We obtained mechanical and caloric consistency expressions (5.47) and (5.48). We may also consider self consistency relations for a hypothetical substance based on the Gibbs equation.
(a) Show that equality of cross partial derivatives in the Gibbs equation demands that [∂ (1/T)/∂V] U = [∂ (P/T)/∂U] V for any substance.
(b) Is a substance satisfying U = C1PV and PV2 = C2T possible?
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(c) Is the substance satisfying U = C1PV2 and PV2 = C2T possible?