Fermi and Bose Gas Pressure
المؤلف:
Sidney B. Cahn, Gerald D. Mahan And Boris E. Nadgorny
المصدر:
A GUIDE TO PHYSICS PROBLEMS
الجزء والصفحة:
part 2 , p 36
26-8-2016
1366
Fermi and Bose Gas Pressure
For a photon gas the entropy is
(i)
where ωi is the angular frequency of the ith mode. Using (i):
a) Show that the isothermal work done by the gas is
where ni 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)
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