1549380323-Statistical Mechanics Theory and Molecular Simulation

442 Quantum ideal gases

3/2

T 0 T

CV

<N>k

Fig. 11.6CVas a function ofTfrom eqn. (11.6.45). ForT < T 0 , the curve increases as
T^3 /^2.

boson gas exhibits a discontinuous change at the transition temperature, signifying a
first-order phase transition (see also Section 16.1). However, using the mass and den-
sity of liquid He^4 in the expression forT 0 in eqn. (11.6.33), we obtainT 0 of about 3.14
K from the ideal gas, which is not far off the experimental transitiontemperature of
2.18 K for real liquid helium.

11.7 Problems

11.1. Derive eqn. (11.5.18). What is the analogous term for bosons?

11.2. a. Can Bose–Einstein condensation occur for an ideal gas of bosons in one

dimension? If so, determine the temperatureT 0. If not, prove that it is

not possible.

b. Can Bose–Einstein condensation occur for an ideal gas of bosons in two

dimensions? If so, determine the temperatureT 0. If not, prove that it is

not possible.