2.4 Thermodynamic Temperature

Take-home message: The fact that the efficiency of a Carnot engine depends only on the operating temperatures provides an alternative way of defining a temperature scale.

If we hadn't already developed a good temperature scale, we could use the fact that the efficiency of a Carnot engine depends only on the operating temperatures to develop one.

Given a heat bath at a reference temperature (eg a very large triple-point cell) we could use the efficiency of a Carnot engine working between it and another body to label that other body's temperature.

By considering compound Carnot engines it can be shown that the dependence of the efficiency on temperature $\theta$, no matter what scale is used for the latter, has the form

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

where $\Theta(\theta)$ is some function of $\theta$. Thus $\Theta$ itself is a perfectly good temperature scale, which vanishes at absolute zero: this is called the thermodynamic temperature. (See here for details.)

The reason you haven't heard of it is that, if we compare it with the efficiency expressed in terms of the ideal gas temperature scale,

$$\eta_{\rm carnot}=1-{T_C\over T_H}$$

we see that $\Theta$ is simply proportional to the ideal gas temperature, and will therefore be identical if we set $\Theta_{\rm triple}=273.15$. Since a nearly ideal gas is more achievable than a reversible heat engine, that's the name that has finally stuck.

References

• Bowley and Sánchez 2.3-4
• Adkins 4.6
• Zemansky 7.5