Carnot's theorem (thermodynamics)

Carnot's theorem states that all heat engines operating between the same two thermal or heat reservoirs cannot have efficiencies greater than a reversible heat engine operating between the same reservoirs. A corollary of this theorem is that every reversible heat engine operating between a pair of heat reservoirs is equally efficient, regardless of the working substance employed or the operation details. Since a Carnot heat engine is also a reversible engine, the efficiency of all the reversible heat engines is determined as the efficiency of the Carnot heat engine that depends solely on the temperatures of its hot and cold reservoirs.

The maximum efficiency (i.e., the Carnot heat engine efficiency) of a heat engine operating between hot and cold reservoirs, denoted as H and C respectively, is the ratio of the temperature difference between the reservoirs to the hot reservoir temperature, expressed in the equation

${\displaystyle {\eta }_{\text{max}}=\frac{T_{\mathrm{H}}-T_{\mathrm{C}}}{T_{\mathrm{H}}},}$

where ⁠${\displaystyle T_{\mathrm{H}}}$⁠ and ⁠${\displaystyle T_{\mathrm{C}}}$⁠ are the absolute temperatures of the hot and cold reservoirs, respectively, and the efficiency ⁠${\displaystyle \eta }$⁠ is the ratio of the work done by the engine (to the surroundings) to the heat drawn out of the hot reservoir (to the engine).

⁠${\displaystyle {\eta }_{\text{max}}}$⁠ is greater than zero if and only if there is a temperature difference between the two thermal reservoirs. Since ⁠${\displaystyle {\eta }_{\text{max}}}$⁠ is the upper limit of all reversible and irreversible heat engine efficiencies, it is concluded that work from a heat engine can be produced if and only if there is a temperature difference between two thermal reservoirs connecting to the engine.

Carnot's theorem is a consequence of the second law of thermodynamics. Historically, it was based on contemporary caloric theory, and preceded the establishment of the second law.

Proof

The proof of the Carnot theorem is a proof by contradiction, based on a situation like the right figure where two heat engines with different efficiencies are operating between two thermal reservoirs at different temperature. The relatively hotter reservoir is called the hot reservoir and the other reservoir is called the cold reservoir. A (not necessarily reversible) heat engine ${\displaystyle M}$ with a greater efficiency ${\displaystyle {\eta }_{{}_M}}$ is driving a reversible heat engine ${\displaystyle L}$ with a less efficiency ${\displaystyle {\eta }_{{}_L}}$, causing the latter to act as a heat pump. The requirement for the engine ${\displaystyle L}$ to be reversible is necessary to explain work ${\displaystyle W}$ and heat ${\displaystyle Q}$ associated with it by using its known efficiency. However, since ${\displaystyle {\eta }_{{}_M}>{\eta }_{{}_L}}$, the net heat flow would be backwards, i.e., into the hot reservoir:

${\displaystyle Q_{\text{h}}^{\text{out}}=Q<\frac{{\eta }_{{}_M}}{{\eta }_{{}_L}}Q=Q_{\text{h}}^{\text{in}},}$

where ${\displaystyle Q}$ represents heat, ${\displaystyle \text{in}}$ denotes input to an object, ${\displaystyle \text{out}}$ for output from an object, and ${\displaystyle h}$ for the hot thermal reservoir. If heat ${\displaystyle Q_{\text{h}}^{\text{out}}}$ flows from the hot reservoir then it has the sign of + while if ${\displaystyle Q_{\text{h}}^{\text{in}}}$ flows from the hot reservoir then it has the sign of -. This expression can be easily derived by using the definition of the efficiency of a heat engine, ${\displaystyle \eta =W/Q_{\text{h}}^{\text{in}}}$, where work and heat in this expression are net quantities per engine cycle, and the conservation of energy for each engine as shown below. The sign convention of work ${\displaystyle W}$, with which the sign of + for work done by an engine to its surroundings, is employed.

The above expression means that heat into the hot reservoir from the engine pair (can be considered as a single engine) is greater than heat into the engine pair from the hot reservoir (i.e., the hot reservoir continuously gets energy). A reversible heat engine with a low efficiency delivers more heat (energy) to the hot reservoir for a given amount of work (energy) to this engine when it is being driven as a heat pump. All these mean that heat can transfer from cold to hot places without external work, and such a heat transfer is impossible by the second law of thermodynamics.

• It may seem odd that a hypothetical reversible heat pump with a low efficiency is used to violate the second law of thermodynamics, but the figure of merit for refrigerator units is not the efficiency, ${\displaystyle W/Q_{\text{h}}^{\text{out}}}$, but the coefficient of performance (COP), which is ${\displaystyle Q_{\text{c}}^{\text{out}}/W}$ where this ${\displaystyle W}$ has the sign opposite to the above (+ for work done to the engine).

Let's find the values of work ${\displaystyle W}$and heat ${\displaystyle Q}$ depicted in the right figure in which a reversible heat engine ${\displaystyle L}$ with a less efficiency ${\displaystyle {\eta }_{{}_L}}$ is driven as a heat pump by a heat engine ${\displaystyle M}$ with a more efficiency ${\displaystyle {\eta }_{{}_M}}$.

The definition of the efficiency is ${\displaystyle \eta =W/Q_{\text{h}}^{\text{out}}}$ for each engine and the following expressions can be made:

${\displaystyle {\eta }_M=\frac{W_M}{Q_{\text{h}}^{\text{out},M}}=\frac{{\eta }_{M}Q}{Q}={\eta }_M,}$

${\displaystyle {\eta }_L=\frac{W_L}{Q_{\text{h}}^{\text{out},L}}=\frac{-{\eta }_{M}Q}{-\frac{{\eta }_M}{{\eta }_L}Q}={\eta }_L.}$

The denominator of the second expression, ${\displaystyle Q_{\text{h}}^{\text{out},L}=-\frac{{\eta }_M}{{\eta }_L}Q}$, is made to make the expression to be consistent, and it helps to fill the values of work and heat for the engine ${\displaystyle L}$.

For each engine, the absolute value of the energy entering the engine, ${\displaystyle E_{\text{abs}}^{\text{in}}}$, must be equal to the absolute value of the energy leaving from the engine, ${\displaystyle E_{\text{abs}}^{\text{out}}}$. Otherwise, energy is continuously accumulated in an engine or the conservation of energy is violated by taking more energy from an engine than input energy to the engine:

${\displaystyle E_{\text{M,abs}}^{\text{in}}=Q=(1-{\eta }_M)Q+{\eta }_{M}Q=E_{\text{M,abs}}^{\text{out}},}$

${\displaystyle E_{\text{L,abs}}^{\text{in}}={\eta }_{M}Q+{\eta }_{M}Q\left(\frac{1}{{\eta }_L}-1\right)=\frac{{\eta }_M}{{\eta }_L}Q=E_{\text{L,abs}}^{\text{out}}.}$

In the second expression, $\left|Q_{\text{h}}^{\text{out},L}\right|=\left|-\frac{{\eta }_M}{{\eta }_L}Q\right|$ is used to find the term ${\eta }_{M}Q\left(\frac{1}{{\eta }_L}-1\right)$ describing the amount of heat taken from the cold reservoir, completing the absolute value expressions of work and heat in the right figure.

Having established that the right figure values are correct, Carnot's theorem may be proven for irreversible and the reversible heat engines as shown below.

Reversible engines

To see that every reversible engine operating between reservoirs at temperatures ${\displaystyle T_1}$ and ${\displaystyle T_2}$ must have the same efficiency, assume that two reversible heat engines have different efficiencies, and let the relatively more efficient engine ${\displaystyle M}$ drive the relatively less efficient engine ${\displaystyle L}$ as a heat pump. As the right figure shows, this will cause heat to flow from the cold to the hot reservoir without external work, which violates the second law of thermodynamics. Therefore, both (reversible) heat engines have the same efficiency, and we conclude that:

All reversible heat engines that operate between the same two thermal (heat) reservoirs have the same efficiency.

The reversible heat engine efficiency can be determined by analyzing a Carnot heat engine as one of reversible heat engine.

This conclusion is an important result because it helps establish the Clausius theorem, which implies that the change in entropy ${\displaystyle S}$ is unique for all reversible processes:

${\displaystyle \mathrm{\Delta }S={\int }_a^b\frac{d{Q}_{\text{rev}}}{T}}$

as the entropy change, that is made during a transition from a thermodynamic equilibrium state ${\displaystyle a}$ to a state ${\displaystyle b}$ in a V-T (Volume-Temperature) space, is the same over all reversible process paths between these two states. If this integral were not path independent, then entropy would not be a state variable.

Irreversible engines

Consider two engines, ${\displaystyle M}$ and ${\displaystyle L}$, which are irreversible and reversible respectively. We construct the machine shown in the right figure, with ${\displaystyle M}$ driving ${\displaystyle L}$ as a heat pump. Then if ${\displaystyle M}$ is more efficient than ${\displaystyle L}$, the machine will violate the second law of thermodynamics. Since a Carnot heat engine is a reversible heat engine, and all reversible heat engines operate with the same efficiency between the same reservoirs, we have the first part of Carnot's theorem:

No irreversible heat engine is more efficient than a Carnot heat engine operating between the same two thermal reservoirs.