Dieterici equation of state

The Dieterici equation of state [1] is given by

${\displaystyle p={\frac {RT}{v-b}}e^{-a/RTv}}$

where (Eq. 8 in [2]):

${\displaystyle a={\frac {4R^{2}T_{c}^{2}}{P_{c}e^{2}}}}$

and

${\displaystyle b={\frac {RT_{c}}{P_{c}e^{2}}}}$

where ${\displaystyle p}$ is the pressure, ${\displaystyle T}$ is the temperature and ${\displaystyle R}$ is the molar gas constant. ${\displaystyle T_{c}}$ is the critical temperature and ${\displaystyle P_{c}}$ is the pressure at the critical point.

Sadus [3] proposed replacing the repulsive section of the Dieterici equation with the Carnahan-Starling equation of state, which is often used to describe the equation of state of the hard sphere model, resulting in (Eq. 5):

${\displaystyle p={\frac {RT}{v}}{\frac {(1+\eta +\eta ^{2}-\eta ^{3})}{(1-\eta )^{3}}}e^{-a/RTv}}$

where ${\displaystyle \eta =b/4v}$ is the packing fraction.

This equation gives:

${\displaystyle a=2.99679RT_{c}v_{c}}$

and

${\displaystyle \eta _{c}=0.357057}$