Problems and Solutions on Thermodynamics and Statistical Mechanics

Statistical Physics 307

When Vl = V2, AS = 0 as expected.

2128
(a) Calculate the partition function z of one spinless atom of mass M
moving freely in a cube of volume V = L3. Express your result in terms of
the quantum concentration

MkT 3/2

n,=(.> *

Explain the physical meaning of n,.

(b) An ideal gas of N spinless atoms occupies a volume V at temper-
ature T. Each atom has only two energy levels separated by an energy
A. Find the chemical potential, free energy, entropy, pressure and heat
capacity at constant pressure.

Solution:

(SVNY, Buflulo)

(a) The energy eigenvalues are given by

h2

2mL2

2M

s = -(n2 + np + n;) ,

2

1 2 2-P

= - (P: + Py + P,) - 2M 9

where n,,ny,nl =O,fl, ....

The energy levels can be thought of as quasi-continuous, so that the

number of quantum states in the range p -+ p+ dp is --p2dp, whence the

number of states in the energy interval E + E + ds is -(2M)3/2fide.

Hence

4sv

h3

2sv

h3

is the average number of quantum states in unit

MkT 3/2
where n, = (7)

volume.

(b) The classical ideal gas satisfies the non-degeneracy condition. The

partition function of a sub-system is z = exp(-PsI) + exp(-Ps2), e2 =