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I have read that Kelvin-Planck statement and Clausius statement are equivalent. The problem I have is that Kelvin-Planck statement does not deny the possibility of a non-cyclic process where the net result is complete conversion of heat $Q_H$ into work $W$, like the case of an isothermal ideal gas expansion. Couldn't the output work of that process $W$ be used to run a refrigerator that takes $Q_c$ from a colder reservoir and deliver heat $Q'_H$ at the original hot reservoir where $Q'_H = Q_c + W = Q_c + Q_H$, thus having a net process that violates the Clausius statement even thoough it doesn't violate the Kelvin-Planck statement due to the cyclic requirement of the latter?

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  • $\begingroup$ Please clarify your question to show mathematically what violation is claimed to occur. Make sure to include the entropy gain from the one-time expansion. $\endgroup$ Commented Mar 26, 2023 at 16:32
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    $\begingroup$ Both the Clausius and the Kelvin-Planck versions require that the engine operate in a cycle, otherwise you cannot be sure that the processes return to a state of "no other changes than"! $\endgroup$
    – hyportnex
    Commented Mar 26, 2023 at 17:16

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I have read that Kelvin-Planck statement and Clausius statement are equivalent.

The Kelvin-Planck statement is: No heat engine can operate in a cycle while transferring heat with a single reservoir. The Clausius' statement is: No refrigeration or heat pump cycle can operate without a net work input. They are equivalent to the extent that the heat engine cycle becomes a refrigeration or heat pump cycle when operated in reverse. See the Figure below.

The problem I have is that Kelvin-Planck statement does not deny the possibility of a non-cyclic process where the net result is complete conversion of heat $Q_H$ into work $W$, like the case of an isothermal ideal gas expansion.

What is the problem with that? The statement only applies to a cycle. And the net result of the cycle is not the complete conversion of heat $Q_H$ into work $W$ by the isothermal expansion. The net work done in the cycle is the work done by the isothermal expansion minus the work done by the isothermal compression, or equivalently $W=Q_{H}-Q_c$.

Couldn't the output work of that process $W$ be used to run a refrigerator that takes $Q_c$ from a colder reservoir and deliver heat $Q'_H$ at the original hot reservoir where $Q'_H = Q_c + W = Q_c + > Q_H$,

It appears you are thinking that the net work done by the heat engine cycle is the work done by the isothermal expansion, which it is not. $Q^{'}_{H}=Q_H$ and $W=Q_{H}-Q_c$.

Hope this helps.

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