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

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

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, 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 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

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} = - k_B T \left[ \frac{3}{2} \ln \left(\frac{2\pi m k_BT}{h^2}\right) + \frac{\partial \ln Q_N}{\partial N} \right]}

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

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 $Q_{N}$ 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

The Kirkwood charging formula is given by

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

• Constant chemical potential molecular dynamics (CμMD)
• Ideal gas: Chemical potential
• Overlapping distribution method
• Widom test-particle method

References

Related reading

• G. Cook and R. H. Dickerson "Understanding the chemical potential", American Journal of Physics 63 pp. 737-742 (1995)
• T. A. Kaplan "The Chemical Potential", Journal of Statistical Physics 122 pp. 1237-1260 (2006)
• Federico G. Pazzona, Pierfranco Demontis, and Giuseppe B. Suffritti "Chemical potential evaluation in NVT lattice-gas simulations", Journal of Chemical Physics 137 154106 (2012)
• E. A. Ustinov "Efficient chemical potential evaluation with kinetic Monte Carlo method and non-uniform external potential: Lennard-Jones fluid, liquid, and solid", Journal of Chemical Physics 147 014105 (2017)
• Claudio Perego, Omar Valsson, and Michele Parrinello "Chemical potential calculations in non-homogeneous liquids", Journal of Chemical Physics 149, 072305 (2018)