Osmotic equilibrium, free energy, and chemical potential

3 Osmotic equilibrium, free energy, and chemical potential 

Let us express the osmotic equilibrium condition by starting from the thermodynamic functions of the system. The number N of molecules being fixed, the system may be in various states i, to which the energy and the probability pt are attributed. The system is in equilibrium when the system entropy 

Let us consider the function S( , V) which, for and fixed, is the maximum of S. The temperature of the corresponding canonical ensemble is defined by introducing the Lagrange multiplier P and the Massieu function 

The maximization of this expression with respect to gives 

The equilibrium, at V and P fixed, is determined by minimizing the function 3F It is also useful to consider the equilibrium of the system at fixed pressure P. For that, we associate the Gibbs free energy 

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