Editing Chemical potential

==Classical thermodynamics==
Definition:

:<math>\mu=\left. \frac{\partial G}{\partial N}\right\vert_{T,p}</math>

where <math>G</math> is the [[Gibbs energy function]], leading to 

:<math>\mu=\frac{A}{Nk_B T} + \frac{pV}{Nk_BT}</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 volume.

==Statistical mechanics==
The chemical potential is the derivative of the [[Helmholtz energy function]] with respect to the 
number of particles

:<math>\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}</math>
where <math>Z_N</math> is the [[partition function]] for a fluid of <math>N</math>
identical particles
:<math>Z_N= \left( \frac{2\pi m k_BT}{h^2} \right)^{3N/2} Q_N</math>
and <math>Q_N</math> is the [[configurational integral]]
:<math>Q_N = \frac{1}{N!} \int ... \int \exp (-U_N/k_B T) dr_1...dr_N</math>
==See also==
*[[Ideal gas: Chemical potential]]
@@ Line 2: / Line 2: @@
 Definition:
-:<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=\left. \frac{\partial G}{\partial N}\right\vert_{T,p}</math>
 where <math>G</math> is the [[Gibbs energy function]], leading to
-:<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>
 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.
@@ Line 16: / Line 16: @@
 number of particles
-:<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>
+:<math>\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}</math>
 where <math>Z_N</math> is the [[partition function]] for a fluid of <math>N</math>
 identical particles
 :<math>Z_N= \left( \frac{2\pi m k_BT}{h^2} \right)^{3N/2} Q_N</math>
-and <math>Q_N</math> is the
+and <math>Q_N</math> is the [[configurational integral]]
-[http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 configurational integral]
 :<math>Q_N = \frac{1}{N!} \int ... \int \exp (-U_N/k_B T) dr_1...dr_N</math>
-==Kirkwood charging formula==
-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>
-:<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>
-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]].
 ==See also==
-*[[Constant chemical potential molecular dynamics (CμMD)]]
 *[[Ideal gas: Chemical potential]]
-*[[Overlapping distribution method]]
-*[[Widom test-particle method]]
-==References==
-<references/>
-'''Related reading'''
-*[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)]
-*[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)]
-*[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)]
-*[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)]
-*[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)]
-[[category:classical thermodynamics]]
-[[category:statistical mechanics]]