## Extensive properties::Intensive and extensive properties

### ::concepts

Center::system    Volume::property    Energy::density    Which::redlich    Amount::specific    Example::molar

Extensive properties The IUPAC Green Book defines an extensive property as a physical quantity that is the sum of the properties of separate non-interacting subsystems that compose the entire system.<ref name=IUPAC/> The value of such an additive property is proportional to the size of the system it describes, or to the quantity of matter in the system. Taking on the example of melting ice, the amount of heat required to melt ice is an extensive property. The amount of heat required to melt one ice cube would be much less than the amount of heat required to melt an iceberg, so it is dependent on the quantity.

Extensive properties are the counterparts of intensive properties, which are intrinsic to a particular subsystem. Dividing one type of extensive property by a different type of extensive property generally gives an intensive value—for example: mass (extensive) divided by volume (extensive) gives density (intensive).

### Combined extensive properties

If a set of parameters $\{a_i\}$ are intensive properties and another set $\{A_j\}$ are extensive properties, then the function $F(\{a_i\},\{A_j\})$ is an extensive property if for all $\alpha$,

$F(\{a_i\},\{\alpha A_j\})=\alpha F(\{a_i\},\{A_j\}).\,$

Thus, extensive properties are homogeneous functions (of degree 1) with respect to $\{A_j\}$. It follows from Euler's homogeneous function theorem that

$F(\{a_i\},\{A_j\})=\sum_j A_j \left(\frac{\partial F}{\partial A_j}\right),$

where the partial derivative is taken with all parameters constant except $A_j$. The converse is also true: any function that obeys the above relationship is extensive.{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }} ### Examples Examples of extensive properties include{{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }}:

Intensive and extensive properties sections
Intro  Intensive properties  Extensive properties  Related extensive and intensive properties  Generality of classification  References

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