Thermodynamic Properties in Canonical Ensemble

Since in Canonical Ensemble

\[ \begin{aligned} Z(T, V, N) &= \frac{1}{N! h^{3N}} \int d^{3N}q\, d^{3N}p \; \exp\left(-\beta \sum_{i=1}^N \frac{p_i^2}{2m}\right) \\ &= \frac{V^N (2 \pi m k_B T)^{3N/2}}{N! h^{3N}} \\ F(T, V, N) &= -k_B T \ln Z \\ S &= -\left(\frac{\partial F}{\partial T}\right)_{V, N} \\ P &= -\left(\frac{\partial F}{\partial V}\right)_{T, N} \\ \mu &= \left(\frac{\partial F}{\partial N}\right)_{T, V} \end{aligned} \]

Partition Function:

\[ \begin{aligned} Z &= \sum g_{i} e^{-\frac{E_{i}}{k T}} \quad \text{(with degeneracy)} \\ &= \sum e^{-\frac{E_{i}}{k T}} \quad \text{(without degeneracy)} \end{aligned} \]

Entropy:

\[ S = k \ln Z + \frac{3}{2} N k \]

Helmholtz Free Energy:

\[ F = E - T S = -k T \ln Z \]

Pressure:

\[ P = k T \left(\frac{\partial}{\partial V} \ln Z \right)_{T} \]

Entropy (Alternative Expression):

\[ S = k \frac{\partial}{\partial T}\left(T \ln Z\right) \]

Internal Energy:

\[ U = k T^{2} \frac{\partial}{\partial T} \left(\ln Z\right) \]

Specific Heat at Constant Volume:

\[ \begin{aligned} C_V &= 2kT \frac{\partial}{\partial T}(\ln Z) \\ &\quad + kT^2 \frac{\partial^2}{\partial T^2}(\ln Z) \end{aligned} \]

Total Energy:

\[ E = -NkT^2 \frac{\partial}{\partial T}(\ln Z) \]

Average Energy:

\[ \langle E \rangle = -kT^2 \frac{\partial}{\partial T}(\ln Z) \]

Gibbs Free Energy:

\[ G = R T - N k \ln Z \]

Enthalpy:

\[ \begin{aligned} H &= U + P V \\ &= E + R T \\ &= N\langle E\rangle + R T \\ &= N k T^2 \frac{\partial}{\partial T}(\ln Z) + R T \end{aligned} \]