Second Law of Thermodynamics - Corollaries - Thermodynamic Temperature

Thermodynamic Temperature

For an arbitrary heat engine, the efficiency is:

where A is the work done per cycle. Thus the efficiency depends only on qC/qH.

Carnot's theorem states that all reversible engines operating between the same heat reservoirs are equally efficient. Thus, any reversible heat engine operating between temperatures T1 and T2 must have the same efficiency, that is to say, the efficiency is the function of 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 T2 andT3. 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).

Now consider the case where is a fixed reference temperature: the temperature of the triple point of water. Then for any T2 and T3,

Therefore if thermodynamic temperature is defined by

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

and the reference temperature T1 will have 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.)

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