This article presents a methodology for checking the existence of the azeotrope and computing its composition, density, and pressure at a given temperature by integrating chemical engineering insights with molecular simulation principles. Liquid-vapor equilibrium points are computed by molecular simulations using the Gibbs ensemble Monte Carlo (GEMC) method at constant volume. The appearance of the azeotropic point is marked by a shift of the equilibrium constant from one side of the unity to the other. After each GEMC simulation, an identity change move is derived in the grand canonical ensemble to progress towards the azeotrope along the equilibrium curve. The effectiveness of the proposed methodology is successfully tested for several binary Lennard-Jones mixtures reported in the literature.