$\begin{array}{ll} a_i(T) = \Psi \frac{\alpha(Tr_i, \omega_i)\cdot R^2 T_{ci}^2}{P_{ci}} & b_i = \Omega \frac{R T_{ci}}{P_{ci}}\\ q_i=\frac{a_i(T)}{b_i R T} & \beta_i=\frac{b_i P}{R T} \\ \Rightarrow q_i \beta_i = \frac{a_i(T) P}{(R T)^2}\\ \end{array}$
$\begin{array}{lll} \text{Phase dimensionless params.}& \beta^p=\frac{b^p P}{R T} & q^p=\frac{a^p}{b^p R T}\\ \text{Mixing rule: linear} &a^p=\sum\limits_i{\sum\limits_j{x_i^p x_j^p a_{ij}}}& b^p=\sum\limits_i{x_i^p b_i}\\ \text{Combining rule: geom. mean} & a_{ij}=(a_i a_j)^{1/2} & b_i \end{array}$
Expressions
$\begin{array}{ccc} \hline Ref& \text{P - explicit}& \text{Z - explicit}& Reparam. \\ \hline [1]& P=\frac{R T}{V-b}-\frac{\theta(V-\eta)}{ (V-b)(V^2+\kappa V +\lambda)}& Z=\frac{V}{V-b}-\frac{\theta}{ (V^2+\kappa V +\lambda)} \frac{V}{R T} & \begin{array}{l} \eta=b, \theta=a(T)\\ 0 = V^2+\kappa V + \lambda \\ \Rightarrow \kappa=b\cdot(\epsilon+\sigma), \lambda=b^2\cdot\epsilon \sigma \end{array} \\ \Rightarrow & P=\frac{R T}{V-b}-\frac{a(T)}{ (V+\varepsilon b)(V+\sigma b)} & \begin{array}{l} \text{v-like: } Z=1+\beta-q \beta\cdot\frac{Z-\beta}{ (Z+\epsilon \beta)(Z+\sigma \beta)}\\ \text{l-like: } Z=\beta+(Z+\epsilon \beta)(Z+\sigma \beta) \left[\frac{1+\beta-Z}{q \beta}\right]\\ \begin{array}{lrl} \text{poly: } 0= & +1 & Z^3\\ & +(\beta(\epsilon+\sigma)-\beta-1)& Z^2\\ & +(q \beta + \epsilon\sigma\beta^2- \beta(\epsilon+\sigma)(1+\beta)& Z \\ & -(\epsilon \sigma \beta^2(\beta+1)+q\beta^2) \\ \end{array} \end{array} \\ \hline [2]& P=\frac{R T}{V-b}-\frac{\Theta(V-\eta)}{ (V-b)(V^2+\delta V +\varepsilon)} & Z=\frac{V}{V-b}-\frac{(\Theta/R T)V(V-\eta)}{ (V-b)(V^2+\delta V +\varepsilon)} & \begin{array}{ll} \epsilon \neq \varepsilon !\\ B'=\frac{b P}{R T}, \delta'=\delta\frac{P}{R T}\\ \Theta'=\frac{\Theta P}{(R T)^2}, \varepsilon' = \varepsilon \left(\frac{P}{R T} \right)^2 \\ \eta' =\eta \frac{P}{R T} \end{array} \\ \Rightarrow & & \begin{array}{lrl} \text{poly: } 0= & +1 & Z^3\\ & +(\delta'-B'-1)& Z^2\\ & +(\Theta' + \epsilon'-\delta'(1+B'))& Z \\ & -(\epsilon'(B'+1)+\Theta'\eta') \\ \end{array} \\ \hline [3]& P=\frac{R T}{V-b}-\frac{a}{ (V^2+u b V +w b^2)} & Z=\frac{V}{V-b}-\frac{a}{ (V^2+u b V +w b^2)}\frac{V}{R T} & \begin{array}{l} B = b P / (R T)\\ A = a P / (R T)^2 \\ \end{array}\\ & & \begin{array}{lll} &\text{poly: } 0 &= Z^3+a_1 Z^2+a_2 Z+a_3 \\ & a_1 &= u B-B-1\\ & a_2 &= w B^2-u B^2-u B+A\\ & a_3 &= -(w B^3+w B^2+ A B) \end{array}\\ \hline \end{array}$
Parameters from ref. [1], [2]
$\begin{array}{ll} \hline [1] & [2] (\epsilon \neq \varepsilon !)\\ \hline \beta=\frac{b P}{R T} & B'=\frac{b P}{R T} \\ \kappa=b\cdot(\epsilon+\sigma) & \delta=b\cdot(\epsilon+\sigma) \\ \lambda=b^2\cdot\epsilon \sigma & \varepsilon=b^2\cdot\epsilon \sigma\\ \kappa'=\kappa\frac{P}{R T}= \beta(\epsilon+\sigma) & \delta'=\delta\frac{P}{R T}= B'(\epsilon+\sigma) \\ \theta' = q b \frac{P}{R T} = q \beta=\frac{\theta P}{(R T)^2} & \Theta'=\frac{\Theta P}{(R T)^2} \\ \lambda' = \lambda \left(\frac{P}{R T} \right)^2 = \beta^2\cdot\epsilon\sigma & \varepsilon' = \varepsilon \left(\frac{P}{R T} \right)^2 = B'^2\cdot\epsilon\sigma\\ \eta' =\eta \frac{P}{R T} \text{ ; } \eta=b & \eta' =\eta \frac{P}{R T} \\ \hline \end{array}$
Cross-equivalences by nomenclature in Ref.
$\begin{array}{cccc} \hline [1] & [2] & [3] &\\ \hline b\cdot(\epsilon + \sigma) = \kappa &= \delta &= u b & \Rightarrow u = \epsilon + \sigma\\ b^2 \epsilon \sigma = \lambda &= \varepsilon &= w b^2 & \Rightarrow w = \epsilon \sigma&\\ \hline \end{array}$
Examples
$\begin{array}{cccccccccc} \hline Eq. & u & w & \delta & \varepsilon & \kappa & \lambda & \sigma & \epsilon\\ \hline RK & 1 & 0 & b & 0 & b & 0 & 1 & 0\\ PR & 2 & -1 & 2b & -b^2 & 2b & -b^2 & 1+\sqrt{2} & 1-\sqrt{2}\\ \hline \end{array}$
[1] Smith, Joseph M. ; Van Ness, Hendrick C. ; Abbott, Michael M.: Introduction to chemical engineering thermodynamics. New York: McGraw-Hill, 2005.
[2] Poling, Bruce ; Prausnitz, John ; Connell, John O': The Properties of Gases and Liquids. Madison: McGraw Hill Professional, 2000.
[3] Zhao, Ensheng, and Sanjoy Saha. "Applications of Complex Domain in Vapor− Liquid Equilibrium Calculations Using a Cubic Equation of State." Industrial & engineering chemistry research 37.5 (1998): 1625-1633.
[4] Gmehling, Jürgen ; Kolbe, Bärbel ; Kleiber, Michael ; Rarey, Jürgen: Chemical Thermodynamics for Process Simulation. New York: John Wiley & Sons, 2012.
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