Energy equation: Difference between revisions
Carl McBride (talk | contribs) No edit summary |
Carl McBride (talk | contribs) No edit summary |
||
| (One intermediate revision by the same user not shown) | |||
| Line 1: | Line 1: | ||
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 [[ | [[excess internal energy]] per particle, and <math>{\rm g}(r)</math> is the [[radial distribution function]]. | ||
[[category:statistical mechanics]] | [[category:statistical mechanics]] | ||
[[category: classical thermodynamics]] | |||
Latest revision as of 13:31, 29 June 2007
The energy equation is given, in classical thermodynamics, by
- 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 \left. \frac{\partial U}{\partial V} \right\vert_T = T \left. \frac{\partial p}{\partial T} \right\vert_V -p }
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.