Definition of an intensive variable

The IUPAC definition of an [intensive variable](https://goldbook.iupac.org/html/I/I03074.html) is:



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> Physical quantity whose magnitude is independent of the extent of the system.
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Why is it defined for "a system", and not for a homogenous system, as for a heterogenous system the quantity (e.g. density, temperature) might be different for the different extents/parts of the system? 




The intensivity of a variable doesn't depend on the homogeneity of the system.


Consider system A, a metal rod of length 2 with a gradient of temperature :


[![Temperature across a system of length 2](https://i.stack.imgur.com/Jlv4H.png)](https://i.stack.imgur.com/Jlv4H.png)


This metal rod has an average temperature, which you could calculate.


Now consider system B, a metal rod of length 4 with the same gradient of temperature :


[![Temperature across a system of length 4](https://i.stack.imgur.com/rN5FL.png)](https://i.stack.imgur.com/rN5FL.png)


This metal rod also has an average temperature. Furthermore, it is the same temperature as system A. (if you are unsure about that try to pick a gradient (I arbitrarily chose T = exp(-x) and T = exp(-x/2) for the graphs) and do the calculations yourself)


System B is bigger (it would be twice as massive) but its temperature is the same. Ergo the temperature doesn't depend on the size of the system and is an intensive variable.


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For a homogeneous material, an intensive variable is independent of the amount of material. Even non-homogeneous materials approach homogeneity in each small local region. And in these regions, intensive variables have the same values as if you had if you had a large homogeneous region under the same local conditions.


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