Chemical potential: Difference between revisions

Revision as of 12:14, 23 August 2013

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_{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 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} = -\frac{3}{2} k_BT \ln \left(\frac{2\pi m k_BT}{h^2}\right) + \frac{\partial \ln Q_N}{\partial N}</math>
+:<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>
 where <math>Z_N</math> is the [[partition function]] for a fluid of <math>N</math>
 identical particles