Abstract
In the preceding chapter, the mathematical nature of the state has been left totally unspecified. Without such a specification, no results can be obtained in addition to those given in the previous chapter. However, it should be borne in mind that, when a specific mathematical structure is assigned to the state, the applicability of the resulting theory is restricted to classes of materials and/or phenomena for which the assigned state is a realistic one.
The considerations we have presented appear to better conform to natural effects, and to provide the right conception of the phenomena considered by evidencing the restrictions under which the results can be applied.
M. Navier
Sylvester wrote Gilman that... (Gibbs) at $5,000 would be dirt cheap... Until that time Gibbs had served Yale as an honorary professor, unpaid. The powers there seem to have been astonished to learn that some other place might find worthy of pay such an eccentric fellow, who just sat and thought and wrote equations....
C. A. Truesdell
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Literature
The concept of “site,” as used in this chapter, was introduced in G. Astarita and G. C. Sarti, Chim. Ind. (Milan) 57, 680, 749 (1975).
The technique of deriving consequences of the Clausius-Duhem inequality used in this chapter makes crucial use of the concept that external state variables, as well as their time derivatives, can be imposed arbitrarily and independently of each other. A formal proof of this is presented in G. Astarita, An Introduction to Nonlinear Continuum Thermodynamics, SpA Editrice di Chimica, Milan (1975).
The question of the constant pressure specific heat at phase transitions is discussed in more detail in Section 4.5.
The question of admissibility of processes and transformations, discussed in the Appendix to Section 2.1, is conceptually very important. See in this regard the paper by Feinberg and Levine cited in the Literature section of Chapter 1.
Dissipation in a Maxwellian gas, and the lack of it in pure expansions and compressions, is discussed in detail in the following two references: H. Grad, “Principles of the Kinetic Theory of Gases,” in Encyclopaedia of Physics, Vol. XII, pp. 205–294, Springer-Verlag, Berlin (1958).
C. A. Truesdell, Rational Thermodynamics, 2nd ed., Chapters 8, 9, and 10, Springer-Verlag, Berlin (1984).
The concept of an absolute temperature is one which has a long tradition and a very rich literature. Possibly the best recent discussion of the subject, which includes a historical perspective, is M. Pitteri, in: C. A. Truesdell, Rational Thermodynamics, 2nd ed., Appendix G6, Springer-Verlag, Berlin (1984).
We have used the idea that entropy is a perfectly legitimate concept also when the system considered is not at equilibrium. This is always a moot point, but the reader is simply asked to consider a mixture of hydrogen and oxygen at ambient conditions, a system for which nobody would deny an entropy is a perfectly acceptable concept, and yet the system is not at equilibrium, since water would be present at equilibrium in such a quantity as to make the concentration of either oxygen or hydrogen negligibly small. That entropy is a perfectly legitimate concept in nonequilibrium situations was so obvious to the pioneers that they didn’t feel the need to make the point explicitly: the only one who made the point explicitly was J. W. Gibbs, “Graphical methods in the thermodynamics of fluids,” Trans. Conn. Acad. 2, 309 (1873).
The question of invertibility of the constitutive mappings is delicate. If one considers the mapping delivering the entropy, s(V, T), this is invariably invertible for temperature (even in highly more complex cases), and hence entropy, rather than temperature, could be used as an independent variable; this has, e.g., to do with the speed of sound, to be discussed in Chapter 8. Should one use the pair V, S as the state, temperature can easily be shown to be (∂U/∂S)V, which of course guarantees that the derivative (∂U/∂S)V exists and is always positive. Indeed, Truesdell and Toupin, The Classical Field Theories, Springer-Verlag, Berlin (1960), develop the classical thermodynamic theory by requiring that internal energy depends, in addition to mechanical variables such as V, on another variable S which is dimensionally independent of the mechanical variables; the latter is then identified with entropy. In the mechanical engineering literature, often the pair S, T is used as the state; this implies, albeit implicitly, that the mapping s(V, T) is invertible for volume, which is almost always, but not invariably, true. In fact, at constant T, S is generally an increasing function of V; however, for instance, when ice melts S increases and so does 1/ V—a very simple counterexample.
The question of invertibility of the f() function with respect to temperature, which has been discussed in the Appendix to Section 2.7, casts doubt on the significance of the axiomatic approach of C. Carathéodory, Math. Ann. 67, 355 (1909), who regards the pair V, p (or generalizations thereof) as identifying the state. Other difficulties with Carathéodory’s approach have been discussed by.
G. Whapples, J. Ratl. Mech. Anal. 1, 302 (1952)
B. Bernstein, J. Math. Phys. 1, 222 (1960)
J. B. Boyling, Proc. R. Soc. London Ser. A 329, 35 (1972)
J. L. B. Cooper, J. Math. Anal. Appl. 17, 172 (1967).
The related problem of the anomalies in Carnot cycles resulting from adiabates along which the rates of change of pressure and temperature have different signs is discussed by C. Truesdell and S. Bharatha, The Concepts and Logic of Classical Thermodynamics as a Theory of Heat Engines, Springer-Verlag, Berlin (1977), and by.
J. S. Thomson and T. J. Hartka, Am. J. Phys. 30, 26, 388 (1962).
The concepts of affinity and of “extent of reaction” were first put forward by De Donder and van Rysselberghe, Thermodynamic Theory of Affinity, Stanford University Press (1936).
A purely thermostatic theory, which is strictly constrained to equilibrium values of all functions of state, has in fact been developed by J. W. Gibbs within the restricting assumption that the state is V,S: Trans. Conn. Acad. 3–16, 108, 343 (1876).
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Astarita, G. (1989). State and Equilibrium. In: Thermodynamics. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-0771-4_3
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