Construction of thermodynamic temperature

Below we have two engines (one consisting of two more in series) working between heat reservoirs at $\theta_H$ and $\theta_C$. Remember $\theta_H$ etc refer to temperatures on some unspecified scale. We know that the efficiency of a Carnot engine working between $\theta_H$ and $\theta_C$ depends only on $\theta_H$ and $\theta_C$, but we don't know how: let us define a function $f(\theta_H,\theta_C)$ such that

$$Q_C=f(\theta_C,\theta_H)Q_H \quad\hbox{so}\quad \eta=1-f(\theta_C,\theta_H).$$

Now considering the two engines in series,

$$\begin{eqnarray*} Q_3=f(\theta_3,\theta_H)Q_H\quad&&\hbox{and }\quad W_A=\left(1... ...B=\left(1-f(\theta_C,\theta_3)f(\theta_3,\theta_H)\right)Q_H && \end{eqnarray*}$$

But everything within the green oval in the diagram can be considered as a single, composite, Carnot engine, so its output must be the same as the simple one for the same heat flow $Q_H$: $W_A+W_B=W$. Thus we can match the efficiencies of the simple and composite engines to get

$$f(\theta_C,\theta_3)f(\theta_3,\theta_H)=f(\theta_C,\theta_H).$$

This has to be true independently of the value of $\theta_3$, which can only be true if $f$ factorises:

$$f(\theta_C,\theta_H)={\Theta(\theta_C)\over \Theta(\theta_H)}$$

where $\Theta$ is some function of $\theta$

Thus we have the desired result,

$$\eta_{\rm carnot}=1-{\Theta(\theta_C)\over \Theta(\theta_H)}.$$

References

• Bowley and Sánchez 2.3
• Adkins 4.5-6
• Zemansky 7.5