
 [ i )'" ~rA ~i ) T ( ~r~ ) uvclt, 
 \dv ) \-dt\dv ) dv \ dt ) J 
 
 was given up to the body B. The latter magnitude is, there- 
 fore, the quantity of heat transferred. Since we may neglect 
 the term of the second order with respect to the one of the 
 first order, we retain simply 
 
 The work produced at the same time was 
 Rdv.dt 
 
 v 
 
 and we can thus at once form the equation 
 Rdv.dt 
 
 '^*> 
 
 or, 
 
 If, in the second place, we make a similar application to the 
 process represented in Fig. 4 relating to vaporization, we have 
 for the quantity of heat carried from A to B 
 
 or rdm i^-4-c h \dmdt, 
 
 \dt I 
 
 for which, by neglecting the term of the second order, we may 
 set simply 
 
 rdm. 
 91
 
 .MEMOIRS ON 
 The work produced was 
 
 and we therefore get the equation 
 
 or, 
 
 (v.) rs -a ( ,_.)& 
 
 These are the two analytical expressions of Carnot's principle, 
 as they are given by Clapeyron in his memoir, in a somewhat 
 different form. For vapors he stops with this equation (V.) 
 and some immediate applications of it. For gases, on the other 
 hand, he makes the equation (IV.) the basis of a more extended 
 development. It is in this development that the partial dis- 
 agreement appears between his results and ours. 
 
 We shall now connect these two equations with the results of 
 the first principle, first considering equation (IV.) in connec- 
 tion with the consequences formerly deduced for the case of 
 permanent gases. 
 
 If we restrict ourselves to that result which depends only on 
 the fundamental principle that is, to equation (II. a) we can 
 use equation (IV.) to further define the magnitude /', which 
 appears there as an arbitrary function of -v ami /, and our equa- 
 tion becomes 
 
 (II.c) d 
 
 where B is now an arbitrary function of / only. 
 
 If we also accept as correct the subsidiary hypothetic, then 
 equation (IV.) is not necessary for the further definition of 
 (Il.a) ; since the same end is more completely attained by 
 e. | nation (9), which followed as an immediate consequence of 
 this hypothesis in connection with the first principle. \\ , 
 gain, however, an opportunity to subject the results of the two 
 principles to a comparative test. Equation (9) reads : 
 dQ\ R.A(a+t) 
 
 and if wo compare this with (IV.), we see that they both ex- 
 
 press the same result, only the one in a more definite way than 
 
 92
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 the other, since for the general temperature function denoted 
 in (IV.) by C, the equation. (9) gives the special expression 
 A (a + t). 
 
 To this striking agreement it may be added that equation 
 (V.), in which also the function C appears, confirms the view 
 that A (a + t) is the correct expression for this function. This 
 equation has been used by Clapeyron and Thomson to calculate 
 the values of C for several temperatures. Clapeyron chose as 
 the temperatures the boiling-points of ether, alcohol, water, 
 and oil of turpentine, arid by substituting inequation (V.) the 
 
 values of -~, s, and r for these liquids, determined by experi- 
 ments at these boiling-points, he obtained for C the numbers 
 contained in the second column of the table which follows. 
 Thomson, on the other hand, considered water vapor only, but 
 at different temperatures, and thence calculated the value of C 
 for every degree between and 230 Cent. For this purpose 
 Kegnault's series of observations have given him an admissible 
 
 basis so far as the magnitudes -j- and r are concerned ; but the 
 
 magnitude s is not so well known for other temperatures as for 
 the boiling-point, and about this magnitude Thomson felt him- 
 self compelled to make an assumption, which he himself rec- 
 ognized as only approximately correct, and considered as a 
 temporary aid, to be employed until more exact data are de- 
 termined namely, that water vapor at its maximum density 
 follows the M. and G. law. The numbers which follow from 
 his calculation for the same temperatures as those used by 
 Clapeyron are given in the third column reduced to French 
 units : 
 
 I 
 
 1 
 
 t IN CENT. DEGRKE8 
 
 2 
 
 C ACCORDING TO 
 CLAPEYKON 
 
 3 
 
 C ACCORDING TO 
 THOMSON 
 
 35.5 
 
 78.8 
 100 
 156.S 
 
 0.733 
 
 0.828 
 0.897 
 0.930 
 
 0.728 
 0.814 
 0.855 
 0.952 
 
 It appears that the values of found in both cases in- 
 crease slowly with the temperature, similarly to the values
 
 IfKMOIBS OH 
 
 of A (a+t). They are in the ratio of tlic numbers in the fol- 
 lowing rows : i:i.i:>:i.- :!.; 
 
 1 : 1.12 : 1.17 : 1.31 
 
 and if we determine the ratios of the values of A (a + /) corre- 
 sponding to the same temperatures, we obtain 
 
 1 : 1.14 : 1.21 : 1.39. 
 
 This series of relative values diverges from the two others only 
 so far as can be accounted for by the uncertainty of the data 
 which underlie them. The same agreement will be shown 
 later in connection with the determination of the constant A, 
 in respect to the absolute values. 
 
 Such an agreement between results which are obtained from 
 entirely different principles cannot be accidental ; it rather 
 serves as a powerful confirmation of the two principles and the 
 first subsidiary hypothesis annexed to them. 
 
 Returning again to the application of equations (IV.) and ( V. i. 
 we may remark that the former, so far as relates to the per- 
 manent gases, has only served to confirm conclusions already 
 obtained. In the consideration of vapors, and of all other sub- 
 stances to which Carnot's principle will In- applied in the future. 
 it furnishes, however, an essential improvement, in that it per- 
 mits us to replace the function C, which recurs everywhere, by 
 the definite expression A (+/). 
 
 Uy this substitution equation (V.) becomes 
 
 and we therefore obtain for a vapor a simple relation between 
 the temperature at which it is formed, the pressure, the vol- 
 ume, and the latent heat. This we can use in drawing further 
 conclusions. 
 
 If the M. and G. law were accurate for vapors at their maxi- 
 mum density, wo should have 
 
 <,, / = /?(<! + /). 
 
 Kliminating the magnitude * from (\ .a) by the use of this 
 (((nation, and neglecting the magnitude r. which vanishes in 
 ' -mparidon with * if the temperature is not very high, we ob-
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 If we make the further assumption that r is constant, we ob- 
 tain by integration, if p l denotes the tension of the vapor at 
 
 100 ' ,& __ r-100) 
 
 *> p l ^.#(a-HlOO)(rt-M) 
 
 or if we set/ -100 = r, + 100 = a, and -.- 
 
 A. 
 
 This equation cannot, of course, be accurate, since the two 
 assumptions made in its development are not accurate ; but 
 since these, at least to a certain extent, approach the truth, the 
 
 quantity -~- will roughly represent the value of the quantity 
 
 log . .We may explain in this way how it happens that this 
 
 relation, if the constants a and /3, instead of having values 
 given them depending on their definitions, are considered as 
 arbitrary, may serve as an empirical formula for the calculation 
 of vapor tensions, without our being compelled to consider it 
 as fully proved by theory, as is sometimes done. 
 
 The most immediate application of equation (V.) is to 
 water vapor, for which we have the largest collection of experi- 
 mental data, in order to investigate how far it departs, when at 
 its maximum density, from the M. and G. law. The magnitude 
 of this departure cannot be unimportant, since carbonic acid 
 and sulphurous acid, even at temperatures and tensions at 
 which they are still far removed from their condensation points, 
 show noticeable departures. 
 
 Equation (V.) may be put in the following form : 
 
 The expression here found on the left-hand side would be 
 very nearly constant, if the M. and G. law were applicable, 
 since this law would give immediately, from (20), 
 
 and s a can be substituted for s in this equation with approxi- 
 
 mate accuracy. This expression is, therefore, especially suited 
 
 95
 
 M KM "I us ON 
 
 to show clearly any departure from the M. and G. law, from 
 tlie examination of its true values as they may be calculated 
 from the expression on the right-hand side of (22). I ha\e 
 carried out this calculation for a series of temperatures, 
 fur / and p the numbers given by Regnault.* 
 
 First with respect to the Intrnt heat: Regnanlt states f that 
 the quantity of heat X. which must be imparted to a unit of 
 weight of water, in order to heat it from to t and then \<> 
 evaporate it at that temperature, may be represented with 
 tolerable accuracy by the formula : 
 
 (23) X = GOG. 5 +0.305 t. 
 
 But now, from the significance of X, 
 
 (23) \ = r+ I < 
 
 and for the magnitude c, the specific heat of water, which ap- 
 pears in this formula, Regnault has given the formula : J 
 
 By the help of these two equations we obtain for the latent, 
 heat from equal ion (23) the expression : 
 
 (24) r = G06.5 0. 695. *-0. 00002. /' 0.0000003./'. 5? 
 
 Second, with respect to the pressure : in order to obtain 
 from his numerous observations the most probable values. 
 lleirnaultjl made use of a graphic representation, by construct- 
 ing curves, of which the abscissas represented the temperature 
 and the ordinates the pressure />. and which arc drawn in sec- 
 tions from 33 to +230. From 100 to 230 he has al.-o 
 
 * Mem. de VAcad, de I'Intt. de France, vol. xxi. (1847). 
 
 f Ibid.. Mem. ix.; also Fogg Ann.. lid. 98. f Ibid.. M'm. x. 
 
 In most of his investigations Ri-gnaiill lias not so much C>I<TY<-<I un- 
 bent which becomes latent by evaporation of the vapor us that which he 
 COOtCT free by its condensation, ami, therefore, since it lias ln-m slmwii 
 above thai, if tbe principle of the equivalence of heat and work is mtr t, 
 the quantity of beat which a quantity of vapor gives up on coixlciis.-itinu 
 iii-i-il not ulvvays In- the same as tbat wbicli it almorbs during ii> fnrina- 
 tion. tbe question may arise, whether such differences may not ba\< in 
 tered in Ib-cnniilt's experiments, so that tin- formula given for r would 
 IM-COIHC in idinidsiblc. I b-li(-vc that we may answer this question in tin- 
 negative, since Itegnault BO arranged his experiments that tin- >n.|.nvi. 
 tion of the vapor occurred under the same pressure as its formation that 
 U. nearly under the pressure which corresponded as a maximum to th<- 
 observed teni|>erntiire. and in tb is case Just us much ln-at IIIUM hi- volvi-,1 
 by condensation as is absorbed by evaporation. | Ibid., .!/<//. viii. 
 
 M
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 drawn a curve, of which the ordinates represent not p itself, 
 but the logarithms of p. From this presentation the following 
 values have been taken, which are to be considered as the im- 
 mediate results of his observations, while the other more com- 
 plete tables contained in the memoir were calculated from 
 formulas, of which the choice and determination depended in 
 the first instance upon these values : 
 
 II 
 
 t IN DEGREES 
 
 
 t IN DEGREES 
 
 CENTIGKADE 
 
 p IN 
 
 CENTIGRADE 
 
 ON THE AIR- 
 THERMOMETER 
 
 METERS 
 
 ON THE AIR- 
 THERMOMETER 
 
 20 
 
 0.91 
 
 110 
 
 10 
 
 2.08 
 
 120 
 
 
 
 4.60 
 
 130 
 
 10 
 
 9.16 
 
 140 
 
 20 
 
 17.39 
 
 150 
 
 30 
 
 31.55 
 
 160 
 
 40 
 
 54.91 
 
 170 
 
 50 
 
 91.98 
 
 180 
 
 60 
 
 148.79 
 
 190 
 
 70 
 
 233.09 
 
 200 
 
 80 
 
 354.64 
 
 210 
 
 90 
 
 525.45 
 
 220 
 
 100 
 
 760.00 
 
 230 
 
 p IN MILLIMETERS 
 
 FROM THE 
 
 FROM THE 
 
 CURVE OP 
 
 CURVE OP 
 
 NUMBERS 
 
 LOGARITHMS* 
 
 1073.7 
 
 1073.3 
 
 1489.0 
 
 1490.7 
 
 2029.0 
 
 2030.5 
 
 2713.0 
 
 2711.5 
 
 3572,0 
 
 3578.5 
 
 4647.0 
 
 4651.6 
 
 5960.0 
 
 5956.7 
 
 7545.0 
 
 7537.0 
 
 9428.0 
 
 9425.4 
 
 11660.0 
 
 11679.0 
 
 14308.0 
 
 14325.0 
 
 17390.0 
 
 17390.0 
 
 20915.0 
 
 20927.0 
 
 Now in order to carry out with these data the calculation in 
 
 hand, I first determined from these tables the values of r . 
 
 p dt 
 
 for the temperatures 15, 5 P , 5, 15, etc., in the following 
 way. Since the magnitude ~ only diminishes slowly as the 
 
 temperature rises, I have considered as uniform the diminution 
 in each interval of 10, say from -20 to -10, from 10 to 
 0, etc., so that I could look on the value holding, for example, 
 for 25 as the mean of all the values holding between 20 and 
 
 30. On this assumption, since "ji 
 the formula : 
 
 d (log p) 
 dt 
 
 could use 
 
 * Instead of the logarithms obtained immediately from the curve and 
 adopted by Regnault, the numbers corresponding to them are given, in 
 order to facilitate comparison with the numbers in the next coluiuu. 
 G 97
 
 MEMOIRS ON 
 
 _ log p*r -log Ptf 
 10 
 
 Log Pao" 
 10. Jf 
 
 in which Log indicates the Briggsiau logarithms and ^f the 
 
 modulus of this system. By help of these values of --.' and 
 
 the values of r given by equation (24), and of the valu< 
 for a, the values which the expression on the right-hand side 
 
 of (22), and so also the expression Ap (s a} - , take for tilt- 
 
 temperatures 15, 5, 5, etc., were calculated and aiv 
 given in the accompanying table. For temperatures above 100 
 both series of numbers given for p are used separatt-ly. and the 
 two results found in each case given opposite each other. The 
 significance of the third and fourth columns will be indicated 
 in the sequel. 
 
 Ill 
 
 t IX DKHRKKM 
 OOTWUM 
 
 OS THK AIK 
 
 FROM TH1 OBttRTKD 
 VALCW 
 
 nu>M QC ATIOX (37) 
 
 Dirriarccn 
 
 -15 
 -5 
 5 
 15 
 25 
 85 
 45 
 .v, 
 65 
 75 
 OB 
 95 
 105 
 115 
 125 
 100 
 145 
 155 
 165 
 175 
 185 
 195 
 
 m 
 
 215 
 225 
 
 00.S1 
 8098 
 00.00 
 
 ::n in 
 
 :t.Mii 
 00.40 
 
 J'.t -s 
 
 OO.SO 
 
 30^00 
 
 J'.t ss 
 J'.t 7J 
 
 j'.. .;.-, 
 
 29.47 2950 
 00.10 00.00 
 
 j* ss -J'.l 1,1 
 
 J* III 
 
 J'.l 1 7 
 js '.t't 
 
 J* M, 
 
 08.01 J-' i'. 
 
 27.84 27.90 
 
 27.76 27.67 
 
 03 r. WIM 
 
 j.; :,; -jr, 711 
 j.j :,u 
 
 28.14 
 
 J7 -'. 
 
 J7 t;-j 
 
 J7 :u 
 J7 n-j 
 j.; .is 
 
 0.00 
 
 + 1 :::: 
 -11 17 
 
 -0.10 
 
 -0.08 
 
 0.00 
 
 + 002 
 
 0.00 
 
 0.00 
 
 -0.02 
 
 - 0.01 
 
 -0.14-0.17 
 
 + 0.01 +015 
 
 + 0.10 + 006 
 
 -0.08 " Jl 
 
 - 0.05 + 0.20 
 
 + 0.18 
 
 
 
 + 0.05 - 0.01 
 -0.14-0.05 
 -0.12+0.18 
 + 0.18 + 0.08 
 + 0.12-011 
 -0.82-0.18 
 
 H
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 It appears at once from this table that Ap (so) ^-7- is not 
 
 constant as it should be if the M. and Gr. law were applicable, 
 but diminishes distinctly as the temperature rises. Between 
 35 and 90 this diminution appears to be very uniform. Under 
 35, especially in the region of 0, there appear noticeable 
 irregularities, which, however, may be simply explained from 
 the fact that in that region the pressure p and its differential 
 
 coefficient ~ are very small, and therefore small errors, which 
 
 (it 
 
 fall quite within the limits of the errors of observation, may 
 become relatively important. It may be added that the curve 
 by which the separate values of p are determined, as mentioned 
 above, is not drawn in one stroke from 35 to 100, but, to 
 economize space, is broken at 0, so that at this temperature 
 the progress of the curve cannot be determined so satisfactorily 
 as it can within the separate portions below and above 0. 
 From the way in which the differences occur in the foregoing 
 table, it would seem that the value 4.60 mm. taken iorp atO is a 
 
 little too great, since if that were so the values of Ap (sa) - 
 
 for the temperatures just under would come out too small, 
 and for those just over too large. Above 100 the values of 
 this expression do not diminish so regularly as between 35 and 
 95; and yet they show, at least in general, a corresponding 
 progress ; and especially if we use a graphic representation, we 
 find that the curve, which within that interval almost exactly 
 joins the successive points determined by the numbers contained 
 in the table, may be produced beyond that interval even to 230 
 quite naturally, so that these points are evenly distributed on 
 both sides of it. 
 
 Within the range of the table the progress of the curve can 
 be represented with fair accuracy by an equation of the form 
 
 (20) Ap (s - a) - = m - ne", 
 
 where e is the base of the natural logarithms, and m, n, and k 
 are constants. If these constants are calculated from the values 
 which the curve gives for 45, 125, and 205, we obtain ; 
 
 (26fl) i=31.549, 7* = 1.0486, =0.007138,
 
 MKMOIRS ON 
 
 and if for convenience we introduce the Briggsian logarithms, 
 we obtain 
 
 0.0206 + 0.003 100 /. 
 
 Log [31.549- Ap(s-v) -^-1 =0. 
 L .+ *J 
 
 The numbers contained in the third column are calculated 
 from this equation, and in the fourth are given the differences 
 between these numbers and those in the second column. 
 
 From the foregoing we may easily deduce a formula by 
 which we can more definitely determine the way in which the 
 behavior of a vapor departs from the M. and G. law. By as- 
 suming this law to hold, and denoting by px a the value of ps 
 at 0, we would have from (20), 
 
 po a 
 and would have, therefore, for the differential coefficient 
 
 {^) a constant quantity namely, the well-known coeffi- 
 cient of expansion - = 0.003665. Instead of this we have from 
 (26), if we simply replace * a by *, the equation : 
 
 m n 
 
 pg - m _ M 
 and hence follows : 
 
 / 8fl) d .(pL\-*.> 
 
 dt \psj~a 
 
 The differential coefficient is, therefore, not a constant, but a 
 function of the temperature which diminishes as the tempera- 
 ture increases. If we substitute the numerical values of ///. //. 
 and k, given in (26), we obtain, among others, the following 
 values for this function : 
 
 IV 
 
 1 
 
 itta 
 
 t 
 
 i(S) 
 
 t 
 
 5() 
 
 ixg. 
 
 10 
 20 
 80 
 40 
 50 
 60 
 
 0.00842 
 0.00888 
 0.00884 
 0.00839 
 0.00825 
 0.00819 
 0.00814 
 
 i>,, 
 70 
 80 
 
 '.H. 
 
 100 
 110 
 120 
 130 
 
 0.00807 
 0.00800 
 0.00298 
 0.00385 
 0.00276 
 0.00266 
 0.00256 
 
 iiiiiii? 
 
 000244 
 160 
 
 0.<*' 
 
 0.001-7 
 0108 
 
 o.omr.t 
 
 100
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 It appears from this table that at low temperatures the de- 
 partures from the M. and G. law are only slight, but that at 
 higher temperatures for example, at 100, and upwards they 
 can no longer be neglected. 
 
 It may appear at first sight remarkable that the values found 
 
 for 4- ( ) are smaller than 0.003665, since we know that in 
 dt \psj 
 
 the case of gases, especially of those, like carbonic acid and 
 sulphurous acid, which deviate most widely from the M. and 
 G. law, the coefficient of expansion is not smaller, but greater, 
 than that number. We are not, however, justified in making 
 an immediate comparison between the differential coefficients 
 which we have just determined and the coefficient of expan- 
 sion in the ordinary sense of the words, which relate to the 
 increase of volume at constant pressure, nor yet with the num- 
 ber obtained by keeping the volume constant during the heating 
 process, and then observing the increase in the expansive force. 
 We are dealing here with a third special case of the general 
 
 differential coefficient -7-1 ) namely, with that which arises 
 dt \p8j 
 
 when, as the heating goes on, the pressure increases in the 
 same proportion as it does with water vapor when it is kept at 
 its maximum density ; and we must consider carbonic acid in 
 these relations if we wish to institute a comparison. 
 
 Water vapor has a tension of l m at about 108, and of 2 m at 
 129. We will, therefore, examine the behavior of carbonic 
 acid if it is heated by 21, and if the pressure upon it is at 
 the same time increased from l m to 2 m . According to Reg- 
 nault * the coefficient of expansion of carbonic acid at the con- 
 stant pressure 760 mra is 0.003710, and at the pressure 2520 ram is 
 0.003846. For a pressure of 1500 mm (the mean between l m and 
 2 m ), if we consider the increase of the coefficient of expansion 
 as proportional to the increase of pressure, we obtain the value 
 0.003767. If carbonic acid were heated at this mean pressure 
 
 from to 21, the magnitude would increase from 1 to 
 
 1 + 0.003767x21.5 = 1.08099. Now from others of Regnault' 
 researches f it is known that if carbonic acid, taken at a tem- 
 perature near under the pressure l m , is subjected to the 
 
 * Mem. de PAcad., Mem. i. f Ibid. , Mem. vi. 
 
 101
 
 MEMOIRS ON 
 
 pressure 1.98292 m , the magnitude ps decreases in the ratio of 
 1 : 0.99146 ; so that for an increase of pressure from l m to 2 m 
 there would be a decrease of this magnitude in the ratio of 
 1 : 0.99131. If, now, both operations were performed at once 
 that is, the elevation of temperature from to 21$ and the in- 
 
 crease in pressure from l m to 2 m the magnitude would in- 
 
 crease nearly from 1 to 1.08099x0.99131 = 1.071596, and lienee 
 we obtain for the mean value of the differential coefficient 
 
 =0.00333. 
 
 It appears, therefore, that in the case now under consideration. :v 
 value is obtained for carbonic acid which is less than 0.003665, 
 and therefore a similar result for a vapor at its maximum deit*i/i/ 
 should not be considered at all improbable. 
 
 If, on the other hand, we were to determine the real coefficient 
 of expansion of the vapor that is, the number which expresses 
 by how much a quantity of vapor expands if it is taken at a 
 certain temperature at its maximum density, and then removed 
 from the water and heated under constant pressure we should 
 certainly obtain a value which would be greater, and perhaps 
 considerably greater, than o.i"i;;r,t;:,. 
 
 From equation (26) we easily obtain tin- rcla/in- volumes of 
 a unit of weight of vapor at its maximum density for <lilT< rent 
 temperatures, referred to the volume at some definite temper- 
 ature. In order to calculate the n/isnliite volumes from these 
 with sufficient precision, wo must know the value of the constant 
 A with greater accuracy than is as yet the case. 
 
 The question now arises whether any one volume can l>e 
 assigned with sufficient accuracy to permit its use as a starting- 
 point in the calculation of the other absolute values fn.m the 
 relative values. Muny investigations of the specific weight f 
 water vapor have been carried out, the results of which, ln\v- 
 cver. are not, in my opinion, conclusive for the case with which 
 we are now dealing, in which the vapor is at its maximum 
 density. The numbers which arc ordinarily given, especially tin- 
 one obtained by Qay-Lnssac O.f>235 agree very well with the 
 theoretical value obtained lv a>-umin^ that 'I parts of hydrogen 
 102
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 and 1 part of oxygen combine to form 2 parts of water vapor 
 that is, with the value 
 
 2x0.069264- 1.10563 
 
 = U.D^/i. 
 
 These numbers, however, are obtained from observations which 
 were not carried out at temperatures at which the resulting 
 pressure was equal to the maximum expansive force, but at 
 higher temperatures. In this condition the vapor might nearly 
 conform to the M. and G. law, and the agreement with the 
 theoretical value may thus be explained. To pass from this 
 result to the condition of maximum density by the use of the 
 M. and G. law would contradict our previous conclusions, since 
 Table IV. shows too large a departure from this law, at the 
 temperatures at which the determination was made, to make 
 such a use of the law possible. Those experiments in which 
 the vapor was observed at its maximum density give for the 
 most part larger numbers, and Regnault has concluded * that 
 even at a temperature a little over 30, in the case in which the 
 vapor is developed in vacuum, a sufficient agreement with the 
 theoretical value is reached only when the tension of the vapor 
 amounts to no more than 0.8 of that which corresponds to the 
 observed temperature as the maximum. A definite conclusion, 
 however, cannot be drawn from this observation, since it is 
 doubtful, as Regnault remarks, whether the departure is really 
 due to too great a specific weight of the vapor formed, or whether 
 a quantity of water remained condensed on the walls of the glass 
 globe. Other experiments, which were so executed that the 
 vapor did not form in vacuum but saturated a current of air, 
 gave results which were tolerably free from any irregularities,! 
 yet even these results, important as they are in other relations, 
 do not enable us to form any definite conclusions as to the be- 
 havior of vapor in a vacuum. 
 
 In this state of uncertainty the following considerations may 
 perhaps be of some service in filling the gap. Table IV. shows 
 that the vapor at its maximum density conforms more closely 
 to the M. and G. law as the temperature is lower, and it may 
 hence be concluded that the specific weight will approach the 
 theoretical value more nearly at lower than at higher temper- 
 atures. If therefore, for example, we assume the value 0.622 
 
 * Ann. Oe Chim. et de Phys. t III. Ser., t. xv., p. 148. f Ibid., p. 158 ff. 
 103
 
 MEMOIRS OX 
 
 as correct for and then calculate the corresponding value <l 
 for higher temperatures by the help of the following equation 
 deduced from (26), 
 
 (30) (7 = 0.622 m ""*.. 
 
 m nr* 
 
 we obtain much more probable values than if we were to adopt 
 0.622 as correct for all temperatures. The following table 
 presents some of these values : 
 
 / 
 
 
 
 50 
 
 V 
 
 100 
 
 wo 8 
 
 200 
 
 d 
 
 0.622 
 
 0.631 
 
 0.645 
 
 0.666 
 
 0.698 
 
 Strictly speaking, we must go further than this. In Table III. 
 we see that the values of Ap (s a) -- -, as the temperature 
 
 falls, approach a limiting value, which is not reached even for 
 the lowest temperatures of the table, and it is only for this 
 limiting value that we have a right, to assume the applicability 
 of the M. and G. law and so set the specific weight equal to 
 0.622. The question therefore arises what this liniiiinir value 
 is. If we could consider the formula (26) as applicable for 
 temperatures below 15, we would have only to take the value 
 which it approaches asymptotically, m = 31.549, and we could 
 then replace equation (30) by the equation 
 
 (3D 
 
 From this equation we obtain for the specific weight at the 
 value 0.643 instead of (U522. and the other numbers of the 
 preceding table must be increased in the same ratio. We are, 
 however, not justified in so extended an application of formula 
 (26), since it is only obtained empirically from the values given 
 in Table HI., and of these, those which relate to the lowest 
 temperatures are rather uncertain. We must therefore, for 1 1 it- 
 
 present, treat the limiting value of A (9 <r) r as unknown, 
 
 and content ourselves with such an approximation as the num- 
 bers in the preceding tables warrant. We may. however, con- 
 clude that these numbers are rather too small than too great. 
 104
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 If we combine equation (V.o) with equation (III.) deduced 
 from the first principle, we may eliminate A (s a), and obtain : 
 
 <*> *--5T. 
 
 By means of this equation we may determine the magnitude h, 
 which has already been stated to be negative. If we set for c and 
 r the expressions given in (23b) and (24), and for a the number 
 273, we obtain : 
 
 606. - 0. 695 / - 0. 00002 P - 0. 0000003 1 3 
 (33) *=:<UJG5~- 273 + * ~ J 
 
 and hence obtain for //, among others, the values : 
 
 
 
 50 
 
 VI 
 
 100 
 
 -1.916 
 
 -1.465 
 
 -1.133 
 
 150 C 
 
 -0.879 
 
 200 C 
 
 -0.676 
 
 In a way similar to that which we have followed in the case 
 of water vapor, we might apply equation (V.a) to the vapors of 
 other liquids also, and then compare the results obtained for 
 these different liquids, as has been done with the numbers cal- 
 culated by Clapeyron and contained in Table I. "We shall not, 
 however, go into these applications any further at present. 
 
 We must now endeavor to determine, at least approximately, 
 the numerical value of the constant A, or, what is more useful, 
 
 of the fraction , that is, the work equivalent of tlte unit of heat. 
 
 For this purpose we can first use equation (lOa) for the per- 
 manent gases, which amounts to the same thing as the method 
 already employed by Mayer and Helmholtz. This equation is : 
 
 c' = c + AR, 
 
 c ' 
 and if we set for c the equivalent expression -p, we have : 
 
 The value commonly taken for c' for atmospheric air from 
 the researches of De Laroche and Berard is 0.267, and for k 
 from the researches of Dulong is 1.421. Further, to determine 
 
 R = " , we know that the pressure of one atmosphere 
 
 a -f- TO 
 
 (760 mm ) on a square meter is 10333 kilogrammes, and that the 
 105
 
 MEMOIRS ON 
 
 volume of one kilogramme of atmospheric air under that pr> 
 and at the temperature of the freezing-point =0.7733 cubic 
 meters. Hence follows : 
 
 10333. 0.7733 
 
 273 
 and consequently 
 
 _ 1.421.29.26 _ 
 ,4-0.421.0.267- 
 
 that is, by the expenditure of a unit of heat (that quantity of 
 heat which will raise the temperature of 1 kilogramme of water 
 from to 1) 370 kilogrammes can be lifted to the height of 
 l m . Little confidence can be placed in this number, on account 
 of the uncertainty of the numbers 0.267 and 1.421. Holt/maim 
 gives as the limits, between which he is in doubt. :M:J ami I .".i. 
 We may further use the equation (V.a) developed for vapors. 
 If we wish to apply it to water vapor, we can use the determi- 
 nations given in the former part of our work, whose result is 
 expressed in equation (26). If we choose in this equation the 
 temperature 100, for example, and set for p the corresponding 
 pressure of 1 atmosphere = 10333 kilogrammes, we obtain : 
 
 (35) 1=257 (*-*). 
 
 If we now use Gay-Lussac's value of the specific weight of 
 water vapor, 0.6235, we obtain s= 1.699, and hence, 
 
 1=43, 
 
 Similar values are given by the use of the numbers contained 
 in Table I., which Clapeyron and Thomson have calculated 
 for C from equation (V.). For if we consider these as the 
 values of A (a+t) for the temperatures corresponding to them, 
 
 we obtain for-j a set of values which lie between 416 and 462. 
 A 
 
 It has already been mentioned that the specific weight of 
 water vapor given by Gay-Lussac is probably somewhat too 
 small for the case where the vapor is at its maximum density. 
 The same may be said of most of the specific weights which an> 
 ordinarily given for other vapors. We must therefore con- 
 clude that the values of -j calculated from them are for tin- 
 
 A 
 
 most part a little too great. If we take for water vapor the 
 100
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 number 0.645 given in Table V., from which s=1.638, we 
 obtain i 
 
 z= 421 - 
 
 This value is also perhaps a little, but probably not much, 
 too great. We may therefore conclude, since this result 
 should be given the preference over that obtained from atmos- 
 pheric air, that the work equivalent of the unit of heat is the 
 lifting of something over 400 kilogrammes to the height 0/l m . 
 
 We may now compare with this theoretical result those which 
 Joule obtained in very different ways by direct observation. 
 Joule obtained from the heat produced by magneto-electricity, 
 
 1=460;' 
 
 from the quantity of heat which atmospheric air absorbs during 
 its expansion, l 
 
 ^=438, t 
 
 and as a mean of a large number of experiments, in which the 
 heat produced by friction of water, of mercury, and of cast- 
 iron, was observed, -i 
 
 Z =5.t 
 
 The agreement of these three numbers, in spite of the diffi- 
 culty of the experiments, leaves really no further doubt of the 
 correctness of the fundamental principle of the equivalence of 
 heat and work, and their agreement with the number 421 con- 
 firms in a similar way the correctness of Carnot's principle, in 
 the form which it takes when combined with the first principle. 
 
 BIOGRAPHICAL SKETCH 
 
 RUDOLF JULIUS EMAJTUEL CLAUSIUS was born on January 
 2, 1822, at Coslin, in Pomerania. He was educated at Berlin, 
 and became Privat-docent in the University of Berlin and In- 
 structor in Physics at the School of Artillery. In 1855 he was 
 appointed to the Professorship of Physics in the Polytechnic 
 School at Zurich, and in 1857 he was appointed to a similar 
 
 * Phil. Mag., xxiii., p. 441. The number, given in English units, is re- 
 duced to French units. 
 
 f Ibid., xxvi , p. 381. \ Ibid., xxxv., p. 534. 
 
 107
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 position in the University of Zurich. In 1869 he was appointed 
 1'rufessor of Physics in the University of Bonn, where lu- re- 
 mained until his death, on August 24, 1888. 
 
 Clausius was a prolific investigator and writer on physical 
 subjects. The line of thought suggested by the discoveries in 
 heat contained in the memoir given in this volume was fol- 
 lowed out by him in a series of papers on the thermodynamic 
 properties of bodies and on the general theory of thermody- 
 namics. These papers were collected and published in a volume 
 in 1864; and ten years later he recast these papers and others 
 which had appeared after the collection was first published into 
 a systematic treatise on the mechanical theory of heat. The 
 concept of the entropy, which Clausing introduced and de- 
 veloped, is the most important single contribution made by 
 him to science. 
 
 Clausius's investigations also extended into radiant heat, in 
 connection with which he proved that radiance also conforms 
 to the second law of thermodynamics. Clausius was the first 
 to apply the doctrine of probabilities, in any systematic way, to 
 the kinetic theory of gases ; and by so doing he laid the foun- 
 dations for the brilliant applications of that doctrine to the 
 kinetic theories which have been made by Maxwell and Boltz- 
 mann. He also contributed something to the theory of elec- 
 tricity. His writings are characterized by simplicity of form 
 and profundity of thought. They deal much with fundamental 
 questions, but by such direct and simple methods that the 
 ideas under discussion are rarely obscured by the difficulties of 
 the analysis.
 
 ON THE DYNAMICAL THEORY OF HEAT, 
 WITH NUMERICAL RESULTS DEDUCED 
 FROM MR. JOULE'S EQUIVALENT 
 OF A THERMAL UNIT, AND 
 M. REGNAULT'S OBSERVA- 
 TIONS ON STEAM 
 
 BY 
 
 WILLIAM THOMSON (LORD KELVIN) 
 
 (Transactions of the Royal Society of Edinburgh, March, 1851 ; Philosopliical 
 Magazine, iv., 1852; Mathematical and Physical Papers, vol. i., p. 174)
 
 CONTENTS 
 
 MM 
 
 Introductory Notice Ill 
 
 Fundamental Principle* 114 
 
 Carnot'i Cycle 116 
 
 Carnot't Function 128 
 
 Jt Detenni 'nation 183 
 
 Tktrmoilyiuiinic Relation* . . 186
 
 ON THE DYNAMICAL THEORY OF HEAT 
 
 BY 
 
 WILLIAM THOMSON 
 
 INTRODUCTORY NOTICE 
 
 1. SIR HUMPHRY DAVY, by his experiment of melting two 
 pieces of ice by rubbing them together, established the follow- 
 ing proposition : " The phenomena of repulsion are not de- 
 pendent on a peculiar elastic fluid for their existence, or 
 caloric does not exist." And he concludes that heat consists 
 of a motion excited among the particles of bodies. " To dis- 
 tinguish this motion from others, and to signify the cause of 
 onr sensation of heat," and of the expansion or expansive press- 
 ure produced in matter by heat, "the name repulsive motion 
 has been adopted."* 
 
 2. The dynamical theory of heat, thus established by Sir 
 Humphry Davy, is extended to radiant heat by the discovery 
 of phenomena, especially those of the polarization of radiant 
 heat, which render it excessively probable that heat propagated 
 through "vacant space," or through diathermanic substances, 
 consists of waves of transverse vibrations in an all-pervading 
 medium. 
 
 3. The recent discoveries made by Mayer and Joule, f of the 
 
 * From Davy's first work, entitled An Essay on H&it, Light, and the Com- 
 binations of Light, published in 1799, in " Contributions to Physical and 
 Medical Knowledge, principally from the West of England, collected by 
 Thomas Beddoes, M.D .," and republished in Dr. Davy's edition of his 
 brother's collected works, vol. ii., Loud., 1836. 
 
 f In May, 1842, Mayer announced in the Annalen of Wohler and Liebig, 
 that he had raised the temperature of water from 12 to 13 Cent, by agi- 
 tating it. In August, 1843, Joule announced to the British Association 
 111
 
 MEMOIRS ON 
 
 generation of heat through the friction of fluids in motion, and 
 by the magneto-electric excitation of galvanic currents, would 
 either of them be sufficient to demonstrate the immateriality 
 of heat ; and would so afford, if required, a perfect continua- 
 tion of Sir Humphry Davy's views. 
 
 4. Considering it as thus established, that heat is not a sub- 
 stance, but a dynamical form of mechanical effect, we jn-rcrive 
 that there must be an equivalence between mechanical work 
 and heat, as between cause and effect. The first published 
 statement of this principle appears to be in Mayer's J: 
 knnyen ilfar '//' AV<>/"/V <l< r /////"/<///,// \<ttiu\* which contains 
 some correct views regard inir the mutual convertibility of heat 
 and mechanical effect, along with a false analogy between the 
 approach of a weight to the earth and a diminution of the vol- 
 ume of a continuous substance, on which an attempt is founded 
 to find numerically the mechanical equivalent of a given quan- 
 tity of heat. In a paper published about fourteen months 
 later, "On the Calorific Effects of Magneto-Electricity and the 
 Mechanical Value of IIeat,"f Mr. Joule, of Manchester. \- 
 presses very distinctly the consequences regarding the mutual 
 convertibility of heat and mechanical effect which follow from 
 the fact that heat is not a substance but a state of motion ; 
 and investigates on unquestionable principles the >% alolutc 
 numerical relations/' according to which heat is connected 
 with mechanical power; verifying experimentally, that when- 
 ever heat is generated from purely mechanical action, and no 
 other effect produced, whether it be by means of the friction 
 of fluids or by the magneto-electric excitation of galvanic cur- 
 rents, the same quantity is generated by the same amount of 
 work spent; and determining the actual amount of work, in 
 foot-pounds, required to generate a unit of heat, whirh he 
 calls "the mechanical equivalent of heat." Since the publica- 
 
 "Tbat heat is evolved by the passage of water through narrow uiU>s ; ' 
 and that he hat] "obtained one degree of heat per pound of water fn.m a 
 UK < iiaiiic:il force capable of raising 770 pounds to the height of one f.-.t ." 
 and that heat is generated when work is spent in turning a magneto-elec- 
 tric machine, or an electro magnetic engine. (See his paper "On the 
 Calorific Effects of Magm-io KI- ttiriiy, and on tli- M<-< li.mirul Value of 
 Heat." Phil. Mag., vol. xxiii.. 1848.) 
 
 * AnnaUn of Wol.ler and Liebig. May. 1842. 
 
 f British Association, August, 1848; and Phil. Mag., September, 1848. 
 Ill
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 tion of that paper, Mr. Joule has made numerous series of ex- 
 periments for determining with as much accuracy as possible 
 the mechanical equivalent of heat so defined, and has given 
 accounts of them in various communications to the British 
 Association, to the Philosophical Magazine, to the Royal So- 
 ciety, and to the French Institute. 
 
 5. Important contributions to the dynamical theory of heat 
 have recently been made by Rankine and Clausius ; who, by 
 mathematical reasoning analogous to Carnot's on the motive 
 power of heat, but founded on an axiom contrary to his funda- 
 mental axiom, have arrived at some remarkable conclusions. 
 The researches of these authors have been published in the 
 Transactions of this Society, and in Poggendorff's Annalen, 
 during the past year ; and they are more particularly referred 
 to below in connection with corresponding parts of the investi- 
 gations at present laid before the Royal Society. 
 
 6. The object of the present paper is threefold : 
 
 (1) To show what modifications of the conclusions arrived 
 at by Carnot, and by others who have followed his peculiar 
 mode of reasoning regarding the motive power of heat, must 
 be made when the hypothesis of the dynamical theory, con- 
 trary as it is to Carnot's fundamental hypothesis, is adopted. 
 
 (2) To point out the significance in the dynamical theory, 
 of the numerical results deduced from Regnault's observations 
 on steam, and communicated about two years ago to the So- 
 ciety, with an account of Carnot's theory, by the author of 
 the present paper ; and to show that by taking these numbers 
 (subject to correction when accurate experimental data regard- 
 ing the density of saturated steam shall have been afforded), 
 in connection with Joule's mechanical equivalent of a ther- 
 mal unit, a complete theory of the motive power of heat, 
 within the temperature limits of the experimental data, is ob- 
 tained. 
 
 (3) To point out some remarkable relations connecting the 
 physical properties of all substances, established by reasoning- 
 analogous to that of Carnot, but founded in part on the con- 
 trary principle of the dynamical theory.
 
 MEMOIRS ON 
 
 PART I 
 Fundamental Principle* in the Tlteory of tlie Motive Power of 
 
 7. According to an obvious principle, first introduced, how- 
 ever, into the theory of the motive power of heat by Carnot. 
 mechanical effect produced in any process cannot be said to 
 have been derived from a purely thermal source, unless at the 
 end of the process all the materials used are in precisely the 
 same physical and mechanical circumstances as they were at 
 the beginning. In some conceivable " thermo- dynamic en- 
 gines," as, for instance, Faraday's floating magnet, or Barlow's 
 "wheel and axle," made to rotate and perform work uniformly 
 by means of a current continuously excited by heat communi- 
 cated to two metals in contact, or the thermo-electric rotatory 
 apparatus devised by Marsh, which has been actually construct- 
 ed, this condition is fulfilled at every instant. On the other 
 hand, in all thermo -dynamic engines, founded on electrical 
 agency, in which discontinuous galvanic currents, or pieces of 
 soft iron in a variable state of magnetization, are used, and in 
 all engines founded on the alternate expansions and contrac- 
 tions of media, there are really alterations in the condition of 
 materials ; but, in accordance with the principle stated above, 
 these alterations must be strictly periodical. In any such en- 
 gine the series of motions performed during a period, at the 
 end of which the materials are restored to precisely the same 
 condition as that in which they existed at the beginning, con- 
 stitutes what will be called a complete cycle of its operations. 
 Whenever in what follows, the tmrk ilnnc or tin iiit'r/ttinir>tl tf- 
 fect produced by a thermo-dynamic engine is mentioned with- 
 out qualification, it must be understood that the mechanical 
 effect produced, either in a non-varying engine, or in a com- 
 plete cycle, or any number of complete cycles of a periodical 
 engine, is meant. 
 
 8. The source of heat will always be supposed to be a hot 
 body at a given constant temperature put in contact with 
 some part of the engine ; and when any part of the engine is 
 to be kept from rising in temperature (which can only he done 
 by drawing off whatever heat is deposited in it), this will be 
 supposed to be done by putting a cold body, which will be 
 
 114
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 called the refrigerator, at a given constant temperature in con- 
 tact with it. 
 
 9. The whole theory of the motive power of heat is founded 
 on the two following propositions, due respectively to Joule, 
 and to Carnot aild Clausius. 
 
 PROP. I. (Joule). When equal quantities of mechanical ef- 
 fect are produced by any means whatever from purely thermal 
 sources, or lost in purely thermal effects, equal quantities of 
 heat are put out of existence or are generated. 
 
 PROP. II. (Carnot and Clausius). If an engine be such that, 
 when it is worked backwards, the physical and mechanical 
 agencies in every part of its motions are all reversed, it pro- 
 duces as much median ical effect as can be produced by any 
 thermo-dynamic engine, with the same temperatures of source 
 and refrigerator, from a given quantity of heat. 
 
 10. The former proposition is shown to be included in the 
 general "principle of mechanical effect," and is so established 
 beyond all doubt by the following demonstration. 
 
 11. By whatever direct effect the heat gained or lost by a 
 body in any conceivable circumstances is tested, the measure- 
 ment of its quantity may always be founded on a determination 
 of the quantity of some standard substance, which it or any 
 equal quantity of heat could raise from one standard temper- 
 ature to another ; the test of equality between two quantities 
 of heat being their capability of raising equal quantities of any 
 substance from any temperature to the same higher temper- 
 ature. Now, according to the dynamical theory of heat, the 
 temperature of a substance can only be raised by working upon 
 it in some way so as to produce increased thermal motions 
 within it, besides effecting any modifications in the mutual dis- 
 tances or arrangements of its particles which may accompany a 
 change of temperature. The work necessary to produce this 
 total mechanical effect is of course proportional to the quantity 
 of the substance raised from one standard temperature to an- 
 other ; and therefore when a body, or a group of bodies, or a 
 machine, parts with or receives heat, there is in reality me- 
 chanical effect produced from it, or taken into it, to an ex- 
 tent precisely proportional to the quantity of heat which it 
 emits or absorbs. But the work which any external forces do 
 upon it, the work done by its own molecular forces, and the 
 amount by which the half vis viva of the thermal motions of 
 
 115
 
 MEMOIRS ON 
 
 all its parts is diminished, must together be equal to the mc- 
 chanical effect produced from it: and, consequently, to tin- 
 mechanical equivalent of the heat which it emits (which will 
 be positive or negative, according as the sum of those term.- is 
 positive or negative). Now let there be either no molecular 
 change or alteration of temperature in any part of the body. 
 or, by a cycle of operations, let the temperature and physical 
 condition be restored exactly to what they were at the begin- 
 ning ; the second and third of the three parts of tin- \\>ik 
 which it has to produce vanish ; and we conclude that the heat 
 which it emits or absorbs will be the thermal equivalent of the 
 work done upon it by external forces, or done by it against ex- 
 ternal forces; which is the proposition to be pio\ed. 
 
 12. The demonstration of the second proposition is founded 
 on the following axiom : 
 
 It is impossible, by means of intnnint,' mult rial <"j< nry, to 
 i/i-rirr iitirliinnntl cfft-rl fnnn nnij jmrtinii of ///////// //// nmlimj it 
 beloir //it- tt'iiijH'ratnrc of tin- mli/rsf <>/ ///> snrmitnilimj uljn-t*.* 
 
 13. To demonstrate the second proposition, let . I and /' l>c 
 two thermo-dynamic engines, of which B satisfies the condi- 
 tions expressed in the enunciation ; and let, if possible. .1 de- 
 rive more work from a given quantity of heat than />', when 
 their sources and refrigerators are at the same trmpcratu: 
 spectively. Then on account of the condition of complete re- 
 
 /lilittf in all its operations which it fulfils. />' m:iy In- 
 worked backwards, and made to restore any quantity of heat to 
 its source, by the expenditure of the amount of work which, by 
 its forward action, it would derive from the same quantity of 
 heat. If, therefore, B be worked hackuards. and made t.> re- 
 store to the source of A (which we may suppose to be adjust- 
 able to the engine B) as much heat as has been drawn from it 
 during a certain period of the working of A, a smaller amount 
 of work will be spent thus than was gained by tin- working 
 , of .1. Hence, if such a series of operations of ./ forward-; and 
 of // backwards be continued, either alternately or simulta- 
 neously, there will result a continued production of work with- 
 
 * If this axiom be denied for all temp, nitun *. it would him: to be 
 ndmiilcil I hut a self acting mnrhim- miiilit he set to work and produ., m<- 
 chanical effect by cooling the sea or nirth. with tu> limit hut the t<>i:il I- 
 of heat from tbc earth and sea. or, in reality, from the whole n 
 world. 
 
 116
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 out any continued abstraction of heat from the source ; and, 
 by Prop. I., it follows that there must be more heat abstracted 
 from the refrigerator by the working of B backwards than is 
 deposited in it by A. Now it is obvious that A might be 
 made to spend part of its work in working B backwards, and 
 the whole might be made self-acting. Also, there being no 
 heat either taken from or given to the source of the whole, all 
 the surrounding bodies and space except the refrigerator might, 
 without interfering with any of the conditions which have been 
 assumed, be made of the same temperature as the source, what- 
 ever that may be. We should thus have a self-acting machine, 
 capable of drawing heat constantly from a body surrounded by 
 others at a higher temperature, and converting it into me- 
 chanical effect. But this is contrary to the axiom, and there- 
 fore we conclude that the hypothesis that A derives more 
 mechanical effect from the same quantity of heat drawn from 
 the source than B is false. Hence no engine whatever, with 
 source and refrigerator at the same temperatures, can get more 
 work from a given quantity of heat introduced than any en- 
 gine which satisfies the condition of reversibility, which was to 
 be proved. 
 
 14. This proposition was first enunciated by Carnot, being 
 the expression of his criterion of a perfect thermo - dynamic 
 engine.* He proved it by demonstrating that a negation of 
 it would require the admission that there might be a self- 
 acting machine constructed which would produce mechani- 
 cal effect indefinitely, without any source either in heat or the 
 consumption of materials, or any other physical agency ; but this 
 demonstration involves, fundamentally, the assumption that, 
 in "'a complete cycle of operations," the medium parts with 
 exactly the same quantity of heat as it receives. A very strong 
 expression of doubt regarding the truth of this assumption, as 
 a universal principle, is given by Carnot himself ;f and that it 
 is false, where mechanical work is, on the whole, either gained 
 or spent in the operations, may (as I have tried to show above) 
 be considered to be perfectly certain. It must then be admit- 
 ted that Carnot's original demonstration utterly fails, but we 
 cannot infer that the proposition concluded is false. The 
 truth of the conclusion appeared to me, indeed, so probable 
 
 * "Account of Carnot's Theory," 13. f Ibid., 6. 
 
 117
 
 MEMOIRS ON 
 
 that I took it in connection with Joule's principle, on account 
 of which Caruot's demonstration of it fails, as the foundation of 
 an investigation of the motive power of heat in air-cnuincs or 
 steam-engines through finite ranges of tempi -ratuiv, and ob- 
 tained about a year ago results, of which the substance is ijivcn 
 in the second part of the paper at present communicated t<> 
 the Royal Society. It was not until the commencement of the 
 present year that I found the demonstration given above, by 
 which the truth of the proposition is established upon an axiom 
 ( 12) which I think will be generally admitted. It i.s with no 
 wish to claim priority that I make these statements, as tin- 
 merit of first establishing the proposition upon correct princi- 
 ples is entirely due to Clausius, who published his drnnmst ra- 
 tion of it in the month of May last year, in the second part of 
 his paper on the motive power of heat.* I may be allowed to 
 add that I have given the demonstration exactly as it occurred 
 to me before I knew that Clausius had either enunciated or 
 demonstrated the proposition. The following is the axiom on 
 which Clausius's demonstration is founded : 
 
 // /x iiii/HHtxible for it .rlf-<n-tini/ iiuicliinc. nmn'<ti'<l In/ tt/ti/ e.r- 
 l> null it;/i nnj, /o convey heat from one bmly 1<> unot/irr at a liiyln-r 
 
 It is easily shown that, although this and the axiom I have 
 used are different in form, either is a consequence of the other. 
 The reasoning in each demonstration is strictly analogous i.. 
 that which Carnot originally gave. 
 
 15. A complete theory of the motive power of heat \\uuld 
 consist of the application of the two proportions dcm..n>ir:it.-d 
 above to every possible method of producing mechanical elTect 
 from thermal agency, f As yet this has not been clone for tin- 
 electrical method, as far as regards the criterion of a perfect 
 engine implied in the second proposition, and pmliaMy < ann..i 
 be done without certain limitations ; but tin- application <>! tin- 
 first proposition ha.- been very thoroughly investigated, and 
 \crified experimentally by Mr. Joule in his researches "On the 
 
 PogRendorfTs Annalen, referred to above. 
 
 f "Tin-re irc at present known two. and only two. distinct ways in 
 which mechanical effect can br nlnnim-d fr..m lu-at. One of these is by the 
 alterations of volume whiHi \\\, -s , -\\ THMH through the action of in ni . 
 the other is through the medium of electric agency."" Account <>! ( n 
 not's Theory, ' 4. (Transaction*, vol. xvi.. part 5.)
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 Calorific Effects of Magneto-Electricity ;" and on it is founded 
 one of his ways of determining experimentally the mechanical 
 equivalent of heat. Thus from his discovery of the laws of 
 generation of heat in the galvanic circuit,* it follows that when 
 mechanical work by means of a magneto - electric machine is 
 the source of the galvanism, the heat generated in any given 
 portion of the fixed part of the circuit is proportional to the 
 whole work spent ; and from his experimental demonstration 
 that heat is developed in any moving part of the circuit at ex- 
 actly the same rate as if it were at rest, and traversed by a cur- 
 rent of the same strength, he is enabled to conclude : 
 
 (1) That heat may be created by working a magneto-electric 
 machine. 
 
 (2) That if the current excited be not allowed to produce 
 any other than thermal effects, the total quantity of heat pro- 
 duced is in all circumstances exactly proportional to the quan- 
 tity of work spent. 
 
 16. Again, the admirable discovery of Peltier, that cold is 
 produced by an electrical current passing from bismuth to anti- 
 mony, is referred to by Joule, f as showing how it may be proved 
 that, when an electrical current is continuously produced from a 
 
 * That, in a given fixed part of the circuit, the heat evolved in a given 
 time is proportional to the square of the strength of the current, and for 
 different fixed parts, with the same strength of current, the quantities of 
 heat evolved in equal times are as the resistances. A paper by Mr. Joule, 
 containing demonstrations of these laws, and of others on the relations of 
 the chemical and thermal agencies concerned, was communicated to the 
 Royal Society on the 17th of December, 1840, but was not published in the 
 Transactions. (See abstract containing a statement of the laws quoted 
 above, in the Philosophical Magazine, vol. xviii., p. 308.) It was published 
 in the Philosophical Magazine in October, 1841 (vol. xix., p. 260). 
 
 f [Note of March 20, 1852, added in Phil. Mag. reprint. In the intro- 
 duction to his paper "On the Calorific Effects of Magneto-Electricity," 
 etc., Phil. Mag., 1843. 
 
 I take this opportunity of mentioning that I have only recently become 
 acquainted with Helmholtz's admirable treatise on the principle of mechani- 
 cal effect ( Ueber die Erhaltung der Kraft, von Dr. H. Helmholtz. Berlin. 
 G. Reimer, 1847), having seen it for the first time on the 20th of January 
 of this year ; and that I should have had occasion to refer to it on this, and 
 on numerous other points of the dynamical theory of heat, the mechanical 
 theory of electrolysis, the theory of electro -magnetic induction, and the 
 mechanical theory of thermo-electric currents, in various papers communi- 
 cated to the Royal Society of Edinburgh, and to this Magazine, had I been 
 acquainted with it in time. W. T., March 20, 1852.] 
 119
 
 MI: MO i us ox 
 
 purely thermal source, the quantities of heat evolved electri- 
 cally in the different homogeneous parts of the circuit arc nnlv 
 compensations for a loss from the junctions of the different 
 nu'tals, or that, when the effect of the current is entirely ther- 
 mal, there must be just as much heat emitted from the pans 
 not affected by the source as is taken from the soun -. 
 
 17. Lastly,* when a current produced by thermal agency is 
 made to work an engine and produce mechanical effect, there 
 will be less heat emitted from the parts of the circuit not af- 
 fected by the source than is taken in from the source, 1>\ an 
 amount precisely equivalent to tin- mechanical effect produced ; 
 since Joule demonstrates experimentally that a current from 
 any kind of source driving an engine, produces in the enirin.- 
 just as much less heat than it would produce in a fixed win- 
 exercising the same resistance as is equivalent to the mechani- 
 cal effect produced by the engine. 
 
 18. The quality of thermal effects, resulting from equal 
 causes through very different means, is beautifully illustrate'! 
 
 * This reasoning was suggested to me by the following pasture eon. 
 tained in a letter which I received from Mr. Joule on the 8ih of July, is 1 7. 
 "In Peltier's experiment on cold produced at the bismuth ami aniimony 
 solder, we have an instance of the conversion of heat into tin- mechanical 
 force of the current." which must have been meant as an answer to a re- 
 murk I had made, that no evidence could be adduced to show that In MI is 
 ever put out of existence. I now fully admit the force of that answer ; hut 
 it would require a proof that there is more heat put out of existence at iln> 
 heated soldering [or in this and other parts of the circuii] than is < 
 at the eold soldering [and the remainder of the circuit, when a machine is 
 driven by the current] to make the "evidence" be rr/ riim nf<il. That 
 this is the case I think in certain, because the statements <>f i H> in the tr\i 
 lire demonstrated consequences of the first fundamental ju.|Mitinn ; hut 
 it is Mill to he remarked that neither in this nor in any other case of the 
 production of mechanical effect from purely thermal agency, has :!.' 
 ing to exist of an equivalent quantity i>f heat leen dem. .titrated otherwise 
 than theoretically. It would I* a very great step in the e\pei inn -ntal illus 
 t rat ion (or r, ririr<itin. for those who consider such to I.. f the 
 
 dynamical theory of heat, to actually show in any one case a loss of heat ; 
 and il might I*' done by operating through a ffMTJ e..n-idei:,)ile ran.je of 
 temperature* with a good air-engine or steam engine, not allowed to waste 
 rk in friction As will IK? seen in Tart II of this paper, no rxpeii 
 mi tit of iinv kind could show a considerable loss of heat without employ 
 ing bodies differing considerably in temperaiiin- . for n 
 mil' h ta .098. or alxiiit one tenth of the whole heat n-ed. if the temperature 
 of all the bodies used be between and 80 (Ynt 
 130
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 by the following statement, drawn from Mr. Joule's paper on 
 magneto-electricity. * 
 
 Let there be three equal and similar galvanic batteries fur- 
 nished with equal and similar electrodes ; let A } and 2?, be the 
 terminations of the electrodes (or wires connected with the two 
 poles) of the first battery, A 2 and Z? 2 the terminations of the 
 corresponding electrodes of the second, and A 3 and B 3 of the 
 third battery. Let A } and B } be connected with the extremi- 
 ties of a long fixed wire ; let A 2 and z be connected with the 
 "poles" of an electrolytic apparatus for the decomposition of 
 water ; and let A 3 and B 3 be connected with the poles (or ports 
 as they might be called) of an electro-magnetic engine. Then 
 if the length of the wire between A } and /?,, and the speed of 
 the engine between A 3 an*d B 3 , be so adjusted that the strength 
 of the current (which for simplicity we may suppose to be con- 
 tinuous and perfectly uniform in each case) may be the same in 
 the three circuits, there will be more heat given out in any 
 time in the wire between A^ and J5, than in the electrolytic ap- 
 paratus between A 2 and # 2 , or the working engine between A 3 
 and /? 3 . But if the hydrogen were allowed to burn in the oxy- 
 gen, within the electrolytic vessel, and the engine to waste all 
 its work without producing any other than thermal effects (as 
 it would do, for instance, if all its work were spent in continu- 
 ously agitating a limited fluid mass), the total heat emitted 
 would be precisely the same in each of these two pieces of ap- 
 paratus as in the wire between A { and B r It is worthy of re- 
 mark that these propositions are rigorously true, being de- 
 monstrable consequences of the fundamental principle of the 
 dynamical theory of heat, which have been discovered by 
 Joule, and illustrated and verified most copiously in his exper- 
 imental researches. 
 
 19. Both the fundamental propositions may be applied in a 
 perfectly rigorous manner to the second of the known meth- 
 ods of producing mechanical effect from thermal agency. This 
 application of the first of the two fundamental propositions has 
 already been published by Rankine and Clausius ; and that of 
 the second, as Clausius showed in his published paper, is sim- 
 
 * In this paper reference is made to his previous paper "On the Heat of 
 Electrolysis" (published in vol. vii., part 2, of the second series of the Lit- 
 erary and Philosophical Society of Manchester) for experimental demon- 
 stration of those parts of the theory in which chemical action is concerned. 
 121
 
 MKMOIRS OX 
 
 ply Carnot's unmodified investigation of the relation between 
 the mechanical effect produced and the thermal cireiuustamvs 
 from which it originates, in the case of an expansive enirine 
 working within an infinitely small range of temperatures. The 
 simplest investigation of the consequences of the first proposi- 
 tion in this application, which has occurred to me, is the fol- 
 lowing, being merely the modification of an analytical expres- 
 sion of Carnot's axiom regarding the permanence of heat, which 
 was given in my former paper,* required to make it express, 
 not Carnot's axiom, but Joule's. 
 
 20. Let us suppose a massf of any substance, occupying a 
 volume v, under a pressure p uniform in all directions, and at 
 a temperature /, to expand in volume to v + <lr. and to rise in 
 temperature to t + dt. The quantity o*f work which it will pro- 
 duce will be pfo . 
 
 and the quantity of heat which must be added to it to make its 
 temperature rise during the expansion to t + dt may be de- 
 noted by Mdv + Sdt. 
 
 The mechanical equivalent of this is 
 
 ./<.)/'// 4- -V'//). 
 
 if J denote the mechanical equivalent of a unit of heat. Hence 
 the mechanical measure of the total external effect produced in 
 the circumstances is 
 
 (p-JM)dv-J.\<lt. 
 
 The total external effect, after any finite amount of expansion, 
 accompanied by any continuous change of temperature, has 
 taken place, will consequently be, in mechanical ti-rm-. 
 
 where we must suppose / to vary with v, so as to be (lie actual 
 temperature of the medium at each instant, ami the integration 
 with reference to v must be performed between limits fin-re- 
 sponding to the initial and final volumes. Now if, at any sub- 
 sequent time, the volume and temperature of the medium be- 
 come what they were at the beginning, however arbitrarily 
 
 " Account of Carnot's Theory." foot-note on 26. 
 
 f This may have part* consisting of different substances, or of the same 
 substance in different states, provided the temperature of all be the same. 
 See below, pan iii., g 58-56. 
 
 m
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 they may have been made to vary in the period, the total ex- 
 ternal effect must, according to Prop. I., amount to nothing ; 
 and hence (p-JM}dv-JNdt 
 
 must be the differential of a function of two independent varia- 
 bles, or we must have 
 
 d(p-JM) d(-JN) m 
 
 dt dv 
 
 this being merely the analytical expression of the condition, that 
 the preceding integral may vanish in every case in which the 
 initial and final values of v and t are the same, respectively. 
 Observing that J is an absolute constant, we may put the result 
 into the form 
 
 dl- Tl** dN \ (2) 
 
 dt ~ \ dt dv ) 
 
 This equation expresses, in a perfectly comprehensive manner, 
 the application of the first fundamental proposition to the ther- 
 mal and mechanical circumstances of any substance whatever, 
 under uniform pressure in all directions, when subjected to 
 any possible variations of temperature, volume, and pressure. 
 
 21. The corresponding application of the second fundamental 
 proposition is completely expressed by the equation 
 
 !=** <") 
 
 where n denotes what is called " Carnot's function," a quantity 
 which has an absolute value, the same for all substances for 
 any given temperature, but which may vary with the temper- 
 ature in a manner that can only be determined by experiment. 
 To prove this proposition, it may be remarked in the first place 
 that Prop. II. could not be true for every case in which the 
 temperature of the refrigerator differs infinitely little from that 
 of the source, without being true universally. Now, if a sub- 
 stance be allowed first to expand from v to v + dv, its temper- 
 ature being kept constantly t ; if, secondly, it be allowed to 
 expand further, without either emitting or absorbing heat 
 till its temperature goes down through an infinitely small 
 range, to t r; if, thirdly, it be compressed at the constant 
 temperature t T, so much (actually by an amount differing 
 from dv by only an infinitely small quantity of the second or- 
 der), that when, fourthly, the volume is further diminished to 
 123
 
 MKMOIRS <>N 
 
 r without the medium's being allowed to either emit or absorb 
 heat. its temperature may be exactly /: it may ho considered 
 as constituting a thermo-dynamic engine which fulfils Carnot's 
 condition of complete reversibility. Hence, by Prop. II., it 
 must produce the same amount of work for the same quantity 
 of heat absorbed in the first operation, as any other substam 
 similarly operated upon through the same range of temper- 
 
 atures. But T.di' is obviously the whole work done in tin- 
 
 complete cycle, and (by the definition of M in $ 20) Mdv is the 
 quantity of heat absorbed in the first operation. Hence the 
 
 value of ,//, dp 
 
 dt r ' dv ft 
 
 must be the same for all substances, with the same values of / 
 and r; or, since r is not involved except as a factor, we must have 
 
 4 
 
 <// ... 
 
 -="' 
 
 where p depends only on /; from which we conclude the prop- 
 osition which was to be proved. i 
 
 22. The very remarkable theorem that "jT must be the same 
 
 for all substances at the same temperature was first given 
 (although not in precisely the same terms) by Carnot, and de- 
 monstrated by him, according to the principles he adopted. 
 We have now seen that its truth may be satisfactorily cstal.- 
 lished without adopting the false part of his principles. Hence 
 all Carnot's conclusions, and all conclusions derived by others 
 from his theory, which depend merely on equation (';$). require 
 no modification when the dynamical theory is adopted. Tims. 
 all the conclusions contained in Sections I.. 11. .and III. of 
 the Appendix to my "Account of Carnot's Theory." and in the 
 paper immediately following it in the YV/n/>v//7/W*. entitled 
 "Theoretical Considerations on the Effect of Pressure in Lower- 
 ing the Kreexing-point of Water," by my elder brother, still hold. 
 Also, we sec that Carnot's expressii-n for the mechanienl eiT.-.-t 
 derivable from a given quantity of heat lv means of a perfect 
 124
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 engine in which the range of temperatures is infinitely small, 
 expresses truly the greatest effect which can possibly be ob- 
 tained in the circumstances ; although it is in reality only an 
 infinitely small fraction of the whole mechanical equivalent of 
 the heat supplied ; the remainder being irrecoverably lost to 
 man, and therefore " wasted," although not annihilated. 
 
 23. On the other hand, the expression for the mechanical 
 effect obtainable from a given quantity of het entering an en- 
 gine from a "source" at a given temperature, when the range 
 down to the temperature of the cold part of the engine or the 
 "refrigerator" is finite, will differ most materially from that 
 of Carnot ; since, a finite quantity of mechanical effect being 
 now obtained from a finite quantity of heat entering the engine, 
 a finite fraction of this quantity must be converted from heat 
 into mechanical effect. The investigation of this expression, 
 with numerical determinations founded on the numbers de- 
 duced from Regnault's observations on steam, which are shown 
 in Tables I. and II. of my former paper, constitutes the second 
 part of the paper at present communicated. 
 
 PART II 
 
 On the Motive Power of Heat through Finite Ranges of 
 Temperature 
 
 24. It is required to determine the quantity of work which a 
 perfect engine, supplied from a source at any temperature, 8, 
 and parting with its waste heat to a refrigerator at any lower 
 temperature, T, will produce from a given quantity, //, of heat 
 drawn from the source. 
 
 25. We may suppose the engine to consist of an infinite num- 
 ber of perfect engines, each working within an infinitely small 
 range of temperature, and arranged in a series of which the 
 source of the first is the given source, the refrigerator of the 
 last the given refrigerator, and the refrigerator of each inter- 
 mediate engine is the source of that which follows it in the 
 series. Each of these engines will, in any time, emit just as 
 much less heat to its refrigerator than is supplied to it from its 
 source, as is the equivalent of the mechanical work which it 
 produces. Hence if t and t + dt denote respectively the tem- 
 peratures of the refrigerator and source of one of the inter- 
 
 125
 
 MK M(> IRS ON 
 
 mediate engines, and if q denote the quantity of heat which 
 this engine discharges into its refrigerator in any time. ami 
 q + dq the quantity which it draws from its source in the same 
 time, the quantity of work which it produces in that time will 
 be Jdq according to Prop. I., and it will also be qpdt according 
 to the expression of Prop. II., investigated in '! : and there- 
 fore we must have 
 
 Hence, supposing that the quantity of heat supplied from the 
 first source, in the time considered is //. we timl hy integration 
 
 But the value of q, when /= T, is the final remainder dis- 
 charged into the refrigerator at the temperature T; and there- 
 fore, if this be denoted by R, we have 
 
 from which we deduce 
 
 R = m-jf'l>" (r.) 
 
 Now the whole amount of work produced will be the mechani- 
 cal equivalent of the quantity of heat lost ; and, therefore, if 
 this be denoted by II . we have 
 
 \\' = J(/I-R), (7) 
 
 and consequently, by (C), 
 
 ir = J//{l-f-i/J^// ( 
 26. To compare this with the expression II J f lt. for the 
 
 duty indicated by Carnot's theory,* we may expand the e\po. 
 nential in the preceding equation, by the usual series. We thus 
 
 find 
 
 M 
 
 I 
 
 where 6 = , / 
 
 JT 
 
 v 
 
 , 
 
 J 
 
 This shows that the work really produced, which always falls 
 short of the duty indicated by Carnot's theory, approaches 
 
 * "Account," etc.. Equation 7, HI. 
 126
 
 THE SECOND LA^y OF THERMODYNAMICS 
 
 more and more nearly to it as the range is diminished ; and ul- 
 timately, when the range is infinitely small, is the same as jf 
 Carnot's theory required no modification, which agrees with 
 the conclusion stated above in 22. 
 
 27. Again, equation (8) shows that the real duty of a given 
 quantity of heat supplied from the source increases with every 
 increase of the range ; but that instead of increasing indefinitely 
 
 in proportion to / \idt, as Carnot's theory makes it do, it never 
 reaches the value JH, but approximates to this limit, as / * pelt 
 
 is increased without limit. Hence Carnot's remark* regarding 
 the practical advantage that may be anticipated from the use 
 of the air-engine, or from any method by which the range of 
 temperatures may be increased, loses only a part of its impor- 
 tance, while a much more satisfactory view than his of the prac- 
 tical problem is afforded. Thus we see that, although the 
 full equivalent of mechanical effect cannot be obtained even by 
 means of a perfect engine, yet when the actual source of heat 
 is at a high enough temperature above the surrounding objects, 
 we may get more and more nearly the whole of the admitted 
 heat converted into mechanical effect, by simply increasing the 
 effective range of temperature in the engine. 
 
 28. The preceding investigation ( 25) shows that the value 
 of Carnot's function, /u, for all temperatures within the range 
 of the engine, and the absolute value of Joule's equivalent, J, 
 are enough of data to calculate the amount of mechanical effect 
 of a perfect engine of any kind, whether a steam-engine, an air- 
 engine, or even a tbermo - electric engine ; since, according to 
 the axiom stated in 12, and the demonstration of Prop. II., 
 no inanimate material agency could produce more mechanical 
 effect from a given quantity of heat, with a given available 
 range of temperatures, than an engine satisfying the criterion 
 stated in the enunciation of the proposition. 
 
 29. The mechanical equivalent of a thermal unit Fahrenheit, 
 or the quantity of heat necessary to raise the temperature of a 
 pound of water from 32 to 33 Fahr., has been determined by 
 Joule in foot-pounds at Manchester, and the value which he 
 gives as his best determination is 772.69. Mr. Rankine takes, 
 
 * "Account," etc. Appendix, Section iv. 
 127
 
 MEMOIRS ON 
 
 as the result of Joule's determination, 77--J, which he estimates 
 must be within yfa of its own amount, of tin- truth. If \\c 
 take ??2f as the number, we find, by multiplying it by |, i:>'."> as 
 the equivalent of the thermal unit Centigrade, which is taken 
 as the value of J in the numerical applications contained in tin- 
 present paper. 
 
 30. With regard to the determination of the values of /< for 
 different temperatures, it is to be remarkrl that equation (4) 
 shows that this might be done by experiments upon any sub- 
 stance whatever of indestructible texture, and indicates exactly 
 the experimental data required in each case. For instance, l>v 
 first supposing the medium to be air ; and again, by supposing 
 it to consist partly of liquid water and partly of saturated vapor. 
 we deduce, as is shown in Part III. of this paper, the tw<< \- 
 pressions (G), given in 30 of my former paper (" Account of 
 ('arnot's Theory"), for the value of p. at any temperature. A< 
 yet no experiments have been made upon air which afford the 
 required data for calculating the value of ^ through any exten- 
 sive range of temperature; but for temperatures between .~>n 
 and 60 Fahr., Joule's experiments* on the heat evolved 1>\ tin- 
 expenditure of a given amount of work on the compression of 
 air kept at a constant temperature, afford the most direct data 
 for this object which have yet been obtained ; since, if Q be the 
 quantity of heat evolved by the compression of a fluid subject 
 to "the gaseous laws" of expansion and compressibility, IT the 
 amount of mechanical work spent, and / the constant temper- 
 ature of the fluid, we have by (11) of 49 of my former paper. 
 
 n. /: 
 
 which is in reality a simple consequence of the other expression 
 for ft in terms of data with reference to air. Remark! upon 
 the determination of fi by such experiments, and by another 
 class of experiments on air originated liy Joule, are reserved 
 for a separate communication, which I hope to be able to make 
 to the Royal .Society on another occasion. 
 
 31. The second of the expressions (6), in 30 of my former 
 paper, or the equivalent expression (32), given below in the 
 
 * "On the Change* of Temperature produced by Hn- Rarefaction and 
 Condensation of Air," Phil. Mug., vol. xxvi , May, 1845. 
 
 in
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 present paper, shows that /* may be determined for any tem- 
 perature from determinations for that temperature of 
 
 (1) The rate of variation with the temperature, of the press- 
 ure of saturated steam. 
 
 (2) The latent heat of a given weight of saturated steam. 
 
 (3) The volume of a given weight of saturated steam. 
 
 (4) The volume of a given weight of water. 
 
 The last mentioned of these elements may, on account of the 
 manner in which it enters the formula, be taken as constant, 
 without producing any appreciable effect on the probable accu- 
 racy of the result. 
 
 32. Regnault's observations have supplied the first of the 
 data with very great accuracy for all temperatures between 
 32 Cent, and 230. 
 
 33. As regards the second of the data, it must be remarked 
 that all experimenters, from Watt, who first made experiments 
 on the subject, to Regnault, whose determinations are the most 
 accurate and extensive that have yet been made, appear to have 
 either explicitly or tacitly assumed the same principle as that of 
 Carnot which is overturned by the dynamical theory of heat ; 
 inasmuch as they have defined the " total heat of steam" as the 
 quantity of heat required to convert a unit of weight of water 
 at into steam in the particular state considered. Thus Reg- 
 nault, setting out with this definition for "the total heat of 
 saturated steam," gives experimental determinations of it for 
 the entire range of temperatures from to 230 ; and he de- 
 duces the " latent heat of saturated steam " at any temperature, 
 from the "total heat," so determined, by subtracting from it 
 the quantity of heat necessary to raise the liquid to that tem- 
 perature. Now, according to the dynamical theory, the quan- 
 tity of heat expressed by the preceding definition depends on 
 the manner (which may be infinitely varied) in which the speci- 
 fied change of state is effected ; differing in different cases by 
 the thermal equivalents of the differences of the external me- 
 chanical effect produced in the expansion. For instance, the 
 total quantity of heat required to evaporate a quantity of water 
 at 0, and then, keeping it always in the state of saturated va- 
 por,* bring it to the temperature 100, cannot be so much as 
 
 * See below (Part III., 58), where the "negative" specific heat of sat- 
 urated steam is investigated. If the mean value of this quantity between 
 and 100 were -1.5 (and it cannot differ much from this) there would 
 i 129
 
 MEMOIRS ON 
 
 three-fourths of the quantity required, first, to raise the tem- 
 perature of the liquid to loo , ami then evaporate it at that 
 temperature , and yet either quantity is expressed by what is 
 generally received as a definition of the " total heat" of the 
 saturated vapor. To find what it is that is really determined 
 as "total heat" of saturated steam in lleirnault's researches, it 
 is only necessary to remark, that the measurement actually 
 made is of the quantity of heat emitted by a certain weight of 
 water in passing through a calorimetrical apparatus, which it 
 enters as saturated steam, and leaves in the liquid state, the 
 result being reduced to what would have been found if the final 
 temperature of the water had been exactly 0. For there bein^ 
 no external mechanical effect produced (other than that of 
 sound, which it is to be presumed is quite inappreciable), the 
 only external effect is the emission of heat. This must, there- 
 fore, according to the fundamental proposition of the dynam- 
 ical theory, be independent of the intermediate agencies. It 
 follows that, however the steam may rush through the calo- 
 rimeter, and at whatever reduced pressure it may actually he 
 condensed,* the heat emitted externally must be exactly t he 
 
 be 150 units of heat emitted by a pound of saturated vapor in having its 
 temperature raised (by compression) from t<> 1(M . Tin- latent ln-at <>l 
 the vapor at being 606.5. the tiiml quantity of heat required to OOB1 
 pound of water at into saturated steam at 100 , in tin* first <>f tli> 
 mentioned in the text, would consequent])' he 456.5. which is only about J 
 of the quantity 687 found us " the total heat" of the saturated vapor at 
 100, by Regnault. 
 
 * If the steam have to rush through a long flne tulic, or through a small 
 aperture within the calorimctrical apparatus, it- pressure will In dimin- 
 ished l>efore it is condensed ; and there will, therefore, in two parts <>f ihe 
 calorimeter be satuntrd Meani at different trmperatuic* (a. for invfirce. 
 would be the case if steam from a high pressure holler were distilled into 
 the open air); yet. on account of the heat developed l>y the tlui<l friction, 
 winch would he precisely the equivalent f tin- inerhan:ca; elTect of the 
 expansion wasted in tin; rushinp. the heat monsnn-d l.y the calnrim. t. r 
 would he precisely the same MS if the condensation took place at a pressure 
 not appreciably lower than that of ihe entering steam The ciicnm-' 
 of such a case have l*cn overlooked by Clausiu (IWirendorlTs Annul, n. 
 1850. No. 4. p. 510). when lie expresses with some d<>ul>l the opinion that 
 ill.- latent heat of saturated steam will he truly found fr-m K ^naull's 
 "total heat." hy deducting " the sensible heat ;" ami give* as i\ reason that, 
 in the actual experiments, the condensation must have token place "under 
 the same pressure, or nearly under the same pressure." as the evaporation 
 The question is not, Did the condensation take place at a lover prettui-f /',.n, 
 MO
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 same as if the condensation took place under the full pressure 
 of the entering saturated steam ; and we conclude that the 
 total heat, as actually determined from his experiments by Reg- 
 nault, is the quantity of heat that would be required, first to 
 raise the liquid to the specified temperature, and then to evap- 
 orate it at that temperature ; and that the principle on which 
 he determines the latent heat is correct. Hence, through the 
 range of his experiments that is, from to 230 we may con- 
 sider the second of the data required for the calculation of /j. 
 as being supplied in a complete and satisfactory manner. 
 
 34. There remains only the third of the data, or the volume 
 of a given weight of saturated steam, for which accurate exper- 
 iments through an extensive range are wanting ; and no ex- 
 perimental researches bearing on the subject having been made 
 since the time when my former paper was written, I see no 
 reason for supposing that the values of p which I then gave are 
 not the most probable that can be obtained in the present state 
 of science ; and, on the understanding stated in 33 of that 
 paper, that accurate experimental determinations of the den- 
 sities of saturated steam at different temperatures may indicate 
 considerable errors in the densities which have been assumed 
 according to the "gaseous laws," and may consequently render 
 considerable alterations in my results necessary, I shall still con- 
 tinue to use Table I. of that paper, which shows the values of 
 H for the temperatures , H, ... 230$, or, the mean values 
 of p for each of the 230 successive Centigrade degrees of the 
 air-thermometer above the freezing-point, as the basis of nu- 
 merical applications of the theory. It may be added, that any 
 experimental researches sufficiently trustworthy in point of ac- 
 curacy, yet to be made, either on air or any other substance, 
 which may lead to values of p. differing from those, must be 
 admitted as proving a discrepancy between the true densities 
 of saturated steam, and those which have been assumed.* 
 
 that of the entering steam ? but. Did Regnnult maketJie steam work an engine 
 in parsing through the calorimeter, or was there so much noise of steam rush- 
 ing through it as to convert an appreciable portion of tJie total heat into ex- 
 ternal mechanical effect? And a negative answer to this is a sufficient reason 
 for adopting with certainty the opinion that the principle of his determina- 
 tion of the fatent heat is correct. 
 
 * I cannot see that any hypothesis, such as that ndopted by Clausius 
 fundamentally in his investigations on this subject, and leading, as he shows, 
 to determinations of the densities of saturated steam at different temper- 
 131
 
 MEMOIRS ON 
 
 35. Table II. of my former paper, which shows the values of 
 f pit for t = l, t = 2, / = 3, ... / = 231, renders the calcula- 
 
 tion of the mechanical effect derivable from a given quantity 
 of heat by means of a perfect engine, with any given range in- 
 cluded between the limits and 231, extremely easy : since the 
 quantity to be divided by J* in the index of the exponential in 
 the expression (8) will be found by subtracting the number in 
 that table corresponding to the value of T, from that corre- 
 sponding to the value of S. 
 
 36. The following tables show some numerical results which 
 have been obtained in this way, with a few (contained in the 
 
 lower part of the second table) calculated from values of f n<lt 
 
 estimated for temperatures above 230, roughly, according to 
 the rate of variation of that function within the experimental 
 limits. 
 
 37. Explanation of the Tables. 
 
 Column I. in each table shows the assumed ranges. 
 Column II. shows ranges deduced by means of Table II. of 
 
 the former paper, so that the value otf'ndt for each may be 
 
 the same as for the corresponding range shown in column I. 
 
 Column III. shows what would be the duty of a unit of heat 
 if Carnot's theory required no modification (or the actual duty 
 of a unit of heat with additions through the ran^e. to compen- 
 sate forAhe quantities converted into mechanical effect). 
 
 which indicate enormous deviations from the gaseous laws of varia- 
 tion with temperature and pressure, is more probable, or is probably nearer 
 the truth, than that the density of saturated steam does follow these laws 
 as it is usually assumed to do. In the present state of science it would 
 perhaps be wrong to say that either hypothesis is more probable than tin- 
 other [or that the rigorous truth of cither hypothesis is probable at all]. 
 
 * It ought to be remarked, that as the unit of force implied in the deter- 
 minations of ft is the weight of a pound <>f mutter at Paris, and the unit <>f 
 force in terms of which ./ is expressed Is the wcipht of a pound at Man- 
 chester. these numbers ought in strictness to be modified so as to express 
 the values in terms of a common unit of force ; but as the force of gravity 
 at Parii differs by less than ^j of its own value from the force of gravity 
 at Manchester, this correction will be much less Minn the probable errors 
 from other sources, and may therefore be neglected.
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 Column IV. shows the true duty of a unit of heat, and a com- 
 parison of the numbers in it with the corresponding numbers 
 in Column III. shows how much the true duty falls short of 
 Carnot's theoretical duty in each case. 
 
 Column VI. is calculated by the formula 
 
 = t~ 1390/2' /'*#> 
 
 the successive values shown 
 
 where e = 2. 71828, and for / 
 v 
 in Column III. are used. 
 
 Column IV. is calculated by the formula 
 
 W= 1390 (l-R) 
 from the values of 1 R shown in Column V. 
 
 38. Table of the Motive Power of Heat. 
 
 Range of Temperatures 
 
 III 
 
 Duty of a unit 
 of beat 
 through the 
 whole range 
 
 IV 
 
 Duty of a unit 
 of heat sup- 
 plied from 
 the source 
 
 V 
 Quantity of heat 
 converted into 
 mechanical 
 effect 
 
 VI 
 
 Quantity of 
 heat wasted 
 
 I 
 
 II 
 
 S 
 
 T 
 
 S 
 
 r 
 
 fS 
 
 IP 
 
 l-R 
 
 R 
 
 
 
 a 
 
 
 
 
 
 ft. Ibs. 
 
 ft. -Ibs. 
 
 
 
 1 
 
 
 
 31.08 
 
 30 
 
 4.960 
 
 4.948 
 
 .003.56 
 
 .99644 
 
 10 
 
 
 
 40.86 
 
 30 
 
 48.987 
 
 48.1 
 
 .0346 
 
 .9654 
 
 20 
 
 
 
 51.7 
 
 30 
 
 96656 
 
 93.4 
 
 .067 
 
 .933 
 
 30 
 
 (1 
 
 62.6 
 
 30 
 
 143.06 
 
 136 
 
 .098 
 
 .902 
 
 40 
 
 
 
 73.6 
 
 30 
 
 188.22 
 
 176 
 
 .127 
 
 .873 
 
 50 
 
 
 
 84.5 
 
 30 
 
 232.18 
 
 214 
 
 .154 
 
 .846 
 
 60 
 
 
 
 954 
 
 30 
 
 27497 
 
 249 
 
 .179 
 
 .821 
 
 70 
 
 
 
 106.3 
 
 30 
 
 316.64 
 
 283 
 
 .204 
 
 .796 
 
 80 
 
 
 
 117.2 
 
 BO 
 
 357.27 
 
 315 
 
 .227 
 
 .773 
 
 90 
 
 
 
 128.0 
 
 30 
 
 396.93 
 
 345 
 
 .248 
 
 .752 
 
 100 
 
 
 
 138.8 
 
 30 
 
 435.69 
 
 374 
 
 .269 
 
 .731 
 
 110 
 
 
 
 149.1 
 
 30 
 
 473.62 
 
 401 
 
 .289 
 
 .711 
 
 120 
 
 
 
 160.3 
 
 30 
 
 510.77 
 
 427 
 
 .308 
 
 .692 
 
 130 
 
 
 
 171.0 
 
 80 
 
 547.21 
 
 452 
 
 .325 
 
 .675 
 
 140 
 
 
 
 181.7 
 
 BO 
 
 582.98 
 
 476 
 
 .343 
 
 .657 
 
 150 
 
 
 
 192.3 
 
 30 
 
 618.14 
 
 499 
 
 .359 
 
 .641 
 
 160 
 
 
 
 2030 
 
 30 
 
 652.74 
 
 521 
 
 .375 
 
 .625 
 
 170 
 
 
 
 213.6 
 
 30 
 
 686.80 
 
 542 
 
 .390 
 
 .610 
 
 180 
 
 Q 
 
 224.2 
 
 3(1 
 
 720.39 
 
 562 
 
 .404 
 
 .596 
 
 190 
 
 
 
 190 
 
 
 753.50 
 
 582 
 
 .418 
 
 .582 ' 
 
 200 
 
 
 
 200 
 
 (1 
 
 786.17 
 
 600 
 
 .432 
 
 .568 
 
 210 
 
 
 
 210 
 
 
 
 818.45 
 
 619 
 
 .445 
 
 .555 
 
 220 
 
 
 
 220 
 
 
 
 850.34 
 
 636 
 
 .457 
 
 .542 
 
 230 
 
 (1 
 
 230 
 
 
 
 88187 
 
 653 
 
 .470 
 
 .530
 
 MKMnlKS ON 
 
 39. Supplementary Table of the Motive Power of 11 ///. 
 
 
 III 
 
 IT 
 
 V 
 
 VI 
 
 Range or Temperatures 
 
 Duty of a unit IMHV of unit 
 of beat or beat sup- 
 
 Quantity of lie*l 
 convened Into 
 
 Quantity of 
 
 I 
 
 II 
 
 whole range 
 
 the source 
 
 effect 
 
 L- M! \\ i.-tr-: 
 
 
 
 / 
 
 s 
 
 T 
 
 ** 
 
 W 
 
 \-R 
 
 R 
 
 
 
 
 
 
 - 
 
 ft Ita. 
 
 ft It*. 
 
 
 
 101.1 
 
 
 
 140 
 
 M 
 
 ::; 11 
 
 877 
 
 .271 
 
 
 105.8 
 
 II 
 
 280 
 
 LOO 
 
 446.2 
 
 882 
 
 .275 
 
 
 800 
 
 
 
 800 
 
 (i 
 
 1099 
 
 757 
 
 .64B 
 
 .455 
 
 400 
 
 
 
 400 
 
 n 
 
 1 :!'.>:, 
 
 879 
 
 .682 
 
 M 
 
 500 
 
 
 
 500 
 
 n 
 
 1690 
 
 979 
 
 .704 
 
 M 
 
 600 
 
 
 
 600 
 
 
 
 1980 
 
 LOW 
 
 .762 
 
 888 
 
 00 
 
 
 
 00 
 
 (1 
 
 30 
 
 1890 
 
 1.000 
 
 .000 
 
 40. Taking the range 30 to 140 as an example suitable to 
 the circumstances of some of the best steam-engines that have 
 yet been made (see Appendix to " Account of Carnot's Theory," 
 sec. v.), we find in Column III., of the supplementary table, 377 
 ft. -11)8. as the corresponding duty of a unit of heat instead of 
 440, shown in Column III., which is Carnot's theoretical duty. 
 We conclude that the recorded performance of the Fowey-Con- 
 sols engine in 1845, instead of being only ."?i per cent, amounted 
 n-ally to 67 per cent., or \ of the duty of a perfect engine with 
 the same range of temperature; and this duty being .Mil 
 (rather more than J ) of the whole equivalent of the heat used ; 
 we conclude further, that ^ or 18 per cent, of the whole lu-at 
 supplied was actually converted into mechanical effort In that 
 steam-engine. 
 
 41. The numbers in the lower part of the supplementary 
 table show the great advantage that may be anticipated from 
 the perfecting of the air-engine, or any other kind of thermo- 
 iynamic engine in which the range of the temperature can be 
 increased much beyond the limits actually attainable in steam- 
 engines. Thus an air-engine, with its hot part at 600, and it- 
 cold part at Cent., working with perfect economy, would 
 convert 76 per cent, of the whole heat used into median i( -al t f- 
 
 or working with such economy as has been estimated for 
 the Fowey-Consols engine that is. producing <>7 per cent, of 
 the theoretical duty corresponding to its range of temperature 
 LM
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 would convert 51 per cent, of all the heat used into mechanical 
 effect. 
 
 42. It was suggested to me by Mr. Joule, in a letter dated 
 December 9, 1848, that the true value of p. might be "inversely 
 as the temperatures from zero ;" * and values for various tem- 
 peratures calculated by means of the formula, 
 
 were given for comparison with those which I had calculated 
 from data regarding steam. This formula is also adopted by 
 Clausius, who uses it fundamentally in his mathematical inves- 
 tigations. If p. were correctly expressed by it, we should have 
 
 and therefore equations (1) and (2) would become 
 
 ir = j^, (12) 
 
 43. The reasons upon which Mr. Joule's opinion is founded, 
 that the preceding equation (11) may be the correct expression 
 
 TJT 
 
 * If we take /*=A;T - where k may be any constant, we find 
 
 1 "T~ J^t 
 
 k 
 
 J I 
 
 '(&' 
 
 which is the formula I gave when this paper was communicated. I have 
 since remarked that Mr. Joule's hypothesis implies essentially that the co- 
 efficient k must be as it is taken in the text, the mechanical equivalent of a 
 thermal unit. Mr. Rankine, in a letter dated March 27, 1851, informs me 
 that he has deduced, from the principles laid down in his paper communi- 
 cated last year to this Society, an approximate formula for the ratio of the 
 maximum quantity of heat converted into mechanical effect to the whole 
 quantity expended, in an expansive engine of any substance, which, on 
 comparison, I find agrees exactly with the expression (12) given in the 
 text as a consequence of the hypothesis suggested by Mr. Joule regarding 
 the value of p at any temperature. [April 4, 1851.] 
 135
 
 MEMOIRS ON 
 
 for Carnot's function, although the values calculated by means 
 of it differ considerably from those shown in Table I. of my 
 former paper, form the subject of a communication which I 
 hope to have an opportunity of laying before the Royal Society 
 previously to the close of the present session. 
 
 PART III. 
 
 Applications of lite Dynamical '/'// ///// fn >*/n/tli./i Hrtations be- 
 lii'fi'ii the ritysicul J'n>/rfi>s of all >W/*A- 
 
 44. The two fundamental equations of the dynamical theory 
 of heat, investigated above, express relations between quanti- 
 ties of heat required to produce changes of volume and tem- 
 perature in any material medium whatever, subjected to a uni- 
 form pressure in all directions, which lead to various remarkable 
 conclusions. Such of these as are independent of Joule's prin- 
 ciple (expressed by equation (2) of g SJO), being also indepen- 
 dent of the truth or falseness of Carnot's contrary assumption 
 regarding the permanence of heat, are common to his theory 
 and to the dynamical theory ; and some of the most important 
 of them* have been given by Carnot himself, and other writers 
 who adopted his principles and mode of reasoning without 
 modification. Other remarkable conclusions on the .-a me MI!>- 
 
 / I/ / V 
 
 ject might have been drawn from the equation ' - '-j- o. 
 
 expressing Carnot's assumption (of the truth of which experi- 
 mental tests might have been thus suggested); hut 1 am nt 
 aware that any conclusion deducible from it, not included in 
 Carnot's expression for the motive power of heat through finite 
 ranges of temperature, has yet been actually obtained and pub- 
 lished. 
 
 45. The recent writings of Rankine and Clausing contain 
 some of the consequences of the fundamental principle of the 
 'lyiiiiinical theory (expressed in the first fundamental proposi- 
 tion above) regarding physical properties of various substances ; 
 among which may be mentioned especially a \< t \ remarkable 
 discovery regarding the specific heat of saturated strain (in- 
 vestigated also in this paper in g 58 below), made independent ly 
 
 See aboTr 
 
 in
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 by the two authors, and a property of water at its freezing- 
 point, deduced from the corresponding investigation regarding 
 ice and water under pressure by Clausius ; according to which 
 he finds that, for each ^ Cent, that the solidifying point of 
 water is lowered by pressure, its latent heat, which under at- 
 mospheric pressure is 79, is diminished by .081. The investi- 
 gations of both these writers involve fundamentally various 
 hypotheses which may be or may not be found by experiment 
 to be approximately true ; and which render it difficult to 
 gather from their writings what part of their conclusions, es- 
 pecially with reference to air and gases, depend merely on the 
 necessary principles of the dynamical theory. 
 
 46. In the remainder of this paper, the two fundamental 
 propositions, expressed by the equations 
 
 -=& - 
 
 and .. , 
 
 Jf=l.f, (8) of g SI 
 
 are applied to establish properties of the specific heats of any 
 substance whatever ; and then special conclusions are deduced 
 for the case of a fluid following strictly the " gaseous laws " of 
 density, and for the case of a medium consisting of parts in 
 different states at the same temperature, as water and saturated 
 steam, or ice and water. 
 
 47. In the first place it may be remarked, that by the defi- 
 nition of J/and N in 20, JVmust be what is commonly called 
 the " specific heat at constant volume" of the substance, pro- 
 vided the quantity of the medium be the standard quantity 
 adopted for specific heats, which, in all that follows, I shall 
 take as the unit of weight. Hence the fundamental equation 
 of the dynamical theory, (2) of 20, expresses a relation be- 
 tween this specific heat and the quantities for the particular 
 substance denoted by Jfand p. If we eliminate M f rom this 
 equation, by means of equation (3) of 21, derived from the 
 expression of the second fundamental principle of the theory cf 
 the motive power of heat, we find 
 
 dN _ \fidt 1 ]_dp 
 
 < u > 
 
 137
 
 M K M I R S OX 
 
 which expresses a relation between the variation in the specific 
 heat at constant volume, of any substance, produced l>y an al- 
 teration of its volume at a constant temperature, and tin- vari- 
 ation of its pressure with its temperature when the volume 
 is constant; involving a function, , of the temperature, which 
 is the same for all substances. 
 
 48. Again, let K denote the specific heat of the substance 
 under constant pressure. Then, if dv and dt he so related that 
 the pressure of the medium, when its volume and temperature 
 are v + dr and / -|- dt respectively, is the same as when they are 
 r and t that is, if 
 
 we have 
 
 Kdt = M<lr + Ml. 
 
 Hence we find 
 
 _^ 
 
 jf = - ( A'-T). (i;,) 
 
 dt 
 
 which merely shows the meaning in terms of the two specific 
 heats, of what I have denoted by M. Using in this for .17 its 
 value given by (3) of 21, we find 
 
 ' 1 ' 1 
 
 an expression for the difference between the two specific heats, 
 derived without hypothesis from the second fundamental prin- 
 ciple of the theory of the motive power of In at. 
 
 r.i. These results maybe put into forms more convenient for 
 ii-'-. in applications to liquid and solid media, by introducing 
 the notation: 
 
 tfe 
 
 where .- will he the reciprocal of the compressibility, and 
 coefficient of expansion with heat.
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 Equations (14), (16), and (3) thus become 
 
 K-N = v, (19) 
 
 M = -. K e; (20) 
 
 the third of these equations being annexed to show explicitly 
 the quantity of heat developed by the compression of the sub- 
 stance kept at a constant temperature. Lastly, if denote the 
 rise in temperature produced by a compression from v + dv to 
 v before any heat is emitted, we have 
 
 = ^- r . -.<&> = -_, - s rf. (21) 
 
 N fi nK VKe* 
 
 50. The first of these expressions for shows that, when the 
 substance contracts as its temperature rises (as is the case, for 
 instance, with water between its freezing-point and its point of 
 maximum density), its temperature would become lowered by a 
 sudden compression. The second, which shows in terms of its 
 compressibility and expansibility exactly how much the tem- 
 perature of any substance is altered by an infinitely small alter- 
 ation of its volume, leads to the approximate expression 
 
 if, as is probably the case, for all known solids and liquids, e be 
 so small that B.VKB is very small compared with pK. 
 
 51. If, now, we suppose the substance to be a gas, and intro- 
 duce the hypothesis that its density is strictly subject to the 
 "gaseous laws," we should have, by Boyle and Mariotte's law 
 of compression, 
 
 d /=-l, (22) 
 
 dv v 
 
 and by Dal ton and Gay-Lussac's law of expansion, 
 dv Ev . 
 
 di^r+m' 
 
 from which we deduce
 
 MKMOIRS ON 
 Equation (14) will consequently become 
 
 r/.V _ d {(! + A7)~7 
 
 ~fo- ~7t 
 
 a result peculiar to the dynamical theory and equation (1C), 
 
 which agrees with the result of 53 of my former paper. 
 
 If V be taken to denote the volume of the gas at the tem- 
 perature under unity of pressure, (25) becomes 
 
 52. All the conclusions obtained by Clausius, with reference 
 to air or gases, are obtained immediately from these equations 
 by taking 
 
 which will make j- = 0, and by assuming, as he does, that N, 
 
 thus found to be independent of the density of the gas, is also 
 independent of its temperature. 
 
 53. As a last application of the two fundamental equations 
 of the theory, let the medium with reference to which M and 
 \ are defined consist of a weight 1 a; of a certain substance 
 in one state, and a weight x in another state at the same trm- 
 perature, containing more latent heat. To avoid circumlocu- 
 tion and to fix the ideas, in what follows we may suppose the 
 former state to be liquid and the latter gaseous ; but tin- in- 
 vestigation, as will be seen, is equally applicable to the case of 
 ft solid in contact with the same substance in the liquid or 
 gaseous form. 
 
 54. The volume and temperature of the whole medium !<- 
 ing, as before, denoted respectively by / and /. we shall have 
 
 if X and y be the volumes of unity of weight of the substance 
 in tlu> liquid and the gaseous states respectively: ami /-. tin- 
 pressure, may be considered as a function of /, depending solely 
 on the nature of the substance. To express M and .V for this 
 mixed medium, let L denote the latent heat of a unit of weight of 
 140
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 the vapor, c the specific heat of the liquid, and li the specific heat 
 of the vapor when kept in a state of saturation. We shall have 
 
 Mdv = L-r-dv, 
 Ndt = c(l-x)dt + lixdt + L C ^dt. 
 
 Now, by (27), we have 7 
 
 (y-\)~ = l, (28) 
 
 dv 
 
 and ( y -X)~ + (l-s)^ + *|| = 0. (29) 
 
 Hence M = ^, (30) 
 
 N = c(l-x) + hx-L, 
 
 y-\ 
 
 55. The expression of the second fundamental proposition in 
 this case becomes, consequently, 
 
 / = I- (32) 
 
 which agrees with Carnot's original result, and is the formula 
 that has been used (referred to above in 31) for determining 
 p. by means of Eegnault's observations on steam. 
 
 56. To express the conclusion derivable from the first funda- 
 mental proposition, we have, by differentiating the preceding 
 expressions for M and N with reference to t and v respectively, 
 
 ^__1_ ^_ L d(y-\] 
 dv ~-\ dt \' dt 
 
 _ 
 dN (.. di~didx 
 
 - ( y _ \ (y _ \) 
 
 Hence equation (2) of 20 becomes 
 dL 
 
 Jdt
 
 MEMOIRS ON 
 
 Combining this with the conclusion (32) derived from tho sec- 
 ond fundamental proposition, we obtain 
 
 +<-= <"> 
 
 The former of these equations agrees precisely with one 
 which was first given by Clausius, and the preceding in 
 gation is substantially the same as the investigation by which 
 he arrived at it. The second differs from another given by 
 Clausius only in not implying any hypothesis as to the form of 
 Carnot's function ^i. 
 
 :>',. If we suppose p and L to be known for any temperature. 
 
 equation (32) enables us to determine the value of -f- for that 
 temperature ; and thence deducing a value of dt, we have 
 
 dt = *^<I/: (85) 
 
 pL 
 
 which shows the effect of pressure in altering the "boiling- 
 point" if the mixed medium be a liquid and its vapor, or tho 
 melting-point if it bo a solid in contact with the >ame sub- 
 stance in the liquid state. This agrees with the conclusion ar- 
 rived lit by my elder brother in his " Theoretical Investigation 
 of the Effect of Pressure in Lowering the Pressing-point of 
 Water." His result, obtained by taking as the value for / that 
 derived from Table I. of my former paper for the temperature 
 0, is that the freezing-point is lowered by .0075 Cent, by an 
 additional atmosphere of pressure. Clausius, with the other 
 data the same, obtains .oo^:53 as the lowering of temperature 
 by the same additional pressure, which differs from my brother's 
 result only from having been calculated from a formula which 
 
 J0 
 implies the hypothetical expression J ' for /i. It was by 
 
 applying equation (33) to determine -tr for the same case that 
 
 ClansiiH arrived at the curious result regarding the latent heat 
 of water muler pressure mentioned above (i 
 
 58. Lastly, it may be remarked that every quantity which 
 appears in equation (33), exeept //. is known with tolerable ac- 
 curacy for saturated steam through a wide ranije of tempera- 
 ture; and we may therefore use this equation to liu.l //. which 
 has never yet been made an object of experimental research. 
 142
 
 THE SECOND LAW OF THERMODYNAMICS 
 Thus we have ^ /z 
 
 For the value of y the best data regarding the density of sat- 
 urated steam that can be had must be taken. If for different 
 temperatures we use the same values for the density of saturated 
 steam (calculated according to the gaseous laws, and Regnault's 
 observed pressure from y^, taken as the density at 100), the 
 values obtained for the first term of the second member of the 
 preceding equation are the same as if we take the form 
 
 derived from (34), and use the values of p shown in Table I. of 
 my former paper. The values of h in the second column in 
 the following table have been so calculated, with, besides, the 
 following data afforded by Regnault from his observations on 
 the total heat of steam, and the specific heat of water 
 
 L = 006.5 + .305/ - (.00002/ 5 + .OOOOOOO- 
 The values of h shown in the third column are those derived 
 by Clausius from an equation which is the same as what (34) 
 
 ri 
 
 would become if /- -- =- were substituted for /u. 
 1 4- &t 
 
 t. 
 
 - A nccordingto Table 
 I. of ' Account of 
 Carnot's Theory " 
 
 -ft according to 
 Clausius 
 
 
 
 1.863 
 
 1.916 
 
 50 
 
 1.479 
 
 1.465 
 
 100 
 
 1.174 
 
 1.133 
 
 150 
 
 0.951 
 
 0.879 
 
 200 
 
 0.780 
 
 0676 
 
 59. From these results it appears, that through the whole 
 range of temperatures at which observations have been made, 
 the value of h is negative ; and, therefore, if a quantity of sat- 
 urated vapor be compressed in a vessel containing no liquid 
 water, heat must be continuously abstracted from it in order 
 that it may remain saturated as its temperature rises ; and con- 
 versely, if a quantity of saturated vapor be allowed to expand 
 143
 
 MEMOIRS ON 
 
 in a closed vessel, heat must be supplied to it to prevent any 
 part of it from becoming condensed into the liquid form as the 
 temperature of the whole sinks. This very remarkable conclu- 
 sion was first announced by Mr. Runkine, in his paper com- 
 municated to this Society on the 4th of February last year. It 
 was discovered independently by Clausing, and published in 
 his paper in Poggendorff's Annalen in the months of April and 
 May of the same year. 
 
 60. It might appear at first sight, that the well-known fact 
 that steam rushing from a high -pressure boiler through a small 
 orifice into the open air does not scald a hand exposed to it, is 
 inconsistent with ^,he proposition, that steam expanding from 
 a state of saturation must have heat given to it to prevent any 
 part from becoming condensed ; since the steam would scald 
 the hand unless it were dry, and consequently above the boil- 
 ing-point in temperature. The explanation of this apparent 
 difficulty, given in a letter which I wrote to Mr. Joule last Oc- 
 tober, and which has since been published in the Philfisnjihintl 
 Magazine, is, that the steam in rushing through the orifice pro- 
 duces mechanical effect which is immediately wasted in fluid 
 friction, and consequently reconverted into heat ; so that the 
 issuing steam at the atmospheric pressure would have to part 
 with as much heat to convert it into water at the teni pi-rat im- 
 100 as it would have had to part with to have been condensed 
 at the high pressure and then cooled down to loo . which for 
 a pound of steam initially saturated at the temperature / is, by 
 Regnault's modification of Watt's law, .305 (t 100) more heat 
 than a pound of saturated steam at 100 would have to part 
 with to be reduced to the same state ; and the issuing steam 
 must therefore be above 100 in temperature, and dry. 
 
 PAW IV 
 
 On a Method of discovering txpfrimfntnllij t1i>< AV//f//;/ ////./< 
 
 the Mechanical Work spent untl the JIif ]<></ nrctl by 
 
 the Compression of a Gaseous Flit i<l. * 
 
 61. The important researches of Joule on the thermal cir- 
 cumstances connected with the expansion and compression of 
 
 Prom Hie Trantaction* of the Royal Society of Edinburgh, vol. we., part 2, 
 April 17. 1851. 
 
 144
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 air, and the admirable reasoning upon them expressed in his 
 paper * " On the Changes of Temperature produced by the Rare- 
 faction and Condensation of Air," especially the way in which he 
 takes into account any mechanical effect that may be externally 
 produced, or internally lost, in fluid friction, have introduced an 
 entirely new method of treating questions regarding the phys- 
 ical properties of fluids. The object of the present paper is to 
 show how, by the use of this new method, in connection with the 
 principles explained in my preceding paper, a complete theoret- 
 ical view may be obtained of the phenomena experimented on 
 by Joule ; and to point out some of the objects to be attained by 
 a continuation and extension of his experimental researches. 
 
 62. The Appendix to my "Account of Carnot's Theory "f 
 contains a theoretical investigation of the heat developed by 
 the compression of any fluid fulfilling the lawsj of Boyle and 
 Mariotte and of Dalton and Gay-Lussac. It lias since been 
 shown that that investigation requires no modification when. 
 the dynamical theory is adopted, and therefore the formula 
 obtained as the result may be regarded as being established 
 for a fluid of the kind assumed, independently of any hypothe- 
 sis whatever. We may obtain a corresponding formula applica- 
 ble to a fluid not fulfilling the gaseous laws of density, or to a 
 solid pressed uniformly on all sides, in the following manner : 
 
 (53. Let Mdv be the quantity of heat absorbed by a body kept 
 at a constant temperature t, when its volume is increased from 
 v to v + dv ; let p be the uniform pressure which it experiences 
 from without, when its volume is v and its temperature t ; and 
 
 let p -f -j-dt denote the value p would acquire if the temper- 
 
 ature were raised to t -f dt, the volume remaining unchanged. 
 Then, by equation (3) of 21 of my former paper, derived 
 from Clausius's extension of Carnot's theory, we have 
 
 * Philosophical Magazine, May, 1845, vol. xxvi., p. 369. 
 
 f Transactions, vol. xvi., part 5. 
 
 j To avoid circumlocution, these laws will, in what follows, be called 
 simply the gaseous laws, or the gaseous laws of density. 
 
 Throughout this paper, formulae which involve no hypothesis what- 
 ever are marked with italic letters ; formulae which involve Boyle's and 
 Dulton's laws are marked with Arabic numerals ; and formulae involving, 
 besides, Mayer's hypothesis, are marked with Roman numerals. 
 K 145
 
 MEMOIRS ON 
 
 where p denotes C&rnot's function, the same for all substances 
 at the same temperature. 
 
 Now let the substance expand from any volume Fto I". :m !. 
 being kept constantly at the temperature /. let it absorb a 
 quantity, //, of heat. * Then 
 
 " 'fr 
 
 But if H' denote the mechanical work which the substance 
 does in expanding, we have 
 
 and therefore 
 
 , r IrfW 
 
 *-tw 
 
 This formula, established without any assumption admitting 
 of doubt, expresses the relation between the heat developed by 
 the compression of any substance whatever, and the mechanical 
 work which is required to effect the compn-ssion. as far as it 
 can be determined without hypothesis by purely theoretical 
 considerations. 
 
 C4. The preceding formula leads to that which 1 formerly 
 gave for the case of fluids subject to the gaseous laws; since 
 for such we have ;// . , /v . o(1 + /;/)f (1) 
 
 from which we deduce, by (c), 
 
 ir=;i r (l + Ay)logy. 
 
 I 1 1 ' J7t 1 
 
 log F = j^-^ll: (a) 
 
 and 
 
 and therefore, by (d), 
 
 which agrees with equation (11) of g 40 of the former paper. 
 
 <;.". llt-nce we conclude, that the heat rv.>l\v<l by any lluid 
 fulfilling the gaseous laws is proportional to the work spent in 
 compressing it at any given constant temperature: Inn that 
 the quantity of work required to produce a unit of heat is not 
 constant for all temperatures, unless Carnot's function for dif- 
 ferent temperatures vary inversely as 1 4- AY; and that it is not 
 the simple mechanical equivalent of the heat, as it was nnwar- 
 140
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 mutably* assumed by Mayer to be, unless this function have 
 precisely the expression ^ 
 
 This formula was suggested to me by Mr. Joule, in a letter 
 dated December 9, 1848, as probably a true expression for ^, 
 being required to reconcile the expression derived from Car- 
 not's theory (which I had communicated to him) for the heat 
 evolved in terms of the work spent in the compression of a gas, 
 with the hypothesis that the latter of these is exactly the me- 
 chanical equivalent of the former, which he had adopted in 
 consequence of its being, at least approximately, verified by his 
 own experiments. This, which will be called Mayer's hypothe- 
 sis, from its having been first assumed by Mayer, is also assumed 
 by Clausius without any reason from experiment ; and an expres- 
 sion for yu the same as the preceding, is consequently adopted by 
 him as the foundation of his mathematical deductions from 
 elementary reasoning regarding the motive power of heat. The 
 preceding formulas show, that if it be true at a particular tem- 
 perature for any one fluid fulfilling the gaseous laws, it must 
 be true for every such fluid at the same temperature. 
 
 [The remaining section* are omitted. They deal with tlte experimental 
 verification of Mayer's hypotJiesis, with the mecfuinwal energy of n fluid, and 
 trith the applications of thermodynamics to electrical plienomena\\ 
 
 BIOGRAPHICAL SKETCH 
 
 WILLIAM THOMSON, now Lord Kelvin, was born at Belfast 
 in June, 1824. His father, James Thomson, an eminent math- 
 ematician and student of science, removed in 1832 to Glasgow, 
 where he occupied a position as professor in the university. 
 His son studied under him at Glasgow and at St. Peter's Col- 
 lege, Cambridge, and was graduated in 1845 at Cambridge, as 
 Second Wrangler and Smith's Piizeman. He was called to the 
 
 * In violation of Carnot's important principle, that thermal agency and 
 mechanical effect, or mechanical agency and thermal effect, cannot be re- 
 garded in the simple relation of cause and effect, when any other effect, 
 sucli as the alteration of the density of a body, is finally concerned. 
 147
 
 THK SECOND LAW OF THERMODYNAMICS 
 
 Chair of Natural 1'hilnsophy at (Jlasirow University in 
 and ho has since been connected with that university. On tin- 
 completion of the Atlantic Cable in I860, to the success of 
 which Thomson had materially contributed by his study of the 
 theoretical questions involved and by his numerous inventions, 
 he was knighted, and on New-Year's day. lv.i\!. he was raised 
 to the peerage as Lord Kelvin. 
 
 His contributions to physics range over the whole domain of 
 the science. The most important of these relate to electricity 
 and magnetism, and to the mechanical theory of heat, of 
 great value also are the improvements made by him in electri- 
 cal measurements, by his invention of the mirror galvanometer 
 and of the various electrometers which bear his name. Mis 
 work is characterized by its great versatility and by a peculiarly 
 happy combination of profound theoretical knowledge and 
 power of analysis with the ability to invent and execute impor- 
 tant experimental investigations.
 
 BIBLIOGRAPHY 
 BOOKS OP REFERENCE 
 
 Clausius. Die Mechanische Warmetheorie. 
 
 Maxwell. Theory of Heat. 
 
 Tail. Thermodynamics. 
 
 Verdet. Tlie&rie Mecanique de la Chaleur. 
 
 Rank inc. The Steam-engine. 
 
 Briot. La Chaleur. 
 
 Poincare. Thennodynamique. 
 
 Planck. Tliermodyamik. 
 
 Muck. Die Principien der Wdrmclehre. 
 
 Tail. Recent Advances in Physical Science. 
 
 ARTICLES 
 On the Validity of the Second Law 
 
 Rankine. Phil. Mag. (IV.), 4, p. 358. 
 
 Holtzmaiin. Pogg. Ann., 82, p. 445. 
 
 Decker. Dingler's Pol. Jour., 148, pp. 1, 81, 161, 241. 
 
 Him. Cosmos, 22, pp. 283, 413. (Also in Exposition Analytique et 
 
 Experimental de la Theorie Mecaniqite dc la Chaleur.) 
 
 Wand. Carl's Repertorium. 4, p. 281, 369. 
 
 Tail. Phil. Mag. (IV.), 43. 
 
 The foregoing articles contain criticisms of Clausius's form of tke Second 
 Law. They are answered by Clansius in Appendices to kis book, Die 
 Mechanische WdrmetJieorie, 2d ed. , 187U. 
 
 On Mechanical Analogies to the Second Law 
 
 Clausius. Pogg. Ann., 93, p. 481. 
 
 Phil. Mag. (IV.), 35, p. 405. 
 
 Pogg. Ann., 142, p. 433. 
 
 Ibid., 146, p. 585. 
 
 Rankine. Phil. Mag. (IV.), 30, p. 241. 
 149 ,
 
 THE SECOND LAW OF THERMODYNAMICS 
 
 Boltzmann. Per. der Wiener Akatl., 58, II., p. 188, 195. 
 
 I bid.. 68. II., pp. :.-V 71-j. 
 
 Ibid., 76, II., p. 
 
 Ibid., 78. II., pp. 1, 738. 
 
 Szily. Pogg. Ann . 14-1, pp. -J9o, 302 ; 149, p. 74 
 
 Burbury. Phil. Mag. (V.), 1, p. 61. 
 
 Of these the later papers by Bolt/.munn and that by But bury <: 
 the relation between the Second Law and the Theory of Probability. The 
 same subject is developed by Boltzmann in his book, I'vrUsungen uber 
 Gattlieorie, 2d part.
 
 INDEX 
 
 Caloric, Action of, 7 ; Conservation 
 
 of, Assmmd by Carnot, 20. 
 Carnot, Biographical Sketch, 60. 
 Carnot's Cycle, 11, 18. 73, 78. 
 Carnot's Function, 91, 93, 123, 128, 
 
 135. 
 Carnot's Principle, 13, 20 : Modified 
 
 by Clausius, 90; Modified by 
 
 Thomson. 115, 117. 
 Clausius, Biographical Sketch, 107. 
 Cycle. See Carnot's Cycle. 
 
 E 
 
 Efficiency of Engines, in Terms of 
 Carnot's Function, 36, 126 ; Com- 
 pared, 42, 58. 
 
 Engine, Ideal, 11, 18, 74, 114. 
 
 Gases, Heat Relations of (Carnot), 
 23, 25, 28, 31, 38 ; (Clausius), 71, 
 77, 84, 85, 86, 87, 88, 91 ; (Thom- 
 son), 139, 146. 
 
 11 
 
 Hent. See Caloric. Effects of, 3; 
 Total and Latent, 69 ; Mechanical 
 Equivalent of, 105. 127. 
 
 Heat, Relations of, to Work ; (Car- 
 not), 8 ; (Cbiusius), 66, 68 ; (Thom- 
 son), 111, 115,122, 126,133. 
 
 Liquids and Vapors. See Vapors. 
 
 Motive Power, Carnot's Definition 
 of, 6 ; Produced by Suitable Trans- 
 fer of Heal, 7, 9/10 ; Relation of, 
 to Difference of Temperature, 36; 
 Produced by Several Agents Com- 
 pared, 41. 
 
 S 
 
 Second Law of Thermodynamics 
 (Clausius),90, 118 ; (Thomson),116. 
 
 Steam-Engine, 3, 7, 50. 
 
 Substances, Heat Relations of, 137, 
 138, 139, 146. 
 
 T 
 
 Thomson (Lord Kelvin), Biograph- 
 ical Sketch, 147. 
 
 Vapors, Heat Relations of (Carnot), 
 43 ; (Clausius). 78. 81, 92, 96, 105 ; 
 (Thomson), 140, 142. 
 
 w 
 
 Water Vapor, Departure of, from 
 Mariotte's and Gay-Lussac's Laws, 
 95 ; Negative Specific Heat of, 
 105, 143. 
 
 Work. See Motive Power. Equiva- 
 lent to Heat. 66, 115 ; Relation ol, 
 to Unit of Heat, 105, 127. 
 
 151
 
 Univer 
 
 Sou 
 
 Lit