In local composition models the focus is, usually, on the distribution of molecules or molecular segments and not on the distribution of the free volume of the system. At conditions, however, where the free volume is a significant percentage of the total volume of the system, as in the case of critical or supercritical conditions, its non-random distribution may not be negligible even for non-polar substances. In this work an attempt is made to account for the non-random distribution of the free volume in fluids and their mixtures. The classical quasi-chemical approach is used in connection with the lattice fluid model. The basic formalism is first developed for pure fluids and subsequently extended to multi-component mixtures by preserving its simplicity. The model is used for the calculation of vapour pressures and orthobaric densities of representative non-polar fluids and for the calculation of compositions and densities of liquid and vapour phases at equilibrium in mixtures of non-polar fluids. These calculations are compared with experimental data and with the corresponding calculations of the original lattice fluid model.