In the search for understanding of several types of abnormal thermodynamic behaviour in the vicinity of critical lines of binary liquid mixtures, we have revisited an apparently forgotten relationship between the pressure dependence of the critical temperature and the second derivatives with respect to the composition of the volumetric and enthalpic properties of the mixture. We refer to an equation originally developed in the fifties by Prigogine and Defay and soon afterwards analysed by others. Under some restrictive assumptions, the T-p slope of the critical locus can simply be inferred from the ratio between vE and hE. The interest and usefulness of this approximate relation is self-evident. Values for any one of the three properties involved, (dT/dp)c, vE or hE, can be assessed based on the availability of the other two. Moreover, the amplitude of the divergence of thermodynamic response functions to criticality are intimately associated with the slope of the critical locus. A link between critical behaviour and solution excess properties is thus established. For instance, double critical points tend to occur if one of the excess properties changes its sign as the temperature or pressure is varied. In this work, we have started a detailed study of the practical limits of validity of the approximate relation. Five binary liquid mixtures were tested, all of them sharing a UCST/LCSP-type of phase transition. Although, from a theoretical perspective, the original second-derivatives approach should perform better, in practice, the direct ratio of the excess properties constitutes a superior strategy for obtaining (dT/dp)c values. The underlying reasons for this are discussed in detail. The T-p critical slope is normally found to play a secondary role in assessing the critical amplitudes of diverging thermodynamic functions.