Curvature of the Weinhold metric for thermodynamical systems with 2 degrees of freedom

In this work the curvature of Weinhold (thermodynamical) metric is studied in the case of systems with two thermodynamical degrees of freedom. Conditions for the Gauss curvature $R$ to be zero, positive or negative are worked out. Signature change of the Weinhold metric and the corresponding singular behavior of the curvature at the phase boundaries are studied. Cases of systems with the constant $C_{v}$, including Ideal and Van der Waals gases, and that of Berthelot gas are discussed in detail.