Chemical potential: Difference between revisions
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Carl McBride (talk | contribs) No edit summary |
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Definition: | Definition: | ||
:<math>\mu=\frac{\partial G}{\partial N}</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 | where <math>G</math> is the [[Gibbs energy function]], leading to | ||
| Line 16: | Line 16: | ||
number of particles | 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> | :<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 | ||
Revision as of 16:42, 22 May 2007
Classical thermodynamics
Definition:
where is the Gibbs energy function, leading to
where is the Helmholtz energy function, is the Boltzmann constant, is the pressure, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} is the temperature and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is the volume.
Statistical mechanics
The chemical potential is the derivative of the Helmholtz energy function with respect to the number of particles
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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}}
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z_N} is the partition function for a fluid of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N} identical particles
and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_N} is the configurational integral
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q_N = \frac{1}{N!} \int ... \int \exp (-U_N/k_B T) dr_1...dr_N}