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In Thermodynamics (1st ed.) by James Luscombe (2018, p. 12), the zeroth law is used to show the existence of empirical temperature (of fluids in thermal equilibrium) as a function of pressure and volume. This notion of temperature is a logical consequence of the zeroth law. Am I right?

However, in An Introduction to Thermal Physics (1st ed.) by Daniel Schroeder (2021, p. 89), the temperature $T$ of a system is defined as \begin{equation*} T=\left(\frac{\partial S}{\partial U}\right)^{-1}, \end{equation*} where $S$ is the entropy and $U$ is the internal energy of the system, and the system's volume and number of particles are fixed. The author then asks us to use the definition of temperature to prove the zeroth law. This notion of temperature doesn't depend on the zeroth law. Am I right?

Are there two ways to define temperature?

Dumb Koala
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1 Answers1

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Yes, there are different notions of temperature and they are not the same.

The notion of temperature arising from the zeroth law is that you can assign some number to objects describing the direction of heat flows. This is not sufficiently precise to lead to a unique quantity temperature – even beyond the problem of deciding a unit or zero point (Celsius vs. Fahrenheit vs. Réaumur vs. …). This is because the zeroth law cannot tell you anything about how this temperature scales. For example the square of everyday temperature (measured in Kelvin) or the logarithm would be a temperature complying with the zeroth law. Only when we impose further empirical constraints such as quantifying the heat flow or thermal expansion scaling nicely with temperature differences, we get a unique physical quantity (and can then fight about units). Let’s call this empirical temperature.

The entropy-based definition coincides with empirical temperature in many applications, but not all. For example, it allows for negative temperatures, which do not comply with the zeroth law when it comes to the direction of heat flows: Heat would be flowing from any negative-entropy-temperature object to any positive-entropy-temperature object. Thus negative-entropy-temperature objects have a higher empirical temperature than positive-entropy-temperature objects.

Finally, a note regarding the last paragraph of your second text: The formulation of the zeroth law used in that text wouldn’t induce a notion of temperature as it is not about the direction of heat flows. Therefore, it is not in conflict with the entropy definition of temperature.

Also see this answer of mine on another question.

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