Chemical potential: Difference between revisions
Carl McBride (talk | contribs) (New page: Definition: :<math>\mu=\frac{\partial G}{\partial N}</math> where <math>G</math> is the Gibbs energy function, leading to :<math>\mu=\frac{A}{Nk_B T} + \frac{pV}{Nk_BT}</math> whe...) |
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==Classical thermodynamics== | |||
Definition: | Definition: | ||
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is the [[Boltzmann constant]], <math>p</math> is the pressure, <math>T</math> is the temperature and <math>V</math> | 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. | ||
==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]] | |||
Revision as of 16:21, 22 May 2007
Classical thermodynamics
Definition:
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 G} is the Gibbs energy function, leading to
- 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=\frac{A}{Nk_B T} + \frac{pV}{Nk_BT}}
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 A} is the Helmholtz energy function, 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 k_B} is the Boltzmann constant, 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 p} is the pressure, 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= \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}}
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
- 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= \left( \frac{2\pi m k_BT}{h^2} \right)^{3N/2} Q_N}
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}