A curved-surface spline function is used to describe the reduced excess Gibbs energy for multicomponent liquid mixtures in a model-free thermodynamic method for calculating vapor compositions from experimental temperatures, pressures and liquid compositions. A Newton-Raphson iteration procedure modified by Broyden is used for solving the method’s nonlinear equations. The method is illustrated with calculations for a variety of binaries, ternaries and two quaternaries at low or moderate pressures; these calculations indicate that this new method is both stable and reliable. Calculated vapor compositions are in good agreement with experiment. The method converges rapidly and has no special requirements for selecting initial guesses. The method may be used for either isothermal or isobaric data.