Question

I'm trying to verify that all reversible heat engines have equal efficiencies by using this diagram. Here, the second heat engine (HE2) is assumed to has higher efficiency than the first heat pump (HP1). Since W2 > W1, some work produced by the HE2 can be used for other work, which is denoted as W2 - W1. This concludes that the overall engine will be extracting heat energy from the low temperature reservoir and completely convert it to work (W2 - W1 = Q2 - Q2'), which violates Kelvin-Planck's statement. What I'm having trouble is, I don't get how Clausius statement (heat doesn't flow spontaneously from cold to hot regions) is violated here? Can someone help?

Answer 1

The Clausius statement is violated in the described system because the excess work from the more efficient heat engine effectively transfers heat from the cold to the hot reservoir, contradicting the natural flow of heat.

Step-by-step explanation:

The scenario described in the question violates the Kelvin-Planck statement of the second law of thermodynamics because it describes a system that seemingly transforms all heat from a cold reservoir into work, which is not possible.

To understand why the Clausius statement is violated, consider this: while it seems that no heat is being ‘transferred’ from the cold to the hot reservoir directly, the work done by the high-efficiency engine (HE2), which exceeds the work input into the heat pump (HP1), effectively acts as a transfer of energy from the cold to the hot reservoir.

This 'transfer' caused by the work difference, in a sense, ‘spontaneously’ moves heat from a cold to a hot body when viewed in terms of the overall system, hence breaking the Clausius statement.