Question

Use gibbs distribution to calculate the average entropy, energy and pressure of a single classical particle of mass m, with no internal degrees of freedom, free to move in volume V at temperature T. For that you will need to know the volume Vd of an d-dimensional hypersphere of the unit radius. (Vd=pi^(d/2)/gamma(d/2+1))

Answer 1

The Gibbs distribution function for the single particle is given by . So the partition function for the single particle free to move in volume V is given by

using the standard formula of Gaussian we can write this as

ones we have the partition function we can calculate the average values of E,S,P as follows