is the total entropy change in the external thermal reservoirs (surroundings),
is an infinitesimal amount of heat that is taken from the reservoirs and absorbed by the system (
In principle, the closed integral can start and end at an arbitrary point along the path.
This is because in a cyclic process the variation of a state function is zero per cycle, so the fact that this integral is equal to zero per cycle in a reversible process implies that there is some function (entropy) whose infinitesimal change is
as an infinitesimal change in entropy of a system (denoted by sys) under consideration applies not only to cyclic processes, but to any process that occurs in a closed system.
The Clausius inequality is a consequence of applying the second law of thermodynamics at each infinitesimal stage of heat transfer.
[3] Equivalently, heat spontaneously flows from a hot body to a cooler one, not the other way around.
[4] The Clausius theorem is a mathematical representation of the second law of thermodynamics.
Clausius developed this in his efforts to explain entropy and define it quantitatively.
In more direct terms, the theorem gives us a way to determine if a cyclical process is reversible or irreversible.
The Clausius theorem provides a quantitative formula for understanding the second law.
Clausius was one of the first to work on the idea of entropy and is even responsible for giving it that name.
Clausius sought to show a proportional relationship between entropy and the energy flow by heating (δQ) into a system.
Clausius writes that "The algebraic sum of all the transformations occurring in a cyclical process can only be less than zero, or, as an extreme case, equal to nothing."
In other words, the equation with 𝛿Q being energy flow into the system due to heating and T being absolute temperature of the body when that energy is absorbed, is found to be true for any process that is cyclical and reversible.
Clausius then took this a step further and determined that the following relation must be found true for any cyclical process that is possible, reversible or not.
This is in contrast to the amount of energy added as heat (𝛿Q) and as work (𝛿W), which may vary depending on the path.
In irreversible cases, the net entropy is added to the system reservoirs
per thermodynamic cycle while in reversible cases, no entropy is created or added to the reservoirs.
At each instant of the process, the system is in contact with an external reservoir.
Because of the Second Law of Thermodynamics, in each infinitesimal heat exchange process between the system and the reservoirs, the net change in entropy of the "universe", so to say, is
, where Sys and Res stand for System and Reservoir, respectively.
When the system takes heat from a hotter (hot) reservoir by an infinitesimal amount
forces us to have: This means the magnitude of the entropy "loss" from the hot reservoir,
, signifying that heat is actually transferring (leaving) from the system to the cold reservoir, with
is equal to or greater than the magnitude of the entropy loss of the system
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
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), 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, 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.
are work done by the heat engine and heat transferred from the hot thermal reservoir to the engine, respectively, can be derived by the first law of thermodynamics (i.e., the law of conservation of energy) and the Clausius theorem or inequality.