Abstract
This chapter will present an alternative route to the thermodynamics of simple fluids at a slightly higher mathematical level, allowing for a generalization of the subject of thermodynamics of uniform fluids. The most important difference to the previous development has to do with what we assume to know about the nature of heat. By introducing the law of balance of heat, and the relation between currents of heat and energy in heating (Equation (13) of Chapter 1), we have directly identified heat with entropy, and have made the relation between entropy and energy the cornerstone of our development. This approach has afforded us a great simplification which is important in an introductory course on the foundations of thermodynamics. Still, it somewhat oversimplifies the matter in that it assumes too much about the nature of heat at the start, and it leaves open the question as to the historical development of the subject. In this chapter, we will therefore not assume any knowledge of the relation between heat and energy. Rather, we will start with what is known as Carnot’s Axiom (an assumption about the power of heat in ideal engines), and with a statement about the existence of a heat function. On this basis we will be able to derive the relation between heat and energy.
I suppose implicitly in my proof that if a body has suffered any changes whatever and that if after a certain number of transformations it is brought back to its original state,... [it will be] found to contain the same quantity of heat as it contained at first, or in other words, that the quantities of heat absorbed or emitted in its various transformations are exactly compensated.
S. Carnot, 1824
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References
S. Carnot (1824): Reflections on the Motive Power of Fire.
In his work, The Tragicomical History of Thermodynamics,C. Truesdell gave a detailed and critical account of the historical development of thermodynamics (Truesdell, 1980). He and Bharatha worked out the logical foundations of classical thermodynamics (The Concepts and Logic of Classical Thermodynamics,1977), where the nature of heat is left to be determined almost up to the end. Following their reasoning, you can see what is needed to make heat a quantity of energy. If you take a slightly different approach, however, heat turns out to be entropy. This line of reasoning was demonstrated in a paper by Callendar (1911).
See Truesdell (1979) for a discussion of this matter.
In Section I.2, the adiabatic compressibility will be introduced.
Proof of Equation (19). Remember the equation of state in the two forms P = p (V, T),and V = N(P, T),and the definition of a and p. We can express the partial derivatives of the two functions P and N by Equations (7a) and (7b).This immediately leads to the relationship between the coefficients proposed above.
That heat can be transferred is the fundamental assumption underlying both the caloric and the mechanical theories of heat. The caloric theory, however, also assumes the existence of a heat function. We may interpret this graphically as meaning that heat also resides in bodies. Therefore, in the context of the caloric theory, we are justified in speaking of a quantity of heat for which a law of balance must hold. Researchers in Carnot’s time never expressed their assumption in this way.
Since our assumptions will lead to the identification of caloric with entropy, we shall use the symbol S for heat.
In earlier times the term specific heat was used for what we now call heat capacity. The relationship between the heating and the constitutive quantities expressed below has been called the doctrine of latent and specific heats by Truesdell (1980).
The proof of this statement is not quite so simple, and it is impossible to give at this point. (It is related to the change of sign of ap/aT which we discussed in Section I.1.2, but we need another assumption for the proof to be possible.)
Truesdell (1980).
S. Carnot: Réflexions sur la Puissance Motrice du Feu et sur les Machines Propres a développer cette Puissance. Paris, Bachelier, 1824.
See J.S. Thomson and T.J. Hartka (1962); J.E. Trevor ( 1928 ); L.A. Turner (1962).
For more information see, for example, Marsden and Weinstein: Calculus III (1985).
For more details, see Truesdell (1979).
Often, absolute is taken to mean a scale which has a point of absolute zero. We have taken this for granted by assuming that T 0 all the time. In Chapter 1 we saw that this should be the case because otherwise the relationship between caloric created and the energy used would not be defined.
The problem is not quite so simple, however. For example, for a given empirical scale, F(T) is independent of the particular fluid used. We cannot be sure, however, that we get the same function F for other scales as well. Then, the axioms and equations should be independent of the particular scale used; they should be invariant under transformation from one scale to another. (This imposes some restrictions on what kind of fluids can be used for thermometry: water is unsuitable.) Finally, Carnot’s function may not vanish; otherwise, the absolute scale chosen above does not make any sense.
See for example Callen (1985), Chapters 5–7.
For more information on the subject of the formalism of classical thermodynamics, see Callen (1985), Chapters 5–7.
Just ask small children before they have been told by teachers that heat is a kind of energy.
R. Fox: The Caloric Theory of Gases. Clarendon Press, Oxford, 1971.
J.B. Fenn: Engines, Energy, and Entropy. Freeman and Company, New York, 1982.
C.E. Mortimer: Chemistry. A conceptual approach. D. Van Nostrand Company, 1979.
While a recent text on introductory physics calls heat the energy of the irregular motion of atoms and molecules, its accompanying study guide calls heat a form of energy transfer. (H. Ohanian: Physics. W.W. Norton & Company, New York, 1985. Van E. Neie and P. Riley: Study guide, Ohaninan’s Physics. W.W. Norton, 1985.)
Note that we do not include in the general properties of caloric its conservation or nonconservation. Historically, however, caloric was assumed to be a conserved quantity.
Using the symbol dH/dt for the heating leaves room for misunderstandings. Is it the rate of flow of heat across the surface of the body, or is it the rate of change of the heat content? In the mechanical theory of heat, there is no “heat content.” There, heating may mean only the rate of flow of heat. In the caloric theory of heat, on the other hand, there is a heat content, but heat might not be conserved.
C. Truesdell (1980), Chapter 8.
S. Carnot: quoted from C. Truesdell (1980), p. 81.
C. Truesdell (1980), Chapter 3.
C. Truesdell (1980), Chapter 7.
R. Clausius: quoted from C. Truesdell (1980), p. 187.
Indeed, it is wrong in the mechanical theory of heat, since there heat may not be thought of as residing in bodies. Clausius’ results contradict the very prejudice they were built upon.
See Stacey(1992), p. 305 and appendixes therein.
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Fuchs, H.U. (1996). Heat Engines and the Caloric Theory of Heat. In: The Dynamics of Heat. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-2542-1_4
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