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 TT and VV (∂U/∂V)T=T(∂p/∂T)V−p (\partial U / \partial V)_ {T}=T(\partial p / \partial T)_ {V}-p where U(T,V)U(T, V) is the internal energy of the substance. Instruction. Consider the infinitesimal Carnot cycle in the variables p,Vp, V

Reveal Answer
2.129. According to the Carnot theorem δA/δQ1=dT/T.\delta A / \delta Q_{1}=d T / T . Let us find the expressions for δA\delta A and δQ1\delta Q_{1}. For an infinitesimal Carnot cycle (e.g. parallelogram 1234 shown in Fig. 14) δA=dp⋅dV=(∂p/∂T)vdT⋅dVδQ1=dU12+pdV=[(∂U/∂V)T+p]dV \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. }
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