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