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

\[a = \frac{1}{9(2^{1/3}-1)} \frac{R^2T_c^{5/2}}{p_c} \approx 0.4274802336 \frac{R^2T_c^{5/2}}{p_c}\]

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 acentric factor (\(\omega\)):

\[\alpha(T)=\left(1+\left(0.48508+1.55171\omega-0.15613\omega^2\right)\left(1-\sqrt\frac{T}{T_c}\right)\right)^2 \]

where \(T_c\) is the critical temperature. This leads to an equation of state of the form:

\[ \left[p+\frac{a\alpha(T)}{v(v+b)}\right]\left(v-b\right)=RT\]

or equivalently:

\[ p=\frac{RT}{v-b}-\frac{a\alpha(T)}{v(v+b)}\]