Mixing rules for EOS parameters

The \(b\) parameter, which for a van der Waals EOS represents the finite size of molecules, is obtained for a mixture by: \[b = \sum x_ib_i\] where \(x_i\) is the mole fraction of component \(i\) in the mixture and \(b_i\) is the \(b\) parameter for pure component \(i\).

The \(a\) parameter, which represents the average attraction between molecules in a mixture, is determined from \[a = \sum \sum x_ix_ja_{ij}\] For a binary mixture \[a = x_i^2a_{11} + 2x_1x_2a_{12} + x_2^2a_{22}\] where \(a_{11}\) represents the interactions between the two “1” molecules
\(a_{22}\) represents the interactions between two “2” molecules
\(a_{12}\) represents the interactions between a “1” molecule and a “2” molecule

In the simplest form: \[a_{12} = (a_{11}a_{22})^{0.5}\]

Fugacity coefficient definition \[ \hat{\, \phi _i} \equiv \frac{\hat {f_i}}{y_iP}\]

where \( \hat {\, \phi_i}\) is the fugacity coefficient of component \(i\)
\(\hat {f_i}\) is the fugacity of component \(i\)
\(y_i\) is the mole fraction of \(i\) in the gas phase
\(P\) is the total pressure

Fugacity equality using an EOS \[y_i\hat {\,\phi_i^v} P = x_i\hat{\,\phi_i^l}P\] where \(\hat {\,\phi_i^v}\) is the fugacity coefficient of component \(i\) in the vapor phase
\(\hat{\,\phi_i^l}\) is the fugacity coefficient of component \(i\) in the liquid phase
\(x_i\) is the mole fraction of \(i\) in the liquid phase.