Problems and Solutions on Thermodynamics and Statistical Mechanics

5latiatical Phyaica 177

Solution:

tions:

distinguished from one another.

a quantum state.

st at ist ics.

(a) FD, as compared with MB, statistics has two additional assump-

1. The principle of indistinguishability: identical particles cannot be

2) Pauli’s exclusion principle: Not more than one particle can occupy

In the limit of non-degeneracy, FD statistics gradually becomes MB

(b) P(E) gives the number of particles in unit interval of energy or at
energy level E. Figure 2.5 gives rough plots of the energy distributions ((a)
MB, (b) FD).

P(&)h-P(E)l ft&

72 ’ T1

I

E &F

(a) MB statistics (b) FD statistics

Fig. 2.5.

(c) According to MB statistics (or the principle of equipartition of

energy), the contribution of an electron to the specific heat of a metal

should be 1.5 K. This is not borne out by experiments, which shows that

the contribution to specific heat of free electrons in metal can usually be

neglected except for the case of very low temperatures. At low temperatures

the contribution of electrons to the specific heat is proportional to the

temperture 7’. FD statistics which incorporates Pauli’s exclusion principle

can explain this result.

2016

State which statistics (classical Maxwell-Boltzmann; Fermi-Dirac; or

Bose-Einstein) would be appropriate in these problems and explain why

(semi-quantit atively) :

(a) Density of He4 gas at room temperature and pressure.

(b) Density of electrons in copper at room temperature.

(c) Density of electrons and holes in semiconducting Ge at room tem-

(VC, Berkeley)

perature (Ge band-gap w 1 volt).