10.11: K- Useful Formulas
( \newcommand{\kernel}{\mathrm{null}\,}\)
Isothermal compressibility:
κT=−1V∂V∂p)T,N=1ρ∂ρ∂p)T=1ρ2∂2p∂μ2)T
Master thermodynamic equation:
dF=−SdT−pdV−MdH+∑iμidNi
Gibbs-Helmholtz equation:
E(T,V,N)=∂(F/T)∂(1/T))V,N=−∂lnZ∂β)V,N
Free energy from partition function:
F(T,V,N)=−kBTlnZ(T,V,N)
Classical pure point-particle partition function:
Z(T,V,N)=1h3NN!∫d3Np∫d3Nre−H(r,p)/kBT
Z(T,V,N)=1λ3N(T)∫d3Nre−U(r)/kBT where λ(T)=h√2πmkBT
Quantal ideal gases (+ for fermions, − for bosons):
Ξ(β,μ)=∏Mr=1[1±e−β(ϵr−μ)]±1
⟨nr⟩=1eβ(ϵr−μ)±1