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To provide an analytic verification of the nature of phase transition at thecritical point of $P-V$ criticality, the original expressions of Ehrenfestequations have been introduced directly. By treating the cosmological constantand its conjugate quantity as thermodynamic pressure and volume respectively,we carry out analytical check of classical Ehrenfest equations. To show thatour approach is universal, we investigate not only higher-dimensional chargedAdS black holes, but also rotating AdS black holes. Not only are the examplesof Einstein gravity shown, but also the example of modified gravity ispresented for Gauss-Bonnet AdS black holes. The specific heat at constantpressure $C_P$, the volume expansion coefficient $\alpha$ and the isothermalcompressibility coefficient $\kappa_T$ are found to diverge exactly at thecritical point. It has been verified that both Ehrenfest equations hold at thecritical point of $P-V$ criticality in the extended phase spaces of AdS blackholes. So the nature of the critical point of $P-V$ criticality of AdS blackholes has been demonstrated analytically to be a second-order phase transition.These results are consistent with the nature of liquid-gas phase transition atthe critical point. In this sense, our research would deepen the understandingof the relations of AdS black holes and liquid-gas systems. Moreover, oursuccessful approaches to introduce the original expressions of Erhenfestequations directly into black hole phase transition research demonstrate againthat black hole thermodynamics is closely related to classical thermodynamics,which allows us to borrow techniques from classical thermodynamics toinvestigate the thermodynamics of black holes.

Ehrenfest scheme for $P-V$ criticality of higher dimensional charged black holes, rotating black holes and Gauss-Bonnet AdS black holes