This paper presents an approach based on the saddle-point approximation to study the equilibrium interactions between small molecules and macromolecules with a large number of sites. For this case, the application of the Darwin-Fowler method results in very simple expressions for the stoichiometric equilibrium constants and their corresponding free energies in terms of integrals of the binding curve plus a correction term which depends on the first derivatives of the binding curve in the points corresponding to an integer value of the mean occupation number. These expressions are simplified when the number of sites tends to infinity, providing an interpretation of the binding curve in terms of the stoichiometric stability constants. The formalism presented is applied to some simple complexation models, obtaining good values for the free energies involved. When heterogeneous complexation is assumed, simple expressions are obtained to relate the macroscopic description of the binding, given by the stoichiomeric constants, with the microscopic description in terms of the intrinsic stability constants or the affinity spectrum.