It is challenging to compute the partition function (Q) for systems with enormous configurational spaces, such as fluids. Recently, we developed a Monte Carlo technique (an energy partitioning method) for computing Q [J. Chem. Phys. 2011, 135, 174105]. In this paper, we use this approach to compute the partition function of a binary fluid mixture (carbon dioxide + methane); this allows us to obtain the Helmholtz free energy (F) via F = -k(B)T In Q and the Gibbs free energy (G) via G = F + pV. We then utilize G to obtain the coexisting mole fraction curves. The chemical potential of each species is also obtained. At the vapor-liquid equilibrium condition, the chemical potential of methane significantly increases, while that of carbon dioxide slightly decreases, as the pressure increases along an isotherm. Since Q is obtained from the density of states, which is independent of the temperature, equilibrium thermodynamic properties at any condition can be obtained by varying the total composition and volume of the system. Our methodology can be adapted to explore the free energies of other binary mixtures in general and of those containing CO2 in particular. Since the method gives access to the free energy and chemical potentials, it will be useful in many other applications.