Chemical potential

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Classical thermodynamics[edit]


\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}{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.

Statistical mechanics[edit]

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 m k_BT}{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 m k_BT}{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[edit]

The Kirkwood charging formula is given by [1]

\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[edit]


Related reading