Energy equation: Difference between revisions
Carl McBride (talk | contribs) (New page: The '''energy equation''' is given by :<math>\frac{U^{\rm ex}}{N}= \frac{\rho}{2} \int_0^{\infty} \Phi(r)~{\rm g}(r)~4 \pi r^2~{\rm d}r</math> where <math>\Phi(r)</math> is a ''central'' ...) |
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The '''energy equation''' is given by | The '''energy equation''' is given, in [[classical thermodynamics]], by | ||
:<math>\left. \frac{\partial U}{\partial V} \right\vert_T = T \left. \frac{\partial p}{\partial T} \right\vert_V -p </math> | |||
and in [[statistical mechanics]] it is obtained via the [[thermodynamic relations | thermodynamic relation]] | |||
:<math>U = \frac{\partial (A/T)}{\partial (1/T)}</math> | |||
and making use of the [[Helmholtz energy function]] and the canonical [[partition function]] one arrives at | |||
:<math>\frac{U^{\rm ex}}{N}= \frac{\rho}{2} \int_0^{\infty} \Phi(r)~{\rm g}(r)~4 \pi r^2~{\rm d}r</math> | :<math>\frac{U^{\rm ex}}{N}= \frac{\rho}{2} \int_0^{\infty} \Phi(r)~{\rm g}(r)~4 \pi r^2~{\rm d}r</math> | ||
where <math>\Phi(r)</math> is a ''central'' potential, <math>U^{\rm ex}</math> is the | where <math>\Phi(r)</math> is a ''two-body central'' potential, <math>U^{\rm ex}</math> is the | ||
[[excess internal energy]] per particle, and <math>{\rm g}(r)</math> is the [[radial distribution function]]. | |||
[[category:statistical mechanics]] | |||
[[category: classical thermodynamics]] | |||
Latest revision as of 14:31, 29 June 2007
The energy equation is given, in classical thermodynamics, by
and in statistical mechanics it is obtained via the thermodynamic relation
and making use of the Helmholtz energy function and the canonical partition function one arrives at
- 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 \frac{U^{\rm ex}}{N}= \frac{\rho}{2} \int_0^{\infty} \Phi(r)~{\rm g}(r)~4 \pi r^2~{\rm d}r}
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 \Phi(r)} is a two-body central potential, 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 U^{\rm ex}} is the excess internal energy per particle, 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 {\rm g}(r)} is the radial distribution function.