Fermi and Bose Gas Pressure

For a photon gas the entropy is

(i)

where ω_i is the angular frequency of the ith mode. Using (i):

a) Show that the isothermal work done by the gas is

where n_i is the average number of photons in the ith mode.

b) Show that the radiation pressure is equal to one third of the energy density:

c) Show that for a nonrelativistic Fermi gas the pressure is

SOLUTION

a) Using F = E - τS and substituting the entropy from the problem, we obtain

(1)

But

Therefore,

We may then find the pressure P of the gas:

(2)

The isothermal work done by the gas

(3)

b) For a photon gas in a cuboid box

(4)

where ξ is a constant.

(5)

So,

(6)

The same is true for a relativistic Fermi gas with dispersion law E = cp.

c) For a nonrelativistic Fermi gas the energy is

(7)

where η is a constant. So,

(8)

and

(9)