When I've studied Thermodynamics I did so in Callen's book and there the author talks about temperature as a single thing, which mathematically is simply defined as:
$$T = \dfrac{\partial U}{\partial S}.$$
Now, currently I'm needing to learn a more conceptual approach to Thermodynamics and in some books I see the authors talk about "two kinds of temperature".
One is called the empirical temperature and the other is called the thermodynamic temperature. As far as I understood the thermodynamic temperature is the one I've always known, which can be defined by the equation I've mentioned.
Now there's this empirical temperature and I have no idea about what it is. The authors introduce it by talking about lots of experiments relating properties of systems to thermodynamic equilibrium. In one of the books it is said:
Let $X$ represent the value of any thermodynamic property such as the emf $\mathcal{E}$ of a thermocouple, the resistance $R$ of a resistance thermometer, or the pressure $P$ of a fixed mass of gas kept at constant volume, and $\theta$ the empirical temperature of the thermometer or of any system with which it is in thermal equilibrium. The ratio of two empirical temperatures $\theta_2$ and $\theta_1$, as determined by a particular thermometer, is defined as equal to the corresponding ratio of the values of $X$:
$$\dfrac{\theta_{2}}{\theta_{1}}=\dfrac{X_2}{X_1}.$$
I don't really get what is going on here. What is this empirical temperature? What is its relation to the usual temperature? And why would anyone define something like that anyway?
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