Is the Clausius statement violated by a non-cyclic isothermal ideal gas expansion?

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?

• 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. Commented Mar 26, 2023 at 16:32
• 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"! Commented Mar 26, 2023 at 17:16

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.