Proportionality of Pressure and Temperature in defining Absolute Temperature scale

Question

I am studying thermodynamics and came across this kind of loop:

First, an absolute temperature scale is attempted to be defined as follows

Assume that temperature is proportional to the pressure of the gas in a constant volume thermometer.

So the book starts by assuming $$P=cT\tag{$i$},$$ $c$ being a constant.

Again it says:

The temperature of triple point of water is assigned a value of 273.16 K.

Let the pressure, at triple point of water , of the gas be $ P_{tr}$. Then, $$P_{tr} = c\cdot273.16K$$

Using it in in equation $(i)$ , we get: $$ T = \frac{P}{P_{tr}}\cdot273.16K\tag{$ii$} $$ It also corrects it saying that this would give absolute temperature only at low density and hence puts a limit ( $P_{tr}$ tends to $0$ ) .But in further use it neglects this limit.

Now a result from Kinetic Theory is derived: $$PV = \frac{1}{3}mNv_{rms}^2, $$

where
$V$ = Volume

$m$ = mass of single molecule of the gas

$N$ = total no. of molecules.

$v_{rms}$ = rms speed of the gas molecules.

It further uses this result along with eq $(ii)$ to prove the equation of states: $$PV=Nk_BT$$ or $ PV = nRT$ , with $n$ being the no. of moles of the gas and $R$ is the gas constant.

It could be shown using the state equation above that at constant volume and moles , pressure is directly proportional to absolute temperature , as assumed in the beginning .

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My questions are :

1. In defining the ideal gas temperature scale, it is assumed that the pressure of the gas at constant volume is proportional to the temperature $T$. How can we verify whether this is true or not? Are we using the kinetic theory of gases? Are we using the experimental result that the pressure is proportional to temperature?

Ironically this is one of the question given at the end of the text in book.

2. There is a loop that first assumes that pressure is proportional to temperature and finally proves that using that assumption. Isn't that like moving in a circle. How exactly is the absolute temperature scale defined.

3. Can the equation of state be proved or it is just on experimental basis . If it can be proved, then please provide the proof or just some hints .

It am totally okay with mathematical answer ( if required even Maxwell distribution and things except the canonical or other ensemble's mathematics ).

Answer 1

In defining the ideal gas temperature scale, it is assumed that the pressure of the gas at constant volume is proportional to the temperature $T$. How can we verify whether this is true or not? Are we using the kinetic theory of gases? Are we using the experimental result that the pressure is proportional to temperature?

There is no need for that. You can use the Van der Waals's equation of state, replacing the ideal gas law, and obtain the same results. Those will just be a complication.

Can the equation of state be proved or it is just on experimental basis . If it can be proved, then please provide the proof or just some hints.

Actually, you can side step all of these issues. By definition of the ideal gas, there is only kinetic energy, no potential energy. This means that any gas ought to become more and more like ideal gas if you make it sparser and sparser.

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In the following, for simplicity, I am only ever using the ideal gas law, but the results are the same if you use the Van der Waals's equation of state, or any better equation of state. All gases converge towards the ideal gas law in the limit $N\to0$

The ideal gas law is $$\tag1pV=Nk_BT$$ A fixed-volume gas thermometer, where we really don't care what $V$ specifically is, and in thermal equilibrium with a heat bath that is set at the triple point of water $T_{tr}=273.16\,$K (in practice, you simply bathe the thermometer in it, where you have a whole system just to keep the heat bath doing its freezing-boiling insanity the whole time), has its pressure $p_{tr}$ being actually a measure of the number of gas molecules inside the thermometer. That is, $$\tag2p_{tr}V=Nk_BT_{tr}\qquad\implies\qquad p_{tr}\propto N$$ Now, if you put this thermometer at an unknown temperature, you will measure a pressure, such that $$\tag3\frac p{p_{tr}}=\frac T{T_{tr}}$$ which we get by dividing Equation (1) by Equation (2).

If we do one such measurement, we will get a pair of values $(p_{tr},\dfrac p{p_{tr}})$. Starting with a somewhat high $p_{tr}$, you can finish one measurement, and then pump a bit of gas out of the thermometer. This decreases $p_{tr}$ and so you can measure another set. After 5 or more sets like these, you can plot $p_{tr}$ on the $x$-axis and $\dfrac p{p_{tr}}$ on the $y$-axis.

If you have an ideal gas, this plot will be a horizontal line. It never will be, because no real gas is ideal. You also do not care about the deviations from ideality when $p_{tr}$ is big. We also may never do an experiment such that $p_{tr}=0$; in fact, experiments become increasingly difficult to do, and wait times for equilibration get absurdly longer, when $p_{tr}\to0$. However, if you do this experiment, for any set of low $p_{tr}$, and any choice of gas molecule, you will get a slanted straight line, the starting part of a curve. The $y$-intercept limit of this line can thus be found.

The important property is this: Ideal gas law always holds for this $p_{tr}\to0$ limit. From which you can thus find out $T$

You can change different gas molecules, and find out that while the lines and their slopes can change, even have positive or negative slopes, but this limit is always unique and well-defined, equal for all different choices of gas molecules.

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The above experimental procedure can be justified by replacing Equations (1), (2) and (3) by their Van der Waals's counterparts. In Equation (2), the $p_{tr}$ can stay strictly linearly proportional to $N$, or pick up higher order corrections, it does not matter. Equation (3) will stop being an equality, but rather will turn into a slanted line, or curve that has a linear expansion when $p_{tr}\to0$, but all corrections will tend to zero with $p_{tr}\to0$, and then you can see that this proves that the ideal gas limit works.

That is, only the $y$-intercept values, i.e. of $\left.\dfrac p{p_{tr}}\right|_{p_{tr}=0}$ are actually well-defined. All the $\dfrac p{p_{tr}}$ values away from the $y$-axis, that we actually measure, are fictitious stepping stones just to get the $y$-intercepts that we actually want.

Notice that, since we have to keep waiting for the gas to get to equilibrium with the heat baths, determining one unknown temperature can easily take upwards of months. This is a price you gladly pay if you really want to get things correct.