  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.
It is found that the contribution of the charge of the black hole to the
chemical potential is always positive and the existence of charge make 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. It is
found 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 fixed number of colors and the
vanishing of the specific heat with fixed chemical potential.
