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 $A$ 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

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

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

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

and $Q_{N}$ is the configurational integral

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

Kirkwood charging formula

The Kirkwood charging formula is given by

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

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