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

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• zenterix
zenterix
TL;DR Summary
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

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

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?

## What does it mean to take a limit in a thermodynamics equation?

Taking a limit in a thermodynamics equation typically means examining the behavior of a system as a certain parameter approaches a particular value. This could involve temperature approaching absolute zero, volume approaching infinity, or time approaching steady-state conditions. Limits help in understanding idealized scenarios and in simplifying complex equations to more manageable forms.

## Why are limits important in thermodynamics?

Limits are crucial in thermodynamics because they allow scientists to derive fundamental laws and principles under idealized conditions. For example, the concept of reversible processes and the definition of thermodynamic quantities such as entropy and enthalpy often rely on taking the limit of certain parameters. This helps in developing a deeper understanding of the underlying physical phenomena.

## Where is the limit taken when defining the thermodynamic limit?

The thermodynamic limit is taken as the number of particles in a system approaches infinity while keeping the density constant. This means taking the limit as the volume of the system goes to infinity, but the ratio of the number of particles to the volume remains constant. This concept is essential for deriving macroscopic properties from microscopic laws.

## How is the limit used in the context of the Carnot cycle?

In the Carnot cycle, limits are often taken to describe an idealized process. For instance, the efficiency of a Carnot engine is derived by taking the limit as the processes become infinitely slow, ensuring they are reversible and there is no entropy production. This helps in defining the maximum possible efficiency that any heat engine can achieve.

## What role do limits play in phase transitions?

Limits play a significant role in understanding phase transitions. For example, the limit is taken as temperature approaches the critical temperature from either side of a phase boundary. This helps in examining the behavior of physical properties such as specific heat, magnetization, and susceptibility, which often show singular behavior at the transition point.

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