hot reservoir, is equal to or less than the magnitude of the entropy "gain" () by the system, so the net entropy change is zero or positive. Similarly, when the system at temperature expels heat in magnitude () into a colder (cold) reservoir (at temperature ) in an infinitesimal step (called the step 2), then again, for the Second Law of Thermodynamics to hold, one would have, in a very similar manner:Here, the amount of heat 'absorbed' by the system is given by , signifying that heat is actually transferring (leaving) from the system to the cold reservoir, with . The magnitude of the entropy gained by the cold reservoir is equal to or greater than the magnitude of the entropy loss of the system , so the net entropy change is also zero or positive in this case. Because the total change in entropy for the system is zero in a thermodynamic cyclic process where all state functions of the system are reset or returned to initial values (values at the process starts) upon the completion of each cycle, if one adds all the infinitesimal steps of heat intake from and heat expulsion to the reservoirs, signified by the previous two equations, with the temperature of each reservoir at each instant given by , one gets In particular, which was to be proven (and is now proven). In summary, (the inequality in the third statement below, being obviously guaranteed by the second law of thermodynamics, which is the basis of our calculation), (as a cyclic process), For a reversible cyclic process, there is no generation of entropy in each of the infinitesimal heat transfer processes since there is practically no temperature difference between the system and the thermal reservoirs (I.e., the system entropy change and the reservoirs entropy change is equal in magnitude and opposite in sign at any instant.), so the following equality holds, (as a cyclic process), The Clausius inequality is a consequence of applying the second law of thermodynamics at each infinitesimal stage of heat transfer, and is thus in a sense a weaker condition than the Second Law itself. Heat engine efficiency In the heat engine model with two thermal reservoirs (hot and cold reservoirs), the limit of the efficiency of any heat engine , where and are work done by the heat engine and heat transferred from the hot thermal reservoir to the engine,