Carnot Principle


A temperature
scale which is independent of the properties of the
substances that are used to measure temperature is called a thermodynamic
temperature scale. Its derivation is given below using some Carnot heat
engines.
The
Carnot principle states that the reversible heat engines have the
highest efficiencies when compared to irreversible heat engines working
between the same two reservoirs. And the efficiencies of all reversible
heat engines are the same if they work between the same two reservoirs.
That is, the efficiency of a reversible
heat engine is independent on the working fluid used and its properties,
the way the cycle operates, and the type of the heat engine. The efficiency
of a reversible heat engine is a function of the reservoirs' temperature
only.
η_{th} = 1  Q_{L}/Q_{H} = g(T_{H},T_{L})
or
Q_{H}/Q_{L} = f(T_{H},T_{L})
where
Q_{L} = heat transferred to the lowtemperature
reservoir which
has a
temperature of T_{L}
Q_{H} = heat transferred from the hightemperature
reservoir which
has a
temperature of T_{H}
g, f = any function

Relations between Reversible
Heat Engines A, B , and C


Consider three reversible heat engines working between a hightemperature
reservoir at temperature T_{1} and a lowtemperature reservoir
at temperature T_{3}. Engine A receives Q_{1} from
the hightemperature reservoir,and rejects Q_{2} to engine
B at temperature T_{2}. Engine B receives Q_{2} from
engine A at temperature T_{2}, and rejects Q_{3} to
the lowtemperature reservoir. Engine C receives Q_{1} from
the hightemperature reservoir,and rejects Q_{3} to the lowtemperature
reservoir. Applying the equation above to the three engines separately
yields,
Engine A: Q_{1}/Q_{2} = f(T_{1},T_{2})
(1)
Engine B: Q_{2}/Q_{3} = f(T_{2},T_{3}) (2)
Engine C: Q_{1}/Q_{3} = f(T_{1},T_{3}) (3)
Multiplying equations (1) and (2) gives
(4)
Comparing (4) and (3) yields,
f(T_{1},T_{3}) = f(T_{1},T_{2})f(T_{2},T_{3})
5)
The lefthand side of equation (5) is only a function of T_{1} and
T_{3}, and thus the righthand side of equation (5) is also
a function of T_{1} and T_{3} only. This condition
will be satisfied only if the function f has the following form:
and
Substituting them to equation (5) gives,
or,
(6)
Equation (6) provides a basis to define a thermodynamic temperature
scale. There are several choices for the function φ. The
Kelvin scale is obtained by making a simple choice as
φ(T) = T
Hence, equation (6) becomes,
Q_{1}/Q_{3} = T_{1}/T_{3}
To bring uniformity to the treatment of heat engines, H is used to denote the hightemperature reservoir, and L to denote the lowtemperature reservoir. The above equation becomes,
Q_{H}/Q_{L} = T_{H}/T_{L}

Determine Absolute Temperature by
Measuring Heat Transfers Q_{H} and Q_{L}


The Kelvin scale temperature is called absolute temperature.
At the International Conference on Weights and Measures held in 1954,
the triplepoint of water was assigned the value 273.16 K. Then, if
a reversible cycle is operated between a reservoir at 273.16 K and
another reservoir at temperature T, temperature T can be expressed as
T = 273.16(Q/Q_{tp})_{rev. cycle}
where Q and Q_{tp} are heat transfer between the cycle and
the reservoirs at temperature T and at temperature 273.16 K, respectively.
The magnitude of a Kevin scale is defined as 1/273.16 of the temperature
interval between the absolute zero and the tripepoint of water. The
magnitude of temperature units on Kevin scale is the same as that of
Celsius.
1 K 1 ^{o}C
When numerical values of the thermodynamic temperature are to be determined,
it is not possible to use a reversible cycle since that does not exist
in practice. Absolute temperatures can be measured by other means,
such as the constantvolume ideal gas thermometer.

A Carnot Heat Engine has the
Highest Efficiency of any
Heat Engines Working between
the Same Source and Sink


A
heat engine is called a Carnot heat engine if it operates
on the reversible Carnot cycle. The thermal efficiency of heat engine,
regardless reversible or irreversible, is given as
η_{th} = 1 Q_{L}/Q_{H}
where Q_{L} and Q_{H} are heat transfer between the
cycle and the lowtemperature reservoir at temperature T_{L} and
the hightemperature reservoir at temperature T_{H}, respectively.
For reversible heat engines, the heat transfer ratio can be replaced
by the absolute temperatures of the two reservoirs.
η_{th} = 1 T_{L}/T_{H}
This efficiency is called the Carnot efficiency since the Carnot
heat engine is one of the wellknown reversible engines. It is the
highest efficiency that a heat engine working between a hightemperature
reservoir at temperature T_{H} and a lowtemperature reservoir
at temperature T_{L} can reach. All irreversible heat engines
working between the same two reservoirs have lower efficiencies.
η_{th} 
< η_{th,rev} 
irreversible heat engine 
= η_{th,rev} 
reversible heat engine 
> η_{th,rev} 
impossible heat engine 
