Problems and Solutions on Thermodynamics and Statistical Mechanics

Statistical Physics 303

The relation between entropy and temperature is shown in Fig. 2.28.

T

Fig. 2.28.

2125

N weakly coupled particles obeying Maxwell-Boltzmann statistics may

each exist in one oi the 3 non-degenerate energy levels of energies -E, 0,

+E. The system is in contact with a thermal reservoir at temperature T.

(a) What is the entropy of the system at T = 0 K?

(b) What is the maximum possible entropy of the system?

(c) What is the minimum possible energy of the system?

(d) What is the partition function of the system?

(e) What is the most probable energy of the system?

(f) If C(T) is the heat capacity of the system, what is the value of

Irn C(T)dT?

oT

(UC, Berkeley)

Solution:

(a) At T = 0 K, the entropy of the system is S(0) = 0.

(b) The maximum entropy of the system is

S,,,,, = klnn,,,,, = kln3N = Nkln3.

(c) The minimum energy of the system is -NE.

(d) The partition function of the system is

= (p/kT + 1 + ,-EIkT)N.