Thermodynamics pp 207-242

# Equations of State

• Gianni Astarita

## Abstract

Let us consider a one-component system such as discussed in Section 4.1. At any given temperature, the a(V) curve has a last phase transition region; at volumes larger than those, no more phase transitions occur and the system is said to be a gas. At temperatures above the critical temperature, this definition implies that the gas phase is reached by sublimation from the solid phase, and the pressure at the sublimation point may be extremely high, so that the resulting gas may well have a density comparable to that of liquids.

## Keywords

Mole Fraction Constitutive Equation Activity Coefficient Ideal Solution Absolute Zero
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## Literature

1. The best presentation of Nernst’s argument on the third law of thermodynamics is in his book, The New Heat Theorem, published originally in German in 1917; the first English translation was published by Methuen and Co. in London in 1926. A good discussion of the possible experimental validations of the third law is given by Taylor and Glasstone, Physical Chemistry, Chapman and Long, London (1953).Google Scholar
2. That the velocity of sound is (∂p/∂̤)l/2 was deduced as early as 1687 by Newton, Philosophiae Naturalis Principia Mathematica, J. Straeter, London (1687). Newton used (in the first edition of the Principia) the isothermal value of ∂p/∂Ф, thus underestimating the speed of sound in ambient air by about 30%. The issue was resolved in 1822 by Laplace [Oeuvres, Imprimerie Royale, Paris (1846)] who, in order to obtain the correct result, simply made use of the known fact that the pressure of air, when compressed very rapidly (and hence adiabatically), is proportional to the 5/3 power of density. Since at those times there was no thermodynamic theory whatsoever available, the result is clearly a purely mechanical one, with no thermodynamic content at all. The thermodynamic analysis (i.e., the application of the first law to the phenomenon considered) was developed by W. J. M. Rankine, Phil. Trans. R. Soc. London 160, 277 (1870).
3. and more completely by H. Hugoniot, J. Ecole Polit. 57, 3 (1887); 58, 1 (1888); J. Math. Pures Appl. 3, 477 (1887); 4, 153 (1887). The whole matter is discussed in detail in any textbook on gas dynamics.Google Scholar
4. Propagation of discontinuities in elastic systems is discussed in general by C. Truesdell and R. A. Toupin, “The Classical Field Theories,” in: Encyclopaedia of Physics, Vol. 3/1, Springer-Verlag, Berlin (1960); this theory is still a thermodynamically reversible one. For the more general case of dissipative systemsGoogle Scholar
5. see B. D. Coleman et al, Wave Propagation in Dissipative Materials, Springer-Verlag, Berlin (1965).
6. The statement in Section 8.3 that no gas-gas phase transitions are known to occur should be qualified. In the case of a shock wave through which a combustion reaction proceeds to completion, a surface of discontinuity exists which separates two gases which are at different pressure, temperature, and composition, and hence should, by all reasonable definitions, be recognized as two phases. However, this is not an equilibrium condition. The existence of interfaces between phases sustained by a nonequilibrium phenomenon is discussed by G. Astarita and R. Ocone, Adv. Chem. Eng. (to be published).Google Scholar
7. The Van der Waals analysis has been discussed in the literature section of Chapter 4. The idea that parameter a may be a decreasing function of temperature was considered as long ago as 1880 by Clausius himself, and a simpler version of Clausius’ result was given by Berthelot in 1900; the Redlich-Kwong form is in O. Redlich and J. N. S. Kwong, Chem. Rev. 44, 233 (1949). Several later variations are due to Wilson; Soave; Lee, Erbar, and Edminster; Peng and Robinson; and Tang. The latter in Huagong Xuebao 2, 149 (1984) [English Transi.: Int. Chem. Eng. 27, 148 (1987)] gives a general overview of equations of state for single-component systems of the Van der Waals type.
8. There is a very ample literature on constitutive equations for nonideal mixtures. A good general presentation is given by J. H. Hildebrand, J. M. Prausnitz, and R. L. Scott, Regular and Related Solutions, Van Nostrand, New York (1970); Perry’s handbook is also a good, if very concise source.Google Scholar
9. Constitutive equations for nonideal mixtures are best written as equations giving G EX as a function of composition. Equation (8.4.14) then produces equations for the activity coefficients which automatically satisfy the Gibbs-Duhem equation, as well as the so-called Gibbs-Helmholtz equation, i.e., δ(G EX/T)/δ(1/T) = H EX. The generality of this procedure is discussed by K. Wohl, Trans. Am. Inst. Chem. Eng. 42, 215 (1946).Google Scholar
10. Special constitutive equations which have been commonly used are in G. M. Wilson, J. Am. Chem. Soc. 86, 127 (1964).
11. The two-parameter Wilson equation has only one major disadvantage, namely, its inability to predict liquid-liquid phase separation. This is overcome by the nonrandom, two-liquid (NRTL) model; see H. Renon and J. M. Prausnitz, Ind. Eng. Chem. 57(5), 18 (1965). Other equations which have been used extensively are discussed in J. Am. Chem. Soc. 57, 1805 (1935); Ind. Eng. Chem. 40, 341 (1935); Ind. Eng. Chem. 50, 391 (1958); Ind. Eng. Chem. 51, 211 (1959); AIChE J. 5, 249 (1959).
12. A model which has been used extensively is the UNIQUAC (Universal Quasi Chemical Theory); for an introduction to the basic concepts of UNIQUAC, see D. S. Abrams and J. M. Prausnitz, AIChE J. 21, 116 (1975).

© Springer Science+Business Media New York 1989

## Authors and Affiliations

• Gianni Astarita
• 1
• 2
1. 1.University of DelawareNewarkUSA
2. 2.University of NaplesNaplesItaly