Thermodynamics and Microcanonical Ensemble

Abstract

We know that the free energy F(T,V,N) of a thermodynamic system is extensive. Show that

$$N{\left( {\frac{{\partial F}}{{\partial N}}} \right)_{T,V}} + V{\left( {\frac{{\partial F}}{{\partial V}}} \right)_{T,N}} = Nf = F$$

with f the free energy density expressed in suitable variables. Given this result, from the differential properties of F(T,V,N), show that

$$\Phi = N\mu$$

with Φ the Gibbs potential defined as Φ=F+PV. In the above expression, μ is the chemical potential properly defined in terms of F(T,V,N).