Chemical potential

Classical thermodynamics

Definition:

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 \mu=\left. \frac{\partial G}{\partial N}\right\vert_{T,p}}

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

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 \mu=\frac{A}{Nk_B T} + \frac{pV}{Nk_BT}}

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 A} is the Helmholtz energy function, 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 k_B} is the Boltzmann constant, 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 p} is the pressure, $T$ is the temperature 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 V} is the volume.

Statistical mechanics

The chemical potential is the derivative of the Helmholtz energy function with respect to the number of 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 \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_BT \ln \left(\frac{2\pi m k_BT}{h^2}\right) + \frac{\partial \ln Q_N}{\partial N}}

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

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

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 ${\rm {g}}(r)$ 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)