Problems and Solutions on Thermodynamics and Statistical Mechanics

368 Pwblema tY Sdutiona on "'hermodynam'ca d Stati~tical Mechanic8

is much smaller than the wavelength of sound waves. Therefore, the propa-

gation of sound through air can be considered adiabatic, i.e., a quasi-static

process.

2178
Consider a non-interacting relativistic Fermi gas at zero temperature.
(a) Write down expressions for the pressure and the energy density in
the rest frame of the gas. What is the equation of state?
(b) Treating the system as a uniform static fluid, derive a wave equation
for the propagation of small density fluctuations, and hence deduce an
expression for the velocity of sound in the gas.

Solution:

particle is given by E = pc. The energy density is

(SVNY, Bufulo)

(a) The relation between the momentum and energy of a relativistic

u = (g) (25 + 1) IPF sp2dp = (25 + 1)T- CP; }

0 h3

where J is the spin quantum number of Fermions and p~ is given by the

equation

N = (25+ l)--TpF v4 3.

h3 3

Hence

The pressure is

au

P=-y a(uv) =-u+VaV=3,

with E = uV.

(b) Let p = po + 6p and p = po + 6p, where po and po are the density

and pressure of the fluid respectively, and 6p and 6p are the corresponding

fluctuations. For a static fluid, v = 6v, and the continuity equation is

__ at + pov ' 6v = 0.