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We study the thermodynamics and thermodynamic geometry of a five-dimensionalReissner-Nordstr\"om-AdS black hole in the extended phase space by treating thecosmological constant as being related to the number of colors in the boundarygauge theory and its conjugate quantity as the associated chemical potential.It is found that the contribution of the charge of the black hole to thechemical potential is always positive and the existence of charge make thechemical potential become positive more easily. We calculate the scalarcurvatures of the thermodynamical Weinhold metric, Ruppeiner metric and Quevedometric, respectively, in the fixed $N^2$ case and the fixed $q$ case. It isfound that in the fixed $N^2$ case the divergence of the scalar curvature isrelated to the divergence of the specific heat with fixed electric potential inthe Weinhold metric and Ruppeiner metric, and the divergence of the scalarcurvature in the Quevedo metric corresponds to the divergence of the specificheat with fixed electric charge density. In the fixed $q$ case, however, thedivergence of the scalar curvature is related to the divergence of the specificheat with fixed chemical potential in the Weinhold metric and Ruppeiner metric,while in the Quevedo metric the divergence of the scalar curvature correspondsto the divergence of the specific heat with fixed number of colors and thevanishing of the specific heat with fixed chemical potential.

Phase transition and Thermodynamical geometry of Reissner-Nordstr\"om-AdS Black Holes in Extended Phase Space