10.11: K- Useful Formulas

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Isothermal compressibility:

\( \kappa_{T}=-\frac{1}{V} \frac{\partial V}{\partial p} )_{T, N}=\frac{1}{\rho} \frac{\partial \rho}{\partial p} )_{T}=\frac{1}{\rho^{2}} \frac{\partial^{2} p}{\partial \mu^{2}} )_{T}\)

Master thermodynamic equation:

\( d F=-S d T-p d V-M d H+\sum_{i} \mu_{i} d N_{i}\)

Gibbs-Helmholtz equation:

\( E(T, V, N)=\frac{\partial(F / T)}{\partial(1 / T)} )_{V, N}=-\frac{\partial \ln Z}{\partial \beta} )_{V, N}\)

Free energy from partition function:

\( F(T, V, N)=-k_{B} T \ln Z(T, V, N)\)

Classical pure point-particle partition function:

\( Z(T, V, N)=\frac{1}{h^{3 N} N !} \int d^{3 N} p \int d^{3 N} r e^{-H(r, p) / k_{B} T}\)

\( Z(T, V, N)=\frac{1}{\lambda^{3 N}(T)} \int d^{3 N} r e^{-U(r) / k_{B} T} \quad \text { where } \quad \lambda(T)=\frac{h}{\sqrt{2 \pi m k_{B} T}}\)

Quantal ideal gases (+ for fermions, − for bosons):

\( \Xi(\beta, \mu)=\prod_{r=1}^{M}\left[1 \pm e^{-\beta\left(\epsilon_{r}-\mu\right)}\right]^{ \pm 1}\)

\( \left\langle n_{r}\right\rangle=\frac{1}{e^{\beta\left(\epsilon_{r}-\mu\right)} \pm 1}\)