The particle exchange method is used to compute the exchange chemical potential for binary mixtures of linear polymers in Monte Carlo lattice model simulations. The internal energy and the specific heat are also calculated for each thermodynamic state. Simulations are performed for nine blend compositions, with the total monomer volume fraction held fixed at phi =0.8 or phi =0.7. Both polymer species have the same polymerization indices, M=40, 50, or 100. The two species are distinguished by their van der Waals interaction strengths, epsilon (11), epsilon (12), and epsilon (22). Several ratios epsilon (22)/epsilon (11) are considered, and the simulation temperatures range from almost athermal conditions to near-critical states. The parameter epsilon (12) is either computed as the geometrical average root epsilon (11)epsilon (22) or is assigned an independent value. Three approximate solutions to the model are compared to the simulations: Flory-Huggins theory, Guggenheim's random mixing approximation, and the lattice cluster theory (LCT). The results confirm the higher accuracy of the LCT. However, some predictions for the specific heat are again found to be inaccurate because of the low order cutoff of the high temperature perturbative expansion.