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.

Table of contents

1 Theorem 2 Proof 3 Corollary 4 Products of Variables 5 Example

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.

The following variables are neither extensive nor intensive: length, time, area, force, angular momentum.