Natural hydrocarbon mixtures are composed of very many components. Usually, several components are lumped into pseudo-components to reduce the dimensionality of the equilibrium flash calculations. An interesting alternative to lumping is the use of semi-continuous thermodynamics, which enables a better characterization of mixture compositions. Traditionally, in the semi-continuous approach, the feed composition is approximated by a standard distribution function. Then, a Gaussian quadrature is used to find discrete carbon numbers representing the "pseudo-components", for which a discrete flash can be performed. However, the feed composition can be random and/or far from classical distributions. In the quadrature method of moments (QMoM), developed by Lage (Lage, Comput. Chem. Eng. 31 (2007) 782-799), an optimal quadrature rule is used, considering the feed composition distribution as the weight function. In this work, a hidden feature of QMoM is revealed, by identifying a distribution function used to approximate a newly considered dependence, which involves the feed composition and the component liquid mole numbers. This dependence is extremely smooth, and, the most important, the shape of the function remains identical for any composition, pressure and temperature. The smoothness of the distribution enables a really good accuracy in solving the quadrature. The calculation of the quadrature is based on a procedure which avoids problems due to the ill-conditioned nature of the problem, even for a larger number of pseudo-components (unlike the original QMoM formulation). An analytical delumping procedure can be used (with an extremely small computational effort) to recover detailed composition in each equilibrium phase. Comparisons between a full discrete flash and the semi-continuous description proved the QMoM to be highly accurate. Various test problems, including two-phase and multiphase flash calculations, showed that Lage's methodology is generally applicable, even in cases when no distribution function can model the feed composition, or several distribution functions are needed to model different portions of the mixture. (C) 2013 Elsevier B.V. All rights reserved.