Derive the expression for fugacity coefficient from a cubic equation of state.

Starting from fundamental relation: ln(phi) = integral from 0 to P of (Z-1)/P dP. For cubic EOS (PR as example): ln(phi) = (Z - 1) - ln(Z - B) - A/(2*sqrt(2)*B) * ln((Z + (1+sqrt(2))*B)/(Z + (1-sqrt(2))*B)), where A = aP/R^2T^2, B = bP/RT. For mixtures, use mixing rules for a_mix, b_mix and partial properties: ln(phi_i) = (b_i/b_mix)*(Z-1) - ln(Z-B_mix) - A_mix/(2*sqrt(2)*B_mix) * (2*sum(y_j*a_ij)/a_mix - b_i/b_mix) * ln((Z + (1+sqrt(2))*B_mix)/(Z + (1-sqrt(2))*B_mix)). This enables VLE calculation using EOS for both phases.