Why is volume an extensive property but molar volume an intensive one?

Question

If we take volume of a system, then it's often defined as volume of container and is an extensive property but it's often said that molar volume is an intensive property. How exactly does dividing the volume by number of moles turn an extensive property into an intensive one?

reference: 21:36 of this of this video

see https://physics.stackexchange.com/questions/464840/rigorous-definition-of-intensive-and-extensive-quantities-in-thermodynamics — Andrew Steane, Sep 03 '20 at 11:36

Archisman Panigrahi · Accepted Answer · 2020-09-03T12:00:41.297

3

The number of moles is proportional to the number of atoms/molecules in the system.

Suppose you bring another identical copy of the system, and consider the two copies as a whole.

The number of atoms/molecules will double, so will the number of moles.

For usual solids and liquids, the volume will double (assuming external conditions like pressure, temperature remains the same).

For gases, the volume does not depend on the mass (it will consume the whole volume of the container). However, if another identical container (having the same mass of gas, and same $P$, $T$) is brought, the volume of the whole system will double. So, both these are extensive properties.

However, the ratio $\frac{\text{volume}}{\text{no of moles}}$ will remain the same. So it is an intensive property.

edited Sep 03 '20 at 12:00

answered Sep 03 '20 at 11:08

Archisman Panigrahi

3,130

@Buraian I don't think intensivity and extensivity of volume and moles become different for different kind of phase this happens. Why are you saying so ? – Ankit Sep 03 '20 at 11:14
I mean, whether the property is intensive or extensive. As you told, the increase of volmn when you increase moles only happens if you have liquid / solid phase – tryst with freedom Sep 03 '20 at 11:15
I think an analogy would help for last part – tryst with freedom Sep 03 '20 at 11:23
@AndrewSteane Thanks for the heads up, I have updated the answer. – Archisman Panigrahi Sep 03 '20 at 11:53
OK; I removed the down-vote by putting an up-vote. – Andrew Steane Sep 03 '20 at 13:21
@ArchismanPanigrahi What if we keep the volume constant and increase the no of moles? The definition is just $V/n$ so it doesn't say anything about "copies". Are all molar quantities defined with the assumption we take "copies" of the the system? – Anton Apr 18 '21 at 11:10
@Anton what exactly is your question? I did not get it. What if we keep the volume constant and increase the no of moles? If so, then what? – Archisman Panigrahi Apr 18 '21 at 12:27
@ArchismanPanigrahi If you have a system with a volume $V$ and amount of substance $n$ then $V_m = V/n$. Change $n$ and you have changed $V_m$ as well. – Anton Apr 21 '21 at 19:18
@Anton Yes, but so what? – Archisman Panigrahi Apr 21 '21 at 21:26

Protein · Answer 2 · 2020-09-03T11:30:21.603

The ratio of an extensive property to another extensive property is an intensive property. $$\frac{extensive \;property}{another \;extensive\; property}={\text extensive\, property\, per \,unit \,}$$

The extensive properties depend on amount and the amount is fixed by the property taken in denominator.

Molar volume =$\frac{volume\, of\, gas}{no.\, of\, moles}$

In the case of molar volume , dividing by no of moles fixes the amount to 1 mole.Thus molar volume is the volume of 1 mole of gas , a property which doesn't depend on the amount under observation.

Bob D · Answer 3 · 2020-09-03T11:53:41.940

The ratio of two extensive properties of the same system is an intensive property. Mass, volume, and moles are extensive properties. Mass divided by volume (density) and volume divided by moles (molar volume) are intensive properties.

An extensive property depends on the amount of stuff, an intensive property doesn’t. Take a room full of gas. It has volume, mass, pressure and temperature. If you divide the room in half each half has only half the original volume and mass of gas. But each half has the same density, pressure, molar volume and temperature as the original room full of gas.

Hope this helps

Why is volume an extensive property but molar volume an intensive one?

3 Answers3