Carnot's Theorem (thermodynamics) - Definition of Thermodynamic Temperature

Definition of Thermodynamic Temperature

The efficiency of the engine is the work divided by the heat introduced to the system or

,

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, any reversible heat engine operating between temperatures T1 and T2 must have the same efficiency, meaning, the efficiency is the function of the temperatures only:

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 T2andT3. This can only be the case if

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).

Specializing to the case that is a fixed reference temperature: the temperature of the triple point of water. Then for anyT2and T3,

Therefore, if thermodynamic temperature is defined by

then the function f, 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

Substituting Equation 3 back into Equation 1 gives a relationship for the efficiency in terms of temperature:

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