Clausius statement of the second law

The Clausius statement is an operational form of the second law :

It is impossible to construct a device operating in a cycle whose sole net effect is to transfer heat from a colder body to a hotter body.

In other words, a refrigerator or heat pump cannot move heat “uphill in temperature ” without consuming work from a work source .

Physical interpretation

When two bodies are placed in thermal contact through a diathermal wall , heat flows spontaneously from higher temperature to lower temperature until thermal equilibrium is reached. The Clausius statement asserts that reversing this natural direction requires compensating changes elsewhere—most simply, an input of work.

Key relations

• Refrigerator energy balance. In a cyclic refrigerator exchanging heats with a cold reservoir ($Q_c$ extracted) and a hot reservoir ($Q_h$ delivered), the first law gives

  $$Q_h = Q_c + W_{\text{in}},$$

  so moving heat from cold to hot requires positive work input $W_{\text{in}}>0$.

• Equivalence to other statements. The Clausius statement is equivalent to the Kelvin–Planck statement : if either were false, one could combine devices to violate the other. Both are summarized quantitatively by the Clausius inequality .

• Entropy viewpoint. For a closed system exchanging heat with reservoirs, the Clausius statement reflects that total entropy production is nonnegative, so spontaneous heat flow increases the entropy of the combined “system + reservoirs.”