# Puzzled: why is this even linear? Must it be? Looking for clarification

• nonequilibrium
In summary, the Celsius temperature scale was initially defined by using the freezing and boiling points of water as reference points. However, this definition was not based on any scientific understanding of temperature and was later reversed. The use of water as a medium for thermometers was not a good idea due to its non-linear expansion. The equal division of the temperature scale was derived from the ideal gas equation and was found to be a "good enough" assumption for practical purposes. However, this assumption was not made by early scientists and was only discovered later.
D H said:
That is an outstanding example, but not of students being fed stuff that is meaningless. It is an outstanding example of modern science at work during its very infancy.

They are not explained that either.

Aha, thank you epenguin, that helps.

So if I understand well, the underlying key understanding is that the ideal gas was very understandable concept and it made the natural suggestion to define temperature as to make it linear with its expansion. Because the ideal gas is also an (exteme) idealized model for most gases and fluids, its linearity became a good general approximation.
Is the fact that "if we have two identical objects at different temperatures, they equilibrate at the temperature that is the mathematical mean of the two initial temperatures" (if the temp.differences are not too big) also a result of the same principle? Namely that this happens to be true for an ideal gas and that the ideal gas is a basic rudimentary approximation for gases and fluids? Ah yes I see that these two characteristics of the ideal gas are logically equivalent: if temperature is a linear measure for energy content and if two identical boxes at different T's equilibrate, the energy content of one of either boxes will be the mathematical mean :)

Hm, afterwards we noticed we could define a temperature $$\theta$$ independent of any material using the 2nd law (and 0th and 1st) by choosing an arbitrary reference temperature and put it in $$f(\theta,\theta_0) = Q/Q_0$$ if Q is the heat absorbed at $$\theta$$ and Q_0 the heat deposited at $$\theta_0$$ by a reversible cycle. The funny thing is that... the most simple choice for a function, namely $$\theta / \theta_0$$ makes this general definition coincide with that of the ideal gas temperature T... Is this truly a coincidence?

I was about to log off, here it's very late, I cannot concentrate.

For your first part I'd roughly agree. Temperature is a bit like time. At first the rotation of the Earth was the standard of time, the clock, your mercury thermometer. But like as you first said, different thermometers didn't necessarily agree nor do different clocks. If the rotation of the Earth is the clock, then it makes no sense to say the Earth is slowing down or varying in any way. You can only say that if there is a better clock. How do you know another clock is better? Well because you have a theory, like for gases, that makes you think another clock has simpler, better understood physics, is less influenced e.g. by environment outside the clock. The rotating Earth on second approximation is quite complicated.

I have not followed you last para today. Perhaps you will re-examine it. You talk of heat - but to measure heat e.g. in a calorimeter requires you measure temperature.

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