Difference between revisions of "Pressure equation"

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:<math>p^*=\frac{\beta p}{\rho}= \frac{pV}{Nk_BT} = 1 - \beta \frac{2}{3} \pi  \rho \int_0^{\infty} \left( \frac{{\rm d}\Phi(r)} {{\rm d}r}~r \right)~{\rm g}(r)r^2~{\rm d}r</math>
 
:<math>p^*=\frac{\beta p}{\rho}= \frac{pV}{Nk_BT} = 1 - \beta \frac{2}{3} \pi  \rho \int_0^{\infty} \left( \frac{{\rm d}\Phi(r)} {{\rm d}r}~r \right)~{\rm g}(r)r^2~{\rm d}r</math>
  
where <math>\beta = 1/k_BT</math>,
+
where <math>\beta := 1/k_BT</math>,
 
<math>\Phi(r)</math> is a ''central'' [[Intermolecular pair potential | potential]] and <math>{\rm g}(r)</math> is the [[pair distribution function]].
 
<math>\Phi(r)</math> is a ''central'' [[Intermolecular pair potential | potential]] and <math>{\rm g}(r)</math> is the [[pair distribution function]].
 +
==See also==
 +
*[[Virial pressure]]
 
==References==
 
==References==
 
[[category: statistical mechanics]]
 
[[category: statistical mechanics]]

Revision as of 18:04, 27 February 2008

For particles acting through two-body central forces alone one may use the thermodynamic relation

p = -\left. \frac{\partial A}{\partial V}\right\vert_T

Using this relation, along with the Helmholtz energy function and the canonical partition function, one arrives at the so-called pressure equation (also known as the virial equation):

p^*=\frac{\beta p}{\rho}= \frac{pV}{Nk_BT} = 1 - \beta \frac{2}{3} \pi  \rho \int_0^{\infty} \left( \frac{{\rm d}\Phi(r)} {{\rm d}r}~r \right)~{\rm g}(r)r^2~{\rm d}r

where \beta := 1/k_BT, \Phi(r) is a central potential and {\rm g}(r) is the pair distribution function.

See also

References