An increment represents the process of heat exchange between two systems. Neither system can be said to possess a quantity of heat . In the usual terminology, is not a "function of state" for a thermodynamic system. However, turns out to represent the differential of a new function of state, , where is known as the entropy. A derivation of this relation is given in Details for the case of an ideal gas.

A compact statement of the second law of thermodynamics is that the entropy of an isolated system (or of the whole universe) always increases in a spontaneous process, . The reverse process is unattainable—as Bertrand Russell put it, "You cannot unscramble eggs." The limiting case pertains to reversible processes (themselves an idealization).

An operational statement of the second law is due to Clausius: "It is impossible in a cyclic process to produce no other effect than the transfer of heat from a cooler to a warmer body." Of course, refrigerators do this all the time, but this requires coupling to other devices, such as motors and condensers.

In this Demonstration, you can vary the temperatures and and the quantity of heat transferred, . The total entropy change of the system is then calculated using . Recall that is counted as negative when a reservoir gives off heat. The system of two reservoirs is presumed to be isolated from its surroundings.