Chemical potential

Classical thermodynamics

Definition:

$\mu ={\frac{\partial G}{\partial N}|}_{T,p}$

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

$\mu =\frac{A}{N{k}_{B}T}+\frac{pV}{N{k}_{B}T}$

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

$\mu ={\frac{\partial A}{\partial N}|}_{T,V}=\frac{\partial (-k_{B}T\ln Z_N)}{\partial N}=-\frac{3}{2}{k}_{B}T\ln (\frac{2\pi m{k}_{B}T}{h^2})+\frac{\partial \ln Q_N}{\partial N}$

where $Z_N$ is the partition function for a fluid of $N$ identical particles

$Z_N={(\frac{2\pi m{k}_{B}T}{h^2})}^{3N/2}{Q}_N$

and $Q_N$ is the configurational integral

$Q_N=\frac{1}{N!}\int ...\int \exp (-U_N/k_{B}T)d{r}_1...d{r}_N$

Kirkwood charging formula

See Ref. 2

$\beta {\mu }_{\mathrm{e}\mathrm{x}}=\rho \underset{0}{\overset{1}{\int }}d\lambda \int \frac{\partial \beta {\mathrm{\Phi }}_{12}(r,\lambda )}{\partial \lambda }\mathrm{g}(r,\lambda )dr$

where ${\mathrm{\Phi }}_{12}(r)$ is the intermolecular pair potential and $\mathrm{g}(r)$ 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)