Phase transition and thermodynamical geometry of Reissner-Nordström-AdS black holes in extended phase space

We study the thermodynamics and thermodynamic geometry of a five-dimensional Reissner-Nordstr\"om-AdS black hole in the extended phase space by treating the cosmological constant as being related to the number of colors in the boundary gauge theory and its conjugate quantity as the associated chemical potential. We find that the contribution of the charge of the black hole to the chemical potential is always positive, and the existence of charge makes the chemical potential become positive more easily. We calculate the scalar curvatures of the thermodynamical Weinhold metric, Ruppeiner metric, and Quevedo metric, respectively, in the fixed ${N}^{2}$ case and the fixed $q$ case. We find that in the fixed ${N}^{2}$ case, the divergence of the scalar curvature is related to the divergence of the specific heat with fixed electric potential in the Weinhold metric and Ruppeiner metric, and the divergence of the scalar curvature in the Quevedo metric corresponds to the divergence of the specific heat with fixed electric charge density. In the fixed $q$ case, however, the divergence of the scalar curvature is related to the divergence of the specific heat with fixed chemical potential in the Weinhold metric and Ruppeiner metric, while in the Quevedo metric, the divergence of the scalar curvature corresponds to the divergence of the specific heat with a fixed number of colors and the vanishing of the specific heat with a fixed chemical potential.

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