Chemical potential
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[edit] 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
\[\mu=\frac{A}{Nk_B T} + \frac{pV}{Nk_BT}\]
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.
[edit] Statistical mechanics
The chemical potential is the derivative of the Helmholtz energy function with respect to the number of particles
\[\mu= \left. \frac{\partial A}{\partial N}\right\vert_{T,V}=\frac{\partial (-k_B T \ln Z_N)}{\partial N} = -\frac{3}{2} k_BT \ln \left(\frac{2\pi m k_BT}{h^2}\right) + \frac{\partial \ln Q_N}{\partial N}\] 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 \[Q_N = \frac{1}{N!} \int ... \int \exp (-U_N/k_B T) dr_1...dr_N\]
[edit] 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.
[edit] See also
[edit] References
- ↑ John G. Kirkwood "Statistical Mechanics of Fluid Mixtures", Journal of Chemical Physics 3 pp. 300-313 (1935)
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