# Cubic Equations of State (EOS)

$$P = \frac{RT}{V_m - b} - \frac{a}{V_m(V_m + b)}$$

$$a_i = \alpha_i 0.42747 \frac{R^2 T_{C,i}^2}{P_{C,i}}$$

$$b_i = 0.08664 \frac{R T_{C,i}}{P_{C,i}}$$

## alpha-Function

The attractive term of any cubic equation of state is expressed as the product of its value at the critical temperature by the so-called alpha function. To get accurate and physically meaningful behaviour in the subcritical and supercritical regions, it is necessary to define a alpha-function which is positive, decreasing, convex and with a negative third derivative. [Source: Jaubert] The set of constraints is according to ... et al.:

$$\alpha(T_C) = 1, \quad \alpha(T) > 0 \quad \forall T$$

$$\frac{d\alpha}{dT} < 0, \quad \frac{d^2 \alpha}{ dT^2} > 0, \quad \frac{d^3 \alpha}{d T^3} < 0 \quad \forall T$$

Definition by (Mathias, 1983):

$$\alpha_i(T) = [1 + m_i(1-T_{r,i}^{1/2})]^2$$

$$m_i = 0.48 + 1.574 \omega_i - 0.176 \omega_i^2$$

## References

Smooth cubic extension for cubic EOS