Comments on the Closure of the Carnot Cycle Levi Tansjo University of Lund, P.O. Box 124, S-221 00 Lund, Sweden In modern teaching of chemical thermodynamics, the Carnot cycle is not, as it used to he, the starting point when the Second Law of Thermodynamics is introduced. Instead, most teachers feel obliged to describe Carnot's classical thought experiment for their students. Teachers know that if a Carnot cycle is to he closed in a pressure-volume diagram with only four operations, one must start with one of the two adiabatic changes of volume. If one starts with one of the two isothermal changes of volume one does not know where to end the other one. Carnot's first commentators neglected this banal fact and closed the cycle in a confusing way. Sadi Carnot's own procedure, however, for establishing the famous, reversible cvcle that hears his name was simple and it was rrjectrd hy his t h incontr&rtihle ( I ).~~rverthelms, interpreters and fell into ohlivion for u long [:me. In this -paper the following questions will he answeredWhy did Carnot set up the cycle as he did? Why did not his first commentators adopt his procedure? ~
First, however, Carnot's closure of the cycle will he descrihed in some detail and compared with the procedures used by his early expositors and developers. This will lead us into the classical works that led to the Second Law of Thermodynamics and will give some insight into the confusion concerning the nature of heat in the years about 1850. Carnot's Closure The reauisites in Carnot's illustrious thouzht - ex~eriment were an arbitrary elastic fluid, "atmospheric air for example," shut un in a cvlindrical vessel with a heat-condnctinz bottom and nonconducting lateral walls, provided with anonconducting, weightless, frictionless piston, and there were two unlimited reservoirs of heat (A and B), each kept at a constant temperature, t~ heing higher than ts, and finally facilities for thermally insulating the cylinder (Fig. 1).
In his description of the sequence of operations on the fluid in the cylinder, Carnot used a sketch like that in Figure 1, showing besides the two reservoirs (A and B), the cylinder and four nositions of its niston. i.e.. four different volumes of the enclosed fluid. I willhenote the'positionsa, b, c, and d and let these notations represent the corresponding volumes of the fluid as well, the position and volume d heing also called ai. Carnot's performance of the thought experiment is independent of any assumption as to the nature of heat and can he summarized in the followinewav (the terms isothermal and adiahiatic were not yet coinez ini824, and Carnot used another numbering of the operations). Initial state: The air in the cylinder is in thermal contact with the
-
reservoir A and occupies the volume oi. bat the temperature t ~Heat . is taken from the reservoir A to the air. S t e p 2: Adiabatic expansion b c, during which the air cools from
S t e p 1: Isothermal expansion oi
t.4 to t g . S t e p 3: Isothermal compression e d, where d = = ai, at the temperature t e . Heat is given from the air to the reservoir B. S t e p 4: Adiabatic compression d a, during which the air warms from t s t ~ . b at the temperature t a . Heat is S t e p 5: Isothermal expansion a
-
taken from the reservoir A to the air. Carnot finds that the cyclic process obtained gives "excess of motive power" (work) and is reversible. The characteristic feature of Carnot's procedure for setting up the cycle is the following. He started with an isothermal expansion on the hot reservoir and used fiue operations, ai b c d -a b, toclose it, hut, since he stopped the isothermal compression on the cold reservoir just when the piston returns to the initial position, he nevertheless got a sketch containing only four piston positions. His procedure looks less lucid when represented in a P V diagram (Fig. 2), because there his choice of volume for the stop of the isothermal compression becomes less intelligible.
-
-- -
Maxwell's Closure J. Clerk Maxwell was the first one after Carnot who set up the cycle in an incontrovertible, yet clear and simple way. In
I Figure 1. Carnot's sketch showing piston pasitions at variws points on Uw cycle (Carnot's notations have been modified).
Heat
Figure 2.Pressurpvolumerepresentation of a Camot cycle f a a gas. (The dntd line shows Carnal's procedure tor sening it up.)
Volume 62
Number 7 July 1985
585
his "Theory of Heat" (2), the first edition of which appeared in 1871, he started (Fig. 2) the cycle with the cylinder standing on the insulator, and he presupposed that the enclosed fluid had first been brought to the temperature of the cold reservoir, t ~He . could then easily trace the cycle through the four opb c d (Fig. 21, and this procedure was erations d -a adonted hv manvwriters.,ex.. - . P. G. Tait (3).Max Planck (.4.,) . and, among 20tlkentnry writers, M. emansk an sky (5). I t should he noted that Zemansky's description of a Carnot cycle is not the prevalent one in 20th-century teaching on elementary thermodynamics. That one is found in Enrico Fermi's published lectures in 1936 a t Columbia University (6). Fermi considered a fluid and drew in the usual wav two adia b a t i c ~and two isothermals for it in a P V diagram.
---
These four curves intersect each other in the four points [a, b, e, and d (Fig. 211 . . .The reversible cyclic transformation [ a b e d a] is called a Cornot cycle (6). Fermi then performed in thouaht the already defined cycle, using Carnot's equipment andstarting withthe isothermal b (Fip. 2). The same technique is used when expansion a &ideal gas is takenas the agent and the students first are taught that the volume d = a(clb). Thousands of teachers before and after Fermi did it that way. It is indeed an excellent way of introducing the Carnot cycle, hut it is of no interest in the present study, which deals with early writers, who in contrast to Fermi did not define the Carnot cycle in advance, hut, so to speak, created i t during the very experiment.
-
Effect of Pressure in Lowering the Freezing Point of Water" (9), by his older brother James was presented. I t contained three thought experiments, all based on Carnot's reasoning. The first two led to the physical basis of skating, i.e., the freezing point of substances, which like water melt under contraction, is lowered when the pressure increases. In the third one, which gave the magnitude of the predicted freezing-point depression from available data, J. Thomson had set up a Carnot cycle. This thought experiment may he summarized as in Figure 3. When J. Thomson specified the third operation, the isod (Fig. 3), in the cold lake, correthermal expansion, c d in Figure 2, sponding to the isothermal compression, c he used the same trick as his brother and Clapeyron had done before, i.e., he referred to the axiom of the conservation of heat, writing:
-
-
-
Continue the motion till all the heat has been eiven out to the second lake at - t o , which was taken in during Process [a b]', from the first lake at 0 ' (9, p. 578). H e was, however, a t that time so skeptical about the caloric theory of heat that he added in a footnote: v
~~
~
~~~~
This step, as well as the corresponding one in Camat's investigation, it must be observed, involves difficult questions, which cannot as
From Clapeyron to Clausius
When the first expositors and developers of Carnot's work, Emile Claoevron (7). William Thomson (Lord Kelvin). (8). . ., his brother J&;S ~ h o m s o n(91, and ~ u d o l Clausius f ( l o ) ,set up the cvcle. - . thev. heean. like Carnot. with an isothermal chanee of volume, a b ?Fig. 2), but in cbntrast to Carnot, who usld five onerations to close it. thev used onlv four. which introduced the problem of finding the volumed a t which the isothermal change of volume on the cold reservoir must ston and the final adiabatic one begin. Clapeyron solved the problem in his 1834 account of Carnot's work, "MBmoire sur la puissance motrice de la chaleur," by taking refuge in the prevailing caloric axiom of the conservation of heat and specified the compression c d (Fig. 2) in the following way:
-
-
Let us suppose that the compression is continued till the heat released by the compression of the gas and absorbed by the body B is exactly equal to the heat communicated by the source A to the
gas during its expansion in contact with it in the first part of the operation (11.0. 76). as if such a definition of the point d automatically were a distinctive mark of it as well. If Clapeyron had not left it to his readers to design the far-from-simple extra thought experiments by which the actual heat quantities could have been measured but instead had tried to do it himself in a footnote, he would ~ r o h a h l vhave noticed how much easier i t would have been to determine the volume d in thought by first cooling the air, being in the state a , adiabatically from t~ to t ~i.e., , he would then have arrived directly at Maxwell's procedure for establishing the cycle. When William Thomson in 1849 described Carnot's thought experiment in "An Account of Carnot's Theory of Heat; with Numerical Results deduced from Regnault's Experiments on Steam," he adopted Clapeyron's specification d (Fig. 2), pushing the piston of the third operation, c down
-
till the air gives out to the body B as much heat as it had taken in from A, during the first operation (8,p. 551). William Thomson's "Account" paper was read before the Royal Society of Edinburgh on January 2,1849, and a t the same meeting the paper, "Theoretical Considerations on the 586
Journal of Chemical Education
Figure 3. Ressure-volume representation of James Thomson's third thought experiment (from ref. (9)).One imagines a Carnot cylinder (Fig. I) and two unlimited heat reservoirs ("lakes"). one at 0% (the freezing point of water under a pressure of 1 atm) and one at -PC (the freezing paint of water under a pressure of 2 atm).
l mntact Initblstafe: A black of ice under lte piston of the cylinder is in m with the lake at 0% and occupies the volume a, which is bigger than 1.09 1.
-
Step 1: isothermal compressim a b (I atm. O°C)until 1.09 1 of the ice has melted to give 1.00 1 water. Heat. 80 kcal. is taken from the lake. Step 2: Adiabatic compression, b ice and water cools to -1%
c,from 1 atm to 2 atm. The mixture of
Step 3: Isothermal expansion. c d (2 atm. freezes and heat is given to the cold lake. Slep 4: Adiabatic expansion d
+
PC) (where is d?).
Water
a closes the cycle.z
The changes of volume during the adiabatic operations are "extremely small compared to" 0.09 1. "Hence lte wark developed during an entire s m e " duing Which 80 kcal are taken from the lake at 0% is 0.09 1 aim. From the value of "Carnot's Cmfficient'' at 0% which was given in W. Thomson's "Account" paper. J. Thornson could then calculate that the freezing point depression of water by pressure should be 0.0075 "Clatm. This predicted value was a few months later on experimentally confirmed by W. Thomson (15, pp. 165-169). Brilliant brothers! Readers wondering how the calculation of the freezing point depression could be made with no reference at all to the first law of thermodynamics are referred to the refs. (8.9).
yet be satisfactorilyanswered, regarding the possibility of absolute formation or destruction of heat as an equivalent for the destruction or formation of other agencies, such as mechanical work; but, in taking it, I go on the almost universally adopted supposition of the perfect conservation of heat (9,p. 578). Clausius brought, in his 1850 paper, "Ueher die hewegende Kraft der Wkme und die Gesetze, welche sich daraus fur die Wirmelehre selhst ahleiten lassen," Carnot's ideas into harmony with the new theory of the convertibility of heat and work. He had then not been able to find a conv of Carnot's memoir and knew it only through Clapeyron will hhng the fluid hack to its initial state. In fact, this was done by Emst Mach (19) when he wrote that the point d in Figure 2 "ist so bestimmt zu denken," that a final adiahatic compression brings the fluid hack t o its initial state. I t was also done to some extent hut less clearlv bv Clausius in the second and third editions of h i s ' . ~ h h a i d l ~ n ~ e( In2") ,which are written as ordinary textbooks on thermodynamirs (20,. Alternatively. the first commentxors (:odd haw made their procedure quite clear, ttut less simple, if they had explicitly stopped the iso. thermal compression and begun the adiahstic one anywhtw on the isotherm (13.When the fluid then rmrhes its original temperature, the eversible cycle a b c d a closes (Fig. 4a), or a i (Fig. is easily closed by a final isothermal expansion, a 4h), and once the cycle is established the starting point a i is of no further interest. Carnot's own method to set up the cycle is the one illustrated in Figure 4h hut with the peculiaritv that he stopped the isother& compression justwhen the-piston returns to the starting position ai. This does not result in any advantage when we use a P V diagram, hut Caruot used a sketch and saved one piston position in it.
- . ~ ~
-
Continue the motion till so much heat has been given out to the second lake at - t o , as that if the wholemass contained in the cylinder were allowed to return to its original volume without anv introduction or abstraction of heat, if would assume its original temperature and pressure (13, p. 253). in other words, without reference to the perfect conservation of heat, hut we are again expected to know where the isothermal change of volume on the cold reservoir is to he stopped. This specification of the third operation is in fact the same one as Clausius had introduced half a year before in a very noticed paper (10). A few months later-in the September issue of Poggendorff's Annalen-Clausius had also demonstrated that J. Thomsou's conclusions from the thought experiment in Figure 3 remain valid even in the light of the new theory of the equivalence of heat and work (14). J. Thomson, however, was not yet ready for a definite break with the caloric theory of heat, so he continued: This, if Carnot's principles he admitted, as they are supposed to be throughout the present investigation, is the same as to say,Continue the motion tillall the heat has been givenout to thesecond lake at -to, which was taken in during Process [a b]', from the first lake at 0" (13, p. 253). The footnote added to the orieinal oaner . . was omitted. I t reappeared for smne unknown reason when the 1850 version of.1. Thomson's classiral . oaoer . was reouhlished in 1882 115. . , pp. 156-164). When William Thomson's "Account" paper of 1849 was to
-
'
-
J. Thomson actually wrote "Process 2 here. This was apparently a lapse. "Process 2 should be read "Process 1," i.e.. a b.
-
Volume 62
Number 7 July 1985
587
Carnot's Closure into Oblivion and out from It Most 19th-century commentators on Carnot's work followed Clausius and-the Thomson brothers in attributing Clapeyron's description of the cycle to Carnot. Maxwell, for instance, wrote along these lines just before the presentation of his own incontrovertible method of setting up the cycle As our object is to understand the theory of heat, and not to give a historical account of the theory, we shall avail ourselves of the
important step which Carnot made, while we avoid the error into which he fell (2, p. 139). After he had established the ryrle starting with the adiaharic compression d n (Fig. 'L), Mnxwell compared it with what he called Carnot's method and wrote:
-
-
Carnot himself began with the operation which we have placed second, the expansion at the upper temperature [a b], and he directs us to continue the fourth aoeration. comoression at the lower temprature [c d] till emrrly as much heat has left the suhshlnre ns rnrrrrd during the expnnston at the upper tPmperaturr (2, p. 118). These misleadina assertions hv Maxwell in the numerous editions of his influential " ~ h e o &of Heat" were probably one reason whv Carnot's orocedure fell into oblivion. It is remarkable that not even translators of the "RBflexions" such as W. Ostwald (21), R. H. Thurston (22),and W. F. Magie (23) in the 1890's commented on the fact that Carnot's description of the cycle differed greatly from the one currently ascribed to him. In 1894,70 years after the publication of Camot's memoir, Preston finally called attention to the fact that Carnot's description of the cycle is independent of all theory of heat and noted:
-
The corrections to Carnot's work introduced by James Thomson and Maxwell are consequently not only unnecessary, hut are an iniustice to the illustrious author of "The Motive Power of Heat". a#;dno douht they wcrr pwpased at a time when Cornor's work x i Icsrned hy report rarlwr than try consultntion oi [he original ~
W,.
~
-
Preston, however. erroneouslv rewrted that Carnot heean the c. cycle with the adiahatic expinsion b In 1910 Callendar (25) reproduced Carnot's description of the cycle verbatim to demonstrate that it does not contain any assumption as to the nature of heat. He then modified Carnot's words in the subsequent deduction of "Carnot's Theorem" until the caloric theory of heat was no longer - there either. and he added in a footnote: Carnot's description of his cycle and statement of his principle ["Carnot's Theorem"] have been given as nearly as possible in his own words, because some injustice has been done to him by erroneous descriptions and statements (25), as if Carnot's statement of his theorem had ever been erroneously reported.
The Source of Carnot's Procedure: His First Thought Experlment Gillispie analyzed Carnot's procedure to trace out the cycle (26, 27), in both cases focusing the discussion upon the question as to why in the last step Carnot returned the piston past the initial point ai to b and then had the steps of the cycle b. Gillispie aprepeating in the sequence b c d -a oarentlv means that Carnot had t o do so to he ahle to dem~~~~~onstrate that the cycle gives an excess of motive power. This is probably a misunderstanding. I t is more probable that Carnot proceeded past ai to b for the simple reason that he could then describe the established cvcle as comnosed of two isothermal and two adiahatic changesbf volume eien if he had used five ooerations to trace it out. Klein was next to examine Carnot'sdescription of thecycle and he cotnoared it with thr orocedures used bv his first c x positors and developers (28): Klein attached special importance to the fact that, in contrast to Clapeyron, Carnot did not link the very definition of the cycle to the caloric theory of heat, and he oointed out that even if Camot's oroeedure is less clear and elegant than Maxwell's, it neverthel&s gives another plain illustration of Carnot's ability to avoid theoretically questionable elements in his reasoning. MY study can he seen as a oarallel to that of Klein.. SUD. plementing his study on some points and emphasizing the merits of Carnot's procedure in comoarison not onlv with Clapeyron's hut with that of ~ l a u s i u s a well. s ~ o w e v kI, do not, with Klein, believe that Carnot's procedure for settine up the cycle should he seen mainly as a r&ect.ion of his critic2 attitude toward the caloric theory of heat. What was it then, that made Carnot find it so natural to stop the isothermal compression on the body B just when the piston comes hack to the initial position that he did not make any comments on it a t all? The simple answer may, of course, he the subtlety that by so doing he saved one piston position in his sketch (Fig. 1). However, would Camot, being soanxious to let us follow the wavs of his thought " throuehout the ~ - "R6- ~~ flexions," then not have inserted a "for example" or "for the sake of simolicitv" in "Ooeration 4" (11. . o. la)? I t is in fact much more probable that ~ a m onever t reflectadupon the best wav to describe the cvcle. that his nresentation of it m i t e simply gave itself, when he modified his first, often forgotten, thouaht exoeriment into the famous one. Let us follow Carnot's-reasoning up to that point in the memoir (11, pp. 315). After praising the steam engine Carnot raises the question whether there is an assianahle limit for the oroduction of rnutive poww by heat, an& if so. what factors determtne this ltmit. Are there, for instance, "in existence arents - nreferahle . to the vapor of water for developing the motive power?" (11, P 5). He then examines all cases of work generation by heat that he can think of and finds more and more confirmation of a principle, which was to he called "Carnot's Principle":
-- -
~
-
,.
-
588
Journal of Chemical Education
~
~
~
~
~~~
~
~
~~~~~
~.~ ~~
Wherever there erists o difference of can be produced (1I , p. 9).
Figure 4. Two procedures far setting up a Carnot cycle starting with the isothermal expansion a, b.
~
temperature motive power
and its converse Wherever we can consume this power, it is possible to produce a differenceof temperature, it is possible to occasion destruction of equilibrium in the caloric (11, p. 9). The word "can" in the principle is an important one. On the next few pages Carnot reminds us several times that the principle shall he read in the following way: Wherever there exists a difference of temperature, we have the chance, the opportunity, to produce work, and wherever we allow a difference of temperature to he equalized-by conduction or radiation-without generation of work, we lose forever the chance that we had to obtain the work, that we just, according to this principle, had the chance to obtain owing to the difference of temperature.
Heat
1
Heat
Figre 5. Ressvevolume repesewation of Carnot's first UmgM experiment. imagines a Camot cylinder and two unlimited heat reservoirs,A, hot, and 8. cold, as in Figure 1. OM)
hhklstate: Liquid water underthe piston of tke cylinder is in k m a l ccntact with me reserwir A and occupies the volume a.
-
bat the temperature t until all the water Stsp I: l x ) m l expanslml a has ~ p ~ i z under e d constant pressure. Heat is taken from the reservoir A to tke agent. Step2: Adiabatic expansion b t.
-
c. during which the agent cwis from t~ to
-
srep 3: isothermal mpresslon c d at the temperature t. Ths steam conden-s Mder constant presswe to water. Heat Is given fromthe agent lo tke reservoir 8.
If we wish to begin again an operation similar to the first, if we wish to develop a new quantity of motive power with the same instrument, with thesame steam, it isnecessary first tore-establishthe original condition-to restore the water to the original temperature. This can undoubtedly be done by at once putting it again in contact with the body A; but there is then contact between bodies of different temperatures,and lws of motive power (11, p. 14) [emphasis added] that is, when he closes the actual process, he loses the reversibility. He wins it back again
hy supping the differenceof temperature between the body A and the body B indefinitely small (11, p. 14), but when he tries to return t o a finite difference of temverature between AandB, he has to take refuge in astrained and mathematically questionable maneuver. Now he is ready to describe his second thought experiment, the famous one, which eives the reversible cvcle and then bv the same reductio a d absirdum as above " ~ a r n o t ' s he or em: The motive power of heat is independent of the agents employed to realize it;its quantity is fixedsolely by the temperature of the bodies between which is effected,f i nally, the transfer of the caloric (11, p. 20). We see that Camot's central nrohlem when be invented the reversible cycle was to restore r&ersibly the initial stateof an aeent. water, which already had undergone, in consecutive o;der; an isothermal expansion, an adiabatic expansion, and an isothermal comvression to a state of lower temperature, lower pressure, andiess caloriccontent t h w in the initial state hut a state of practically unchanged volume. This distinct problem forced him, so to speak, IJI invent the reversible cycle. I think that it is primarily this fact that is reflected when Carnot. ~.in the descrintion of his second. famous. thoueht experiment, found it s i natural to begin the cyclewith& isothermal exnansion on the bodv A and to stov the isothermal compressidn on B just when tge volume of the air equals the initial volume that he did not comment on i t a t all. The first description that occurred to him just happened to be a simple and clear one as well. ~~~
Carnot is now ready for his first thought experiment in which water is used for developing work in the same way as in a simple steam engine. I t may, in a somewhat simplified form, be illustrated and summarized as in F i e 5. We should note that in this preliminary thought experiment the agent, water, after three operations, a b c d occupies approximately the same volume, d , as in the initial state, a (Fig.
---
5 ) "1.
Carnot finds that the process in his thought experiment gives an excess of motive power and is reversible: The operations which we have just described might have been oerformed in an inverse direction of order and the quantit~ofcaioric pasred in the first cape from the body A to thebdy B would be equal w the quantity which passes back again in thesecond from thehdyBwthebdyAtll,p.111. Carnot then exploits the reversibility of the process in the following way. He imagines two reversibly working engines, one operating as a heat engine, one as a heat pump. His famous reductio ad absurdum, with perpetuum mobile as the excluded, on the proposition that there existed means superior to the vapor of water for developing the motive power produced by the action of heat, then leads him to the conclusion the maximum of motive power resulting from the employment of steam is also the maximum of matiue power realizable by any means whoteuer (11, p. 12). Carnot's nroof is unaccentable to us in the light of the f i t law of therm&ymamies. NO; is it adequate to carnot himself, hut i t is nevertheless a decisive sten in his reasoning because now he knows exactly what a rigorous proof shouldlook like and he asks us to consider his demonstration as an "apercu," a glimpse, an outline, for a while. 1'0carnot the demonstration has the flaw that the steam and the Eent to he compared with steam are not brought back to the; initial states after the three operations. I t strikes him that for a perfect reductio a d absurdum, reversibility of the prwess is not sufficient; a closed process, a cycle, is needed as well:
From Clapeyron to Clauslus Once More Clapeyron exchanged in his 1834 memoir Carnot's sketch (Fig. 1) for a P V diagram, a type of representation that had not been seen in the literature before. I t is true that Carnot's nrocedure for settine un a Camot cvcle looks less lucid in a P V hiagram (Fig. 2) t h i in his sketch Fig. I), but this was scarcely Clanevron's and William Thomson's reason for reiecting .. Carnot's procedure in their accounts of his work. T o find the; main reason we have to go hack aaain - to Camot's first thourhc . experiment. Clanevron did not describe Carnot's first thought experi. * ment (Fig. 5) a s a preliminary experiment that led Carnot to the subseauent. theoreticallv interesting one. In fact, he reported the'two experiments in the reverse order and modified Camot's preliminary one in such a way that the liquid water (Fig. 5) is reversibly brought back to its initial state (Fig. 6) (11, pp. 7S-81). Since, however, the point a in Figure 6 is to represent pure liquid, Carnot's procedure for setting up a Camot cycle is inapplicable. Clapeyron had to find another one. There is in fact only one incontrovertible way to trace out the cycle in Figure 6 when the point a is to represent pure liquid. We must first cool the fluid adiabatically from t~ to t~ to find the volume d and go clockwise from there, i.e., perform the adiabatic change of volume a-d in both directions. The onlv wav to escane this clumsv nrocedure is to modifv Camot's a repfirst thought experiment stiilkore and have the resent not pure liquid but a mixture of the liquid and its vapor. Carnot's procedure, and Maxwell's as well, are then applicable aeain if we assume that the amount of vavor in the state d s;ffices to raise the temperature from tB to t A by adiabatic
-
Volume 62 Number 7 July 1965
589
PI
Heat
Heat Figure 8. P r e s s u r e v o l u m e representation of aCarnot c y c l e for a m i d u r e 01 a liquid and its vapor. In Clapeyron's and W. Thomsan's descriptions of this cycb t h e point a represented pure liquid.
compression. Clausius introduced this extra modification of Carnot's first thought experiment in his 1850 paper (11, p. 122), hut he did not mention why he did it. I t is not likely that Clapeyron or W. Thomson ever considered these procedures for setting up the cycle in Figure 6. They felt perfectly sure that Carnot's "Wflexions" were based on the caloric theory of heat and saw no reason t o avoid this theory in the very description of a reversible cycle. Clapeyron quite simply closed the cycle by defining the point d (Fig. 6) as the point where
-
.. .all the heat provided by the hody A in the first part of the operation [a b ] has been given to the body B (11, p. 86), neglecting the fact that the two actual heat quantities would be rather difficult to measure. Perhaps he thought: What does i t matter in a thought experiment? In William Thomson's 1849 "Account," Carnot's two thought experiments are reported in the orooer order but the W. first-one fs described with ~ l a ~ e ~ r o modification. n's Thomson was aware of his deviation from the orieinal and added in a footnote:
Also, the fact that Clapeyron in 1834 and William Thomson in 1849 in their accounts of Carnot's "RBflexions" described Carnot's famous thought experiment quite differently seems to have its source in C¬9sfirst thouiht experiment. There is no evidence at all that Clape~ronand W. Thomson disliked Carnot's way of estahlishingthe cycle when a gas is the agent, hut they modified his first thought experiment in such a way that they had to diverge from his procedure and find a new one. When once they had diverged from Carnot's procedure, what reason had they then to use i t when the agent is a gas, a case which they already looked upon as a special one. Hypothetical (luestion
We can only guess how Carnot would have handled the situation if he, too, had shut up pure water or pure ice in the cylinder and found that the procedure which had occurred to him when air was the agent now would have heen inapplicable. My guess is: Carnot would then have searched for a procedure free from any reference to the axiom of the perfect conservation of heat in order to spare us complicated measurements, in thought, of heat quantities taken from and given to unlimited heat reservoirs. I find it difficult to imagine that his growing skepticism (29) regarding the validity of the axiom of the conservation of heat would have been a decisive factor. He knew too well, however skeptical he may have been of the axiom, that he already had allowed it to enter his reductio ad absurdurn (30-32); then to exclude it from the very description of a cycle could scarcely have been a goal in itself. ~
~~~
Acknowledgmeni
The author wishes to thank George B. Kauffman, California State University, Fresno, who kindly polished the Enplish in an early version of this paper. Literature Cited
(31 T&. P. C.."Heet." ~ a e m i l l ~and " Co., London. 1884, pp. 62-63. 141 Planek, M., "Vorlpaungm uberThcrmdynamik." 2ndod.. Voit & Co., Leipmg, 1905.
"" f i e 1 -*...
In Carnot's work some perplexity is introduced with reference to the temperature of the water [the initial and fmal ones] ...No such difficultvoccurs with reference to the evcle described in the text. far which I am indebted to Mons. ~lapiyron(8,p. 547).
15) Zemansky, M. W.,"HestandThemdynami~: 2nd ed.,MeCrsw-Hill BmkCo.,lnc., New Yark, L943,pp. 187-140, (61 Fermi, E., "Thermodynamics: Dover Publications, Inc., New York, 1937, w.12-
James Thomson's situation was similar to that of his younger brother and of Clapeyron. He set up a cycle (Fig. 3) starting with a large piece of pure ice in the cylinder, and he could not possibly use Carnot's procedure. He should then first have cooled the ice adiabatically from ODCto -t°C to find the ooint d (Fie. . " 3). ., hut he started instead with the isothermal compression a b in the warm lake and consequently got into trouble in findine-the ooint . d. Alternativelv... he could have shut up a mixture of ice and water in the cylinder instead of pure ice and used Carnot's ~rocedure(or Maxwell's) for tracing out the cycle, assuming the amount of liquid water of the mixture in the state d (Fig. 3) to suffice for raising the temperature from -t°C to O0C by an adiabatic expansion. Clausius introduced alreadv in 1850 this modification of J. Thomson's thought experiment (14), but he did not mention why he did it.
in ref. (1IJ.I Mendom, E. (Editor), '"Refledionson the Motive Power a1Fire by Sadi Camot and Other Papers on the Second Law of Thormadynarnics by E. Clapeyron and R. Clausius." Dover Publications. Inc.. New Ynrk. 1960. Clamiv8, d., '"Ahhandlungen hbkr die mochanische Warrnetheorie: Abteilung I. F. View- 8 Sohn, Braunschweig, 1864, pp. 1 6 7 8 . Thomson, J., Cambridm Dublin Math. J ,5,248 (18M). ClP.usius, R.,Poggcndorfi An,l.. 81,168 118501. Thornsun, W.. "Mathematical and Physical Papes." Vol I, University Press. Csmbridge, 1882. Helm. G.,"Die Energetik: veit & Comp.. 1898. p. 55. Helmholu, H. M","Vorl~ungeni l k Thearetisehe Physik," Band VI. (Editor: Rirahrz. F1,Theorieder Wbrme, Verlag J. A. Bsrth. Inipzig. 1903, pp. 203.206. Ostwsld, W., "Crundriss der Allgemeinen Chemie." 3rd ed.. Wilhelrn Enpelmann.
-
34.
(71 Clapeyron. E., J. d e I ' E c o l ~Pol~lechNque.14,153 (18341. (English translstion!~E. Mendoza in ref. (1Ih.I
.
....,.. ..
... .,
(21) Ostwald, W. (Editor),''Betrachtuneen ilber die Beweeende Kraft des Feuen und die
Results
Carnot took only one agent, a gas, though a complete, finite Carnot cycle. His preliminary, often forgotten, thought experiment, in which water was vaporized and the steam adiabatically cooled and finally condensed to water (Fig. 5) as in a simple steam engine, allows us to understand why Carnot described his subsequent, famous, thought experiment in the way he did. 590
Journal of Chemical Education
124) Preston, T., "TheTheowof Heat." Mamillan and Co., London, 1894, pp. 606608. (25) Callendsc. H. L., "Heat," in "EneyelapediaBritannica: 11th ad., Vol. XIII, Cantbridge, 1910.0.142. Princeton Univ~rsityPres, Princeton, 1960, (261 Cilhsplk: C. C.."TheEdgeofObj&ivity," pp. 363366.
127) Gillkpie. C. C.. "LarareCarnat, Savant" Princeton University Prers, Princeton, 1971. pp. 97-99. 128) K1ein.M. J.. "Closing theCarnotCycle,"in Taton,R., lEditor1,"Sadi Carnotetl'oesor de lsthemdynamique,"Centre National dels&hercheScientifique,Paris. 1976, pp. 213-219. 1291 Mendoul.E.,Arehlnt. Hist. Sci, 12,377 (1959).
(30) Kuhn,T.S.,Amer J. Phys,23,91I19551. 1311 Lervig. P.,Arch. Hist. Exp. Sci.,9,222 119721. I821 Lorvig, P.. "The E x k t e n a of s Work Function in Carnof's Theory," in Tafon. R., IEdilorl, "Ssdi Carnot et l'essor de la therrndynarnique," Centre National de L, Recherche Scientifique. Paris, 1976. pp. 199-212.
Volume 62
Number 7
July 1985
591
