A free volume theory of diffusion coefficients is formulated for binary mixtures of simple liquids. The free volume is defined by means of the generic van der Waals equation of state for mixtures, which is developed in this work, and computed in terms of the pair correlation function obtained by means of Monte Carlo simulations with a square-well potential model. The free volume thus computed is used to investigate the composition and temperature dependence of the binary diffusion coefficient of argon-krypton mixtures as well as the tracer diffusion coefficients of argon in liquid nitrogen and krypton in liquid argon. The present theoretical predictions compare very well with the experimental or simulation results available in the literature. The size and mass dependence of the ratio of the tracer diffusion to the solvent self-diffusion coefficients is also presented. This ratio is found to be almost independent of temperature and density. It therefore can be used to calculate the tracer diffusion coefficient from the self-diffusion coefficient and vice versa.