# Why is density an intensive property?

I am still trying to understand what are intensive and extensive properties. Possibly someone can give a pointer to a decent text (preferably on the web), as I am not too happy (to say the least) with what I found so far on the web. I already asked here one question on this, which I finally answered myself.

My new problem (among several others) is that density seems to be one of the first properties taken as example of an intensive property. While it seems a good approximation of what I know about solids and liquids, it seems to me a lot more problematic with gas, as they tend to occupy all the available space you give them.

But none of the documents I found seems to make any resriction regarding density of gas. It seems to me that my opinion (apparently contested) that velocity is an intensive property, may be easier to support than the intensiveness of density in the case of gas. Or to put it differently, I do not see why pressure should be more intensive than volume, while wikipedia lists pressure as intensive, but not volume. Ideal gas law states that $PV=nRT$, which apparently gives a pretty symmetrical role to $P$ and $V$. And density depends on pressure (actually using this same formula and molecular weight).

If it were not for the fact that some principles seem to be based on the concept, such as the state postulate which I found on wikipedia, I would start wondering whether these are real concept in physics.

The best way to understand the nature of intensive and extensive quantities in thermodynamics is like this: Take a system of your interest. Make it into two portions (one large portion and the other a small portion) by using a partition, for example. Then see the property of interest of the two samples. Density of the two portions will be the same as the density of the total system we started with; so is the case with temperature, refractive index, which are also intesive properties. However, the volumes of the portions and the total system will have different values; so is the case with mass and energy. Such properties are extensive properties.

Mathematically, Extensive property is a homogeneous equation of first degree, in mass, mole numbers etc and intensive property is a homogeneous equation of zero degree in mass, mole numbers etc.

• From my point of view, you did not really answer my question. I do not know what is the volume of a gaz, say one mole of oxygen. Could you develop your last sentence? I do not understand it. Possibly with an example. What are those equations? - - - - I feel like I am missing an important concept that explains things. Commented Sep 16, 2014 at 20:41

From the ideal gas law $PV = nRT$ we can develop: $$PV = \frac {m}{M}RT \rightarrow PM= \frac {m} {V} RT \quad$$ and since $\frac {m}{V}= \rho \quad$ where $\rho$ is the density of the gas and $M$ the molar mass then we have $$PM = \rho RT \rightarrow \rho = \frac {PM}{RT}$$ So density is dependable only of intensive properties.

Let's prove that the ratio of intensive properties is also intensive. There are 3 properties $a, b, c \quad$ which relate by $a= \frac {b}{c}$ Suppose $b$ and $c$ intensive and $a$ extensive, that would lead to $ac = b$ Which is a contradiction because LHS depends on system's size and the RHS of the equation does not.

PS: You seem confused with pressure being intensive, if true check this and this out.

The definition I use are the following.

An extensive quantity is proportional to the number of components in the system it qualifies. If you double the number of components of the system (by doubling the number of atoms, the volume of liquid...), its extensive quantities will double too.

On the opposite, an intensive quantity is always in a way or another a quantity you can express as something "per component".

If you double the size of, say, an elephant, you will multiply the number of atoms by 8 = 2x2x2, but the number of atoms per cubic meter (the density) remains the same. The number of atoms is an extensive quantity while the density is intensive.

Does this help ?

• I know what you explained (thanks anyway). The issue is that, for a gas, I do not see that you have to change the volume when you double the number of molecules. It will simply double the density. I have, yet, no problem with liquids. Commented Aug 20, 2014 at 0:48
• I improved the question to make this clearer. Commented Aug 20, 2014 at 1:07
• @babou - I see your point. The thing is, there is an idea of "replication keeping every other property unchanged" when you increase an extensive quantity. If you double the number of gas molecules in the same volume, you can't consider that you replicate the system without changing anything else. Obviously you changed another property of the system that is density. Commented Aug 20, 2014 at 1:07
• Precisely, density and pressure which is related. Commented Aug 20, 2014 at 1:09
• That's it. Actually there is not much more to intensive/extensive that this, even if it might look like the contrary! Commented Aug 20, 2014 at 7:58

Consider $10~\mathrm{ kg}$ of a substance. Take a few $\mathrm{kg}$ of the substance and measure the mass density. The density is same as before. So we can say that from the above explanation, density is an intensive property.

• Hi Chetan Nagose and welcome to Physics.SE! This is really a comment, not an answer. With a bit more rep, you will be able to post comments. Commented Feb 7, 2016 at 14:33
• @gonenc To the contrary, I think this does answer the question.
– Danu
Commented Feb 7, 2016 at 14:41
• @Danu I would suggest that you both read the question. If you have 1kg of gas, and you add another to it, what do you get? Twice the volume, twice the density, or what? Commented Feb 7, 2016 at 18:46
• @babou Note that answers on SE do not directly have to appeal to the asker of the question. Others may still benefit from it. I think this answer addresses the question (in the title).
– Danu
Commented Feb 7, 2016 at 19:12
• @Danu I am fully aware that the question and its answers are for the benefit of everyone interested. But the question is not its title. The title gives the general context, but is to be considered with the further details described in the body. I made some effort to explain the reason for that question, and why I consider there is a problem. If one does not address the issues precisely raised in the body of the question, one is simply out of topic. This is actually downvoted on some SE sites. Consider 10 kg of gas and leak part of it out of the container: the remaining gas has a lower density Commented Feb 7, 2016 at 22:38