Critical behaviour and microscopic structure of charged AdS black holes via an alternative phase space

It has been argued that charged Ads black holes have similar thermodynamic behavior as the Van der Waals fluid system, provided one treats the cosmological constant as a thermodynamic variable in an extended phase space. In this paper, we disclose the deep connection between charged AdS black holes and Van der Waals fluid system without extending the phase space. We keep the cosmological constant as a fixed parameter and instead, treat the square of the charge of black hole as a thermodynamic variable. Therefore, we write the equation of state as $Q^{2}=Q^{2}(T,\Psi)$ where $\Psi$ (conjugate of $Q^{2} $) is the inverse of the specific volume, $\Psi=1/v$. This allows us to complete the analogy of charged AdS black holes with Van der Waals fluid system and derive the phase transition as well as critical exponents of the system. We identify a thermodynamic instability in this new picture with real analogy to Van der Waals fluid with physically relevant Maxwell construction. We therefore study the critical behavior of isotherms in $% Q^2-\Psi$ diagram and deduce all the critical exponents of the system and determine that the system exhibits a small-large black hole phase transition at the critical point $(T_c,Q^2_c, \Psi_c)$. This alternative view is important as one can imagine such a change for a given single black hole i. e. acquiring charge which induces the phase transition. Finally, we disclose the microscopic properties of charged AdS black holes by using thermodynamic geometry. Interestingly, we find that scalar curvature has a gap between small and large black holes, and this gap becomes exceedingly large as one moves away from the critical point along the transition line. Therefore, we are able to attribute the sudden enlargement of the black hole to the strong repulsive nature of the internal constituents at the phase transition.