A new model. is derived for lattice systems (lattice gas and binary mixtures of monomers). This model is based on a generalization to three dimensions of the Ono-Kondo equations for the density profile near a flat surface. The internal energy is calculated and compared with previous models. Unlike many previous theories, this new model has the correct limiting behavior at infinite dilution, at high densities, when the interchange energy goes to zero and for the lattice gas. In addition, it displays the correct behavior for systems with very strong interactions (such as hydrogen bonds) in that it predicts that the energy saturates to a constant value at a low density. For one-component, monomer systems, the new theory also describes simulation data for square-well (off-lattice) molecules better than previous theories.