Chemical potential

Classical thermodynamics[edit]

Definition:

μ = {\frac{\partial G}{\partial N} |}_{T, p} = {\frac{\partial A}{\partial N} |}_{T, V}

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

\frac{μ}{k_{B} T} = \frac{G}{N k_{B} T} = \frac{A}{N k_{B} T} + \frac{p V}{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.

The chemical potential is the derivative of the Helmholtz energy function with respect to the number of particles

μ = {\frac{\partial A}{\partial N} |}_{T, V} = \frac{\partial (- k_{B} T \ln Z_{N})}{\partial N} = - k_{B} T [\frac{3}{2} \ln (\frac{2 π 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 π m k_{B} T}{h^{2}})}^{3 N / 2} Q_{N}

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

The Kirkwood charging formula is given by ^[1]

β μ_{e x} = ρ \int_{0}^{1} d λ \int \frac{\partial β Φ_{12} (r, λ)}{\partial λ} g (r, λ) d r

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