$\alpha$-functions


cubic

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) [1]


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


2 G. M. Wilson (1964) [2]


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

3 G. Soave (1972) [3]


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

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


4 E. Usdin and J. C. McAuliffe (1976) [4]


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

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) [5]


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


References

[1] O. Redlich and J. N. S. Kwong, “On the Thermodynamics of Solutions. V. An Equation of State. Fugacities of Gaseous Solutions.” Chemical Reviews, vol. 44, no. 1, pp. 233–244, Feb. 1949.

[2] G. M. Wilson, “Vapor-Liquid Equilibria, Correlation by Means of a Modified Redlich-Kwong Equation of State,” in Advances in Cryogenic Engineering, K. D. Timmerhaus, Ed. Boston, MA: Springer US, 1964, pp. 168–176.

[3] G. Soave, “Equilibrium constants from a modified Redlich-Kwong equation of state,” Chemical Engineering Science, vol. 27, no. 6, pp. 1197–1203, Jun. 1972.

[4] E. Usdin and J. C. McAuliffe, “A one parameter family of equations of state,” Chemical Engineering Science, vol. 31, no. 11, pp. 1077–1084, 1976.

[5] G. Soave, “Application of a cubic equation of state to vapor-liquid equilibria of systems containing polar compounds,” Inst. Chem. Eng. Symp. Ser, no. 56, 1979.