# $\alpha$-functions

A list of $\alpha$-functions available in the literature classified with respect to their publication year. For component dependent $\alpha$-functions, adjustable parameters are written in the form $c_i$. For generalized $\alpha$-functions, the associated equation of state is also given in the $\alpha$-function description.

# 1 O. Redlich and J. N. S. Kwong (1949) 

$\begin{equation} \alpha\left(T_r\right) = \frac{1}{\sqrt{T_r}} \end{equation}$

# 2 G. M. Wilson (1964) 

$\begin{equation} \alpha\left(T_r, c_1\right) = \left(1-c_1\right) \times T_r + c_1 \end{equation}$

# 3 G. Soave (1972) 

$\begin{equation} \left\{ \begin{array}[ll] & \alpha & = \left[1+m\left(\omega\right)\left(1-\sqrt{T_r}\right)\right]^2 \\ m & = 0.480 + 1.574\omega - 0.176\omega^2 \\ \end{array} \right. \end{equation}$

$\begin{equation} \text{For the SRK equation of state} \end{equation}$

# 4 E. Usdin and J. C. McAuliffe (1976) 

$\begin{equation} \left\{ \begin{array}[ll] & \alpha & = \left[1+m\left(\omega\right)\left(1-\sqrt{T_r}\right)\right]^2 \\ m\left(T_r \leq 0.7\right) & = 0.48049 + 4.516 \omega c_1 + \left[0.67713 \left(\omega - 0.35\right) - 0.02\right] \times \left(T_r - 0.7\right) \\ m\left(0.7 \lt T_r \leq 1.0\right) & = 0.48049 + 4.516 \omega c_1 + \left[37.7846 \omega \left(c_1 \right)^3 + 0.78662\right] \times \left(T_r - 0.7\right)^2 \\ m\left(T_r \gt 1.0\right) & = \displaystyle m\left(\frac{1}{T_r}\right) \end{array} \right. \end{equation}$

If the compound is liquid at 1 atm and 60°F (which is equal to 288.706 K):

$\begin{equation} c_1 = \displaystyle \frac{\left(p = 1 atm\right) \times \left[v_{liq}^{exp}\left(T = 60°F, p = 1 atm\right)\right]}{R \times \left(T = 60°F\right)} \end{equation}$

Else:

$\begin{equation} c_1 = \displaystyle \frac{\left(p = 1 atm\right) \times \left[v_{liq}^{exp}\left(T = T_b\left(p = 1 atm\right), p = 1 atm\right)\right]}{R \times \left(T = 60°F\right)} \end{equation}$

With $v_{liq}^{exp}$ expressed in $lb.ft^{-3}$

$\begin{equation} \text{For the SRK equation of state} \end{equation}$

# 5 G. Soave (1979) 

$\begin{equation} \alpha\left(T_r\right) = 1 + \left(1 - T_r\right)\left(c_1 + \frac{c_2}{T_r}\right) \end{equation}$