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.

Statistical mechanics

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_N= \left( \frac{2\pi m k_BT}{h^2} \right)^{3N/2} Q_N}

and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_N} is the configurational integral

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_N = \frac{1}{N!} \int ... \int \exp (-U_N/k_B T) dr_1...dr_N}

Kirkwood charging formula

See Ref. 2

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi_{12}(r)} is the intermolecular pair potential and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\rm g}(r)} is the pair correlation function.

See also

• Ideal gas: Chemical potential

References

1. T. A. Kaplan "The Chemical Potential", Journal of Statistical Physics 122 pp. 1237-1260 (2006)
2. John G. Kirkwood "Statistical Mechanics of Fluid Mixtures", Journal of Chemical Physics 3 pp. 300-313 (1935)