Problems and Solutions on Thermodynamics and Statistical Mechanics

Statiatical Phyaica 249

(c) The theorem of equipartition of energy gives the constant volume
1
specific heat of a molecule as c, = -k where 1 is the number of degrees
2
of freedom of the molecule. For a monatomic molecule in a space of n
dimensions, 1 = n. With cp = c, + k, we get

2080

(a) Suppose one carries out a measurement of the specific heat at

constant volume, C,, for some solid as a function of temperature, TI and

obtains the results:

T C, (arbitrary units)

lOOOK 20

500 K 20

40 K 8

20 K 1

Is the solid a conductor or an insulator? Explain.

(b) If the displacement of an atom about its equilibrium position in a

harmonic solid is denoted by U, then the average displacement squared is

given by

where M is the mass of the atom, g(E) is a suitably normalized density of

energy states and n(c) is the Bose-Einstein occupation factor for phonons

of energy E. Assuming a Debye model for the density of states:

g(E) = ~E~/(AW~)~

g(E) = 0

for E < ~WD ,

for E > hw~ ,

where WD is the Debye frequency, determine the temperature depeudence

of (U2) for very high and very low temperatures. Do your results make

sense?

( Chica g 0)