Problems and Solutions on Thermodynamics and Statistical Mechanics

Statistical Physics 271

2098
Sketch the specific heat curve at constant volume, C,, as a function
of the absolute temperature, T, for a metallic solid. Give an argument
showing why the contribution to C, from the free electrons is proportional
to T.
( wis c 0 ns i n)
Solution:
As shown in Fig. 2.22, the specific heat of a metal is

C, = rT + AT3

where the first term on the right hand side is the contribution of the free
electrons and the second term is the contribution of lattice oscillation.

T2

Fig. 2.22.

For a quantitative discussion of the contribution to C, of the free

electrons see answer to Problem 2097(a).

2099
(a) Derive a formula for the maximum kinetic energy of an electron in
a non-interacting Fermi gas consisting of N electrons in a volume V at zero
absolute tempcrature.
(b) Calculate the energy gap between the ground state and first excited
state for such a Fermi gas consisting of the valence electrons in a 100A cube
of copper.
(c) Compare the energy gap with kT at 1 K.
The mass density and atomic weight of copper are 8.93 g/cm3 and 63.6
respectively.
(UC, Berkeley)