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Hooman Fatoorehchi Assistant Professor School of Chemical Engineering University of Tehran, Tehran, Iran  Formulas in this page are typed in $\LaTeX$. The Peng-Robinson Equation of State Original references: [1] Ding-Yu Peng and Donald B. Robinson, A New Two-Constant Equation of State, Industrial & Engineering Chemistry Fundamentals, 15(1) (1976) 59-64. DOI: 10.1021/i160057a011 [2] Donald B. Robinson, Ding-Yu Peng, The characterization of the heptanes and heavier fractions for the GPA Peng-Robinson programs. Gas Processors Association, 1978. Pressure-explicit form $ P=\frac{RT}{v-b}-\frac{a}{v^2+2bv-b^2} $ $ b=0.07780 \frac{R\hspace{0.1cm}T_c}{P_c} $ $ a=0.45724 \frac{R^2 T_c^2}{P_c} \left[1+f_{\omega}\left(1-\sqrt{T_r}\hspace{0.1cm} \right)\right]^2 $ $\omega < 0.49: f_{\omega}=0.37464+1.54226\hspace{0.1cm}\omega-0.26992\hspace{0.1cm}\omega^2$ $\omega \ge 0.49: f_{\omega}=0.379642+1.48503\hspace{0.1cm}\omega-0.164423\hspace{0.1cm}\omega^2+0.016666\hspace{0.1cm}\omega^3$ → This is from Ref [2] $R=8.31446261815324 \frac{J}{mol\hspace{0.1cm} K}$ $ T_r=\frac{T}{T_c}$ Units to be used: $P\left[=\right]$ Pa, $\hspace{0.1cm}T\left[=\right]$ K, $\hspace{0.1cm}v\left[=\right]\frac{m^3}{mol}$ Represented in terms of compressibility factor $ z^3-\left(1-B\right)z^2 +\left( A-3B^2-2B\right)z-\left( AB-B^2-B^3\right)=0 $ $ A=\frac{a\hspace{0.1cm}P}{R^2\hspace{0.1cm}T^2} $ $ B=\frac{b\hspace{0.1cm}P}{R\hspace{0.1cm}T} $ Fugacity (of pure component) based on the P-R EOS $ \ln{\frac{f}{P}}=z-1-\ln{\left(z-B \right)}-\frac{A}{2\sqrt{2}\hspace{0.1cm}B} \ln{\left( \frac{z+\left(1+\sqrt{2}\hspace{0.1cm}\right)B}{z+\left(1-\sqrt{2}\hspace{0.1cm}\right)B} \right)} $ Enthalpy departure function based on the P-R EOS $ h-h^{idl}=RT\left[ z-1 -\frac{A}{2\sqrt{2}\hspace{0.1cm}B} \left(1+\frac{f_{\omega}\hspace{0.1cm}\sqrt{T_r}}{1+f_{\omega}\left(1-\sqrt{T_r}\hspace{0.1cm} \right)} \hspace{0.1cm} \right) \ln{\left( \frac{z+\left(1+\sqrt{2}\hspace{0.1cm}\right)B}{z+\left(1-\sqrt{2}\hspace{0.1cm}\right)B} \right)} \right] $ Entropy departure function based on the P-R EOS $ s-s^{idl}=R \ln{\left( z-B \right) } - \frac{A \hspace{0.1cm} R}{2\sqrt{2}\hspace{0.1cm}B} \frac{f_{\omega}\hspace{0.1cm} \sqrt{T_r}}{1+f_{\omega}\left(1-\sqrt{T_r}\hspace{0.1cm} \right)} \ln{\left( \frac{z+\left(1+\sqrt{2}\hspace{0.1cm}\right)B}{z+\left(1-\sqrt{2}\hspace{0.1cm}\right)B} \right)} $ Mixture parameters $ a=\sum \sum y_i y_j \left( 1- \delta_{ij} \right) \sqrt{a_i} \sqrt{a_j} $ $ b=\sum y_i b_i $ Note that the $\delta_{ij}$ are empirically determined binary interaction coefficients. Fugacity coefficient of component k in a mixture $ \ln{\hat{\phi_k}}=\ln{\left( \frac{\hat{f_k}}{y_k P} \right)} =\frac{b_k}{b}\left(z-1 \right)-\ln{\left( z-B \right)}-\frac{A}{2\sqrt{2}\hspace{0.1cm}B}\left( \frac{2\sum_i{y_i a_{ik}}}{a} - \frac{b_k}{b}\right) \ln{\left(\frac{z+\left(1+\sqrt{2}\hspace{0.1cm} \right)B}{z+\left(1-\sqrt{2}\hspace{0.1cm} \right)B} \right)} $ Note that in this equation, $a$, $b$, and $B$ are mixture parameters. In another notation, for example, $B=\frac{b_{mix}\hspace{0.1cm} P}{R \hspace{0.1cm}T}$ and $b_{mix}=\sum{y_i b_i}$. The Redlich-Kwong Equation of State Pressure-explicit form $ P=\frac{RT}{v-b}-\frac{a}{\sqrt{T} \hspace{0.25cm}v \hspace{0.25cm}(v+b)} $ $ a=\frac{1}{9\sqrt[3]{2}-9} \frac{R^2 T_c^{2.5}}{P_c}=0.42748 \frac{R^2 T_c^{2.5}}{P_c} $ $ b=\frac{\sqrt[3]{2}-1}{3} \frac{R T_c}{P_c}=0.08664 \frac{R T_c}{P_c} $ $R=8.31446261815324 \frac{J}{mol\hspace{0.1cm} K}$ Represented in terms of compressibility factor $ z=\frac{1}{1-\xi}-\frac{a}{b\hspace{0.05cm}R\hspace{0.05cm}T^{\frac{3}{2}} }\frac{\xi}{1+\xi} $ $\xi=\frac{bP}{RTz}$ Useful Relations Difference of specific heat capacities $C_P-C_V=-T\frac{\left( \frac{\partial v}{\partial T} \right)_P^2}{\left( \frac{\partial v}{\partial P} \right)_T} =-T\frac{\left( \frac{\partial P}{\partial T} \right)_v^2}{\left( \frac{\partial P}{\partial v} \right)_T}$ Note that the first equation is more useful as in many cases we have $P$ and $T$ as known values and $v$ is to be found. |