Currently available models to simulate naturally occurring mineral-melt equilibria use mineral components limited to tholeiitic basalt compositions and thus they cannot be used for alkali-rich basalts and basanites. To expand mineral-melt equilibria calculations to alkali-rich composition space at low pressures, we have derived equations that describe chemical equilibria between olivine-melt, pyroxene-melt, plagioclase-melt, nepheline-melt and leucite-melt components. Excess free energies of reactions between the end-member mineral and melt components at equilibrium have been expressed as a function of melt composition, temperature and f(O2). The database used to calculate the mineral-melt expressions consists of a total of >350 anhydrous experiments conducted under controlled oxygen fugacity defined by the quartz-fayalite-magnetite (QFM) oxygen buffer. Rocks used in these experiments range from basanites, nephelinites and alkali olivine basalts, to tholeiitic basalts and basaltic andesites. Using bulk compositions of starting materials both in this experimental database and in others that were not incorporated into the regression of modeled parameters, modeled equations successfully predict, at a given temperature and fo, compositions of multiply saturated melts as well as the compositions of coexisting minerals. Standard deviations of the calculated male fractions of mineral components (sigma) are as follows: anorthite 0.02; forsterite 0.02; clinoenstatite 0.02; enstatite 0.003; nepheline 0.02; and leucite 0.01. Standard deviations (sigma) of the calculated melt compositions in terms of weight percent of oxides are: SiO2 0.96; Al2O3 1.32; Fe2O3 0.23; FeO 1.21; MgO 0.84; CaO 0.79; Na2O 0.58; and K2O 0.69. All calculations were carried out using a non-linear Newton-Raphson numerical procedure.