Absolute Temperatures as a Consequence

of Carnot's General Axiom C. TRUESDELL

Dedicated to JAMES SERRIN

Contents 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 1I.

Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The General Carnot-Clapeyron Theorem . . . . . . . . . . . . . . . . . Apparatus from Concepts and Logic . . . . . . . . . . . . . . . . . . . Hotness and Empirical Temperature, I. Continuity . . . . . . . . . . . . . Hotness and Empirical Temperature, II. Differentiability . . . . . . . . . . Kelvin's Absolute Temperatures . . . . . . . . . . . . . . . . . . . . . Consequences of Part II of Carnot's General Axiom . . . . . . . . . . . . Thermometric Axiom . . . . . . . . . . . . . . . . . . . . . . . . . The Physical Dimension of Absolute Temperature . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix. Axioms of Classical Thermodynamics . . . . . . . . . . . . . .

357 360 360 361 367 369 372 373 375 377 377

1. P r o g r a m In our book, Concepts and Logic of Classical Thermodynamics, 1 Mr. BHARATHA and I developed classical thermodynamics on the basis of Part I of CARNOT'S General Axiom, namely, the motive power of a Carnot cycle is positive and is determined by its operating temperatures and by the amount of heat it absorbs. For a giv.en body, then, there is a function G such that for any Carnot cycle cg

Li~) = G(0 +, 0-, c +(~')) > o.

(1)

The domain of G is the set of operating temperatures and heats absorbed that may appertain to C a r n o t cycles for the body in question. It is part of the definition o f a Carnot cycle that 0 ÷ > 0 - and that C ÷ ( ~ ) > 0 . This definition and t C. TRUESDELL& S. BHARATHA,Concepts and Logic of Classical Thermodynamics as a Theory of Heat Engines. Rigorously Constructed upon the Foundation Laid by S. Carnot and F. Reech, N.Y. etc., Springer-Verlag, 1977.

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this axiom make sense in terms of underlying axioms accepted tacitly or expressly by the pioneers regarding piecewise smooth processes undergone by a body which exerts pressure p and receives positive or negative or null heating Q in response to its temperature 0, its volume v. and their time-rates of change P and 0: grv p = ~ ( K 0), yg- < 0, (2)

Q=Av(V,O) P+Kv(V,O)O,

Kv>0.

(3)

The time rates are assumed to exist at all but a finite number of times in the interval of time over which the process is defined. The three constitutive functions ra, Av, and K v are defined and.continuously differentiable* over the body's constitutive domain ~, which is a non-empty, connected, open subset of a real half-plane. The names of rv, Av, and K v are pressure function, latent heat with respect to volume, and specific heat at constant volume. By definition, the heat absorbed C + in any process is the value of the integral of Q with respect to time over the set of times on which Q >0. For future reference I remark here that by substituting (2) into (3) we obtain Q = Av{V, 0)/~ + Kp(V,. 0) 0, /&v ?r~/&v

A , = A v / ~--~,

K,-Kv=-Av--~/

(4)

~V.

Kp is the specific heat at constant pressure. In our book we took 0 as being what is sometimes called "the ideal-gas temperature". Our attitude toward it I later found described well and with evident disapproval by MACH 2: It is remarkable how much time passed before it was understood that to represent hotness by a number rested upon a convention. In nature there are hotnesses, but the concept of temperature exists only through our arbitrary definition, which might have turned out otherwise. Until very recently those who worked in this field seem to have looked more or less unconsciously for a natural measure of temperature, for a real temperature, for a sort of Platonic idea of temperature, of which the temperature read on a thermometer would be only an incomplete, inexact expression. As among the proponents of this to him reprehensible idea MACH was able to cite CLAUSIUS, I do not think we were at fault in holding to it, especially since by imputing to one ideal gas a simple property long known to hold very nearly for many natural gases we were able to prove as theorems not only a traditional "Second Law" but also a traditional "First Law". 2 E. MACH. Die Prinzipien der W'drmelehre, Historisch-kritisch entwickelt, Leipzig, Barth, 1896. Most of the quotations below are from the chapter called "Kritik des Temperaturbegriffes", hereinafter referred to a Temperaturbegriff The quotation above is from § 14. * See the note added in proof, p. 380.

Absolute Temperatures from Carnot's Axiom

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Nevertheless, it was not necessary for us to introduce the concept of ideal gas at all. After having seen Mr. SERRIN'S beautiful work on hotness and absolute temperature 3 and after having been stimulated by his letters and lectures and by our conversations on the subject for three years, I have perceived that to develop the concept of absolute temperature upon the basis of CARNOT's ideas would be not only easy but even worthwhile. KELVIN'S work 4, also, I have studied in detail, and I give a full account of it in my Tragicomical History of Thermodynamics, 1822-1854, now being polished for the press. Here I wish to remark only that KELVIN'S work rests essentially upon CARNOT's ideas alone but is neither clear nor explicit nor entire nor right in all details. CLAUSIUS attempted to persuade his readers that he had established an absolute temperature in circumstances far more general than what can be described in CARNOT's terms; his work, too, is described in my Tragicomical History, but I cannot follow the arguments that he presents; indeed, I cannot even distinguish what he assumes from what he claims to prove. It is my impression that Mr.SERRIN's theory achieves what CLAUSIUS attempted, or, rather, the achievable part of CLAUSIUS' inchoate program. The treatment I give below seems to me to clarify, consolidate, confirm, correct, and complete KELVIN's. Because I offer it only as a modest prohistorical study, I present it informally, using only such mathematical concepts as were accessible to any competent mathematical scientist of the 1870s-concepts, however, some of which were not at the disposal of KELVIN and CLAUSIUS in 1848-1854 and hence are not to be found in textbooks of thermodynamics today. Use of the modern theory of manifolds would effect briefer deduction of results more general than those below, but all I attempt here is to motivate the assumptions and to show that elementary arguments, calling upon no general theorems about manifolds, would have sufficed to prove everything that the pioneers sought in this domain. For a fully mathematical treatment meeting modern standards of rigor and generality the reader should consult Mr. SERRIN's work and studies by others which I expect to be complete soon. 3 j.B. SERRIN,Foundations of Classical Thermodynamics, Lecture Notes, Department of Mathematics, University of Chicago, 1975, and "The concepts of thermodynamics", pp. 411-451 of Contemporary Developments in Continuum Mechanics & Partial Differential Equations (Proceedings of the International Symposium on Continuum Mechanics and Partial Differential Equations, Rio de Janeiro, August 1977), edited by G.M. DE LA PENHA & L.A. MEDEIROS, Amsterdam, North-Holland Publishing Co., 1978. Mr. SERRIN presented further results in his Bateman lectures at The Johns Hopkins University in May, 1978, and in subsequent lectures elsewhere. His work applies to a class of materials more general than that considered here. 4 W. THOMSON(later Lord KELVIN),"On the absolute thermometric scale founded on Carnot's theory of the motive power of heat; and calculated from Regnault's observations". Proceedings of the Cambridge Philosophical Society 1 (1843/1866), No. 5, 66-71 (1849) =Philosophical Magazine (3) 33 (1849), 313-317=(with added notes) pp. 100-112 of W. THOMSON'S Mathematical and Physical Papers, Volume 1, 1882. J. P. JOULE & W. THOMSON, "On the thermal effects of fluids in motion, Part II", Philosophical Transactions of the Royal Society (london) 144 (1854), 321-364 = (with pp. 362-~-364 excised), pp. 357-400 of W. THOMSON'S Mathematical and Physical Papers, Volume 1, 1882 =pp. 247-299 of JOULE'SScientific Papers, Volume 2, 1887. See Section IV of "Theoretical Deductions".

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C. TRUESDELL

Essential to my argument is the apparatus provided by Concepts and Logic. Of course only those parts of it that survive when all mention of ideal gases is excised can be used for the purpose. Thus I have phrased (1)-(4)in terms of some one empirical scale of temperature, values of which are denoted by 0, 0 -~. and 0-. The range of this scale is some open real interval. Until §7 all reasoning will be local, will refer only to temperatures in some subinterval, perhaps very small, and only to some one body.

2. The General Carnot-Clapeyron Theorem

CLAPEYRON, applying to (1) an argument that CARNOT had invented but had used only for ideal gases, determined as follows the work done by a Carnot cycle ~ with infinitesimal difference A O of operating temperatures: L((ff)~/~(0) C + (~)A 0.

(5)

Here ~t is "Carnot's function", derived from G in (1) through a limiting process. From (1) and (5) we might think to conclude that/1>0, and the early authors. KELVIN among them, seem to have done so. However. examination of the limit process shows that only a weaker statement follows: ~>0.

(6)

We shall see that the distinction, altogether unnoticed by the pioneers, bears upon the concept of absolute temperatures. Essentially equivalent to (5) is the General Carnot-Clapeyron Theorem: 8~73 laA v = -~-ff.

(7)

Hence we at once perceive the constitutive inequality Av 8~> 80 = o"

(8)

Kp > K v.

(9)

It follows from (4)a that

We note that K v = K v if and only if A v 8vo/~O=O.

3. Apparatus from Concepts and Logic Concepts and Logic provides rigorous proofs of delimited statements more precise and more general than CARNOT'S and more specific than those obtained by REECH in a much earlier and incomplete attempt to develop CARNOT's ideas. The normal set ~n of the body represented by the constitutive quantities cj, ~v, A v and K v is the set of points P of ~ such that

Absolute Temperatures from Carnot's Axiom

361

1. P is arbitrarily near to a point of 2 at which At4:0. 2. The isotherm through P contains a point of 2 at which Av 4:0. The normal set need be neither connected nor open; it may be empty. It is the largest subset of ~ for which CARNOT'S ideas seem to be fruitful. The set of temperatures that correspond to points of ~ , is open, but it need not be connected. It is the union of an enumerable collection of disjoint open intervals. In what follows now we shall presume that 0 lies in one of these intervals. Theorem 7ex, in Chapter 9 and Theorems 9his and 10bi s in Chapter 10 of Concepts and Logic tell us that on that interval there are continuously differentiable functions g and h, the former being an increasing function and the latter a positive function, such that in a simple Carnot cycle ~ in ~ . C

. . . . h(O-) t~) = h(--~ C+(~)'

L(rg) =

g(O+)-g(O -) h(O+)

C÷(~),

(10)

(11)

and that at each point of ~n g' &~ ~ Av = - ~ , 0

(12)

8

Moreover, h is unique to within a multiplicative factor; when h is assigned, g is unique to within an additive constant; and g ' > 0 except on a set with empty interior.

(14)

In particular, Carnot's function # in (7) has the form g' #=~-;

(15)

# > 0 except on a set with empty interior.

(16)

it is unique and continuous; and

4. Hotness and Empricai Temperature, I. Continuity I do not regard the contents of this section and the succeeding one as anything but an exposition of ideas so simple that they must have been sensed informally by the pioneers of thermodynamics. Therefore, although I do not find them explicitly in any early source, I do not specifically attribute them to any later author.

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The concept of hotness is natural and primitive, like presence, weight, and place. We perceive heat and cold as we do light and dark, wet and dry. far and near, through our senses. As MACH wrote 5 A m o n g the sensations by which, tt'irough the conditions that excite them, we perceive the bodies around us, the sensations of heat form a special sequence (cold, cool, tepid, warm. hot) or a special class of mutually related elements. ... The essence of this physical behavior connected with the characteristic of sensations of heat (the totality of these reactions) we call its hotness [-'~W ~ r m e z u s t a n d " ] 6. Also, wrote MACH 7, The sensations of heat, like thermoscopic volumes, form a simple series, a

simple continuous manifold. It does not follow therefrom that the hotnesses also form such a manifold. The properties of a system of signs do not determine the properties of that which they designate . . . . The assumption of

a simple continuous manifold of hotnesses is sufficient. MACH justifies this last assertion only by stating that no experience indicates the contrary. The concept of "simple continuous manifold" or "continuous manifold of one dimension" derives from RIEMANN s and has been made precise in more recent works on pure mathematics. MACH assumes that the set fig of all 5 MACH, op. cir., § I of"Historische Ubersicht der Entwicklung der Thermometrie". 6 Mr. SERRIN prefers "hotness level". Earlier modern authors, I being one, have used "temperature" loosely to mean sometimes hotness, sometimes a number representing it. For example A. H. WILSON, Thermodynamics and Statistical Mechanics, Cambridge, Cambridge University Press, 1957, writes in § 1.21 "If we assume that temperature is a primary idea, the methods of measuring it must be to a certain extent empirical." 7 MACH, Temperaturbegriff § 5. s Cf § 14 of the article by F. ENRIOUES, "Prinzipien der Geometrie", Enzyklopiidie der Mathematischen Wissenschaften, Volume 31, Heft 1, 1907:

The fundamental concept in the theory of the continuum is that of the continuous manifold of one dimension. Regarded abstractly, this concept may be identified with the concept of the line. if the point of the line is taken as the element of the manifold and if no regard is had for the relation of the line to the rest of space or for any (metric) concept of the length of segments (or arcs) of the line. Thus only those properties of the line are considered that are connected with geometrical determination of the line through the motion of a point: the natural orderings of the points of the line and their continuity, the segments, etc. Though the "continuous manifold of one dimension" is thus defined partly in terms of what it is not, the explanation is pretty clear. ENRIQUES attributes the c..oncept to RIEMANN'S celebrated Habilitationsrortrag of 1854, first published in 1868: "Uber die Hypothesen, welche der Geometrie zu Grunde liegen', republished in all editions of RIEMANN's Gesammelte Mathematische Werke. RIEMANN's presentation, conditioned by the circumstances, is not explicit, but it sufficed to give mathematicians some essential ideas which they could render, in time, precise. For a history of the concept ofmanifold see the article by D. M. JOHNSON."The problem of the invariance of dimension in the growth of modern topology", Archive for History of Exact Sciences, Volume ~0 (1979), pp. 97-189.

Absolute Temperatures from Carnot's Axiom

363

hotnesses is a continuous manifold. ~ may well be the first manifold to have been suggested by physical experience not expressed in terms of mass, position, and time. This circumstance lets me think it worthwhile to present the simple physical ideas that lead us to impute specializing properties to the hotness manifold. First of all there is our sensation that at a certain time one body is hotter than another. Sensations, indeed, are not always consistent. This fact does not make us reject all the evidence they bring us. Rather, if they sometimes bear conflicting witness, we regard them as being partly in error then, and we do not on that account deny the existence of what they usually suggest. We let the exception prove the rule. Guided by sensory experience, we conceive ~ as being ordered. Using >- to denote "hotter than" and ~ to denote "not cooler than", we assume that ~ is a simple ordering upon J{: If hx sour' and h2e.H then either hl~=h2 or h2~=h1, and those two relations both hold if and only if hi = h 2. If hz~-ht, the set of all h such that h 2 ~ h ~ h ~ we may call a bounded segment ]h~,h2[ of ._,'f. The segments of ~f' consist in the sets of this kind and the sets defined by relations of the forms h>-h~ and h 1~-h. The segments provide a basis for a topology on ~ ' , in terms of which we may define limits and continuity. When we restrict attention to a bounded segment, only its subsegments, also necessarily bounded, are needed for this purpose. Temperature is a numerical indicator of hotness, just as milestones along a road indicate nearness and farness by numbers. As MACH wrote 9, The temperature is ... nothing else then the characterisation, the mark of the hotness by a number. This temperature number has simply the property of an inventory entry, through which this same hotness can be recognized again and if necessary sought out and reproduced.

An empirical-temperature function or empirical-temperature scale is an assignment of co-ordinates to a segment ovgo of ovt' or to ~ itself. The value 00 of a temperature function 0 at a particular hotness h is the empirical temperature of h 0 on the scale 0: If h o e o~o, then 0 o = 0(h o). That is not all. As MACH ~° wrote, This [temperature] number makes it possible to recognize at the same time the order in which the indicated hotnesses follow one another and [to recognize] between which other hotnesses a given hotness lies. That is, the mapping 0 preserves the order ofhotnesses:

hl~=h 2 ~ O(hl)>=O(h2).

(17)

It is to this assumption, I believe, that MACH referred when he wrote that "the sensations of heat ... form a simple series". He merely put in words the silent prejudice all the pioneers of thermometry, calorimetry, and thermodynamics accepted as a matter of course: Hotnesses may be represented faithfully by points 9 MACH, Temperaturbegriff §22. to Temperaturbegriff § 22.

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C. TRL'ESDELL

on tlre real line. Were this not so. the innumerable diagrams in which a 0-axis appears would misrepresent. Temperatures had been introduced and were grown familiar long before the abstract concept of hotness, and it was study of our experience with temperatures that taught us, finally, to isolate and abstract hotness. Indeed, temperature and hotness were confused for a long time. and even in MAXWELL'S magisterial 77wory of Heat the two are not distinguished except by implication x~" The temperature of a body ... is a quantity which indicates how hot or how cold the body is. When we say that the temperature of one body is higher or lower than that of another, we mean that the first body is hotter or colder than the second, but we also imply that we refer the state of both bodies to a certain scale of temperatures. By the use, therefore, of the word temperature, we fix in our minds the conviction that it is possible, not only to feel, but to measure, how hot a body is. When we come to consider how to measure "how hot a body is", it suffices for practical purposes to limit the range of an empirical scale to some bounded interval ]01,02[. For our present, local considerations, that interval is best regarded as contained in one of the intervals mentioned in § 3 in connection with some one body. It is natural to suppose that this real interval is the image of a bounded segment ]hl, h2[ o f ~ . Accordingly we begin by considering only those empirical-temperature scales that are one-to-one, order-preserving mappings of bounded segments ]hl,h2[ of ~,~ onto bounded intervals ]01,0,[ of ~. Thus the function 0 in (17) is a "homeomorphism": a continuous transformation having a continuous inverse. It is plausible and can be proved easily that the segment ]hx,h2[ has the "Hausdorff property": Any two distinct hotnesses in it lie in subsegments whose intersection is empty. That fact makes ]h a, hz[ a continuous manifold in the modern sense of the term; an empirical-temperature function is a

chart on 2It" which preserves the intrinsic order of hotnesses. Suppose that on some one bounded segment ~ there are two empiricaltemperature scales; 0 and 0". Then if h o ~ o,

0o= 0(ho), 0"=0"(/,o)=0"o0-'(0o).

(18)

The composite mapping 0* o0- ~ maps the range of the scale 0 onto the range of the scale 0": Iff=O*oO - 1. then 0* =f(0o),

0osRange 0.

(19)

Because of (17) we see that f is a one-to-one, bicontimtous, increasing mapping of Range 0 onto Range 0". It was these mappings of real intervals onto real intervals that the pioneers of thermometry had to use in comparing the results of using different empirical-temperature scales. 1~ j.C. MAXWELL, Theory of Heat, 10'h (last)edition, 1891, p. 2.

Absolute Temperatures from Carnot's Axiom

365

If 0 and 0* are defined on bounded segments ~ and 9f, whose intersection is the segment ~f'0, the foregoing argument shows that 0 and 0* provide compatible charts which together cover ~ w o ~ , . . Thus ._,ug~w~_, is a continuous manifold. We can go on in this way to any collection of overlapping segments of .~. For the time being, however, we shall revert to the study of a single bounded segment of J~. The homeomorphisms of a bounded segment of .3f with a bounded real interval fall into two classes: those that preserve the order on o~, and those that reverse it. How do we decide which of the two classes of homeomorphisms does consist of empirical-temperature scales and which does not? It is customary here to appeal again to the direct evidence of our senses: We feel that boiling water is much hotter than freezing water, etc. The mappings that assign to boiling water a higher temperature than to freezing water are therefore regarded as faithful to the intrinsic ordering of ~ , and mappings of the other class are rejected. While of course any comparison of theory with experience rests ultimately upon human sensations, we prefer to eliminate as many appeals to them as we can. To discern which of the two classes of homeomorphisms respects (17) and hence consists of empirical-temperature scales, we may use heat engines as they are idealized by Carnot cycles. A Carnot cycle absorbs heat at the higher of its two operating temperatures: 0 + > 0 - . A particular cycle that is a Carnot cycle according to the homeomorphisms of one class is the reverse of a Carnot cycle according to those of the other class. The work done by a cycle is a purely mechanical property of that cycle, independent of the methods of measuring hotness. That work is positive, negative, or null. If a cycle does positive work, its reverse does negative work. CARNOT's General Axiom (1) asserts that every Carnot cycle does positive work. Moreover, this statement is invariant under the transformations (19) defined by increasing functions f Thus CARNOT's General Axiom serves to distinguish the class of empirical-temperature scales from the class of homeomorphisms that reverse the intrinsic order of hotnesses. Suppose that a body undergo a cycle such that 1. it absorbs and emits heat only on two distinct isotherms, with which we have somehow associated numbers 0 + and 0-, respectively; 2. the amount of heat it absorbs is positive; 3. it does positive work. Then the numbers 0 + and 0- cannot result from an empirical-temperature function unless 0 + > 0 - . To give a practical example of this distinction, we proceed now to the study of thermometers. A thermometer is a body used to determine empirical temperatures. Typically it is a body which expands appreciably when it is heated, and its volumes are regarded as the values of an empirical-temperature function. In MAXWELL's words, they make it "possible not only to feel, but to measure, how hot a body is." More than that, just because volumes are measurable they are taken as replacing the sometimes inconsistent evidence of our sensations. To this end we cannot use an arbitrary body. Let us suppose, for example, that our first thermometric body consists of air, and let us measure volumes of a

366

C. TRUESDELL

body of water at atmospheric pressure when the air temperature is near to 4 ° C. Figure 1 shows what we shall find:

p-- 1 arm

I

i

Fig. 1.0 (air thermometer) in degrees C As has been known since the days of the Accademia del Cimento, the body of water at the pressure i a t m . occupies the same volume at two different temperatures according to the air thermometer, one above 4 ° and one below it. Thus a water thermometer operating in this range at constant pressure would indicate the same temperature at hotnesses which according to the air thermometer are different. The mapping from the numbers determined by volumes of water onto the numbers determined by volumes of air is not one-to-one in any open interval that contains 4 °C. That is not all. Suppose we try to use as a thermometer a body of water at atmospheric pressure in the range from 0 ° to 4 ° C. The temperature according to the presumed water thermometer is then an invertible function of the temperature according to the air thermometer, but the temperature that is higher according to the water thermometer will be lower according to the air thermometer. Thus it is impossible that both of these alleged scales of temperature can conform with (17). One must be rejected. In this regard MACH 12 wrote

77~ose hotnesses will be called higher in which bodies give rise to greater volume indications on the thermoscope . . . . Therefore we avoid use of water as thermoscopic substance... MACH went too far here. There is no inherent reason for regarding greater volume of air to be more faithful than greater volume of water. True, most other substances agree with air and disagree with water in their behavior, but resort to majority rule would seem dubious if not dangerous practice in physics. x2 MACH, Temperaturbegriff §4. Modern authors use the term "'anomalous" in this regard. E.g. WILSON, § 1.21 of the book quoted in Footnote 6: "By comparing the results obtained by using different thermometers we can reject as unsuitable those substances, such as water, whose behavior is anomalous .... All that we require of a substance to be used in constructing a thermometer is that the property to be measured ... should be a strictly increasing function of the temperature [i.e. hotness]." The "anomalous" substances are few indeed in the range of hotnesses presently available to experiment, but we are left wondering how to tell which substances are "anomalous" and which are not when new ranges of hotness shall become accessible.

Absolute Temperatures from Carnot's Axiom

367

Under our interpretation of CARNOT'S General Axiom, if any cycle that according to the air thermometer is a Carnot cycle with operating temperatures in the interval between 0°C and 4°C does positive work, then the volumes of a fixed mass of water at atmospheric pressure are not the values of any empiricaltemperature scale for the corresponding range of hotnesses. Experience teaches us that the engine which absorbs heat at the higher temperature according to the air thermometer does positive work. Thus it is the supposed water thermometer t h a t w e reject in the range including 4°C and air temperatures lower than that. The foregoing, I believe, is a precise replacement for MACH'S vague claim. It also justifies use of the term "anomalous behavior" here. Of course, in other circumstances there is no reason at all to reject water as a thermoscopic substance.

5. Hotness and Empirical Temperature, II. Differentiability So far, we have appealed only to the homeomorphism and intrinsic ordering required of empirical-temperature scales. If we are to use them in calorimetry, we demand something more. The very statement (3) of the Doctrine of Latent and Specific Heats refers to the derivative of a function of time whose value is the temperature, and the uses to which the constitutive functions w, A v, and K v are put in the development of thermodynamics also appeal to differentiation. Differentiability itself requires assumptions regarding ._fg. If we approach hotness through use of empirical temperature-scales, we may refer to those assumptions indirectly by requiring that the transformation of one scale to another be differentiable. Then f in (19) is a differentiable function, and we shall write

d0* =f'(0) d0.

(20)

Because f is an increasing function, f'(O) > O. Because the inverse transformation f - i is also differentiable, the possibility that f'(O)= 0 for some 0 is excluded. Thus f'>O.

(21)

The requirement that 0 exist almost always is independent of the choice of scale. Still restricting attention to a single bounded segment Jr0, let us now investigate the rules of transformation for the quantities that enter (2) and (3). First, the chain rule and (20) show that

~0" dO* =-~-dO.

(22)

Here, as usual in works on physics,

~ a ~.(V,f_ 1(0,)). ,~0" -00"

(23)

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C. TRUESDELL

From (22) and similar formulae for the derivatives of the other constitutive functions we see that to render invariant under change of empirical-temperature scale the assumption that vv be continuously differentiable, it is necessary to assume that f ' is continuous. We may sum the effect of adjoining this assumption to the preceding ones by stating that S o is a diffeomorph of an open interval. Likewise, if a segment is the union Q)o~ of a finite number of bounded segments ~,~,, on each of which an empirical-temperature scale is defined, and if each intersection of those segments satisfies the conditions just demanded of .H0, then Q)~¢~, is a diffeomorph of an open interval and has an intrinsic ordering. More general conclusions can be obtained from the modern theory of manifolds but are not needed if our objective is only to clear and specify the pioneers' ideas about scales of empirical temperature. In his treatment of 1975 Mr. SERRIN assumed ~ ' to be a diffeomorph of the real line, equipped with an intrinsic ordering. Such an assumption seems to be implicit in some earlier studies by others. In his later work NERRIN requires of ~¢' merely that it be a continuous manifold. For details the reader should consult his lecture of 1977, cited above in Footnote 2. Mr. C.-S. MAN in a master's thesis accepted by the University of Hong Kong in 1975 provided a far less general but constructive rather than postulational introduction of the hotness manifold. We revert to the consideration of a single bounded segment of Jr, on which an empirical-temperature scale is defined. The quantities Av and Kv in (3) will generally depend upon the choice of empirical-temperature scale, but Q will not. Therefore, in an evident notation, Av.o,(V,O*) V+Kv,o,(V,O*)O*=Av, o(V,O) V + K v , o(V,O)O.

(24)

Av,o, = Av,o, Kv. o*dO* = Kv, o dO.

(25)

Po*dO* = I1o dO.

(26)

Hence

It follows from (7) that In these formulae the subscript 0 may be read "when the scale 0 is used", and likewise for 0", and the arguments 0* on the left-hand sides are related to the arguments 0 on the right-hand sides through (19). In passages where only a single empirical-temperature scale is considered, we drop the subscript. The transformation rules (22) and (25) show that the smoothness assumed for w, A v, and K v and the signs of ~v~/~,V, ~w/gO, A v, and K v are invariant under change of empirical-temperature scale. In particular, the constitutive inequalities (2)2 and (3) z hold for all scales if and only if they hold for one scale. In summary, the basic axioms (2) and (3) hold in terms of one empiricaltemperature scale if and only if they hold in terms of all scales. Since a differential equation satisfied by the adiabats is dO/dV= - A v / K v. the qualitative behavior of the adiabats near a point is the same for all scales. That means that Carnot cycles defined in terms of one empirical-temperature scale

Absolute Temperatures from Carnot's Axiom

369

exist locally if and only if they exist locally according to all such scales, temperatures corresponding to the same hotness being understood in the term "locally". Moreover, ~10,(0*)=0 if and only if ~L0(0) = 0, so the set of temperatures at which IL vanishes on any one scale correspond to hotnesses that make # vanish on all scales. We recall that the set of such temperatures has empty interior.

6. Kelvin's Absolute Temperatures KELVIN used "absolute temperature" in three senses: 1. A scale of empirical temperature independent of the choice of thermometer. 2. A scale of empirical temperature such that the work done per unit heat absorbed in a Carnot cycle depends upon the difference of operating temperatures only. 3. A scale of empirical temperature such that the ratio of heat emitted to heat absorbed in a Carnot cycle shall equal the ratio of the lower operating temperature to the upper. KELVIN'S absolute temperature of 1848 satisfies the first requirement; so does any differentiable function of that temperature having a positive derivative; one such function defines KELVIN's absolute temperature of 1854. KELVIN'S second requirement is satisfied by his first absolute temperature both in the Caloric Theory and in CLAUSIUS' thermodynamics. KELVIN's third requirement is satisfied in CLAUSIUS' thermodynamics by his absolute temperature of 1854. When rendered concrete by reference to the classical axioms (2) and (3), CLAUSIUS' absolute temperature of 1854 reduces to KELVIN'S of the same year. The historical details will be found in my forthcoming Tragicomical History. Here by use of the apparatus just now assembled I will prove all of the foregoing statements that refer to KELVIN'S work and compare the results appropriate to different possibilities. KELVIN's first scale of "absolute temperature" is defined as follows:

-- S # dO.

(27)

Since dr/dO=It, the requirement (21) is satisfied if and only if # > 0 for all 0 in its domain.

(28)

Henceforth we shall assume provisionally that (28) holds, deferring until the next section the analysis of just what that means and why assumptions sufficient for it should be laid down. Since/~ is continuous, our provisional assumption (28) makes r an empirical-temperature fimction. Moreover, from (26) we see that we can adjust the constants of integration in the definitions of ~0, and z0 so as to insure that

ro.(O*)=zo(O).

(29)

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C. TRUESDELL

Thus r is an absolute temperature in the sense that its value is independent of the empirical-ten+perature scale used to obtain it. The physical dimensions of r are work + heat. The zero of r is arbitrary. Any continuously differentiable function of z with positive derivative satisfies all the same requirements. Thus there are infinitely many absolute temperatures in "KELVIN'S first sense. ~ is one of them. KELVIN's second scale of +'absolute temperature" T is defined as follows: T--- e ~/'r.

(30)

J being a constant having the dimensions of work + heat. Values of T are dimensionless and positive; T is absolute in KELVIN'S first sense. The value T = 1 corresponds to the arbitrary 0 of ~. All our considerations so far are restricted to one body and to an interval of empirical temperatures in ..~, for that body on some one empirical-temperature scale. The interval may be small, but it need not be. Without knowing something about/~, we cannot determine the range of ~. If/~ is of the right kind, the range of ~ is the whole real line. Then T = 0 corresponds to r = - oe, and T = ~c corresponds to z = oc. Thus the zero of T has a special status. Whether or not r as defined by (27) can have the value - o c , depends upon the nature of/~. It is easy to give examples of functions # that make z have a finite lower bound. Then T never attains the value 0. The pioneers regarded # as a function to be determined by experiment. KELVIN seems to have taken for granted that the range of T would come out to be ]0, 0¢[. Since z and Tare empirical-temperature scales, we may express the function ~ in terms of them. Calling the results/4 and #r, we see from (26) that dO dr

J

(31)

Thus the General Carnot-Ctapeyron Theorem (7) assumes the following forms when z and T are used: c~p= _ OPT A v ' ,==--, JAv 'r=lOT" (32) oz Of course p,(V,z)-- w(V, 0(z)),

pr(V, r)--- w(V,O(T)).

Use of an absolute temperature enables us to state the Carnot-Clapeyron Theorem without reference to Carnot's function I~. All the foregoing results follow from CARNOT'S work alone. Use of the results in Concepts and Logic enables us to go further. First we note that go. = go,

h o, = h o.

(33)

If g' never vanishes, g is an empirical-temperature function. In general h is not. Because of (15) and (31), h, =g;,

d hr=- rg' r.

(34)

Absolute Temperatures from Carnot's Axiom Hence (10) becomes

C - (if) _ h,(z-) _ h r ( T - ) . C+(¢g) h~(z+) hr(T+) '

(11) becomes

,r+

L(~)

(35)

T+

S h~(x) dx -

C+(~)

371

J S [hr(x)/'x] dx --

T -

hr(T +)

h~(r +)

;

(36)

and (13) becomes (37)

Thus use of an absolute temperature enables us to dispense altogether with the function g. In the Caloric Theory of heat we may always take h as 1. Then (34)1 shows that g, = z + const., while (36) reduces to L(~) r ÷ C+(C~) --z + - z- = J log T - .

(38)

The former of these expressions shows that by introducing z KELVIN obtained, for the Caloric Theory, a temperature "absolute" in his second sense as well as his first. Also (37) becomes OKv,~ = 8Av,~ O K v , T = OAv, r (39)

~V

8z '

8V

~T '

conditions necessary and sufficient that a heat function shall exist at least locally. CLAUSIUS laid down as his fundamental axiom the requirement that in all cycles c~ L( ff) = J C(Cg), (40) in which J is a universal positive constant having the dimensions of heat + work, while C(ff) is the net gain of heat in c~. Corollary 10.3 in § 10 of Concepts and Logic, restated according to the directions on p. 98, makes CLAUSIUS' axiom equivalent, for temperatures that correspond to points of N,, to

g = J h + const.

(41)

Hence if g is an empirical-temperature function, so is h. We recall that ifg' does not vanish, g is an empirical-temperature function in all theories consistent with CARNOT's General Axiom, but h generally is not. On the other hand, h is always a positive function. By making h become an empiricaltemperature function, CLAUSIUS'thermodynamics provides an empirical temperature which is always positive. In the definition (30) of Tthe constant called J was arbitrary. If we choose it as being the J of CLAUSIUS' axiom (40), from (34) we see that h =KT,

(42)

K being an arbitrary positive constant. Then (35) becomes C - (~) _ e- ('+ -~- )/a - T C+(,~) -

- T+.

(43)

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C. TRUESDELL

KELVIN regarded the second of these statements as expressing a desirable property of absolute temperature: for CLAUSIUS' theory T is "absolute" in KELVIN's third sense of the term, and for that reason KELVIN adopted it in 1854. in CLAUSIUS" theory (36) reduces to L(~) -(r + -r-) TJ C + fig) = 1 - exp J = 1 - T---T. (44) The former evaluation of efficiency is due to KELVIN. It shows that even in CLAUSIUS'theory r is a temperature "absolute" in KELVIN'ssecond sense as well as his first. The latter evaluation in (44) is due to RANKINE and CLAUSIUS. more or less. Details may be found in my Tragicomical History. Finally, in CLAUSIUS'theory (13) becomes

cKv,~=cAv.~ ?,V ~r

Av.~ J '

c K v . r = T c_~_[Av,r ~ ~V gT \ T ] "

(45)

In (32) 2 and (45), the reader will recognize the basic constitutive restrictions of classical thermodynamics. We have obtained them here only for the functions w, Av, Kv, and/~ belonging to some one body, and only for an interval of empirical temperature which may be very small. Thus, so far, the range of T m a y be small, and we might obtain different scales Tfor different bodies. We proceed now to remove the limitation and the variety. At the s a m e time we return to the generality of CARNOT'S theory.

7. Consequences of Part II of Carnot's General Axiom CARNOT asserted also that the motive power of a Carnot cycle was independent of the body that underwent it. This is Part II of his General Axiom. Axiom IV of Concepts and Logic expresses it in one way. For our purposes here a somewhat weaker axiom will do: Carnot'sfunction l~ is a universal function in the sense that whatever be the constitutive functions rv and A v used to determine it through (7), the same function p results. Hence the same absolute temperature r is obtained through (27). The value of KELVIN's absolute temperature r is independent not

only of the empirical-temperature scale used but also of the body employed to determine #. In other words, z maps hotnesses onto numbers with no reference to any particular body. Our considerations are still local. There are many ways to extend them to the whole hotness manifold ~ or as large a part as may seem desirable. I will choose one. In particular, let ~ and ~ , be bounded segments of W with a non-empty intersection ~ 2 . Let empirical-temperature functions 01 and 0: be defined on ~1 and )f':, respectively. Then on o'¢g~2we may interconvert 01 and 0: by a rule like (19), restricted by (21). Let bodies B 1 and B z have normal sets whose sets of empirical temperatures according to 01 and 02, respectively, include the ranges of those functions. The corresponding functions p are tL1 and pa, say. On )f12 we may express P2 as a function of 0~. Since/~ for a given scale is a universal function, IL_,so expressed must be the same function as It ~. By choice of the constant of integration in

Absolute Temperatures from Carnot's Axiom

373

(27), we get on ~f~ 2 the same absolute-temperaturefimction r for both bodies. Now considering only Bz, still on J{~,, we may use 02 instead of 0~ and because r is an absolute-temperature scale again calculate r through (27). In terms of 0, the definition of z still makes sense on all the image of ~ , . We have proved the following lemma. On bounded segments ~1 and ~2 having a non-empty intersection let there be empirical-temperature fimction 0 t and Oz; let the ranges of O1 and 0 e be intervals included in the temperature sets of the normal sets of bodies B ~and B 2 . Then the absohtte temperature-function z, unique to within an arbitrary constant, exists on The lemma shows that if a segment of H is the union of a finite number of bounded segments on each of which # exists and is positive for some body, the absolute-temperature function r exists on the whole segment and provides a coordinate system upon it. The reader familiar with the theory of differentiable manifolds will see that it is unnecessary to restrict attention to a segment which is the union of a finite number of bounded segments of H. I have done so because I desire to present an argument using only elementary ideas; I have no wish to make KELVIN's conceptual problem seem trivial by invoking a modern general theorem. Moreover, because experiments by their nature can determine only finitely many numbers, there is no physical gain in weakening the assumption, and if the part of ) f accessible to us through use of empirical-temperature scales could not be covered by a finite number of them, no experiment could detect that fact. Perhaps Y?' itself could be covered similarly, but no experiment could establish that.

8. Thermometric Axiom It remains to consider the condition (28), which all the foregoing constructions have presumed to hold. In order to establish (28), we need a further axiom. Mr. SERRIN, working upon a different framework of ideas, has emphasized the importance of such an axiom, which he calls thermometric. Here I adopt a special case of his axiom of 1975. phrased a little differently: Thermometric Axiom. JY' contains a segment -Hth which is the union of a.finite number oJ bounded segments, on each oJ which an empirical-temperature scale is defined, l f Oo is an empirical temperature of a hotness in ~th, it lies in the temperature interval of the constitutive domain ~ of some body such that at one point on the isotherm 0 = 0 o K v ~ K v. (46)

Let us call P the point which the Thermometric Axiom posits. Then because of (9) we know that Kp > K v at P. (47) By (4), then, Ut27 Av-~->O

at P.

(48)

Hence A v # 0 . Hence P belongs to the normal set of the body that the

374

C. TRUESDELL

Thermometric Axiom posits. Hence (7) holds at P. Because/L is a universal function, it is now determined once and for all at 0o; because of(48)./z(0o) > 0. But according to the Thermometric Axiom, for each h in ~ h there is a 00 of this kind. By appeal to (29) we establish the following Theorem. z as defined by (27) is an empirical-temperature scale not only locally but

also ol7 all of .)f~h ; for each h the value r(O(h)) does not depend upon the empiricaltemperature scale 0 used ro calculate it. Thus r is an absohae-temperarure scale on all of'.Y~,. I have chosen (46) as the critical condition on which to base the T h e r m o m e t r i c Axiom because the inequality (47) reflects c o m m o n experimental knowledge. According to tables of physical quantities, nearly all real materials satisfy (47) almost always. There are some exceptions; indeed, we can see directly from (4)3 that Kp = K v if and only if A N ~w/~0 = 0. Some few materials seem to obey very closely an equation of state of the form p = w ( p ) , and for them K p = K v always. A few others, like water, exhibit curves on which ~ v / ? 0 = 0 ; Figure 1 shows one point on such a curve. Because/~ > 0 in virtue of the Thermometric Axiom~ we conclude that Av=O if and only if ~ w / ~ 0 = 0 , so the curves just mentioned could be defined alternatively as curves on which A v = 0. Counterexamples show that without some assumption such as the Thermometric A x i o m it would be possible for kt to vanish for certain particular temperatures. If p(00)= 0, (7) shows that ~ v / ? 0 = O for all V and for all bodies on the isotherm 0 = 0 0 . Such a statement would abundantly contradict experience. This fact provides further support for the T h e r m o m e t r i c A x i o m 13 Intentionally I have left ,~th unspecified. Mathematical authors usually wish to have a single scale of temperature on all of aft. A n o b v i o u s modification of the thermometric axiom can adjust the foregoing theorem to that requirement. Then ~3 The function of the thermometric axiom is to make sure that /~(0o) exists and #(0o) > 0 for all 0o in range 0. Assumptions sufficient to this end have been made, expressly or tacitly, since the beginnings of thermodynamics. CARNOT restricted his attention to ideal gases, for which of course ~v/gO-p/O > 0. He stated expressly that Av>O. Since he also derived the corresponding special case of the General Carnot-Clapeyron Theorem (7), it was unnecessary for him to state that ~t> 0. His successors extended the scope of the theory, but as they agreed that # was a universal function, the same for all bodies, and as they used freely the concept of ideal gases, they had no need to determine the sign of # afresh, for CARNOT had already done so. They took for granted that ~ > 0, as is shown by their dividing by it whenever convenient. The first time that this argument failed to remain valid was in KELVIN's second theory of absolute temperature. 1854. He could no longer have recourse to evaluating # by use of an ideal gas. and his analysis of the behavior of water in 1853 had shown him that for some substances in some conditions Av 0, but his work in these years was sketchy, We cannot be sure just what he assumed or how carefully he thought out the details. He was the first person to publish (4) 3 in full generality, and certainly neither he nor any other early student would have hesitated to assume that Kp>Kv if ~ra/~0#0, for that was precisely what all available experimental data indicated. It is for this very reason that I have chosen to state a Thermometric Axiom in terms of Kp and K v. Of course an axiom must go beyond experimental data, must extrapolate from the known into the unknown, for otherwise it would be useless in prediction. Later authors have abandoned CARNOT'S approach. Not having the General CarnotClapeyron Theorem (7) on which to base their constructions, they have had to use more

Absolute Temperatures from Carnot's Axiom

375

each hotness lies in the domain o f s o m e empirical-temperature function. While I see no objection to such an assumption, it seems both daring and unnecessary. We might be content to think of .3f~h as the set of all hotnesses so far accessible to experiment. That set may well grow as time goes on.

9. The Physical Dimension of Absolute Temperature A value of KELVIN'S second absolute-temperature scale is a dimensionless number. We are accustomed to temperatures that bear a dimension called "the dimension of temperature". L o o k i n g back at an empirical-temperature scale 0, we m a y if we like think of it as bearing the physical dimension which we m a y call "the dimension of the scale 0". Physical dimensions such as those of mass and length and time are assigned once and for all, and only the mass, length, and time-interval to which the value 1 is assigned may be chosen at will. All scales of mass are proportional. Such is not true of empirical-temperature scales. We may indeed divide them into equivalence classes, all members of any one of which are proportional. Then a physical unit appertains to a given class but has no meaning with respect to any other class. Thus there are infinitely m a n y distinct physical dimensions of empirical temperature. KELVIN'S absolute temperatures ~ and T are conceived in such a way as to correct this unsatisfactory state of affairs. They annul the differences between the physical dimensions of all possible empirical scales. The values of any one of the scales T are dimensionless numbers which do not depend upon the choice of the empirical-temperature scale 0 used to obtain ,u and then to calculate T and T. If we can prove all scales T that correspond to a given J to be proportional, we can think of these pure numbers as coordinates on one and the same ldimensional vector space, the space of absolute-temperature vectors. It is just the same thing to say that absolute temperature has its own physical dimension, the dimension of absolute temperature. Only the choice of unit of absolute temperature distinguishes one absolute scale from another. elaborate reasoning. They also have recourse freely to a "First Law", which I wish expressly to avoid. The earliest explicit thermometric axiom I have found is Postulate V (thermometers) in the paper of J.B. BOYLING "An axiomatic approach to classical thermodynamics", Proceedings of the Royal Society (London) A 329 (1972), 35-70. That postulate reads in part: "There exists a class of simple systems called thermometers with the following' properties:... (e) For every thermometer M... the restriction of [the heat form] ~, to an arbitrary isotherm of M is everywhere non-zero (on that isotherm)." In my notation the axiom j.ust quoted asserts that on each sufficiently short isothermal segment Av*O for some body. To Mr. SERRIN we owe a deeper understanding of the role of a Thermometric Axiom; most of what I know about the matter I have learnt from him. In his notes of 1975 he assumes that for each h there is one body such that AvOw/~O#O locally on some empiricaltemperature scale covering h. In the framework of CARNOT's ideas, his axiom and the one I propose here are equivalent. In his lecture of 1977 he assumes only that for each h there is one body such that A v +-0 locally. To within technical details this axiom seems to be the same as part of BOYLING's.

376

C. TRUESDELL

Coming to the details, we first render explicit the role of the constant of integration in (27) and so express (30) in the form 0 T-Aexp (-J1 JooPO(X)dx ),

A =_exp/79_> 0,

(49)

ro being the value assigned to r when 0=00. If ~o=0, then A = I , and the absolute temperature 1 is assigned to the arbitrary empirical temperature 0 o. If. on the other hand, we make the choice of 0o and ro depend upon some physical phenomenon, we may obtain a value other than 1 for A. By varying the positive number A we obtain all possible absolute-temperature scales T corresponding to a given constant J. It is to the equivalence class so obtained that we assign the physical dimension of absolute temperature. Choice of A defines the unit of a particular scale of absolute temperature Z International convention now takes 0o as an empirical temperature of the triple point of water and assigns the value 273.16K to A. KELVIN himself preferred to assign a particular difference of temperatures rather than particular temperatures. If T1 and T2 are the values of T that correspond to the empirical temperatures 01 and 02, then (49) shows us that

T = (T2_ T1) T/ T1 Tz/ TI - 1'

(50)

exp (~ i l~o(x)dx) =(T2-T l) exp

(,\a Os= o, ~°(x) dx]

1

When 01 and 02 are given, to assign A is one and the same thing as to assign 7"_, - T 1. For example, we may choose for the empirical-temperature scale 0 the Celsius scale: that which is provided by the air thermometer with 0 ° and 100 °, respectively, assigned to the hotnesses at which water at standard atmospheric pressure freezes and boils. If we wish an absolute scale which preserves the difference of 100° in the temperatures assigned to these hotnesses, we simply put T2 - Ta = 100 in (50) and so obtain

lOOexp (l i po(x)dx ) T=

lOO

•

(51)

This formula converts degrees Celsius on the air thermometer to degrees absolute with the same difference of temperatures, namely I00 °, between the boiling point and the freezing point of water. It was suggested by HELMHOLTZ and JOULE that 11o varied inversely as the temperature above absolute cold. That is, on the Celsius scale J ~0-0+0a,

(52)

Absolute Temperatures from Carnot's Axiom

377

- 0 ~ being the Celsius temperature of absolute cold. If this formula were rigorously correct, (51) would reduce to

r=O+O~.

(53)

The experiments of JOULE & KELVIN on the porous plug are interpreted as showing that (52) is very nearly but not exactly true. Therefore the absolute scale determined by (51) is very nearly (53), which is the Celsius scale with its zero shifted to absolute cold. This conclusion requires experimental determination not only of/z 0 but also of 0a. It has long been known that 0o is very nearly -273°C. The foregoing treatment refers only to temperatures, not to hotnesses. We may think of each possible choice of the scale T as corresponding to a particular hotness h o selected by some rule. If we write Tho for the scale that assigns to h o the value 1, Tho(ho)= 1, (54) then it is easy to demonstrate the rule of transformation under change of unit hotness:

L: (h) = L,. (h~) L1 (h).

(55)

Any two absolute scales T that correspond to one choice of the positive constant J are proportional to each other, and if we select two particular hotnesses h: and h 1, the constant Th2(hl) plays the role of a unit of temperature:

Th (.~, , "Fa2( h ) - Th2(h') ,v'9 = ~ .

(56)

The results presented at the beginning of this appendix can be derived from (55) and (56), provided/~ be of the right kind. I think this approach is clearer as well as neater. For example, it shows that if (52) held strictly, then the absolute scale defined by the hotness hI at which water freezes would be given as follows in terms of the Celsius scale 0: Th, =

O(h) + 0a 0a

(57)

10. Conclusion The concept of absolute temperature has always been inherent in CARNOT'S general theory. To construct the absolute temperatures introduced by KELVIN, we require neither the "First Law" nor the "Second Law" of thermodynamics.

11. Appendix. Axioms of Classical Thermodynamics In Concepts and Logic, Appendix to Chapter 15, Mr. BHARATHA & i have pointed out one way to modify our presentation so as to deliver classical

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C. TRUESDELL

thermodynamics quickly: Adopt CLAUSIUS' "'First Law" (40). In §6 of this note I have shown that the corresponding special choice of J in (30) then gives to KELVIN's second absolute temperature T exactly the property he desired for it. namely (43)2. a property which it most certainly does n o t have for other choices of J. However. I think the basic program of Concepts and Logic is more elegant: Prove both the "First Law" and the "Second Law" as theorems based upon axioms which render formal CARNOT'S own assumptions, of course not restricted by his subsidiary and undesirable adherence to the Caloric Theory of heat. In Concepts and Logic we attained this goal by our Axiom l/.• There is a body of ideal gas such that

the whole I7-0 quadrant is a thermodynantic part, and both specific heats are constant. There we used the concept of ideal-gas temperature. While in this note I have excised all use of ideal gases in the sense of the pioneers, if we start from KELVIN'S second absolute temperature T as defined in terms of an arbitrary positive constant J we can by modifying a little the axiomatic structure of Concepts and Logic uphold its program. By using absolute temperatures only we can again prove both the First Law and the Second Law as theorems. I now present the details. We replace the primitive concept of temperature by the primitive concept of hotness and lay down Axiom O. The set of all hotnesses is a diffeomorph of a real interval, equipped with an intrinsic ordering. Definition O. An empiricaI-temperature fun'ction is an order-preserving chart on a bounded segment of the hotness manifold. Axioms I - I I I of Concepts and Logic are then to be understood as referring to some one empirical-temperature scale and hence to all such scales on one b o u n d e d segment. Axiom IV should be modified slightly: Any two fluid bodies which may undergo

Carnot cycles with the same operating temperatures do in those cycles the same amount of work per unit of heat absorbed. Next we lay down the 77wrmometric Axiom stated above in § 8. On that basis we may construct in terms of any positive constant J the absolute-temperature scale T on all of YC~,h-We may then convert all the local restrictions shown to be valid for some one empirical-temperature scale into statements valid for all T. Most of these have been listed above in § 6. Definition. A body of ideal gas is a body such that

for all T.

pV=RT,

R=const. >0,

(58)

The temperature measured by the volumes of such a body at constant pressure provides "a natural measure of temperature . . . . a real temperature . . . . a sort of Platonic idea of temperature," of which the temperature read on an air thermometer is "only an incomplete. inexact expression". The results of JOULE & KELVIN'sexperiment with the porous plug show that the inexactness of the air thermometer is negligible for most purposes. Thus we are not astonished to learn that the idea "'absolute cold" or "'absolute zero" is 150 years older than the idea "absolute temperature". The relations (31)a, (32)~, (34), and (35) were all first obtained by use of the "Platonic idea" of an ideal gas.

Absolute Temperatures from Carnot's Axiom

379

Appealing to (32) and (4)3, we see that the specific heats of an ideal gas referred to the scale T satisfy the relation d(Kp, r - Kv, r) = R,

(59)

Y being the positive constant used to define T. This relation, like (58) itself, is familiar because of its counterpart when ideal-gas temperature is used, but the dimensions of the quantities occurring in the two are not the same. As the absolute temperature T is dimensionless, R has the dimensions of work, and Kv, r has the dimensions of heat. For each choice of the arbitrary constant J we obtain an absolute temperature T through the definition (30), so the definition (58) of" ideal gas" depends upon J. A substance that is an ideal gas for one choice of J is not an ideal gas for another. For any choice of J we see that (59) holds. Thus with any choice of 9" the two specifc heats are distinct; all of the V-T quadrant that corresponds to ~ h is normal; and both specific heats are constant if and only if one of them is. The rule of transformation (25), when applied to (30) for different choices Jl and 9'2 of J shows that

9'2 T~ K

Kv, ra=dl T2 V.T,.

(60)

The example of the Caloric Theory of heat shows that there need be no choice of 9' such as to make Kp, T and Kv, r constant (Concepts and Logic, Historical Scholion after Definition 13 in Chapter 6). We are now ready to lay down a final axiom similar in form to Axiom V of Concepts and Logic. but conceptually quite different: Axiom V. There is a J such that for one of the corresponding bodies of ideal gas Kv, r = const. To such a gas, which Axiom V makes compatible with the general theory, we may apply the reasoning that leads to Theorem 15 in Chapter 15 of Concepts and Logic. Hence the functions gr and h r occurring in (34) 2 have the forms gr = 9"hr + const.,

(61)

h r =MT, and (37) 2 reduces to (45):, which is the local "Second Law" of classical thermodynamics. The other classical constitutive restriction, namely (32)2, we have already for all choices of 9", so it holds for the choice provided by Axiom V. Thus the entire formal structure of classical thermodynamics results. In particular, one demonstrated consequence of the axioms here is the "First Law" in CLAUSIUS' form (40), and the constant J that Axiom V provides turns out to be the uniform and universal mechanical equivalent of a unit of heat in cyclic processes. The Thermometric Axiom of§ 8 has enabled us to construct KELVIN'S second absolute-temperature scale upon the general framework of CARNOT'S ideas. Using it, we have shown that those ideas, duly modified by Axiom V, suffice to construct classical thermodynamics without use of ideal-gas temperature, the "'First Law",

or the "Second Law".

380

C, TRUESDELL

Acknowledgment. I am grateful not only to Mr. SERRIN but also to Messrs. C.-S. ~VIAN and M. PITTERIfor long and patient discussion of the subject and for scrupulous criticism of this note. The work it presents was partially supported by grants from the U.S. National Science Foundation's Programs in the History and Philosophy of Science, in Applied Mathematics, and in Solid Mechanics. The Johns Hopkins University Baltimore, Maryland

(Received December I5, 1978)

Note added in proofi The demonstrations of the theorems in Concepts and Logic on which this paper draws do not require that OKv/~.O exist; (25),_ shows that its existence is not invariant under all changes of empirical-temperature scale. Accordingly, in this paper I have supposed of K v only that it be continuous and that ~Kv/gV exist and be continuous.