On the History and Analytical Expression of the First and Second Laws

ON T H E HISTORY AND ANALYTICAL EXPRESSION OF T H E FIRST AND SECOND LAWS OF THERMODYNAMICS, AND T H E ROLE OF T H E DIFFERENTIALS, dW AND dQ BY GEORGE TUNELL

The fundamental physical facts on which the science of thermodynamics rests were established long ago. According to Gibbs thermodynamics as a science was brought into existence by Clausius. Gibbs’ has summarized this part of the history of the subject in the following notable paragraphs: “The fundamental questions concerning the relation of heat to mechanical effect, which had been raised by Rumford, Carnot, and others, to meet with little response, were now2 everywhere pressing to the front. “‘For more than twelve years,’ said Regnault in 1853, ‘I have been engaged in collecting the materials for the solution of this question:-Given a certain quantity of heat, what is, theoretically, the amount of mechanical effect which can be obtained by applying the heat to evaporation, or the expansion of elastic fluids, in the various circumstances which can be realised in practice?’ The twenty-first volume of the Memoirs of the Academy of Paris, describing the first part of the magnificent series of researches which the liberality of the French government enabled him to carry out for the solution of this question, was published in 1847. In the same year appeared Helmholtz’s celebrated memoir, ‘Ueber die Erhaltung der Kraft.’ For some years Joule had been making those experiments which were to associate his name with one of the fundamental laws of thermodynamics and one of the principal constants of nature. I n 1849 he made that determination of the mechanical equivalent of heat by the stirring of water which for nearly thirty years remained the unquestioned standard. In 1848 and 1849 Sir William Thomson was engaged in developing the consequences of Carnot’s theory of the motive power of heat, while Professor James Thomson in demonstrating the effect of pressure on the freezing point of water by a Carnot’s cycle, showed the flexibility and the fruitfulness of a mode of demonstration which was to become canonical in thermodynamics. Meantime Rankine was attacking the problem in his own way, with one of those marvellous creations of the imagination of which it is so difficult t o estimate the precise value. “The Collected Works of J. Willard Gibbs,” 2, part 2 , pp. 261, 2 6 2 (1928);or Proc. Am. Acad., New Series, 16, 458, 459 (1889). The histor of the first and second laws has been stated by Lord Kelvin in different words but w i d very similar conclusions. Lord Kelvin’s statement is given as an Appendix to this paper. 2 I n the eighteen-forties. (G.T.)

LAWS OF THERMODTNAMICS

I745

“Such was the state of the question when Clausius published his first memoir on thermodynamics: ‘Weber die bewegende Kraft der Warme, und die Gesetze, welche sich daraus fur die Warmelehre selbst ableiten lassen.’s “This memoir marks an epoch in the history of physics. If we say, in the words used by Maxwell some years ago, that thermodynamics is ‘a science with secure foundations, clear definitions, and distinct boundaries,’ and ask when those foundations were laid, those definitions fixed, and those boundaries traced, there can be but one answer. Certainly not before the publication of that memoir. The materials indeed existed for such a science, as Clausius showed by constructing it from such materials, substantially, as had for years been the common property of physicists. But truth and error were in a confusing state of mixture. Neither in France, nor in Germany, nor in Great Britain, can we find the answer to the question quoted from Regnault. The case was worse than this, for wrong answers were confidently urged by the highest authorities. That question was completely answered, on its theoretical side, in the memoir of Clausius, and the science of thermodynamics came into existence. And as Maxwell said in 1878, so it might have been said a t any time since the publication of that memoir, that the foundations of the science were secure, its definitions clear, and its boundaries distinct.” Unfortunately in the formulations of the fundamental principles given in the current text-books full use is not made of the symbols, forms, and theorems in the theory of functions of real variables best adapted to the problems a t issue. These symbols and forms are essentially those of Clausius, and it is the purpose of this paper to point out this close relationship. Clausius used several mathematical theorems the proofs of which he did not introduce into his texts. No proofs of these theorems will be given in this paper but the theorems will be stated and the reader will be referred to clear and rigorous proofs in the literature of the theory of functions of real variables. The first law of thermodynamics was stated by Clausius4 as follows: “Die ganze mechanische Warmetheorie beruht auf zwei Hauptsatzen. . . . Um den ersten Satz analytisch auseudrucken, denken wir uns irgend einen Korper, welcher seinen Zustand andert, und betrachten die Warmemenge, welche ihm wahrend dieser Zustandsanderung mitgetheilt werden muss. Bezeichnen wir diese Warmemenge mit Q, wobei eine vom Korper abgegebene Warmemenge als aufgenommene negative Warmemenge gerechnet werden soll, SO gilt fur das einer unendlich kleinen Zustandsanderung entsprechende Element dQ der aufgenommenen Warme folgende Gleichung : (I.)

dQ = dU

+ AdW.

Read in the Berlin Academv February 18, 1850, and published in the March and April numbers of Poggendorff’s “Armalen.’’ “Abhandlungen uber die mechanische Warmetheorie,” Zweite Abtheilung, Abhandlung IX (1867). The quotations in German may be skipped without disruption of the consecutive development of the subject matter although it is believed that the statements of Clausius are of no inconsiderable interest to the student of thermodynamics.

‘

1746

GEORGE TUlVELL

Hierin bedeutet U die Grosse, welche ich zuerst in meiner Abhandlung von 1850 in die Warmelehre eingefuhrt und als die Summe der hinzugekommenen freien Warme und der zu innerer Arbeit verbrauchten Warme definirt habe. W. Thomson hat fur diese Grosse spater den Namen Energie des Korpers vorgeschlagen, welcher Benennungsweise ich mich, als einer sehr zweckmassig gewahlten, angeschlossen habe, wobei ich aber doch glaube, dass man sich vorbehalten kann, in solchen Fallen, wo die beiden in U enthaltenen Bestandtheile einzeln angedeutet werden mussen, such den Ausdruck Warmeund Werkinhalt zu gebrauchen, welcher meine ursprungliche Definition in etwas vereinfachter Form wiedergibt. W bedeutet die wahrend der Zustandsanderung des Korpers gethane aussere A r h i t , und A das Warmeaquivalent fur die Einheit der Arbeit oder kurzer das calorische Aequivalent der Arbeit. Hiernach ist AW die nach Warmemaasse gemessene aussere Arbeit oder, gemass einer kurzlich von mir vorgeschlagenen bequemeren Benennungsweise, das aussere W e r k . “Wenn man der Kurze wegen das aussere Werk durch einen einfachen Buchstaben bezeichnet, indem man setzt: AW = w,

so kann man die vorige Gleichung folgendermaassen schreiben :

+

dQ = dU dw.” (Ia.) Clausius continued farther on: “[Eine Grosse], welche sich auf den ersten Hauptsatz bezieht, ist die schon im Anfange dieser Abhandlung besprochene, in Gleichung (Ia.) enthaltene Grosse U, welche den Warme- und Werkinhalt oder die Energie des Korpers darstellt. Zur Bestimmung dieser Grosse ist die Gleichung (Ia.) anzuwenden, welche wir so schreiben konnen: (57)

dU = dQ - dw,

oder, wenn mir sie uns integrirt denken: (58)

U

=

U,

+Q -

W.

Hierin stellt U, den Werth der Energie fur einen willkurlich gewahlten Anfangszustand des Korpers dar, und Q und w bedeuten die Warmemenge, welche man dem Korper mittheilen muss, und das aussere Werk, welches gethan wird, wahrend der Korper auf irgend eine Weise aus jenem Anfangszustande in den gegenwartigen Zustand iibergeht. . . .” From the preceding paragraphs Clausius obtained an important conclusion, his equation ( j ) , by the following reasoning: “Wenn die Grossen x und y den Zustand des Korpers bestimmen, SO muss die Grosse U, die Energie des Korpers, welche nur von dem augenblicklich stattfindenden Zustande des Korpers abhangt, sich durch eine Function dieser beiden Veranderlichen darstellen lassen. “Anders verhalt es sich mit den Grossen w and Q. Die Differentialcoefficienten dieser Grossen, welche wir folgendermaassen bezeichnen wollen :

LAWS OF THERMODYNAMICS

I747

sind bestimmte Functionen von x und y. Wenn namlich festgesetzt wird, dass die Veranderliche x in x dx ubergehen soll, wiihrend y unverandert bleibt, und dass diese Zustandsanderung des Korpers in umkehrbarerweise geschehen soll, so handelt es sich um einen vollkommen bestimmten Vorgang, und es muss daher auch das dabei gethane aussere Werk ein bestimmtes sein, woraus weiter folgt, dass der Bruch dw/dx ebenfalls einen bestimmten Werth haben muss. Ebenso verhalt es sich, wenn festgesetzt wird, dass y in y dy ubergehen 9011, wahrend x constant bleibt. Wenn hiernach die Differentialcoefficienten des ausseren Werkes w bestimmte Functionen von x und y sind, so muss zufolge der Gleichung (Ia.) auch von den Differentialcoefficienten der vom Korper aufgenommenen Warme Q dasselbe gelten, dass auch sie bestimmte Functionen von x und y sind. “Bilden wir nun aber fur dw und dQ ihre Ausdriicke in dx und dy, indem wir unter Vernachlassigung der Gleider, welche in Bezug auf dx und dy von hoherer Ordung sind, schreiben: dw = mdx ndy (3 1

+

+

+ +

dQ = Mdx Ndy, so erhalten wir dadurch zwei vollstiindige Differentialgleichungen, welche sich nicht integriren lassen, so lange die Veranderlichen x und y von einander unabhangig sind, indem die Grossen m, n und AI, N der Bedingungsgleichung der Integrabilitat, namlich: dm - dn dM dN resp. - = -, dy dx dy dx nicht genugen. Die Grossen w und Q gehoren also zu denjenigen, . . . . deren Eigenthumlichkeit darin besteht, dass zwar ihre Differentialcoefficienten bestimmte Functionen der beiden unabhangigen Veranderlichen sind, dass sie selbst aber nicht durch solche Functionen dargestellt werden kcinnen sondern sich erst d a n n bestimmen lassen, w e n n noch eine weitere Beziehung zwischen den Veranderlichen gegeben und dadurch der W e g der Veranderungen aorgeschrieben ist.”j (Italics in the last sentence by the author of this paper.) “Kehren wir nun zur Gleichung (Ia.) zuruck und setzen darin fur dw und dQ die Ausdrucke (3) und (4) und zerlegen ebenso dU in seine beiden auf dx und dy bezuglichen Theile, so lautet die Gleichung: (4)

Mdx

+ Ndy = (E -+ m

1

dx

+ :(-+

n

) dy.

5 The importance of this sentence makes it worth while to offer the following translation: The quantities w and Q belong to those quantities. . . . the peculiarity of which is that, while their differential coefficients are definite functions of the two independent variables (x and y), the quantities themselves cannot be represented by such functions and can only then he determined when a further relation is given between the variables and the path of the changes is thereby prescribed.

1748

GEORGE TUNELL

D a diese Gleichung fur alle beliebigen Werthe von dx und dy gultig win muss, so zerfiillt sie in folgende zwei: dU M=-+m dx dU N=-+n. dY Differentiiren wir die erste dieser Gleichungen nach y und die zweite nach x, so erhalten wir dM - d2U +% dy dxdy dy dN dx

-

d2U dydx

dn +-. dx

Nun ist auf U der fur jede Function von zwei unabhangigen Veriinderlichen geltende Satz anzuwenden, dass, wenn man sie nach den beiden Veranderlichen differentiirt, die Ordnung der Differentiationen gleichgtiltig ist, so dass man setaen kann: d’U - d*U ---* dxdy dydx Wenn man unter Beriicksichtigung dieser letzten Gleichung die zweite der beiden vorigen Gleichungen von der ersten abzieht, so kommt:

_ - _ =d N- - - .dm dM dy

dx

dy

dn dx

”

The attempt will next be made to prove that the essentials of Clausius’s analytical expressions of the first law are in accord with the theory of functions of real variables; Planck’s objection to Clausius’s use of dQ and its designation as a differential is based on an argument that will be proved erroneous. The simplest system discussed in thermodynamics consists of a fixed quantity, for example one gram or one mol, of one gaseous phase containing only one component, throughout which the temperature and pressure are uniform. For such a system there exists a characteristic relation that may be expressed as follows: (1) *(P,V,t) = 0j6 where p denotes the pressure; v, the volume; and t , the temperature of the gas on the scale of the constant volume hydrogen thermometer, which has been adopted as standard by the International Bureau of Weights and Measures (at S(.vres, France). On the centigrade scale of this thermometer the temperature of melting ice is taken as oo and that of the vapor of distilled 6 The present discussion applies to systems such as steam, a satisfactor representation aa the perfect of which IS not iven by a simple, highly specialized equation of state gas law or van fer Waals’s equation.


I749

water boiling under normal atmospheric pressure as roo’, the hydrogen being under a pressure of one meter of mercury when the temperature is 0’. The characteristic equation of the gas is in general such that by means of it any one of the variables can be represented as a function of the other two. For reversible processes (continuous series of equilibrium states)’ the work done by the system (work of the path),*W, is defined by the equation

where W is measured in mechanical units. The heat received (heat of the path), Q, is defined by the equation

(3)

where c, and 1, denote some functions of t and p, and Q is measured in thermal units. These definitions correspond to the facts stated by Clausius that the work done and the heat received each depends on all of the intermediate states as well as the initial and final states; the integrals in ( 2 ) and (3) are line integrals that depend on the particular choice of the path of integration. As Clausius says, the quantities Wand Q can only be determined when an additional relation between the variables that determine the state of the system is given and the path of the changes is thereby fixed. The justification for the introduction of heat as a physical quantity in thermodynamics (as contrasted with the justification for the definition utilized in kinetic theory) is well stated by Prestonq as follows: “In order to account for the sensation experienced in presence of a hot body an active agent is postulated, and the name given to this agent is heat. A hot body is regarded as a source of heat just as a luminous body is regarded as a source of light. I n the same way, when two bodies at different temperatures are placed in contact, the temperature of the warmer falls while that of the other rises. To account for this we say that heat passes from one to the other, that the warmer loses heat and the colder gains it. I n this sense heat is regarded as something which may be added to or taken away from matter; 7 “On dit qu’une transformation est operee par voie reversible quand elle est constituke par une succession d’etats d’equilibre. Une transformation ne peut donc se produire reelement par voie rigoureusement reversible; elle est la limite de deux series de transformations realisables et s’effectuant en sen8 inverses.” H. Bouasse: “Cours de Thermodynamique,” D e u x i h e Qdition, Premiere partie, 70 (1913). “Suppose the body to change its state, the points associated with the 8 Gibbs says: states through which the body passes will form a line, which we may call the path of the body. The conception of a path must include the idea of direction, to express the order in which the body passes through the series of states. With every such change of state there is connected in general a certain amount of work done, W, and of heat received, H, which we may call the work and the heat of the path.” “Collected Works,” 1, 3 (1928). 8 “The Theory of Heat,” Third Edition, Edited by J. Rogenon Cotter, 19 (1919).

‘750

GEORGE TUNELL

something which can be communicated to matter, and which can be handed on from one piece of matter t o another. Heat thus possesses the rank of a quanttty, and we are led to seek how much heat a body gains or loses when its temperature changes. On the other hand, temperature is regarded rather as a quality which varies from one body to another, or from one part to another of the same body, when heat is being communicated to or abstracted from it, or which may vary . . . in consequence of actions taking place within the body itself, or performed on it from without. ‘‘It must , however, be distinctly remembered that what we directly observe is temperature and changes of temperature, and when the temperature of a body (free from other actions) rises we say it has received heat. The effect observed is the change of temperature, and the postulated cause is addition or subtraction of heat.” The establishment in thermodynamics of the quantitative character of heat is completed by means of calorimetric mixing experiments as explained by Mach;l0 in such experiments it is always found possible to write a “compensation equation” indicating that the heat received (or given up) by the one body in its change of state is equal to the heat given up (or received) by the second body in its simultaneous change of state. This experimental fact is the basis for the introduction of heat as a physical quantity in thermodynamics. Heat” is then analogous to work since work done by one body is always equal to that done on a second body. The double-ended character of a force of course necessitates the conclusion that work is a physical quantity; when the point of application (that is, the point at which the two bodies touch) of the force moves, the work done by the body exerting the force in one direction is equal in magnitude but opposite in sign to that done by the other body. Now a line integrali2can be evaluated as an ordinary integral; thus U

(tOJpO)

U O

E. Mach: “Die Principien der Warmelehre,” 4. Aufl., 182-194 (1923). The writer is indebted to Professor E. C. Kemble for a clear and co ent development of this topic in his lectures in Physics 6a (a course given in Harvard tollegej, and also for calling the writer’s attention to the exposition of Mach. In numerous books on thermodynamics (for example, Goodenough: “Principles of Thermodynamics,” Third Edition, 20 (1920)) it is r o d that the line integral representing the heat received by a system depends on the pa& This proof is based, either explicitly or tacitly, on the postulates of kinetic theory. The argument is that the internal energy depends only on the state of the body and that the work done depends on the path; therefore the heat received, which is equal to the increaae in internal energy less the work done, must depend on the y t h . At the present time it is unnecessary to base the principles of thermodynamics on t e postulates of kinetic theory and, for most purposes, it is probably undesirable. The experimental facts of calorimetry discussed by Mach suffice to establish the analytical characterization of heat as a line integral that depends on the path without reference to the postulates of kinetic theory, and this statement is one of the indispensable foundation blocks of the science of empirical thermodynamics. The developments in this paper are in accord with P. W. Bridgman’s illuminating discussion, in terms of experimental operations, of the significance of the quantities, temperature, heat, and energy, in thermodynamics. “TheLogic of Modern Physics,” I 17-131(,1927). lZ For the definition of the line integral see W-. F. Osgood: “Lehrbuch der Funktionentheorie,” 5 . Aufl., 1, 134-137 (7928). 10

LAWS OF THERMODYNAMIC8

‘75’

where u denotes the parameter that determines the particular series of states through which the system passes, and the path or curve is represented in parametric form by the equations:

t

(5) ( 6 )

= cp(u), p =

#(u).’S

The line integrals in the definitions of work and heat are functions of the upper limit of integration, u, and may be differentiated with respect to u.14 Thus

dQ = c pd- t+ l p - d P * du du da Multiplying each of these equations by du, one obtains the results,

dQ = c,dt

+ l,dp,

where u is the independent variable of the functions W and Q and thus du = Au is the independent (principal) infinitesimal (W and Q are not functions of the independent variables t and p and thus d t and dp are not independent infinitesimals in these equations). The conclusion follows that dQ is just as truly a differential as dv: dv is the differential of a function of two independent variables while dQ is the differential of a function of a single independent variable. The following statement of Planck concerning dQ has led to much confusion in the literature of thermodynamics, since it is seriously in error and since it has been restated in several text-books. PlancklS writes: If the path be represented by the equation P = f (t), instead of in the parametric form given in the text, then R and Q can be evaluated by the equations,

t Q =!c,(t,f(t))

dt

+ l,(t,f(t))*d t dt.

t.,

These two integrals obviously cannot be used to express the work and heat of a straightline path parallel to the p-axis (change of pressure at constant temperature) and thus these two integrals have a serious disadvantage in comparison with the integrals given in the text. The integrals in the text apply to straight-line paths in all directions as well as to curved paths. I4W. F. Osgood: “{hwnced Calculus,” 215, 216 (192j);E. Goursat: “A Course in Mathematical Analysis, translated by E. R. Hedrick, 1, 154 (1904). ‘5 “Vorlesungen uher Thermodynamik,” 7. Aufl., 5 j, 56 (1922).

GEORGE TUNELL

1752

“Nach Clausius’ Vorgang wird dieser Ausdruck gewohnlich, um seine unendliche Kleinheit anzudeuten, mit dQ bezeichnet. Dies hat jedoch nicht selten zu dem Missverstandnis Anlass gegeben, als ob die zugeleitete Warme das Differential einer bestimmten endlichen Grosse Q ware. Der hierdurch nahe gelegte Trugschluss sei durch folgende kleine Rechnung illustriert. Wahlt man T und V als unabhangige Variable, so ist

Folglich, da d T und dV voneinander unabhangig sind:

und daraus durch Differentiation der ersten Gleichung nach V, der zweiten nach T: Q --a*u - a*u a p -~-Z- -

aTav

aTav

aTav + aT -7

also ap/aT = 0,was sicher unrichtig ist.” I t has been proved in this paper that dQ is the differential of a function, Q, of the parameter, u, that determines the series of states through which the system passes, and that the rigorous definition of a differential is satisfied by dQ.16 The first part of Planck’s statement is therefore erroneous. Moreover in the equation dQ = c,dt 1,dp

+

dt and dp are not independent infinitesimals; likewise in the equation dQ

=

c,dt

+ 1,dv

dt and dv are not independent infinitesimals, and therefore it does not follow from the latter equation that

,yc - -+*. aiu

-atav

atav

at

16 The same definition of the differential is given in the following works: W. F. Osgood: “Introduction to the Calculus,” 91-93 (1922); “Lehrbuch der Funktionentheorie,” s. Aufl., 1, 236 (1928); E. Goursat: “A Course in Mathematical Analysis,” translated by E. R. Hedrick, 1, 19, 2 0 (1904); James Pierpont: “The Theory of Fynctions of Real Variables,” 1, 244 (1905); E. B. Wilson: “Advanced Calculus, 64 (1912); F. S. Woods: “Advanced Calculus,” 2 8 (1926); Cf. also the statement by E. L. Mickelson in his review of D. Humphrey’s “Advanced Mathematics for Students of Engineering and Physics” that: “alterations in two or three objectionable statements, of which the most jmportant is the statement that the differential of f(x) is the actual change when x is increased by dx, would contribute accuracy well within the student’s powers of appreciation.” American Mathematical Monthly, 38, 454 (1931).


I753

In the special cases of straight-line paths parallel to the coordinate axes the reductions of equations (7) and (8) are as follows. I. Let the path be a straight line in the t,p-plane parallel to the t-axis: (11) (12)

t =a, p = K , K, a constant greater than zero.

Then dW - dW du dt and

On account of the relation expressed by equation (14)~ c, is called the heat capacity at constant pressure. 11. Let the path be a straight line in the t,p-plane parallel t o the p-axis: (16)

(1.5)

p = a , t = K ’ , K’, a constant.

Then dW - dW da dp

(17)

and

On account of the relation expressed by equation (18), 1, is called the latent heat of change of pressure a t constant temperature. The derivatives dW dW dQ dQ -and dt ’ d p ’ d t ’ dp are thus total derivatives, as Professor Kemblel’ has rightly pointed out. Therefore one should not write for them

since this notation in mathematics has only one meaning, namely, the partial derivative of a function, W, of the independent variables, t and pl taken with respect to t holding p fast; W and Q are not functions of the independent variables t and p. Clausius’s analytical expression of the first law by means of his equation (j8) can be rendered more explicit for reversible processes in the system under consideration by the following equation :

which is also written in the more familiar but less explicit form: Lectures in Physics 6a.

GEORGE TUNELL

1754

U(t,p) - U(t,,p,) = p d Q - dW,

(20)

(t0,PO) the integrals in each of the equations (19) and (20) being extended over any path connecting the points, (to,po) and (t,p); Q denotes the heat received (heat of the path); W, the work done (work of the path); J = I/A, a constant, the mechanical equivalent of heat; and U, the internal energy per gram measured in mechanical units. The physical hypothesis (experimental fact) embodied in each of the equations, (19) and ( z o ) , may be stated in words as follows: The value of the integral in each of the equations, (19) and (20), is independent of the choice of the path of integration, that is, it depends only on the limits of integration, and the integral may therefore be used to define a function, the internal energy per gram, of the independent variables, t and p. Equation (20) may be brought readily into the form (19) by means of ( 2 ) and (3). From (19) it follows directly that

(g) P

(g)

=

Jcp-p(E)

=

J1,

-

p($).

Two clear and rigorous proofs of this theorem are given by Osgood.l* Substituting the values of (aU/at), and (~3U/ap)~ just obtained in the equation of the total differential of U(t,p),

one obtains

where t and p are the independent variables of the function U and thus dt = A t and dp = A p are the independent (principal) infinitesimals. A necessary and sufficient condition for

is (25)

[d( aP

JC,

- P(;)J]

=

p(

JlP - P ( E ) ) ]

at

P

W. F. Osgood: “Lehrbuch der Funktionentheorie,” 5 . Aufl., 1, 138-150 (1928).


I755

Two clear and rigorous proofs of this theorem are given by Osgood.19 Both of these proofs are invaluable in thermodynamics; the first has connections with geometry that render it easily remembered; the second admits immediate extension to systems with more than two independently variable properties and is carried through purely arithmetically (without presupposition of geometric axioms) although it is also interpreted geometrically for ease of comprehension and memory. Equation ( 2 5 ) becomes identical with Clausius’s equation ( 5 ) if his variable properties y and x that determine the state of the system be identified with t and p. As ClausiusZopointed out, his equation ( 5 ) forms an analytical expression of t,he first law for reversible changes in a system the state of which is determined by two independent variables, y and x. Thus equation ( 2 5 ) constitutes an analytical expression of the first law for reversible processes in the system considered in this paper.21 ‘8

W. F. Osgood: “Lehrbuch der Funktionentheorie,” 5 . Aufl., 1, 138-150(1928).

Op. cit., p. 9. Lord Kelvin wrote the analogous equation with t and v aa the independent variables as the analytical expression of the “first fundamental proposition” or first law of thermodynamics. His statement follows: “Observing that J is an absolute constant, we may put the result into the form 20 21

dp=Js-E.

dt (dt dv) This equation expreases, in a perfectly comprehensive manner, the application of the first fundamental roposition to the thermal and mechanical circumstances of any substance whatever, d e r uniform ressure in all directions, when subjected to any possible variations of temperature, vogme, and pressure.” Trans. Roy. SOC. Edinburgh, 20, 270 (18.51). In the notation of this essay Lord Kelvin’s equation would be written: , - , where 1” denotes the latent heat of change of volume a t constant temperature and cy, the heat capacity at constant volume. Li pmann has made a similar statement aa follows: “8xpresaion gbnbrale du principe de 1’6quivalence.-En g6nbra1, dU sera fonction de deux variables independantes x et y, de sorte que l’on pourra poser dU = Pdx Qdy et le principe de l’bquivalence sera exprim6 par la condition d P = 9.’’ dy dx In the preceding paragraph Lippmann denotes a function of x and y (not the heat received or given up) by Q. He continues farther on: “Applications du principe de 1’6quivalence.-I. Soit un kiloqramme d’un corps quelconque dont on fait varier le volume et la temperature en lui faisant parcourir un cycle ferm6. Formons l’expreasion EdQ - dT. On a dQ = cdt Idv c &ant la chaleur sficifique B volume constant, 1 la chaleur latente, de dilatation du corps (quantitb de chaleur absorb6e par le corps pour que son volume vane d’une quantitb &ale A l’unit6, k temperature constante); de plus d T = pdv, dU = E (cdt ldv) - pdv donc dU = E cdt (El - p)dv. “La condition d’int6grabilit6 est - d ( E 1 - P) dv dt

+

+

+ +

“Cette Qquation est la traduction du principe de l’bquivalence.” Cows de Thermodynamique, Profem6 B la Sorbonne, par M. Lippmann, Rbdig6 par M M . E. Mathias et A. Renault, pp. 4p,45 (188 ) In the paravaph following the headin? “Applications du principe de l’bquivalence” Eippmann denotes the heat received by Q and the work done by T.

$

+Tdy

=

0.

-

I758

GEORGE TUNELL

Wenn das hier an der linken Seite stehende Integral jedesmal, so oft x und y wieder zu ihren urspriinglichen Werthen gelangen, Null werden soll, so muss der unter dem Integralzeichen stehende Ausdruck das vollstandige Differential einer Function von x und y sein, und es muss daher die oben besprochene Bedingungsgleichung der Integrabilitat erfullt sein, welche fur diesen Fall folgendermaassen lautet :

-(-)=-(-). d N

d M dy T

dx

T

Fuhrt man hierin die Differentiationen aus, indem man bedenkt, dass die Temperatur T des Korpers ebenfalls als Function von x und y zu betrachten ist, so kommt : _I . _dM- M d T - I d N N d T T dy T2 dy T dx T2 ds;'

--.

oder anders geordnet :

A more explicit form of the analytical expression of the second law given by Clausius in his equation (60) is the following equation for reversible processes in one-component systems consisting of one gaseous phase of unit mass: (t,P) (28)

S(t,p) - S(t,,pA = -/F d t

+ 1T"dp,

(t.,PO) which is also written in the more familiar but less explicit form: (t,P)

(29)

S(t,P)

- S(t0,PO)

=jT' dQ

(t0,PJ the integrals in each of the equations ( 2 8 ) and (29) being extended over any path connecting the points (to,po)and (t,p); T denotes some function of t alone the same for all systems, T = w(t); and S denotes the entropy per gram measured in thermal units. The physical hypothesis (experimental fact) embodied in each of the equations, (28) and ( 2 9 ) , may be stated in words as follows: The value of each of the integrals in equations ( 2 8 ) and (29) is independent of the choice of the path of integration, that is, it depends only on the limits of integration, and the integral may therefore be used to define a function, the entropy per gram, of the independent variables, t and p. Equation (29) may be brought readily into the form (28) by means of (3).

LAWS O F THERMODYNAMICS

From

(28)

I759

it follows directlyz6that

Substituting the values of (as/&), and (aS/ap), in the equation of the total differential of S(t,p),

one obtains C

dS = 2 dt T

(33)

+ 1T dp,

where t and p are the independent variables of the function S andthus dt =At and dp=Ap are the independent (principal) infinitesimals.

A necessary and sufficient ~ o n d i t i o nfor ?~ S(t,p)

-

S(t,,p,)

=yg + dt

+ 1 dp,

(tOIP0)

If the indicated differentiation be carried out and the result be multiplied by TI 5 =hdT (3 5 ) at dp T

dt’

and if Clausius’s y and x be identified with t and p, his equation (6) is obtained. As C l a u ~ i u s ?pointed ~ out, his equation (6) forms a n analytical expression of the second law for reversible processes in a system the state of which is determined by two independent variables, y and x. Thus equation Cf. footnote (18). Cf. footnote (19). 23 The importance of equations ( 2 5 ) and (34) lies in the fact that from them all of the thermodynamic relations for this system summarized in P. W. Bridgman’s “Condensed Collection of Thermodynamic Formulas” ( 1 9 2j ) are obtained by direct mathematical methods. 29 Op. cit., p. 9. 26

27

I 760

GEORGE TUNELL

(34) constitutes an analytical expression of the second law for reversible processes in the system considered in this ~ a p e r . 3 ~ Equation (zg), like ( z ) , (3), and (zo), may be differentiated with respect to the parameter u. Thus d_S _ - _ I ~_ Q du T d u Multiplying through by du one obtains (37) where u is the independent variable of the functions Q, T, and S and thus d a = Au is the independent (principal) infinitesimal. In this equation T is a function of t, but t is itself a function of u so that T is also a function of a ; similarly S is a function of t and p, but t and p are themselves functions of u so that S is also a function of u ; in this equation dt and dp are not independent infinitesimals. This equation is a necessary condition but is not a sufficient condition to establish the truth of the second law; and therefore it cannot properly be called an expression of the second law. The function T = w(t) was used by Lord Kelvin to define a new temperature scale, the absolute thermodynamic scale, which does not depend on the properties of any particular substance. Lord Kelvin’s later characterization*’ of the absolute thermodynamic temperature scale is as follows: “In a communication to the Cambridge Philosophical Society of June, 1848, it was pointed out that any system of thermometry, founded either on equal additions of heat, or equal expansions, or equal augmentations of Lippmann hm written similarly: “Pour chercher I’expression analytique de 8, ne fixons pas x et y. Supposons simplement

30

ue l’une des deux variables, x, soit une fonction de la temperature seulement, et varie 8ans le mkme sens que la temperature. Par exemple, x sera donnt- par un thermomktre

11 liquide quelconque, inconnu, d’une forme bizarre, irrbgulier, astreint it la seule condition que son indication x croisse avec la tempbrature. On a alors: dQ = Pdx Rdp. “P et R seront d’ailleurs des fonctions qui pourront &re connues exp6rimentalement ii ehaque instant. Alors: dS =dQ=rdx+!!dy.

+

e

e

e

“La condition d’int6grabilit6 qui exprime le principe de Carnot est que:

d’oh, en remarquant que 8 n’est fonction que de la seule variable x, et que y est indbpendant.

de x. (Op. cit., pp. 79, 80.) Lippmann here denotes the absolute temperature by 8. Mathematical and Physical Papers by Sir William Thomson, I , 393, 394 (1882),or J. P. Joule and W. Thomson: On the Thermal Effects of Flulds in Motion, Part 11, Phll. Trans., 144, 350-352 (1854). 31

1761


pressure, must depend on the particular thermometric substance chosen, since the specific heats, the expansions, and the elasticities of substances vary, and, so far as we know, not proportionally with absolute rigour for any two substances. . . . I t appears then that the standard of practical thermometry consists essentially in the refersnce to a certain numerically expressible quality of a particular substance. In the communication alluded to, the question, ‘Is there any principle on which an absolute thermometric scale can be founded?’ was answered by showing that Carnot’s functions? (derivable from the properties of any substance whatever, but the same for all bodies at the same temperature), or any arbitrary function of Carnot’s function, may be defined as temperature, and is therefore the foundation of. an absolute system of thermometry. . and we may define temperature simply as the reciprocal of Carnot’s function. When we take into account what has been proved regarding the mechanical action of heat, and consider what is meant by Carnot’s function, we see that the following explicit definition may be substituted:“If a n y substance whatever, subjected to a perfectly reversible cycle of operations, takes in heat only an a locality kept at a uniform temperature, and emits heat only i n another locality kept at a uniform temperature, the temperatures of these localities are proportional to the quantities of heat taken in or emitted at them in a complete cycle of the operations. “TOfix on a unit or degree for the numerical measurement of temperature, we may either call some definite temperature, such as that of melting ice, unity, or any number we please; or we may choose two definite temperatures, such as that of melting ice and that of saturated vapour of water under the pressure 29.9218 inches of mercury in the latitude of 4 5 O , and call the difference of these temperatures any number we please, IOO for instance. . it becomes a question, what is the temperature of melting ice, if the difference between it and the standard boiling-point be called IOO’?” Lord Kelvin’s definition of the absolute thermodynamic temperature scale is related to equation (29) as follows. For a Carnot cycle of operations of a reversible thermodynamic engine (a closed cycle consisting of two isothermal and two adiabatic transformations), with any working fluid, equation (29) takes the form

..

..

where TI denotes the absolute temperature of the upper isotherm of the Carnot cycle (temperature of the heat reservoir) ;T2, the absolute temperature a Carnot’s function, q, is the amount of work done per unit of heat received by a Carnot engine o rating in cycles between a heat source and a heat sink that differ in temperature by one g g r e e absolute. It may be proved easily from equations (20) and ( 2 9 ) of this paper that v-Q’+Q”- W =T’-T”-f

5

Q’

JQ’

T‘

T”

where Q’ denotes the heat absorbed (a positive quantity) by the working fluid of the Carnot engine a t the temperature T’. Q”, the heat absorbed (a negative quantity) a t the temperature T”;W, the work done. ‘ithe ? absolute , temperature of the heat source; and T”,the absolute temperature of the heat sink. In regard to the definition of Carnot’s function cf. Mathematical and Physical Papers by Sir William Thompson 1, 224, zag, 39’. 397.

GEORGE TUNELL

1762

of the lower isotherm (temperature of the heat sink); Q1,the heat absorbed (a positive quantity) at the temperature TI; and Q2, the heat absorbed (a negative quantity) at the temperature Tz. Thus for the segments of three isotherms at the arbitrary temperature T, and the fixed temperatures To and T,. one has (39) provided all three segments of isotherms connect the same pair of adiabatics (as illustrated in Fig. I ) . The temperature To is that of ice melting under the pressure of one atmosphere and T, that of saturated vapor of water boiling under the pressure of one atmosphere. In the case discussed T is assumed to be greater than To. Now let both sides of the equation

be multiplied by T, T,; the result is

Q, To = - Qo T,. To each side of this equation let QoTo be added (41)

(42)

thus (43) and hence

Qs

To

+

Qo

+

To (8

To = Qo)

- Qo Ts+ Qo

= -

Qo

To;

(T, - TO),

(44)

From (39) one has (45) Therefore

T

To

a = -8'

(46)

The difference T, - To is defined as 100'. Hence finally T=-j 100 Q (47) Qs

+

QO

where Q, QB,and Q,, denote heat absorbed and Q and Q9are thus positive quantities and Qoa negative quantity in the case discussed. This equation determines the absolute temperature in terms of heat quantities; and the heat quantities can be measured, in theory, by means of a hydrogen thermometer and calorimeter. In practice this method of evaluating the absolute temperature is replaced by one founded on the Joule-Thomson coefficient (porous plug effect). The equation required in the latter method is derived in the following paragraphs.

1763


The evaluation of the absolute temperature by the determination of the Joule-Thomson coefficient (porous plug effect), together with data for the heat capacity and specific volume of the thermometric substance, is accomplished by means of equations ( 2 5 ) and (35). From ( 2 5 ) one has

J

(s),

=

J

(g).

(%)t-

P

From (48) is subtracted (3 j) multiplied by J

thus (49)

The Joule-Thomson coefficient, p, is defined by the equation

where x denotes Gibbs’s chi function or the total heat (also called heat content, enthalpy, enkaumy). The function x is defined by the equation

x

(51)

=

u + PV,.

and is thus a function of the temperature and pressure: (52)

x

=

W,P)

+ D v ( ~ , P )= r(t,p).

I t is of course assumed in the definition of the Joule-Thomson coefficient, p , that the equation x=U+pv can be solved for t in terms of p and

x

: t = q ( p , x ) . Then one has

Independent variables of the 1st class, (t,p); Independent variables of the (53) (54) P = P, t = WP,X) 2nd class, ( p , ~ ) . To these equations the theorem for change of variables in partial differentiation is applied and thereby the result obtained:

i

s3 Cf. W. F. Osgood: “Advanced Calculus,” Chapter V, Section 14, “A Question of Notation” and Exercise 3, p. 141. Section I was written with particular reference to problems of thermodynamics such as the one %ere a t issue and is the clearest statement of the solution of this problem known to the present writer.

GEORGE TUNELL

1764

From the equation of definition of follows that (57)

x

and equations

J 1,

($)t

-

($)x=

=

-*

(21)) ( 2 2 ) )

and (56), it

+v

JCP

P

Hence (58)

J 1,

=

-

Jc,

(E) -

v.

Substituting this value of 1, in equation (49) one obtains

(59)

d log, T -dt

v

+ Jc, (e) aP x

Integrating this equation one obtains

The integrand is a function of the temperature, t, alone, and does not vary with the pressure, p; the integral may therefore be evaluated along the path in the t,p-plane followed by the thermometric substance in the constant volume hydrogen thermometer. Let the value of e raised to the power the integral between to = 0°C. and t. = I O O ~ C be. denoted by G:

(61) Since,by definition (62)

T.

- To=

100,

therefore (63)

To =-. IO0 G-I

Hence finally

The integrand and the constant, G, can be determined experimentally. To evaluate the constant, G , that is, the value of e raised to the power the integral in equation (61) between the fixed limits, to and t,, the integrand


1765

must be determined experimentally between to and t,. To evaluate T = w(t) if t lies above t,, the integrand in equation (64) must be determined experimentally in the interval (t,,t) in addition to the interval (to,t,). Similarly to evaluate T = w(t) if t lies below to,the integrand in equation (64) must be determined experimentally in the interval (t,to) in addition to the interval (to,tr).a4 Numerous investigations of one-component systems of one phase are recorded in the literature and it is not necessary to discuss such a system in detail in this paper. Especially notable is the recent work on the one-component system, H20, which is important in applications of thermodynamics in many fields, from power generation to geochemistry. N. F. Osborne3; has stated that: “The most formidable problem in the preparation of a thermodynamic table or chart for a given substance is to obtain data adequate as to kind, range, and accuracy.” The data for the system, HzO, to which Osborne has contributed so much himself, are based on numerous and consistent measurements that have been improved and greatly extended since 1921. Thus J. H. Keenans6 has written: “the demand for steam data at still higher pressures and temperatures [than 1200 lb. per sq. in. and 800 deg. Fahr.] has been steadily growing. The additional experimental data recently obtained a t the Massachusetts Institute of Technology and a t the Bureau of Standards have made possible the development of a complete set of steam tables and diagrams to 3 500 lb. per sq. in. and 1000 deg. Fahr. A year ago the Turbine Engineering Department of the General Electric Company undertook this development. The necessary data and many valuable suggestions were obtained from those engaged in the Steam Research Program, and the successful completion of the work is in no small degree due to their cooperation. “Sources of data. The Harvard Joule-Thomson-effect experiments which extended to 565 lb. per sq. in. and 657 deg. Fahr. had been carefully studied and reformulated by Dr. Davis during 1926 and 1927, and the new formulation was used in this development. It differed only slightly from the older formulation that was used in the work of 192 j,but it embodied certain concepts which guided extrapolation. “The Knoblauch specific-heat measurements extending to 420 lb. per sq. in. were used as in the work of 1925 to supplement the Harvard data. “In 1927 Dr. Keyes and Dr. Smith at the Massachusetts Institute of Technology completed a set of experimental determinations of the specific volume of superheated steam between 1350 lb. per sq. in. and 3850 lb. per sq. in., and between saturation and 7 5 2 deg. Fahr. They constitute the basis of the development between 1000lb. per sq. in. and 3 joo lb. per sq. in. 34 For numerical data and bibliography see Arthur L. Day and R. B. Sosman: Realisation of Absolute Scale of Temperature, “A Dictionary of Applied Physics,” Edited by Sir Richard Glazebrook, 1, 836-871 (1922). I Trans. Am. SOC.Mech. Engineers, Fuels and Steam Power, 52, 221 (1930). 38 Mechanical Engineering, 51, 109, 114 (1929).

I 766

GEORGE TUNELL

“The same experimenters had determined the pressure-temperature relationship a t saturation up to the critical point. These data, which agree closely with the Reichsanstalt experiments, were used in this development. “The Bureau of Standards contributed two very important pieces of data: ( I ) The mechanical equivalent of heat reported by Dr. Osborne in December, 1927,namely, I mean B.t.u. = 778.57 ft-lb. . . ( 2 ) The total heat of saturated water between 3 2 deg. Fahr. and 482 deg. Fahr. (0.09 lb. per sq. in. and 5 7 7 lb. per sq. in.), communicated privately with the permission of the Director of the Bureau. Other experimenters contributed minor parts at various stages of the work, but the development was primarily dependent on those mentioned. . “It is hoped that in the interim [before the goal of an international table is attained] this steam table . . . will fill the need for a tabulation of the properties of steam covering the critical region and offering both thermodynamic consistency and faithfulness to reliable experimental data. “It has served to bring to light the remarkable agreement between the experimental results of the three parts of the A. S. M. E. program, and between these results and other high-grade experimental data. I t has shown the agreement between workdone both by continuous-flow andstaticmethods.” Keenana7has subsequently made a comparison of his table and Mollier chart with more recent experimental data. Concerning this comparison he writes as follows: “The steam table and chart presented . . . . were based on all the experimental data available in April, 1928. Since that time the experimental progress includes a complete experimental check of the Keyes-Smith data which verifies the original values within a few hundredths of I per cent; an extension of the Bureau of Standards liquid total heats from 600 lb. per sq. in. to 800 lb. per sq. in.; the Masaryk total-heat measurements; and, most notably, an extension of the Knoblauch specific heats from 420 Ib. per sq. in. to 1750 lb. per sq. in. In days like these, steam tables must be well founded to stand the test of the next six months’ work. . . . “The Knoblauch measurements almost completely cover a large range of pressures where the properties of steam could be determined six months ago only by extrapolation of existing experimental data. The new tables represent such an extrapolation, and it is very interesting to compare specific heats computed from them with these new measurements from Germany. The agreement is so good as to justify confidence in these tables even through the range not covered by experiment at the time they were computed. It indicates a fundamental agreement between experimental results obtained at Munich by one method with those obtained a t M. I. T. and Harvard by entirely different methods. . . . The evidence at hand shows a general con-

.

..

.

37

Mechanical Engineering, 51, 129 (1929).


I767

vergence on a well-defined grid of data that will ultimately constitute the international standard.”a* I n concluding this paper one may pause briefly at the part of the boundary of thermodynamics where Clausius labored to construct lines of communication into the adjoining region of atomistics, to consider the fact that Clausius and most investigators since his time have not been content with the evaluation of the relations of thermodynamics and the measurement of thermodynamic quantities, but in addition have correlated thermodynamics with atomistics in various ways. For this purpose the well-known explanations of temperature and heat in terms of atomic properties have been developed. Thus H. A. Bumstead3@ has written: “[In his earliest publications Gibbs] had been concerned with the development of the consequences of the laws of thermodynamics which are accepted as given by experience; in this empirical form of the science, heat and mechanical energy are regarded as two distinct entities, mutually convertible of course with certain limitations, but essentially different in many important ways. In accordance with the strong tendency toward unification of causes, there have been many attempts to bring these two things under the same category; to show, in fact, that heat is nothing more than the purely mechanical energy of the minute particles of which all sensible matter is supposed to be made up, and that the extra-dynamical laws of heat are consequences of the immense number of independent mechanical systems in any body,-a number so great that, to human observation, only certain averages and most probable effects are perceptible.” The last work of Gibbs is his elaboration of these problems, entitled “Elementary Principles in Statistical Mechanics developed with especial reference to the Rational Foundation of Thermodynamics.” However, according to Bumstead, “Gibbs has not sought to give a mechanical explanation of heat, but has limited his task to demonstrating that such an explanation is possible.”40 These correlations are undoubtedly of very great importance and value, as are, in general, attempts to connect the various fields of science. The fundamental principles of thermodynamics may still be taken as empirical results of experience, however, since sufficient experimental data are available to establish them without reference to the laws of atomistics. And it is probably fortunate in some ways that this is so, because many problems in thermodynamics and its applications in power generation, geochemistry, and other fields, are therefore capable of solution where adequate atomistic data 38 Among the further develo rnents uith this system are the “Skeleton Steam Tables” prepared a t the International itearn-Table Conference of physicists and engineers from America, Great Britain, Gerrnanv and Czechoslovakia, held in London during July, 1929. I n these Tables are given mean vdues of the pro erties of saturated and superheated steam, with a plus or minus tolerance attached to em\ value, which, with the tolerances of the agreed magnitudes, were unanimously accepted by the delegates. Mechanical Engineering, 52, 120-122 (1930). 39 “The Collected Works of J. Willard Cibbs,” 1, 23, 2 5 (1928),or Am. J. Sci., (4) 16, 197, I99 (1903). 4 0 Footnote on page 1768.

1;68

GEORGE TUNELL

are lacking. Thus temperature as that which is measured by means of the gas thermometer and thermocouple, heat as that which is measured in the calorimeter, and thermodynamics as the science of heat and work, still serve as powerful artillery in several advancing sectors of science and technology. Acknowledgment The author wishes to thank Professor E. C. Kemble of the Department of Physics of Harvard University and Dr. George W. Morey of the Geophysical Laboratory for their kindness in making several valuable suggestions that have been incorporated in this paper. Appendix Statement by Lord Kelvin of the history of the first and second laws of thermodynamics. (Extracts from Lord Kelvin’s paper entitled “On the Dynamical Theory of H e ~ t . ” ) ~ l “3. The recent discoveries made by Mayer and Joule, of the generation of heat through the friction of fluids in motion, and by the magneto-electric excitation of galvanic currents, would either of them be sufficient to demonstrate the immateriality of heat; and would so afford, if required, a perfect confirmation of Sir Humphry Davy’s views. “4. Considering it as thus established, that heat is not a substance, but a dynamical form of mechanical effect, we perceive that there must be an equivalence between mechanical work and heat, as between cause and effect. The first published statement of this principle appears to be in Mayer’s Bemerkztngen uber die Krafte der unbelebten iliatur, which contains some In this connection it seems worth while to quote the following from Gibbs’s “Elementary Principles in Statistical Mechanics” (“The Collected Works of J. Willard Gibbs,” art I , 165, 166, 167): ’ “Tf we wish to find in rational mechanics an a priori foundation for the principles of thermodvnamics, we must seek mechanical definitions of temperature and entropy. . “At [east, we have to show by a priori reasoning that for such systems as the material bodies which nature resents to us, these relations hold yith such approximation that they are sensibly true for guman faculties of observation. This!ndeed,m all that is realty necessary to establish the science of thermodynamics on an a pnorz basis. Yet we will naturally desire to find the exact expression of those prmciples of which the laws of thermodynamics are the approximate expreasion. A very little study of !he statjstical properties of conservative systems of a finite number of degrees of freedom is sufficient to,make it appear, more or less distinctly, that the general laws of thermodynamlcs are the limit toward which t,he exact laws of such systems approximate, when t h e r number of degrees of freedom is indefinitely increased. “The enunciation and proof of these exact laws, for systems of any finite number of degrees of freedom, has been a principal object of the preceding discussion. But it should be distinctly stated that, if the results obtained when the numbers of degrees of freedom are enormous coincide sensibly with the general laws of ,thermodynamics, however interesting and si nificant this coincidence may be, we are still far from,having explained the phenomena of nature with respect to these laws. For, as compared with the case of nature, the systems which we have considered are of an ideal simplicity. . . “The ideal case of systems of a finite number of degrees of freedom remains as a subject which is certainly not devoid of a theoretical interest, and which may serve to point the way to the solution of the far more difficult problems presented to us by nature.” 41 S .+ I! ’ Roy. SOC.Edinburgh, 20, 261-267 (1851), or “Mathematical and Physical Papers, 1, 174-181 (1882). 2

..

...

.

1769


correct views regarding the mutual convertibility of heat and mechanical effect, along wit.h a false analogy between the approach of a weight to the earth and a diminution of the volume of a continuous substance, on which an attempt is founded to find numerically the mechanical equivalent of a given quantity of heat. I n a paper published about fourteen months later, ‘On the Calorific Effects of Magneto-Electricity and the Mechanical Value of Heat,’ Mr. Joule, of Manchester, expresses very distinctly the consequences regarding the mutual convertibility of heat and mechanical effect which follow from t’he fact, that heat is not a substance but a state of motion; and investigates on unquestionable principles the ‘absolute numerical relations,’ according to which heat is connected with mechanical power; verifying experimentally, that whenever heat is generated from purely mechanical action, and no other effect produced, whether it be by means of the friction of fluids or by the magneto-electric excitation of galvanic currents, the same quantity is generated by the same amount of work spent; and determining the actual amount of work, in foot-pounds, required to generate a unit of heat, which he calls ‘the mechanical equivalent of heat.’ . “ 9 . The whole theory of the motive power of heat is founded on the two following propositions, due respectively to Joule, and t o Carnot and Clausius. “Prop. I. (Joule).-When equal quantities of mechanical effect are produced by any means whatever from purely thermal sources, or lost in purely thermal effects, equal quantities of heat are put out of existence or are generated. “Prop. 11. (Carnot and Clausius).-If an engine be such that, when it is worked backwards, the physical and mechanical agencies in every part of its motions are all reversed, it produces as much mechanical effect as can be produced by any thermo-dynamic engine, with the same temperatures of source and refrigerator, from a given quantity of heat. . . . “12. The demonstration of the second proposition is founded on the following axiom:I t i s impossible, by means of inanimate material agency, to derive mechanical e$ect from a n y portion of matter by cooling i t below the temperature o j the coldest oj the surrounding objects. “13. To demonstrate the second proposition, let A and B be two thermodynamic engines, of which B satisfies the conditions expressed in the enunciation; and let, if possible, A derive more work from a given quantity of heat than B, when their sources and refrigerators are a t the same temperatures, respectively. Then on account of the condition of complete reversibility in all its operations which it fulfils, B may be worked backwards, and made to restore any quantity of heat t o its source, by the expenditure of the amount of work which, by its forward action, it would derive from the same quantity of heat. If, therefore, B be worked backwards, and made t o restore to the source of A (which we may suppose to be adjustable to the engine B) as much heat as has been drawn from it during a certain period of the working of A,

.

I

1770

GEORGE TUNELL

a smaller amount of work will be spent thus than was gained by the working of A. Hence, if such a series of operations of A forwards and of B backwards be continued, either alternately or simultaneously, there will result a continued production of work without any continued abstraction of heat from the source; and, by Prop. I., it follows that there must be more heat abstracted from the refrigerator by the working of B backwards than is deposited in it by A. Now it is obvious that A might be made to spend part of its work in working B backwards, and the whole might be made self-acting. Also, there being no heat either taken from or given to the source on the whole, all the surrounding bodies and space except the refrigerator might, without interfering with any of the conditions which have been assumed, be made of the same temperature as the source, whatever that inay be. We should thus have a self-acting machine, capable of drawing heat constantly from a body surrounded by others at a higher temperature, and converting it into mechanical effect. But this is contrary to the axiom, and therefore we conclude that the hypothesis that A derives more mechanical effect from the same quantity of heat drawn from the source than B is false. Hence no engine whatever, with source and refrigerator a t the same temperatures, can get more work from a given quantity of heat introduced than any engine which satisfies the condition of reversibility, which was to be proved. “14. This proposition was first enunciated by Carnot, being the expression of his criterion of a perfect thermo-dynamic engine. He proved it by demonstrating that a negation of it would require the admission that there might be a self-acting machine constructed which would produce mechanical effect indefinitely, without any source either in heat or the consumption of materials, or any other physical agency; but this demonstration involves, fundamentally, the assumption that, in ‘a complete cycle of operations,’ the medium parts with exactly the same quantity of heat as it receives. A very strong expression of doubt regarding the truth of this assumption, as a universal principle, is given by Carnot himself; and that it is false, where mechanical work is, on the whole, either gained or spent in the operations, may (as I have tried to show above) be considered to be perfectly certain. I t must then be admitted that Carnot’s original demonstration utterly fails, but we cannot infer that the proposition concluded is false. The truth of the conclusion appeared t o me, indeed, so probable that I took it in connexion with Joule’s principle, on account of which Carnot’s demonstration of it fails, as the foundation of an investigation of the motive power of heat in air-engines or steam-engines through finite ranges of temperature, and obtained about a year ago results, of which the substance is given in the second part of the paper at present communicated to the Royal Society. It was not until the commencement of the present year that I found the demonstration given above, by which the truth of the proposition is established upon an axiom (0 12) which I think will be generally admitted. It is with no wish to claim priority that I make these statements, a~ the merit of first establishing the proposition upon correct principles is entirely due to Clausius, who published

,


1771

his demonstration of it in the month of May last year, in the second part of his paper on the motive power of heat. I may be allowed to add, that I have given the demonstration exactly as it occurred to me before I knew that Clausius had either enunciated or demonstrated the proposition. The following is the axiom on which Clausius’ demonstration is founded:ILIti s impossible for a self-acting machine, unaaded by a n y external agency, to conuey heat f r o m one body to another at a higher temperature. “It is easily shown, that, although this and the axiom I have used are different in form, either is a consequence of the other. The reasoning in each demonstration is strictly analogous to that which Carnot originally gave.” Geophysical Laboratory, Carnegie Institution of Washington, March, 1932.

On the History and Analytical Expression of the First and Second Laws

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