Section 4.4.8.1: Implicit Differentiation of Redlich-Kwong equation

In general a pressure explicit EOS is

$\begin{displaymath}p = \Pi(T,v)\end{displaymath}$

Differentiate the EOS with respect to p at constant T.

$\begin{displaymath}1 = \frac{\partial \Pi}{\partial v} \left(\frac{\partial v}{\partial p}\right)_T \end{displaymath}$

$\begin{displaymath}\left(\frac{\partial v}{\partial p}\right)_T \equiv \pi_1(T,v) = \frac{1}{\frac{\partial \Pi}{\partial v}} \end{displaymath}$

For $(\partial v/\partial T)_p$ , differentiate with respect to T at constant p.

$\begin{displaymath}0 = \frac{\partial \Pi}{\partial T} + \frac{\partial \Pi}{\partial v} \left(\frac{\partial v}{\partial T}\right)_p \end{displaymath}$

$\begin{displaymath}\left(\frac{\partial v}{\partial T}\right)_p \equiv \pi_2(T,v... ...i}{\partial v}} = - \pi_1(T,v) \frac{\partial \Pi}{\partial T} \end{displaymath}$

For the Redlich-Kwong equation

$\begin{displaymath}\Pi(T,v) = \frac{RT}{v-b} - \frac{a}{T^{\frac{1}{2}} v(v+b)} \end{displaymath}$

$\begin{displaymath}\frac{\partial \Pi}{\partial v} = \frac{-RT}{(v-b)^2} + \frac{a}{T^{\frac{1}{2}}} \frac{(2 v + b)}{v^2(v+b)^2} \end{displaymath}$

$\begin{displaymath}\pi_1(T,v) = \frac{1}{\frac{-RT}{(v-b)^2} + \frac{a}{T^{\frac{1}{2}}} \frac{(2 v + b)}{v^2(v+b)^2}} \end{displaymath}$

$\begin{displaymath}\frac{\partial \Pi}{\partial T} = \frac{R}{v-b} + \frac{a}{2 T^{\frac{3}{2}} v(v+b)} \end{displaymath}$

$\begin{displaymath}\pi_2(T,v) = - \frac{ \frac{R}{v-b} + \frac{a}{2 T^{\frac{3}{... ...)^2} + \frac{a}{T^{\frac{1}{2}}} \frac{(2 v + b)}{v^2(v+b)^2}} \end{displaymath}$

Though rather complex to look at, these expressions are easy to evaluate given v. Non-integer powers of T = T₁ would be calculated before the integration over p, to avoid repetition of the more expensive computations.

Return to Section 4.4.8 Index

Return to Section 4.4 Index

Course Organiser

Last modified: Wed Aug 5 16:10:30 BST