The one-dimensional diffusion equation is solved numerically to calculate the spreading profile of a molecularly-thin liquid on a solid surface. The thickness-dependent diffusion coefficient of the liquid is derived from the thickness dependence of the dispersive and polar surface free energies. Changes in the gradient and curvature of the total free energy with respect to film thickness give rise to distinct features in the calculated spreading profile. In the sub-monolayer thickness regime, a rapid spreading front develops as a result of the steep gradient in the dispersive force with film thickness. For polar end-group terminated perfluoropolyethers, a shoulder follows behind the rapid spreading front. At the top of the shoulder, a step increase in the film thickness is observed. This vertical step results from film thicknesses that correspond to negative disjoining pressures and hence are thermodynamically unstable. The profiles calculated from the thickness dependence of the total surface energy account for all of the features observed experimentally during the spreading of perfluoropolyethers onto solid surfaces.