All Problems Second law of thermodynamics, entropy
Problem 2.129
Making use of the Carnot theorem, show that in the case of a physically uniform substance whose state is defined by the parameters \(T\) and \(V\) \[ (\partial U / \partial V)_ {T}=T(\partial p / \partial T)_ {V}-p \] where \(U(T, V)\) is the internal energy of the substance. Instruction. Consider the infinitesimal Carnot cycle in the variables \(p, V\)
2.129. According to the Carnot theorem \(\delta A / \delta Q_{1}=d T / T .\) Let us find the expressions for \(\delta A\) and \(\delta Q_{1}\). For an infinitesimal Carnot cycle (e.g. parallelogram 1234 shown in Fig. 14) \[ \begin{array}{c} \delta A=d p \cdot d V=(\partial p / \partial T)_{v} d T \cdot d V \\ \delta Q_{1}=d U_{12}+p d V=\left[(\partial U / \partial V)_{T}+p\right] d V \end{array} \] \text { It remains to substitute the two latter expressions into the former one. }