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 w_cy is the work done per cycle. Thus, the efficiency depends only on q_C/q_H.

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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