The reaction Gibbs ensemble Monte Carlo (RGEMC) computer simulation method [J. Phys. Chem. B 103 (1999) 10496] is used to predict the vapour-liquid equilibrium (VLE) behaviour of binary mixtures involving water, methanol, ethanol, carbon dioxide, and ethane. All these mixtures contain molecularly complex substances, and accurately predicting their VLE behaviour is a considerable challenge for molecular-based approaches, as well as for traditional engineering approaches. The substances are modelled as multi-site Lennard-Jones (LJ) plus Coulombic potentials with standard mixing rules for unlike site interactions. No adjustable binary-interaction parameters and no mixture experimental properties are used in the calculations; only readily-available pure-component vapour-pressure data are required. The simulated VLE predictions are compared with experimental results and with those of two typical semi-empirical macroscopic-level approaches. These latter are the UNIFAC liquid-state activity-coefficient model combined with the simple truncated virial equation of state, and the hole quasi-chemical group contribution equation of state. The agreement of the simulation results with the experimental data is generally good and also comparable with and in some cases better than those of the macroscopic-level empirical approaches.