Difference between revisions of "Pressure equation"

From SklogWiki
Jump to: navigation, search
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>,
 
where <math>\beta = 1/k_BT</math>,

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_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.

References