Chemical potential: Difference between revisions

Revision as of 14:11, 16 April 2015

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

{\frac {\mu }{k_{B}T}}={\frac {G}{Nk_{B}T}}={\frac {A}{Nk_{B}T}}+{\frac {pV}{Nk_{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.

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 mk_{B}T}{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 mk_{B}T}{h^{2}}}\right)^{3N/2}Q_{N}

Q_{N}={\frac {1}{N!}}\int ...\int \exp(-U_{N}/k_{B}T)dr_{1}...dr_{N}

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.

Related reading

@@ Line 32: / Line 32: @@
 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]]
-*[[Overlapping distribution method]]
 ==References==