Problems and Solutions on Thermodynamics and Statistical Mechanics

232 Problems EI Sdutiom on Therdyam'ca EI Statistical Mechanics

2065

A gas of N spinless Bose particles of mass m is enclosed in a volume
V 'at a temperature T.

(a) Find an expression for the density of single-particle states D(&) as

a function of the single-particle energy E. Sketch the result.

(b) Write down an expression for the mean occupation number of a
single particle state, E, as a function of E,T, and the chemical potential
p(T). Draw this function on your sketch in part (a) for a moderately high
temperature] that is, a temperature above the Bose-Einstein transition.
Indicate the place on the &-axis where E = 1.1.

(c) Write down an integral expression which implicitly determines

p(T). Referring to your sketch in (a), determine in which direction p(T)

moves as T is lowered.

(d) Find an expression for the Bose-Einstein transition temperature,

T,, below which one must have a macroscopic occupation of some single-

particle states. Leave your answer in terms of a dimensionless integral.

(e) What is p(T) for T < T,?

Describe E(E, 7') for T < Tc?

(f) Find an exact expression for the total energy, U(TIV) of the gas

for T < Tc. Leave your answer in terms of a dimensionless integral.

(MITI

Solution:

(a) From e = p2/2rn and

47rV

h3

D(&)d& = -p2dp

we find

27rv

h3

D(&) = -(2rn)3/2E1/2.