# Is length an extensive property?

From my experience, volume, surface and length are extensive properties. Indeed :

• the reunion of two cubes of 1 $$m^3$$ leads to a cube of 2 $$m^3$$
• the reunion of two tiles of 1 $$m^2$$ leads to a tile of 2 $$m^2$$
• the reunion of two spaghetti of 1 $$m$$ leads to a spaghetti of 2 $$m$$

But I also read that the ratio of two extensive properties should lead to an intensive property. This is indeed the case for molar volume (ratio of a volume over a mole number) or pressure (ratio of a force over a surface). But if I do the ratio of a volume over a surface I get a length that is not an intensive property.

So should I consider that length is an intesive quantity or that the ratio of two extensive properties does not necessarily leads to an intensive property?

I did some sketches and it seems fairly true that only volume is an extensive property and that their is not absolute result for length and surface. I observed that a pavement of volume $$V = S \times L$$, with $$S$$ its caracteristic surface and $$L$$ its caracteristic length, it is possible to create two subsystems in many different ways. For exemple

• either splitting the pavement into two pavements of caracteristic surface $$S/2$$ and caracteristic length $$L$$ ($$S$$ is extensive while $$L$$ is intensive)
• either splitting the pavement into two pavements of caracteristic surface $$S$$ and caracteristic length $$L/2$$ ($$S$$ is intensive while $$L$$ is extensive)
• either splitting the pavement into two pavements of caracteristic surface $$S$$ and caracteristic length $$L$$ ($$S$$ and $$L$$ are intensive)

So I guess it is a question of context whether these quantites should be considered extensive or intensive. Are there some classical exemples where surface or length are alternatively considered as extensive or intensive?

Surface and length are not well-defined properties for 3d systems. The length of two spaghetti only adds up if you lay them end-to-end, but if you stack them on top of each other, then the "length" of the resulting stack is in no meaningful way double the length of the individual spaghetti. Without an explicit procedure of how to combine two systems with arbitrary shapes, it is not clear how one would ever compute the "length" or "surface" of a 3d system that one just composed from two subsystems.

• I agree. Another way to think about this is if you had one piece of spaghetti and you sliced it in half the long way, you'd get a spaghetti that is twice as long, which doesn't make sense. This is what @ACuriousMind means when he says that length is not well-defined. Jan 23, 2019 at 19:04
• I agree that length seems hard to define but this raises a question: why is it often written that the ratio of two extensive properties leads to an intensive one if this is an obvious counter example? Is their a mathematical reason for such wrong sentance? Jan 24, 2019 at 13:02
• So, the dimension (one inch) of coarse aggregate crushed rock is an intensive property; two tons of aggregate has the same coarseness as one ton. Jan 26, 2019 at 0:20

Length is an extensive property and the ratio of two extensive property is always an intensive property.

An intensive property can be used(not always) as a unit for an extensive property. In the case mentioned in question, the result of ratio of volume to area of a system made of similar and defined subsystems, with each subsystem having a unit area, is the volume of the subsystem and is always constant. Now if the system is halved or doubled(in a way such that there is no change in subsystem, otherwise it will alter the very defination of the subsystem), volume of a subsystem or the ratio of volume to area of system remains constant. This ratio gives a length(which, in some special cases of shapes like cuboid or cylinder, is height of respective shapes. But not in general, like hollow sphere,etc.) for some arbitrary dimensions of volume and area. It is intensive to the subsystem and can be defined as a unit length. "a unit length" is an intensive quantity while a multiple of this unit is extensive. For example, an extensive quantity named 'distance' can be defined, with values which are real multiples of a unit length, say 'meter'. Let a rod of 1 meter be halved. The distance also gets halved and equals 0.5 meter. But the unit does not change because a unit is always an intensive property defined on some other system.

So when we divide volume by area, we get a length which is intensive to our original system from which we measured volume and area, but it naturaly appeal to us to use this length to measure some other quantity of a new system and then when we double or halve the new system we see the length accordingly changing(and wonder why the ratio or the length is extensive, while forgetting that we have moved from our original system). But if we halve or double our original system, both the volume and area accordingly gets halved or doubled such that the ratio or length still remains the same and intesive to system.

Edit: As someone mentioned in the comment an example of cube, here is a simple example.

Let a hollow cube be of volume 1000 m³ and a height 10 m, then volume divided by height is 100 m³ per m. Now this ratio is property of the subcube i.e. volume=100 m³ and height=1 m. Now if the cube is halved, volume=500 m³, height=5 m(note the ratio is still same). What if the cube is halved other way, such that its height remains same. Then it will be against the definition of the subcube because its properties will change i.e. volume= 50 m³, height=1m. Infact this is a whole new cube (a 90° rotation of original halved cube), i just moved from the original system!

So the question arises, why one has to always adhere to the definition?. Because when the ratio of any properties of such shapes are taken, by only looking at the ratio one can't tell the orientation(in 3d space) and configurational properties(whether it is a sphere, cube or cylinder) of shape i.e. some details are lost and the ratio becomes extensive because the properties are not well defined in the 3d space(As someone mentioned in another answere to this question). But but soon as the fundamentals are defined it is intensive. That is what is meant by the statement "The ratio is always intensive to the original system from which it is measured".

• A cube facet area is the ratio of its volume to its depth, but is NOT intensive. Jan 25, 2019 at 9:19
• @Whit3rd You are right. The cube facet area is extensive and not intensive. I've edited my answere to add an example based on the case you mentioned to show that although a cube facet area is extesive but the ratio of its volume to its depth is not. Jan 26, 2019 at 5:51