Problems and Solutions on Thermodynamics and Statistical Mechanics

230 Problems 8 Soldiona on Thermodynamics fY Statistical Mechanics

Solution:

(a) The number of particles is

(b) In the approximation of a dilute gas, we have exp(-p/kT) >> 1,

and the Bose-Einstein distribution becomes the Boltzmann distribution.

We will prove as follows that this limiting condition is just d >> A.

Since

we have

where X = h/dm is the de Broglie wavelength of the particle’s thermal
motion, and d = m.
Thus the approximation exp(-p/kT) >> 1 is equivalent to d >> A.
(c) In the 1st order approximation

the average energy is