Carnot's theorem (thermodynamics)

An impossible situation: A heat engine cannot drive a less-efficient reversible heat engine without violating the second law of thermodynamics. Quantities in this figure are the absolute values of energy transfers (heat and work).

The proof of the Carnot theorem is a proof by contradiction or reductio ad absurdum (a method to prove a statement by assuming its falsity and logically deriving a false or contradictory statement from this assumption), 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, ${\displaystyle {\text{in}}}$ for input to an object denoted by the subscript, ${\displaystyle {\text{out}}}$ for output from an object denoted by the subscript, 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, ${\displaystyle \eta =W/Q_{h}^{out}}$, 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),[2] 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:

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

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

The denominator of the second expression, ${\displaystyle Q_{h}^{{\text{out}},L}=-{\frac {\eta _{M}}{\eta _{L}}}Q}$, 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, ${\textstyle \left|Q_{h}^{{\text{out}},L}\right|=\left|-{\frac {\eta _{M}}{\eta _{L}}}Q\right|}$ is used to find the term ${\textstyle \eta _{M}Q\left({\frac {1}{\eta _{L}}}-1\right)}$ 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.[3]

The efficiency of a heat engine is the work done by the engine divided by the heat introduced to the engine per engine cycle or

${\displaystyle \eta ={\frac {w_{\text{cy}}}{q_{H}}}={\frac {q_{H}-q_{C}}{q_{H}}}=1-{\frac {q_{C}}{q_{H}}}}$

(1)

where ${\displaystyle w_{cy}}$ is the work done by the engine, ${\displaystyle q_{C}}$ is the heat to the cold reservoir from the engine, and ${\displaystyle q_{H}}$ is the heat to the engine from the hot reservoir, per cycle. Thus, the efficiency depends only on ${\displaystyle {\frac {q_{C}}{q_{H}}}}$.[6]

Because all reversible heat engines operating between temperatures ${\displaystyle T_{1}}$ and ${\displaystyle T_{2}}$ must have the same efficiency, the efficiency of a reversible heat engine is a function of only the two reservoir temperatures:

${\displaystyle {\frac {q_{C}}{q_{H}}}=f(T_{H},T_{C})}$.

(2)

In addition, a reversible heat engine operating between temperatures ${\displaystyle T_{1}}$ and ${\displaystyle T_{3}}$ must have the same efficiency as one consisting of two cycles, one between ${\displaystyle T_{1}}$ and another (intermediate) temperature ${\displaystyle T_{2}}$, and the second between ${\displaystyle T_{2}}$ and ${\displaystyle T_{3}}$ (${\displaystyle T_{1}<T_{2}<T_{3}}$). This can only be the case if

${\displaystyle f(T_{1},T_{3})={\frac {q_{3}}{q_{1}}}={\frac {q_{2}q_{3}}{q_{1}q_{2}}}=f(T_{1},T_{2})f(T_{2},T_{3})}$.

(3)

Specializing to the case that ${\displaystyle T_{1}}$ is a fixed reference temperature: the temperature of the triple point of water as 273.16. (Of course any reference temperature and any positive numerical value could be used — the choice here corresponds to the Kelvin scale.) Then for any ${\displaystyle T_{2}}$ and ${\displaystyle T_{3}}$,

${\displaystyle f(T_{2},T_{3})={\frac {f(T_{1},T_{3})}{f(T_{1},T_{2})}}={\frac {273.16\cdot f(T_{1},T_{3})}{273.16\cdot f(T_{1},T_{2})}}.}$

Therefore, if thermodynamic temperature is defined by

${\displaystyle T'=273.16\cdot f(T_{1},T),}$

then the function viewed as a function of thermodynamic temperature, is

${\displaystyle f(T_{2},T_{3})={\frac {T_{3}'}{T_{2}'}}.}$

It follows immediately that

${\displaystyle {\frac {q_{C}}{q_{H}}}=f(T_{H},T_{C})={\frac {T_{C}'}{T_{H}'}}}$.

(4)

Substituting this equation back into the above equation ${\displaystyle {\frac {q_{C}}{q_{H}}}=f(T_{H},T_{C})}$ gives a relationship for the efficiency in terms of thermodynamic temperatures:

${\displaystyle \eta =1-{\frac {q_{C}}{q_{H}}}=1-{\frac {T_{C}'}{T_{H}'}}}$.

(5)

Since fuel cells and batteries can generate useful power when all components of the system are at the same temperature (${\displaystyle T=T_{H}=T_{C}}$), they are clearly not limited by Carnot's theorem, which states that no power can be generated when ${\displaystyle T_{H}=T_{C}}$. This is because Carnot's theorem applies to engines converting thermal energy to work, whereas fuel cells and batteries instead convert chemical energy to work.[7] Nevertheless, the second law of thermodynamics still provides restrictions on fuel cell and battery energy conversion.[8]

A Carnot battery is a type of energy storage system that stores electricity in heat storage and converts the stored heat back to electricity through thermodynamic cycles.[9]

• Chambadal–Novikov efficiency
• Heating and cooling efficiency bounds