1.
The partition function for a crystal (in the Einstein model) is given by:
$Q = q^N = \left\{ \frac{\exp(\hbar\omega / 2k_B T)}{\exp(\hbar\omega / k_B T) - 1} \right\}^N = \left\{ \frac{e^{\Theta / 2T}}{e^{\Theta / T} - 1} \right\}^N$
where $\Theta$, which has units of temperature is called the \"characteristic temperature\",
and is given by $\Theta = \hbar\omega / k_B$. Using the expression for the entropy:
$S = \frac{U}{T} + k_B \ln Q = k_B T \left( \frac{\partial \ln Q}{\partial T} \right)_V + k_B \ln Q$
derive the following:
a. The equation for S for the Einstein crystal.
b. The value for S at T = 0 Kelvin.
c. The equation for dS, the total differential of S.
d. The equation for $\Delta S$ for the heating of a crystal.
