We develop a thermodynamically consistent version of the random phase approximation for multicomponent mixtures. The developed approach makes it possible, in particular, to accurately take into account the free volume effect on the entropy of mixing and steric interactions between the molecules. Further, we apply the developed theory to derive the total structure factor, equation of state and density correlation length of a multicomponent polymer blend. We demonstrate that the developed theory adequately reproduces the results of the small angle neutron scattering experiments for the structure factor of the investigated ternary polymer blend.