I Where are the limits being taken in these thermodynamics equations?

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The discussion centers on the limits in thermodynamics equations, specifically questioning if the limits are taken as pressure approaches zero (P_TP → 0). The author of the book does not clarify whether the limit of (Pv) is also taken as P_TP approaches zero, leading to confusion. The context provided indicates that plotting (Pv) against pressure for different gases shows a consistent limiting value as pressure approaches zero. The explanation attempts to clarify that as the pressure in a constant volume thermometer decreases, the pressure associated with steam also approaches zero, suggesting that the limits of (Pv) for both pressures are equivalent. The inquiry seeks to confirm this understanding of the limits in the equations.
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In a passage in the book Heat and Thermodynamics by Zemansky, the notation omits the variable for which limits are being taken. I would like to understand the limits better.
Here is a passage from a book I am reading

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My question is about the limits.

Are all the limits in the derivation above done for ##P_{TP}\to 0##?

In particular, is it ##\lim\limits_{P_{TP}\to 0} (Pv)## that appears above?

The author omits this information in all but the first limit and it got me confused.

Here is a bit more context now to show why this has me confused.

Just before the equations above, the book writes of the fact that if we plot ##Pv## against ##P## for different gases at a specific temperature, we see that for all of the gases the limiting value of ##Pv## as ##P\to 0## is the same.

Here is an example at the boiling point of water

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Here is my attempt at explaining away the confusion

The ideal-gas temperature definition involves a limit in which we compute the value of ##P/P_{TP}## as ##P_{TP}## is made to approach zero at constant volume.

The way I understand this, a constant volume pressure thermometer is used. We have some particular temperature that we would like to measure, for example that of steam.

Now, in order to make ##P_{TP}## smaller, in each successive measurement we have the same volume of gas in the thermometer but we remove some gas from the thermometer: this way, the triple point of water is reached at a lower pressure for the same constant volume.

As we make these successive measurements, I think that the pressure ##P## associated with the steam will also be lower and will approach zero just like ##P_{TP}## (even though the ratio of these two pressures will approach a non-zero value).

Thus, it seems that ##\lim\limits_{P\to 0} (Pv)## is the same as ##\lim\limits_{P_{TP}\to 0} (Pv)##.

Is this what is happening?
 
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