Pressure equation: Difference between revisions

Revision as of 13:21, 28 June 2007

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_{B}T}}=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 $\beta =1/k_{B}T$ , $\Phi (r)$ is a central potential and ${\rm {g}}(r)$ is the pair distribution function.

@@ Line 1: / Line 1: @@
-The '''pressure equation''', also known as the '''virial equation''' is given by
+For particles acting through two-body central forces alone one may use the [[Thermodynamic relations | thermodynamic relation]]
-:<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^3~{\rm d}r</math>
+:<math>p = -\left. \frac{\partial A}{\partial V}\right\vert_T </math>
+Using this relation, along with the [[Helmholtz energy function]] and the [[partition function | canonical partition function]], one
+arrives at the so-called
+'''pressure equation''' (also known as the '''virial equation'''):
+:<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>,