# Chemical potential

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## Classical thermodynamics

Definition:

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

where ${\displaystyle G}$ is the Gibbs energy function, leading to

${\displaystyle {\frac {\mu }{k_{B}T}}={\frac {G}{Nk_{B}T}}={\frac {A}{Nk_{B}T}}+{\frac {pV}{Nk_{B}T}}}$

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

## Statistical mechanics

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

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

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

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

and ${\displaystyle Q_{N}}$ is the configurational integral

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

## Kirkwood charging formula

The Kirkwood charging formula is given by [1]

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

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

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