Compressibility factor (Z) of the Redlich-Kwong Equation I've been having a problem trying to calculate the compressibility factor of the Redlich-Kwong equation:
\begin{equation}
P = \frac{RT}{v-b}-\frac{a}{\sqrt{T} \cdot (v^2+vb)}
\end{equation}
The compressibility factor is calculated as:
\begin{equation}
z = \frac{P \cdot v}{RT}
\end{equation}
This factor is calculated if we substitute the molar volume ($v$), but I can't express the Redlich-Kwong equation in terms of $P$ and $T$, that's why I would like to know if some of you guys could help me to isolate the $v$ of this Redlich-Kwong equation. Thanks! :)
 A: \begin{align*}
P &= \frac{RT}{v-b} - \frac{a}{\sqrt{T}(v^2+vb)}\\
&=\frac{RT\sqrt{T}(v^2+vb)-a(v-b)}{\sqrt{T}(v^2+vb)(v-b)}\\
\implies P\sqrt{T}(v^3 - b^2 v)
&-(RT\sqrt{T}(v^2+bv)-av+ab)=0\\
\implies  P \sqrt{T} v^3 - b^2 P \sqrt{T} v
&-a b + a v - b R \sqrt{T^3} v - R \sqrt{T^3} v^2=0 \\
\implies  P \sqrt{T} v^3- R \sqrt{T^3} v^2 
&- \big(b^2 P \sqrt{T}+b R \sqrt{T^3} -a\big)v - a b =0 \\
\end{align*}
We now have a cubic of the form
$\quad ax^3+x^2+cx+d=0\quad$ where
$$a=P \sqrt{T}\quad 
b=- R \sqrt{T^3}\quad 
c=- \big(b^2 P \sqrt{T}+b R \sqrt{T^3} -a\big)\quad 
d=-a$$
and these may be plugged into the
The Cubic Formula
to obtain one real root. The cubic will be a product of this "factor" and a quadratic equation. It's not pretty. The cubic formula looks like this.
\begin{align*}
n&=\sqrt[\huge{3}]{\biggl(\frac{-b^3}{27a^3 }+\frac{bc}{6a^2}-\frac{d}{2a}\biggr)+\sqrt{\biggl(\frac{-b^3}{27a^3}+\frac{bc}{6a^2}-\frac{d}{2a}\biggr)^2+\biggl(\frac{c}{3a}-\frac{b^2}{9a^2}\biggr)^3}}\\
&+\sqrt[\huge{3}]{\biggl(\frac{-b^3}{27a^3 }+\frac{bc}{6a^2}-\frac{d}{2a}\biggr)-\sqrt{\biggl(\frac{-b^3}{27a^3}+\frac{bc}{6a^2}-\frac{d}{2a}\biggr)^2+\biggl(\frac{c}{3a}-\frac{b^2}{9a^2}\biggr)^3}}\\
&-\frac{b}{3a}   \end{align*}
A: Determine the critical compressibility factor $$=/$$ for a gas obeying
a) Dieterici equation,
b) Berthelot equation,
c) Redlich-Kwong equation.
a) Dieterici equation $=(/(−))−/$,
For which $=422/2$ and $=/2$.
b) Berthelot equation $(+/2)(−)=$,
For which $=(27/64)23/$ and $=/8$.
c) Redlich-Kwong equation
