Analytical Formulation of the Gibbs Criterion for a System in Equilibrium

2 ANALYTICAL FORMULATION OF THE GIBBS CRITERION FOR A SYSTEM IN EQUILIBRIUM 

According to the Gibbs criterion, the entropy function S is a maximum (with respect to certain allowed variations). Recall that for a general function f(x, y.), the conditions that/be a maximum are  

In order to obtain the analogous conditions for maximization of S with respect to allowed variations, we consider in turn the following questions  

Variations of S can therefore be expressed in terms of its natural variables U, V for a closed system. However, for the more general case of an open system (variable N), we should include the total mass variable N to write 

We can envision these variations by partitioning the system into a large number of cells, as follows  

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