What does it mean to specify the extensive state of an ideal gas?

[zenterix]

TL;DR Summary

What exactly does it mean to describe the extensive state of a system composed of a single one-phase ideal gas?

I was trying to solve the following problem at the end of chapter 1 of the book "Physical Chemistry", 4th Edition, by Silbey, Alberty, and Bawendy:

The intensive state of an ideal gas can be completely defined by specifying (1) T, P, (2) T, V, or (3) P, V. The extensive state of an ideal gas can be specified in four ways. What are the combinations of properties that can be used to specify the extensive state of an ideal gas? Although these choices are deduced for an ideal gas, they also apply to real gases.

Here is the information I collected about this topic in the chapter:

1) Intensive properties of a gas remain the same for any subsystem of a system.

2) Extensive properties change when we consider subsystems of a system.

3) A state equation describes the state of a gas based on the values of a few specific variables.

In the case of an ideal gas, the ideal gas state equation is PV = nRT.

P and T are intensive: if we have a system consisting of a certain amount of gas (n moles) at P, T, and V and we consider only half of the system, then this half-system will have the same P and T but both V and n will be cut in half. Note that V/n will be the same in the half-system.

Thus, P, T, and V/n are intensive properties of the system and V and n are extensive properties.

A certain number of intensive variables describe the intensive state of a system. This "certain number" is $N_S+1$, where $N_S$ is the number of different kinds of species in the system.

On the other hand, we can also speak of an extensive state of a system, but to describe it we need a certain number of intensive variables plus at least one extensive variable. This "certain number" is $(N_S+1)+1$, with the last one being extensive.

Now, in the problem above, we are told that we can describe the intensive state of a system in three different ways, each consisting of two intensive variables. We wish to describe the extensive state.

When I first tried to solve this problem, I could only find three ways

1) P, V, n
2) P, T, n
3) T, V, n

because I understood from the chapter text that only one of the variables should be extensive and two intensive.

However, I looked up the answer and the four ways are

1) P, V, n
2) P, T, n
3) T, V, n
4) P, V, T

So, it's ok to have two extensive variables and one intensive variable to describe the extensive state.

Truth is at this point I have no idea what it means to describe the extensive state.

In PV=nRT, since R is just a constant then if we specify any three of the four remaining variables we can obtain the fourth variable.

It seems that it was we are doing here.

My question is, what exactly does it mean to describe the extensive state?.

 

[Borek]

Extensive properties depend on the amount of substance, any way of describing the system which can be used to calculate number of moles fits.

 

[Chestermiller]

We can express any equation of state of a pure single phase substance in a form involving three intensive variables: F(P,T,v)=0 where the specific molar volume v is given by $$v=\frac{V}{n}$$ so once two of the three intensive variables are specified, the third one is known. But the state is also determined by the amount of substance you have, so we need an additional (extensive) variable to specify the state: n or V separately. So, for example, state 1 is $$1)\ T,v,n=T,\frac{V}{n},n\rightarrow T,V,n$$

 

[vela]

Now, in the problem above, we are told that we can describe the intensive state of a system in three different ways, each consisting of two intensive variables.

I was confused by this statement until I found out that the $V$ in the problem statement you provided is actually $\bar{V}$ (what Chester called $v$) in the actual problem statement in the book, not the $V$ as you used the symbol everywhere else in your post. It would have been helpful if you had pointed that out.