van der Waals Equation of State

The van der Waals (vdW) equation of state (EOS) is a cubic EOS which was developed by Johannes van der Waals in 1873¹ as part of his Ph.D. dissertation. The EOS was the first cubic EOS and had a major impact on the ability of EOS models to predict liquid-vapor phase behavior. This work resulted in van der Waals being rewarded with a Nobel prize in physics i 1910.

van der Waals EOS Structure

The vdW EOS model is defined by

$\begin{equation}\label{eq:ceos_pr} p = \frac{RT}{v-b} - \frac{a}{v^2} \end{equation}$

where $a$ and $b$ are the EOS model parameters. van der Waals attributed the parameters to the non-ideal behavior resulting from attractive and repulsive forces between molecules. The $a$ parameter is attributed to the non-ideal behavior because of attractive forces between molecules while the $b$ parameter is attributed to the non-ideal behavior because of repulsive forces between molecules. The parameters were measures experimentally for pure components and were able to accurately predict a range of different mixtures of both single and two phases.

The description of the two parameters in the vdW EOS model are given by

$\begin{eqnarray}\label{eq:a_vdw} a = \Omega_a \frac{R^2T_c^2}{p_c} \end{eqnarray}$

$\begin{eqnarray}\label{eq:b_vdw} b = \Omega_b \frac{RT_c}{p_c} \end{eqnarray}$

where

$\begin{eqnarray} \Omega_a = \frac{27}{64} = 0.42188... \end{eqnarray}$

$\begin{eqnarray} \Omega_b = \frac{1}{8} = 0.125 \end{eqnarray}$

Mixing Rules

Note

The main article for cubic EOS models describes the mixing rules with some more detail.

The original mixing rule developed by van der Waals is a quadratic mixing rule. The set of equations describing the mixing are given by

$\begin{equation}\label{eq:a_mix_vdw} a = \sum_{i=1}^N \sum_{i=1}^N u_i u_j a_{ij} \end{equation}$

$\begin{equation}\label{eq:b_mix_vdw} b = \sum_{i=1}^N u_i b_i \end{equation}$

where $a_{ij}=\sqrt{a_ia_j}$, $u_i$ can be the liquid composition ($x_i$), vapor composition ($y_i$) or the total composition ($z_i$) and $a_i$ and $b_i$ are the component EOS parameters for component $i$.

Modifications to the $a$ and $b$ mixing rules were developed to enhance the performance of the EOS models. The most common modified mixing rules are given by

$\begin{equation}\label{eq:a_mix_mod} a = \sum_{i=1}^N \sum_{i=1}^N u_i u_j a_{ij} (1-k_{ij}) \end{equation}$

where $k_{ij}$ is often referred to as the binary interaction parameter (BIP) or sometimes referred to as the binary interaction coefficient (BIC).

J.D. Van der Waals. On the continuity of the gaseous and liquid states. Universiteit Leiden, 1873. ↩