Chemical potential: Difference between revisions

Revision as of 15:21, 22 May 2007

Definition:

μ = \frac{\partial G}{\partial N}

where $G$ is the Gibbs energy function, leading to

μ = \frac{A}{N k_{B} T} + \frac{p V}{N k_{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

μ = \frac{\partial A}{\partial N} = \frac{\partial (- k_{B} T \ln Z_{N})}{\partial N} = - \frac{3}{2} k_{B} T \ln (\frac{2 π m k_{B} T}{h^{2}}) + \frac{\partial \ln Q_{N}}{\partial N}

where $Z_{N}$ is the partition function for a fluid of $N$ identical particles

Z_{N} = {(\frac{2 π m k_{B} T}{h^{2}})}^{3 N / 2} Q_{N}

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

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+==Classical thermodynamics==
 Definition:
@@ Line 10: / Line 11: @@
 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= \frac{\partial A}{\partial N}=\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]]