Why is density an intensive property?

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

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 ?

Answer 4

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.