Cubic Equations of State (PR, SRK, PR78)¶

Peng-Robinson, SRK and Peng-Robinson (1978) share one generalized cubic, with constants \((\epsilon_{1},\epsilon_{2})=(1+\sqrt{2},\,1-\sqrt{2})\) for the Peng-Robinson family and \((1,0)\) for SRK. Write \(u=\epsilon_{1}+\epsilon_{2}\) , \(w=\epsilon_{1}\epsilon_{2}\) , \(\delta=\epsilon_{1}-\epsilon_{2}\) . The fugacity coefficient is

\[ \ln\varphi_{i}=\frac{b_{i}}{b}(Z-1)-\ln(Z-B)-D\,q_{i}\,L \]

with the shorthands \(D=\dfrac{A}{B\delta}\) , \(q_{i}=\dfrac{2\bar{a}_{i}}{a}-\dfrac{b_{i}}{b}\) , \(L=\ln\dfrac{Z+\epsilon_{1}B}{Z+\epsilon_{2}B}\) , the mixture parameters \(a=\sum_{i}\sum_{j}x_{i}x_{j}a_{ij}\) , \(b=\sum_{i}x_{i}b_{i}\) , \(\bar{a}_{i}=\sum_{j}x_{j}a_{ij}\) , and \(A=aP/(RT)^{2}\) , \(B=bP/(RT)\) . The compressibility \(Z\) is the selected root of

\[ \begin{aligned}Z^{3}+c_{2}Z^{2}+c_{1}Z+c_{0} & =0\\ c_{2}=(u-1)B-1,\quad c_{1} & =A+wB^{2}-uB(1+B)\\ c_{0} & =-AB-wB^{2}(1+B) \end{aligned} \]

Temperature derivative¶

Temperature enters through the attractive parameter. With the alpha function

\[ \begin{aligned}a_{i} & =\Omega_{a}\frac{R^{2}T_{c,i}^{2}}{P_{c,i}}\alpha_{i},\qquad\alpha_{i}=\left[1+\kappa_{i}\!\left(1-\sqrt{T/T_{c,i}}\right)\right]^{2}\\ a_{ij} & =\sqrt{a_{i}a_{j}}\,(1-k_{ij}) \end{aligned} \]

\[ \begin{aligned}\frac{da_{i}}{dT} & =-\Omega_{a}\frac{R^{2}T_{c,i}^{2}}{P_{c,i}}\frac{\kappa_{i}\sqrt{\alpha_{i}}}{\sqrt{T\,T_{c,i}}}\\ \frac{da_{ij}}{dT} & =\frac{1-k_{ij}}{2\sqrt{a_{i}a_{j}}}\!\left(a_{i}\frac{da_{j}}{dT}+a_{j}\frac{da_{i}}{dT}\right) \end{aligned} \]

so that \(\partial a/\partial T=\sum_{i}\sum_{j}x_{i}x_{j}\,da_{ij}/dT\) and \(\partial\bar{a}_{i}/\partial T=\sum_{j}x_{j}\,da_{ij}/dT\) . The dimensionless groups and the compressibility follow as

\[ \frac{\partial B}{\partial T}=-\frac{B}{T},\qquad\frac{\partial A}{\partial T}=\frac{P}{(RT)^{2}}\!\left(\frac{\partial a}{\partial T}-\frac{2a}{T}\right) \]

\[ \begin{aligned}\frac{\partial c_{2}}{\partial T} & =(u-1)\frac{\partial B}{\partial T}\\ \frac{\partial c_{1}}{\partial T} & =\frac{\partial A}{\partial T}+\left[2wB-u(1+2B)\right]\frac{\partial B}{\partial T}\\ \frac{\partial c_{0}}{\partial T} & =-B\frac{\partial A}{\partial T}-\left[A+w(2B+3B^{2})\right]\frac{\partial B}{\partial T} \end{aligned} \]

\[ \frac{\partial Z}{\partial T}=-\frac{Z^{2}\,\partial c_{2}/\partial T+Z\,\partial c_{1}/\partial T+\partial c_{0}/\partial T}{3Z^{2}+2c_{2}Z+c_{1}} \]

The full temperature derivative of the fugacity coefficient is then

\[ \begin{aligned}\frac{\partial\ln\varphi_{i}}{\partial T} & =\frac{b_{i}}{b}\frac{\partial Z}{\partial T}-\frac{\partial Z/\partial T-\partial B/\partial T}{Z-B}\\ & \quad{}-\frac{\partial D}{\partial T}q_{i}L-D\frac{\partial q_{i}}{\partial T}L-Dq_{i}\frac{\partial L}{\partial T} \end{aligned} \]

with the building-block derivatives

\[ \begin{aligned}\frac{\partial D}{\partial T} & =D\!\left(\frac{1}{A}\frac{\partial A}{\partial T}-\frac{1}{B}\frac{\partial B}{\partial T}\right),\qquad\frac{\partial q_{i}}{\partial T}=\frac{2}{a}\!\left(\frac{\partial\bar{a}_{i}}{\partial T}-\frac{\bar{a}_{i}}{a}\frac{\partial a}{\partial T}\right)\\ \frac{\partial L}{\partial T} & =\frac{\partial Z/\partial T+\epsilon_{1}\partial B/\partial T}{Z+\epsilon_{1}B}-\frac{\partial Z/\partial T+\epsilon_{2}\partial B/\partial T}{Z+\epsilon_{2}B} \end{aligned} \]

Composition derivative¶

On the one-mole basis the partial-molar (Michelsen-Mollerup) derivatives of the mixture parameters are

\[ \begin{aligned}\frac{\partial b}{\partial n_{j}} & =b_{j}-b,\quad\frac{\partial a}{\partial n_{j}}=2\bar{a}_{j}-2a,\quad\frac{\partial\bar{a}_{i}}{\partial n_{j}}=a_{ij}-\bar{a}_{i}\\ \frac{\partial A}{\partial n_{j}} & =\frac{A}{a}\frac{\partial a}{\partial n_{j}},\qquad\frac{\partial B}{\partial n_{j}}=\frac{B}{b}\frac{\partial b}{\partial n_{j}} \end{aligned} \]

The cubic-coefficient and compressibility derivatives have the same form as the temperature case, with the temperature derivative replaced by the mole-number derivative throughout. The full composition derivative of the fugacity coefficient is

\[ \begin{aligned}\frac{\partial\ln\varphi_{i}}{\partial n_{j}} & =\frac{b_{i}}{b}\!\left(\frac{\partial Z}{\partial n_{j}}-\frac{Z-1}{b}\frac{\partial b}{\partial n_{j}}\right)-\frac{\partial Z/\partial n_{j}-\partial B/\partial n_{j}}{Z-B}\\ & \quad{}-\frac{\partial D}{\partial n_{j}}q_{i}L-D\frac{\partial q_{i}}{\partial n_{j}}L-Dq_{i}\frac{\partial L}{\partial n_{j}} \end{aligned} \]

with

\[ \begin{aligned}\frac{\partial D}{\partial n_{j}} & =D\!\left(\frac{1}{A}\frac{\partial A}{\partial n_{j}}-\frac{1}{B}\frac{\partial B}{\partial n_{j}}\right)\\ \frac{\partial q_{i}}{\partial n_{j}} & =\frac{2}{a}\!\left(\frac{\partial\bar{a}_{i}}{\partial n_{j}}-\frac{\bar{a}_{i}}{a}\frac{\partial a}{\partial n_{j}}\right)+\frac{b_{i}}{b^{2}}\frac{\partial b}{\partial n_{j}}\\ \frac{\partial L}{\partial n_{j}} & =\frac{\partial Z/\partial n_{j}+\epsilon_{1}\partial B/\partial n_{j}}{Z+\epsilon_{1}B}-\frac{\partial Z/\partial n_{j}+\epsilon_{2}\partial B/\partial n_{j}}{Z+\epsilon_{2}B} \end{aligned} \]

A useful check is the Maxwell symmetry \(\partial\ln\varphi_{i}/\partial n_{j}=\partial\ln\varphi_{j}/\partial n_{i}\) .