Chemical potential

Classical thermodynamics

Definition:

\mu =\left.{\frac {\partial G}{\partial N}}\right\vert _{T,p}=\left.{\frac {\partial A}{\partial N}}\right\vert _{T,V}

where $G$ is the Gibbs energy function, leading to

\mu ={\frac {A}{Nk_{B}T}}+{\frac {pV}{Nk_{B}T}}

where $A$ is the Helmholtz energy function, $k_{B}$ is the Boltzmann constant, $p$ is the pressure, $T$ is the temperature and $V$ is the volume.

The chemical potential is the derivative of the Helmholtz energy function with respect to the number of particles

\mu =\left.{\frac {\partial A}{\partial N}}\right\vert _{T,V}={\frac {\partial (-k_{B}T\ln Z_{N})}{\partial N}}=-{\frac {3}{2}}k_{B}T\ln \left({\frac {2\pi mk_{B}T}{h^{2}}}\right)+{\frac {\partial \ln Q_{N}}{\partial N}}

where $Z_{N}$ is the partition function for a fluid of $N$ identical particles

Z_{N}=\left({\frac {2\pi mk_{B}T}{h^{2}}}\right)^{3N/2}Q_{N}

Q_{N}={\frac {1}{N!}}\int ...\int \exp(-U_{N}/k_{B}T)dr_{1}...dr_{N}

See Ref. 2

\beta \mu _{\rm {ex}}=\rho \int _{0}^{1}d\lambda \int {\frac {\partial \beta \Phi _{12}(r,\lambda )}{\partial \lambda }}{\rm {g}}(r,\lambda )dr

where $\Phi _{12}(r)$ is the intermolecular pair potential and ${\rm {g}}(r)$ is the pair correlation function.