The general form of correcting the $v$ for "excluded volume" and subtracting a term proportional to $v^{-2}$ (top correct the pressure to account for attractive forces) leads to a class of equations of state called cubic equations of state, that have the interesting property of being able to capture both the liquid and vapor conditions:

In order to use the van der Waals equation of state, we need to determine the material-dependent constants, $a$ and $b$.

Using the principle of corresponding states, we can argue that the constants for all materials may be obtained by recognizing that, at the critical point:

$\left ( \frac{\partial P}{\partial v} \right )_{T_c} = \left ( \frac{\partial^2 P}{\partial v^2} \right )_{T_c} = 0$

We can then use three equations (the vdW EOS and these two derivatives) at the critical point to write $a$ and $b$ in terms of $P_c$ and $T_c$:

$P_c = \frac{RT_c}{v_c-b} -\frac{a}{v_c^2}$

$\left ( \frac{\partial P}{\partial v} \right )_{T_c} = 0 = -\frac{RT_c}{(v_c-b)^2} +\frac{2a}{v_c^3}$

$\left ( \frac{\partial^2 P}{\partial v^2} \right )_{T_c} = 0 = \frac{2RT_c}{(v_c-b)^3} -\frac{6a}{v_c^4}$

this approach leads to (i.e., solving the two equations -- above -- for the two unknowns -- a and b):

$a = \frac{27}{64}\frac{(RT_c)^2}{P_c}$

$b = \frac{RT_c}{8P_c}$

doing a bit more algebra, we can write an expression for $z_c$:

$z_c = \frac{P_cv_c}{RT_c} = \frac{3}{8}$

which is higher than is typically observed experimentally.

Following the same procedure for a few of the more common other cubic equations:

Name | Attractive term | a | b |

van der Waals | $\frac{a}{v^2}$ | $\frac{27}{64}\frac{(RT_c)^2}{P_c}$ | $\frac{RT_c}{8P_c}$ |

Redlich-Kwong | $\frac{a}{\sqrt{T}v(v+b)}$ | $\frac{0.42748R^2T_c^{2.5}}{P_c}$ | $\frac{0.08664RT_c}{P_c}$ |

Soave-Redlich-Kwong | $\frac{a\alpha(T)}{v(v+b)}$ | $\frac{0.42748R^2T_c^2}{P_c}$ | $\frac{0.08664RT_c}{P_c}$ |

Peng-Robinson | $\frac{a\alpha(T)}{v(v+b)+b(v-b)}$ | $\frac{0.45724R^2T_c^2}{P_c}$ | $\frac{0.0778RT_c}{P_c}$ |

The critical compressibility factor for the Redlich-Kwong is improved, at 1/3, and is even more in line with experimental observations for SRK and PR (closer to 0.3)

In the SRK and PR equations, $\alpha(T)$ is a function that replaces the $\sqrt{T}$ from the RK equation. Both functions are similar, and given by:

$\alpha_{SRK}(T) = \left [ 1+(0.48508+1.55171\omega-0.17613\omega^2)(1-\sqrt{T_r})\right ]^2$

$\alpha_{PR}(T) = \left [ 1+(0.37464+1.54226\omega-0.26992\omega^2)(1-\sqrt{T_r})\right ]^2$

Describe how cubic equations of state account for attractive and repulsive interactions

Calculate P, v, or T from non-ideal equations of state (cubic equations, the virial equation, compressibility charts, and ThermoSolver)

Use the vdW EoS to calculate the pressure of 44 g $CO_2$ in a 0.23 liter container at 26C (i.e., do we get P=6 Mpa?)