We systematically study the effect of the nuclear symmetry energy in the basic properties of finite, neutron-rich, heavy nuclei where symmetry energy plays a dominant role. We employ a variational method, in the framework of the Thomas-Fermi approximation, to study the effect of the symmetry energy on the neutron skin thickness and symmetry energy coefficients of various nuclei. The isospin asymmetry function a(r) is directly related to the symmetry energy as a consequence of the variational principle. In addition to this, the Coulomb interaction is included in a self-consistent way. The energy density of the asymmetric nuclear matter that is used, has its origins in a momentum-dependent interaction.