In this work, it is shown that the most common approximations for the vapor-pressure equation of a fluid yield an inappropriate behavior for the second temperature derivative of the saturation pressure in the vicinity of the critical point. This fact gives rise to an unphysical divergent behavior for the change of the specific heat at vaporization (Delta(v)c(nu)) near the critical point. An expression for the vapor-pressure equation with the appropriate critical behavior is proposed in terms of a truncated Renormalization Group (RG) expansion. As an example, it is shown that the well-known Wagner and Pruss vapor-pressure equation and the IAPWS-95 result for water yield an incorrect critical behavior for Delta(v)c(nu), whereas the RG equation, in agreement with the results of a crossover equation of state, provides the appropriate critical behavior. An application to the Ambrose-Walton vapor pressure equation is also presented.