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The phase transition of Reissner-Nordstr\"om black holes in (n+1) -dimensional anti--de Sitter spacetime is studied in detail using the thermodynamic analogy between a RN-AdS black hole and a van der Waals liquid-gas system. We first investigate critical phenomena of the RN-AdS black hole. The critical exponents of relevant thermodynamical quantities are evaluated. We find identical exponents for a RN-AdS black hole and a van der Waals liquid-gas system. This suggests a possible universality in the phase transitions of these systems. We finally study the thermodynamic behavior using the equilibrium thermodynamic state space geometry and find that the scalar curvature diverges exactly at the van der Waals--like critical point where the heat capacity at constant charge of the black hole diverges.
Niu et al. (Thu,) studied this question.