During the past quarter, research was directed toward the kinetic modeling of coal liquefaction reactions. Specifically, a new lumping system, abstract-discrete lumping, was developed and tested for a vapor-liquid equilibrium calculation. The abstract-discrete method combines the advantages of traditional pseudo-component lumping with the advantages of continuous lumping. The method is especially applicable to mixtures for which the analytical data consist of partially-resolved chromatograms, such as for the size-exclusion chromatographic analysis of coal liquids. This report records the development and testing of the theory behind the abstract-discrete approach. Although the derivation of the method utilizes mathematical concepts which may at times seem difficult, the final result consists of simple algebraic equations which may be solved with only a little more difficulty than the classical equations. The theory has first been developed for a vapor-liquid equilibrium calculation. The reason for this is two-fold. First, the vapor-liquid equilibrium equations are generally simpler than kinetic modeling equations, a fact which makes the description of the theory much easier. In addition, vapor-liquid equilibrium effects are of importance to liquefaction reactor modeling, and are to be incorporated into the tubing-bomb reactor model presently under development. We have shown that vapor-liquid equilibrium calculations based upon a thermodynamic equation-of-state can predict the hydrogen composition of the liquefaction-reactor liquid phase with considerable accuracy. 7 refs., 40 figs., 49 tabs.