Consistent calculation of fugacities of fluid mixtures remains as one of the most important subjects in contemporary molecular thermodynamics. In practice, equations of state (EOSs) and g^E-models have been used. However, most EOSs are erroneous for condensed phases at high densities and g^E-models are inapplicable for pressure-sensitive systems. Recently to remedy the shortcomings in both approaches, there has been a surge of new g^E-EOS mixing rules. By equating any set of EOS and g^E-models, the limitations in both approaches could be resolved significantly. However, the self-consistency in the underlying concept of those mixing rules remains controversial. During the last several years, the present authors proposed a new lattice-fluid EOS and its simplification relevant to phase equilibrium calculations. Without employing any g^E-EOS mixing rule and with only two parameters for a pure component and one adjustable interaction energy parameter for a binary mixture, results obtained to date demonstrated that the EOSs are quantitatively applicable to a great variety of phase equilibrium properties of mixtures, especially, for complex and/or macromolecular systems. In the present article we summarize the EOSs and extended the applications to liquid-liquid Equilibria. In part I, we discussed briefly the molecular thermodynamic aspects of general derivation of the EOS and a brief discussion of applying the EOSs to pure fluids while the illustrative application to various real mixture systems is discussed in part II.