Redlich-Kwong equation of state

The Redlich-Kwong equation of state is :

$\left[p+{\frac {a}{T^{1/2}v(v+b)}}\right](v-b)=RT$.

The Redlich-Kwong equation of state has a critical point compressibility factor of :

$Z_{c}={\frac {p_{c}v_{c}}{RT_{c}}}={\frac {1}{3}}$

leading to

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle a={\frac {1}{9(2^{1/3}-1)}}{\frac {R^{2}T_{c}^{5/2}}{p_{c}}}\approx 0.4274802336{\frac {R^{2}T_{c}^{5/2}}{p_{c}}}}

and

$b={\frac {(2^{1/3}-1)}{3}}{\frac {RT_{c}}{p_{c}}}\approx 0.08664034995{\frac {RT_{c}}{p_{c}}}$

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

Soave Modification

A modification of the the Redlich-Kwong equation of state was presented by Giorgio Soave in order to allow better representation of non-spherical molecules. In order to do this, the square root temperature dependence was replaced with a temperature dependent acentricity factor:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \alpha (T)=\left(1+\left(0.48508+1.55171\omega -0.15613\omega ^{2}\right)\left(1-{\sqrt {\frac {T}{T_{c}}}}\right)\right)^{2}}

where $T_{c}$ is the critical temperature and $\omega$ is the acentric factor for the gas. This leads to an equation of state of the form:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left[p+{\frac {a\alpha (T)}{v(v+b)}}\right]\left(v-b\right)=RT}

or equivalently:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle p={\frac {RT}{v-b}}-{\frac {a\alpha (T)}{v(v+b)}}}