The Clausius and the Kelvin statement of the
Second Law are equivalent

Let us first restate both statements (the sequence is arbitrary and does not reflect the relative importance) The Second Law:

• the first states that no cyclic process is possible that does nothing but convert heat into work (the Kelvin statement)
• the second states that no cyclic process is possible that does nothing but transfer heat from a cold body to a hot one (the Clausius statement)

The statements are equivalent, as can be proved by contradiction. Imagine the Kelvin statement holds but not the Clausius statement. Then, with the help of an anti Clausius machine you could devise an engine that would contradict the Kelvin statement.

Consider an ordinary heat engine that converts part of a given heat quantity Q_hot into work and dumps the rest Q_cold into a cold sink. Then use an anti-Clausius engine to pump the dumped heat Q_cold from the cold reservoir back into the hot one, in order to cancel all changes other than the heat to work conversion.

Fig. 1
If the Kelvin statement is true but not the Clausius statement, then a device that proves the Kelvin statement to be false can be devised (heat is coloured and work is grey).

The net effect of this engine is to convert part of the heat from the hot reservoir into work and to cause no other change whatsoever to the universe. This contradicts the Kelvin statement that was assumed to be true to derive this result.

Conversely, imagine the Clausius statement to hold but not the Kelvin statement. Now an anti Kelvin engine could convert the heat Q_hot from a hot reservoir into work without loss. This work could be used to drive an ordinary heat pump to transfer the heat Q_cold from the cold reservoir into the hot one. This would consist in a spontaneous heat flow from the cold reservoir to the hot one; spontaneous, that means causing no changes to the universe other than the heat flow.

Fig. 1
If the Clausius statement were true but not the Kelvin statement, then a engine could be constructed that proved the Clausius statement to be false (heat is coloured and work is grey).

Because of the First Law, a heat pump dumps as heat into a hot reservoir the sum of the energy it takes from the cold reservoir plus the energy it takes up as work. Thus our device dumps Q_hot plus Q_cold into the hot reservoir.

Net effect is a spontaneous flow of heat from the cold into the hot reservoir, contradicting the Clausius statement that was assumed to be true.

\ Hence, if the Clausius statement of the Second Law is true, then the Kelvin statement must also be true. And if the Kelvin statement is true, then so must be the Clausius statement. It follows that the two statements must be equivalent.

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