Carnot's theorem (English: kar-NOH, French: [kaʁno]), also called Carnot's rule or Carnot's law, is a principle of thermodynamics developed by Nicolas Léonard Sadi Carnot in 1824 that specifies limits on the maximum efficiency that any heat engine can obtain.
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
η
max
=
T
H
−
T
C
T
H
,
{\displaystyle \eta _{\text{max}}={\frac {T_{\mathrm {H} }-T_{\mathrm {C} }}{T_{\mathrm {H} }}},}
where
T
H
{\displaystyle T_{\mathrm {H} }}
and
T
C
{\displaystyle T_{\mathrm {C} }}
are the absolute temperatures of the hot and cold reservoirs, respectively, and the efficiency
η
{\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).
η
max
{\displaystyle \eta _{\text{max}}}
is greater than zero if and only if there is a temperature difference between the two thermal reservoirs. Since
η
max
{\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.
Contents
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
M
{\displaystyle M}
with a greater efficiency
η
M
{\displaystyle \eta _{_{M}}}
is driving a reversible heat engine
L
{\displaystyle L}
with a less efficiency
η
L
{\displaystyle \eta _{_{L}}}
, causing the latter to act as a heat pump. The requirement for the engine
L
{\displaystyle L}
to be reversible is necessary to explain work
W
{\displaystyle W}
and heat
Q
{\displaystyle Q}
associated with it by using its known efficiency. However, since
Reversible engines
To see that every reversible engine operating between reservoirs at temperatures
T
1
{\displaystyle T_{1}}
and
T
2
{\displaystyle T_{2}}
must have the same efficiency, assume that two reversible heat engines have different efficiencies, and let the relatively more efficient engine
M
{\displaystyle M}
drive the relatively less efficient engine
L
{\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
S
Irreversible engines
Consider two engines,
M
{\displaystyle M}
and
L
{\displaystyle L}
, which are irreversible and reversible respectively. We construct the machine shown in the right figure, with
M
{\displaystyle M}
driving
L
{\displaystyle L}
as a heat pump. Then if
M
{\displaystyle M}
is more efficient than
L
{\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.
Definition of thermodynamic temperature
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
where
w
cy
{\displaystyle w_{\text{cy}}}
is the work done by the engine,
q
C
{\displaystyle q_{C}}
is the heat to the cold reservoir from the engine, and
q
H
{\displaystyle q_{H}}
is the heat to the engine from the hot reservoir, per cycle. Thus, the efficiency depends only on
q
C
q
H
{\displaystyle {\frac {q_{C}}{q_{H}}}}
.
Because all reversible heat engines operating between temperatures
T
1
{\displaystyle T_{1}}
and
T
Applicability to electrical systems
Since fuel cells can generate useful power when all components of the system are at the same temperature (
T
=
T
H
=
T
C
{\displaystyle T=T_{H}=T_{C}}
), they are clearly not limited by Carnot's theorem, which states that no power can be generated when
T
H
=
T
C
{\displaystyle T_{H}=T_{C}}
. This is because Carnot's theorem applies to engines converting thermal energy to work, whereas fuel cells instead convert chemical energy to work. Nevertheless, the second law of thermodynamics still provides restrictions on fuel cell energy conversion.
A Carnot battery is a type of energy storage system that stores electricity in thermal energy storage and converts the stored heat back to electricity through thermodynamic cycles.