Mauro Sbragaglia

Grand Canonical Ensemble

A gas is in contact with a surface. On the surface we find N 0 localized and distinguishable sites adsorbing N (N≤N 0) molecules of the gas (each site can adsorb zero or one molecule of the gas). Find the grand canonical partition function of the system, and determine the chemical potential as a function of the average number of particles 〈N〉 which are adsorbed by the surface. You can think that the canonical partition function of an adsorbed molecule is a function only of the temperature, Q(T), and that all the adsorbed molecules are non interacting.

Thermodynamics and Microcanonical Ensemble

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

Central Force Field

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

Angular Momentum and Spin

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

Fluctuations and Complements

\Phi = N\mu

Perturbation Theory and WKB Method

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).