Editing Chemical potential
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Latest revision | Your text | ||
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Definition: | Definition: | ||
:<math>\mu=\left. \frac{\partial G}{\partial N}\right\vert_{T,p | :<math>\mu=\left. \frac{\partial G}{\partial N}\right\vert_{T,p}</math> | ||
where <math>G</math> is the [[Gibbs energy function]], leading to | where <math>G</math> is the [[Gibbs energy function]], leading to | ||
:<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> | where <math>A</math> is the [[Helmholtz energy function]], <math>k_B</math> | ||
is the [[Boltzmann constant]], <math>p</math> is the | 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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number of particles | 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} = - | :<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> | where <math>Z_N</math> is the [[partition function]] for a fluid of <math>N</math> | ||
identical particles | identical particles | ||
:<math>Z_N= \left( \frac{2\pi m k_BT}{h^2} \right)^{3N/2} Q_N</math> | :<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]] | ||
[ | |||
:<math>Q_N = \frac{1}{N!} \int ... \int \exp (-U_N/k_B T) dr_1...dr_N</math> | :<math>Q_N = \frac{1}{N!} \int ... \int \exp (-U_N/k_B T) dr_1...dr_N</math> | ||
==Kirkwood charging formula== | ==Kirkwood charging formula== | ||
See Ref. 2 | |||
:<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> | :<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]]. | 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== | ==See also== | ||
*[[Ideal gas: Chemical potential]] | *[[Ideal gas: Chemical potential]] | ||
==References== | ==References== | ||
#[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/10.1063/1.1749657 John G. Kirkwood "Statistical Mechanics of Fluid Mixtures", Journal of Chemical Physics '''3''' pp. 300-313 (1935)] | |||
[[category:classical thermodynamics]] | [[category:classical thermodynamics]] | ||
[[category:statistical mechanics]] | [[category:statistical mechanics]] |