In this work, the exactly integrated form of the Clapeyron equation found by Mosselman et al. has been used in a systematic manner to derive a comprehensive set of equations describing the first-order transition curves of pure substances. The application of each of these equations requires the knowledge of only one (reference) point on the particular equilibrium line, of the corresponding enthalpy of transition, and some ancillary data (molar volumes and heat capacities of the phases at equilibrium). No fitting to (p, T) experimental data is needed. In this respect the equations developed here can be regarded as a source for calculating a priori the phase equilibrium curves. The results have been tested for a number of selected pure substances of variable molecular complexity, and the uncertainties attached to the calculations have been assessed. Empirical equations currently used for first-order transitions are compared with those obtained from the exact integration. As far as we are aware, no equation was previously proposed for solid + solid equilibrium lines.