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Equivalence of thermodynamical fundamental equations

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Eur. J. Phys. 21 (2000) 395–404. Printed in the UK

PII: S0143-0807(00)13142-3

Equivalence of thermodynamical fundamental equations
J Guemez†, C Fiolhais‡ and M Fiolhais‡ ¨´
† Departamento de F´sica Aplicada, Universidad de Cantabria, E-39005 Santander, Spain ı ‡ Departamento de F´sica andCentro de F´sica Computacional, Universidade de Coimbra, ı ı P-3000 Coimbra, Portugal E-mail: guemezj@ccaix3.unican.es, tcarlos@teor.fis.uc.pt and tmanuel@teor.fis.uc.pt Received 5 April 2000, in final form 13 June 2000

Abstract. The Gibbs function, which depends on the intensive variables T and P , is easier to obtain experimentally than any other thermodynamical potential. However, textbooksusually first introduce the internal energy, as a function of the extensive variables V and S, and then proceed, by Legendre transformations, to obtain the Gibbs function. Here, taking liquid water as an example, we show how to obtain the internal energy from the Gibbs function. The two fundamental equations (Gibbs function and internal energy) are examined and their output compared. In both casescomplete thermodynamical information is obtained and shown to be practically the same, emphasizing the equivalence of the two equations. The formalism of the Gibbs function is entirely analytical, while that based on the internal energy is, in this case, numerical. Although it is well known that all thermodynamic potentials contain the same information, usually only the ideal gas is given as anexample. The study of real systems, such as liquid water, using numerical methods, may help students to obtain a deeper insight into thermodynamics.

1. Introduction

Thermodynamic potentials are an important topic in any course of thermodynamics. According to the first and second laws, for a hydrostatic, monocomponent, one-phase and closed system, the equation du = T ds − P dv indicates that thespecific internal energy u = u(s, v) exists as a function of the specific entropy, s, and of the specific volume, v ([1], pp 1–2 and p 33). These are known as natural variables of u ([2], p 41). Other state variables, such as temperature and pressure, are readily obtained from u(s, v): T (s, v) = (∂u/∂s)v and P (s, v) = − (∂u/∂v)s . Eliminating s from T (s, v) and P (s, v), one finds the thermalequation of state P = P (T , v). In fact, all thermodynamical properties may be derived from u = u(s, v) or, in other words, such an equation, the so-called fundamental equation in the energy representation ([3], p 41), contains complete thermodynamical information for a given system. We note that neither the thermal equation of state nor the internal energy equation of state, u = u(T , v), nor even theentropy equation of state, s = s(T , v), which can all be derived from the fundamental equation, contains complete thermodynamical information about a system ([1], pp 33–4). Other fundamental equations, such as the specific enthalpy, h = h(s, P ), the specific Helmholtz function, f = f (T , v), and the specific Gibbs function, g = g(T , P ), can be used instead of u = u(s, v). They are calledfundamental equations in the enthalpy, Helmholtz function and Gibbs function representations, respectively. These thermodynamical potentials are equivalent in the sense that they contain, and may provide, the same (full) thermodynamical
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information. It is well known that any fundamental equation can be...