Empirical thermal cohesion functions, a(T,), are frequently used in conventional equations of state (EOS) for fitting the vapor pressures of pure fluids. Accurate vapor pressure predictions are required for correlating and/or predicting the phase equilibrium and interfacial tension of multicomponent mixtures. This is the case for the Redlich-Kwong-Soave and Peng-Robinson models, two well-established models for engineering applications. In this work, we demonstrate that, in the case of pure fluids, the alpha(T-r) function can potentially predict multiple mechanically stable critical points, thus affecting the global topology of phase equilibrium predictions. A detailed analysis, based on the consistency of the prediction of the Joule-Thomson inversion curve, reveals that these predictions are not reliable from a physical point of view. In fact, conventional cubic EOS are able to predict multiple Joule-Thomson inversion curves, a behavior symptomatic of the prediction of multiple stable critical points for pure fluids. Similar pitfalls have been detected in theoretically based EOS such as SAFT and the model proposed by Johnson et al. (Johnson, J. K.; Zollweg, J. A.; Gubbins, K. E. Mol. Phys. 1993, 78, 591-615) for Lennard-Jones fluids, although beyond the range in which such models are usually employed. In the case of conventional cubic EOS with quadratic mixing rules, another pitfall related to conventional alpha(T-r) functions is the prediction of nondifferentiable critical lines and equilibrium envelopes for mixtures. Such a physical inconsistency might generate a mechanism that predicts closed loops of immiscibility in van der Waals-type EOS that contain a temperature-dependent parameter.