A unified treatment of the phase equilibria and interfacial properties of fluids is presented. This is done through the development of a framework model, which is applicable to nonpolar systems as well as to highly nonideal systems with strong specific interactions, to systems of small molecules as well as to polymers, glasses, and gels, to liquids as well as to vapors and supercritical systems, and to homogeneous as well as to inhomogeneous systems. One key characteristic of this equation-of-state model is its capacity to estimate the nonrandom distribution of the free volume in the system. A quasi-thermodynamic approach of inhomogeneous systems is used for modeling the fluid-fluid interface. The present model is referred to as the nonrandom hydrogen-bonding model. The key differences between this model and the previous quasi-chemical hydrogen-bonding model are the following: (1) The combinatorial term is replaced by the generalized Staverman term. (2) The nonrandomness factor is also generalized. Two alternative expressions are presented in this work. (3) The shape factor, s, is no longer an adjustable parameter. It is set equal to the UNIFAC qlr ratio and obtained from the corresponding UNIFAC compilations in the literature. (4) A most recent quasi-thermodynamic approach is used for the fluid-fluid interface. In the first part of this series of papers, the model is applied for the estimation of basic thermodynamic properties of pure fluids, such as vapor pressures, orthobaric densities, heats of vaporization, surface tensions, and glass transition temperatures.