# Intensive variable

An intensive variable of a substance is such that its value does not depend on the amount of the substance.

It is the counterpart of an extensive variable.

Let there be one piece of substance whose quantity is n and another piece of substance whose quantity is m. Let V be an intensive variable. The value of variable V corresponding to the first substance is V(n) and the value of V corresponding to the second substance is V(m). Then put the two pieces together forming a substance with quantity n+m, then the value of their intensive variable should be

which is a weighted mean. If V(n)=V(m) then
so the intensive variable is independent of the quantity.

Examples of intensive variables are: density, temperature, pressure, specific heat, voltage (electric potential), speed.

## Theorem

Generally, if

where V2 and V3 are extensive variables, then V1 is an intensive variable.

## Proof

(extensivity of V2 and V3)

(hypothesis)
(extensivity of V3)
which is equation (1), therefore V1 is an intensive variable.

## Corollary

If an intensive variable V1 is a ratio of two extensive variables, as in equation (2), then

and

There is one catch though: an intensive variable stays constant with respect to quantity of substance, as long as other variables on which the intensive variable depends stay the same (ceteris paribus).

## Products of Variables

If
and if V2 is extensive and V3 is intensive, then
therefore V1 satisfies extensivity.

But if V2 and V3 are both intensive, then

,
so that V1 is also intensive. Likewise, the reciprocal of an intensive variable is an intensive variable.

## Example

Pressure is force divided by area. But neither force nor area are either extensive or intensive. However, force multiplied by length is work, which is extensive, and area multiplied by length is volume, which is extensive. Therefore pressure actually is a ratio of two extensive variables -- work/volume -- and so it is intensive.

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