Ehrenfest scheme for P-V criticality in the extended phase space of black holes

In this Letter, the nature of phase transition at the critical point of P-V criticality in the extended phase space of RN-AdS black holes has been investigated. By treating the cosmological constant and its conjugate quantity as thermodynamic pressure and volume respectively, we introduce the original expressions of Ehrenfest equations directly into the black hole research instead of utilizing the analogy of Ehrenfest equations. We carry out an analytical check of Ehrenfest equations and prove that both Ehrenfest equations are satisfied. So the black hole undergoes a second order phase transition at the critical point. This result is consistent with the nature of liquid-gas phase transition at the critical point, hence deepening the understanding of the analogy of charged AdS black holes and liquid-gas systems.