Carnot's theorem (thermodynamics)
Carnot's theorem (thermodynamics)
Carnot's theorem, developed in 1824 by Nicolas Léonard Sadi Carnot, also called Carnot's rule, is a principle that specifies limits on the maximum efficiency any heat engine can obtain. The efficiency of a Carnot engine depends solely on the temperatures of the hot and cold reservoirs.
Carnot's theorem states:
All heat engines between two heat reservoirs are less efficient than a Carnot heat engine operating between the same reservoirs.
Every Carnot heat engine between a pair of heat reservoirs is equally efficient, regardless of the working substance employed or the operation details.
The formula for this maximum efficiency is
- is the
- is the
Based on modern thermodynamics, Carnot's theorem is a result of the second law of thermodynamics. Historically, it was based on contemporary caloric theory and preceded the establishment of the second law.[1]
Proof
It is generally agreed that this is impossible because it violates the second law of thermodynamics.
- engine. If the less efficient engine (
Having established that the heat flow values shown in the figure are correct, Carnot's theorem may be proven for irreversible and the reversible heat engines.[3]
Reversible engines
- All reversible engines that operate between the same two heat reservoirs have the same efficiency.
This is an important result because it helps establish the Clausius theorem, which implies that the change in entropy is unique for all reversible processes.,[4]
over all paths (from a to b in V-T space). If this integral were not path independent, then entropy, S, would lose its status as a state variable.[5]
Irreversible engines
- No irreversible engine is more efficient than the Carnot engine operating between the same two reservoirs.
Definition of thermodynamic temperature
The efficiency of the engine is the work divided by the heat introduced to the system or
**(1)** |
where wcy is the work done per cycle. Thus, the efficiency depends only on qC/qH.
Because all reversible engines operating between the same heat reservoirs are equally efficient, all reversible heat engines operating between temperatures T1 and T2 must have the same efficiency, meaning the efficiency is a function only of the two temperatures:
**(2)** |
In addition, a reversible heat engine operating between temperatures T1 and T3 must have the same efficiency as one consisting of two cycles, one between T1 and another (intermediate) temperature T2, and the second between T2 and T3. This can only be the case if
Therefore, if thermodynamic temperature is defined by
then the function viewed as a function of thermodynamic temperature, is
and the reference temperature T1 has the value 273.16. (Of course any reference temperature and any positive numerical value could be used—the choice here corresponds to the Kelvin scale.)
It follows immediately that
**(3)** |
Substituting Equation 3 back into Equation 1 gives a relationship for the efficiency in terms of temperature:
**(4)** |