Problems and Solutions on Thermodynamics and Statistical Mechanics

Statistical Phyaica 301

2123

(a) Give a definition of the partition function z for a statistical system.

(b) Find a relation between the heat capacity of a system and -

a2 Inz

aa2 '

1

kT

where P = -.

(c) For a system with one excited state at energy A above the ground
state, find an expression for the heat capacity in terms of A. Sketch the
dependence on temperature and discuss the limiting behavior for high and
low temperatures.
(UC, Berkeley)
Solution:
(a) The partition function is the sum of statistical probabilities.
For quantum statistics, z = xexp(-PE,), summing over all the
8
quantum states.
For classical statistics, z = exp(-PE)dI'/h7, integrating over the
1
phase-space where 7 is the number of degrees of freedom.
a
E = -- In z ,
ap

1 a- 1 a2

dT kP2 ap kP2 ap2

E = ---lnz,

aE

cv = - =

(c) Assume the two states are non-degenerate, then

A

eA/kT + 1

• 
  
  
  
  • 
    
  
  
  
  


• Ae-A/kT
  1 + e-AlkT
  E=
  dz eA/kT
  (1 + eAlkT)2 *

c, = - = k (&)

dT

The variation of specific heat with temperature is shown in Fig. 2.26.

f cv

Fig. 2.26.