Prove that a violation of the Clausius statement would necessarily lead to a violation of the Kelvin–Planck statement of the Second Law.
(a) Assume hypothetically that a cyclic device exists which transfers heat from a cold reservoir at temperature TC to a hot reservoir at temperature TH without any external work input. Show, with the aid of an appropriate schematic and energy balance, how this device could be coupled to an ordinary heat engine operating between the same two reservoirs to construct a composite system that converts all heat absorbed from the hot reservoir completely into work.
(b) Explain why the existence of such a composite system contradicts the Kelvin–Planck statement, thereby demonstrating the logical equivalence of the two statements of the Second Law.