Clausius theorem

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The Clausius theorem (1855), also known as the Clausius inequality, states that for a thermodynamic system (e.g. heat engine or heat pump) exchanging heat with external thermal reservoirs and undergoing a thermodynamic cycle, the following inequality holds.

${\displaystyle -\oint dS_{\text{Res}}=\oint {\frac {\delta Q}{T_{\text{surr}}}}\leq 0,}$

where ${\displaystyle \oint dS_{\text{Res}}}$ is the total entropy change in the external thermal reservoirs (surroundings), ${\displaystyle \delta Q}$ is an infinitesimal amount of heat that is taken from the reservoirs and absorbed by the system (${\displaystyle \delta Q>0}$ if heat from the reservoirs is absorbed by the system, and ${\displaystyle \delta Q}$ < 0 if heat is leaving from the system to the reservoirs) and ${\displaystyle T_{\text{surr}}}$ is the common temperature of the reservoirs at a particular instant in time. The closed integral is carried out along a thermodynamic process path from the initial/final state to the same initial/final state (thermodynamic cycle). In principle, the closed integral can start and end at an arbitrary point along the path.

The Clausius theorem or inequality obviously implies ${\displaystyle \oint dS_{\text{Res}}\geq 0}$ per thermodynamic cycle, meaning that the entropy of the reservoirs increases or does not change, and never decreases, per cycle.

For multiple thermal reservoirs with different temperatures ${\displaystyle \left(T_{1},T_{2},\dots ,T_{N}\right)}$ interacting a thermodynamic system undergoing a thermodynamic cycle, the Clausius inequality can be written as the following for expression clarity:

${\displaystyle -\oint dS_{\text{Res}}=\oint \left(\sum _{n=1}^{N}{\frac {\delta Q_{n}}{T_{n}}}\right)\leq 0.}$

where ${\displaystyle \delta Q_{n}}$ is an infinitesimal heat from the reservoir ${\displaystyle n}$ to the system.

In the special case of a reversible process, the equality holds,^[1] and the reversible case is used to introduce the state function known as entropy. 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 ${\displaystyle {\frac {\delta Q}{T}}}$.

The generalized "inequality of Clausius"^[2]

${\displaystyle dS_{\text{sys}}\geq {\frac {\delta Q}{T_{\text{surr}}}}}$

for ${\displaystyle dS_{\text{sys}}}$ 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. The Clausius statement states that it is impossible to construct a device whose sole effect is the transfer of heat from a cool reservoir to a hot reservoir.^[3] Equivalently, heat spontaneously flows from a hot body to a cooler one, not the other way around.^[4]