- Problem 1: Phonons
For a phonon gas, starting from the partition function we wrote in class and
using the Debye approxmation for the density of states, derive the expression
for
the heat capacity CV.
- Problem 2: Photons
We saw that the free energy for a gas of photons is, to a good approximation,
F = –(4 /3c) VT 4.
Show all the steps that lead to the equation of state
for photons from this expression and, after verifying that it does
not fit into the general form of a virial expansion, explain the physical
reason why it does not.
- Problem 3: Paramagnetism
The third law of thermodynamics states that in the low-temperature limit, the
entropy of a system tends to a constant, or that the heat capacity of the system
tends to zero. Show that in the classical description of paramagnetism the
third law is not satisfied, while in the quantum description it is satisfied.
- Problem 4: Paramagnetism
Derive an expression for the magnetic susceptibility for a paramagnetic material,
starting from the canonical partition function Z, both in the classical and
in the quantum case. In each case, show that the susceptibility is indeed positive.
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