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Entropic variables and Massieu-Planck functions

 

We consider a simple pure substance under hydrostatic conditions described by the following fundamental equation:
 equation65
where the extensive variables U, V and N are the internal energy, the volume, and the number of particles respectively, and the intensive variables T, p and tex2html_wrap_inline599 are the temperature, the pressure and the chemical potential respectively.

Equation (gif) corresponds to the choice of the variables U, V and N as independent variables of the entropy S(U,V,N). These variables are precisely those which are fixed and determine the macrostate of the members of the Microcanonical Ensemble and consequently S is the relevant potential in this statistical ensemble.

It is useful to define the following quantities: tex2html_wrap_inline611, tex2html_wrap_inline613 and tex2html_wrap_inline615 so that Eq. (gif) can then be written in the dimensionless form:
 equation76
In general, for other thermodynamic systems with tex2html_wrap_inline617 degrees of freedom, one will have:
equation80
where tex2html_wrap_inline619 are extensive variables, and tex2html_wrap_inline621 the corresponding entropic conjugate variables. Massieu-Planck functions [] are entropic thermodynamic potentials defined as Legendre transformations of the entropy. In the case of a pure substance, the following (dimensionless) potentials can be formally defined:
 eqnarray86
The function tex2html_wrap_inline623 was first introduced by Massieu [], and it is called Massieu's potential. The function tex2html_wrap_inline625 was introduced by Planck[] and is called Planck's. potential.

Given the extensivity of tex2html_wrap_inline525, and using Euler's theorem for homogeneous functions, it is easy to see that tex2html_wrap_inline629. Therefore the Legendre transformation of all variables redefines the entropy,
 equation98
Substituting Eq. (gif) into the differentials of the potentials defined above one gets:
      eqnarray103
From Eq. () one obtains:
 equation112
The above equations allow a re-derivation of all the standard thermodynamic equations in terms of tex2html_wrap_inline631, tex2html_wrap_inline633 and tex2html_wrap_inline635. For instance, Maxwell relations can be deduced, by imposing that the equations (gif)-(gif) are exact differentials (equality of crossed derivatives). Moreover, Eq. (gif) is the Gibbs-Duhem equation which states that the complete set of intensive variables of the system are not all independent. On the other hand, the extremal condition of tex2html_wrap_inline525 leads us to deduce that tex2html_wrap_inline631, tex2html_wrap_inline633 and tex2html_wrap_inline635 are homogeneous at equilibrium [].


next up previous
Next: Statistical Mechanics Up: Entropic Formulation of Statistical Previous: INTRODUCTION

Eduard Vives
Tue Oct 24 11:27:45 CEST 2000