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| ==Classical thermodynamics==
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| Definition: | | Definition: |
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| :<math>\mu=\left. \frac{\partial G}{\partial N}\right\vert_{T,p} = \left. \frac{\partial A}{\partial N}\right\vert_{T,V}</math> | | :<math>\mu=\frac{\partial G}{\partial N}</math> |
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| where <math>G</math> is the [[Gibbs energy function]], leading to | | where <math>G</math> is the [[Gibbs energy function]], leading to |
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| :<math>\frac{\mu}{k_B T}=\frac{G}{N k_B T}=\frac{A}{N k_B T}+\frac{p V}{N k_B T}</math> | | :<math>\mu=\frac{A}{Nk_B T} + \frac{pV}{Nk_BT}</math> |
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| where <math>A</math> is the [[Helmholtz energy function]], <math>k_B</math> | | where <math>A</math> is the [[Helmholtz energy function]], <math>k_B</math> |
| is the [[Boltzmann constant]], <math>p</math> is the [[pressure]], <math>T</math> is the [[temperature]] and <math>V</math> | | is the [[Boltzmann constant]], <math>p</math> is the pressure, <math>T</math> is the temperature and <math>V</math> |
| is the volume. | | is the volume. |
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| ==Statistical mechanics==
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| The chemical potential is the derivative of the [[Helmholtz energy function]] with respect to the
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| number of particles
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| :<math>\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]</math>
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| where <math>Z_N</math> is the [[partition function]] for a fluid of <math>N</math>
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| identical particles
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| :<math>Z_N= \left( \frac{2\pi m k_BT}{h^2} \right)^{3N/2} Q_N</math>
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| and <math>Q_N</math> is the
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| [http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 configurational integral]
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| :<math>Q_N = \frac{1}{N!} \int ... \int \exp (-U_N/k_B T) dr_1...dr_N</math>
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| ==Kirkwood charging formula==
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| The Kirkwood charging formula is given by <ref>[http://dx.doi.org/10.1063/1.1749657 John G. Kirkwood "Statistical Mechanics of Fluid Mixtures", Journal of Chemical Physics '''3''' pp. 300-313 (1935)]</ref>
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| :<math>\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</math>
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| where <math>\Phi_{12}(r)</math> is the [[intermolecular pair potential]] and <math>{\rm g}(r)</math> is the [[Pair distribution function | pair correlation function]].
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| ==See also==
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| *[[Constant chemical potential molecular dynamics (CμMD)]]
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| *[[Ideal gas: Chemical potential]]
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| *[[Overlapping distribution method]]
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| *[[Widom test-particle method]]
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| ==References==
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| <references/>
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| '''Related reading'''
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| *[http://dx.doi.org/10.1119/1.17844 G. Cook and R. H. Dickerson "Understanding the chemical potential", American Journal of Physics '''63''' pp. 737-742 (1995)]
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| *[http://dx.doi.org/10.1007/s10955-005-8067-x T. A. Kaplan "The Chemical Potential", Journal of Statistical Physics '''122''' pp. 1237-1260 (2006)]
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| *[http://dx.doi.org/10.1063/1.4758757 Federico G. Pazzona, Pierfranco Demontis, and Giuseppe B. Suffritti "Chemical potential evaluation in NVT lattice-gas simulations", Journal of Chemical Physics '''137''' 154106 (2012)]
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| *[http://dx.doi.org/10.1063/1.4991324 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)]
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| *[https://doi.org/10.1063/1.5024631 Claudio Perego, Omar Valsson, and Michele Parrinello "Chemical potential calculations in non-homogeneous liquids", Journal of Chemical Physics 149, 072305 (2018)]
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| [[category:classical thermodynamics]]
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| [[category:statistical mechanics]]
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