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
Examples of intensive variables are: density, temperature, pressure, specific heat, voltage (electric potential), speed.
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2 Proof 3 Corollary 4 Products of Variables 5 Example |
Generally, if
If an intensive variable V1 is a ratio of two extensive variables, as in equation (2), then
But if V2 and V3 are both intensive, then
The following variables are neither extensive nor intensive: length, time, area, force, angular momentum.Theorem
where V2 and V3 are extensive variables, then V1 is an intensive variable.Proof
which is equation (1), therefore V1 is an intensive variable.
Corollary
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.
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.