In the natural sciences, an extensive quantity (also extensive variable) is a physical quantity, whose value is proportional to the size of the system it describes. Such a property can be expressed as the sum of the quantities for the separate subsystems that compose the entire system.

Extensive quantities are the counterparts of intensive quantities, which are intrinsic to a particular subsystem and remain constant regardless of size. Dividing one type of extensive quantity by a different type of extensive quantity will in general give an intensive quantity (mass divided by volume gives density).

Combined extensive quantities

If a set of parameters $\backslash \{a\_i\backslash \}$ are intensive quantities and another set $\backslash \{A\_j\backslash \}$ are extensive quantities, then the function $F(\backslash \{a\_i\backslash \},\backslash \{A\_j\backslash \})$ is an extensive quantity if for all $\backslash alpha$,

$F(\backslash \{a\_i\backslash \},\backslash \{\backslash alpha\; A\_j\backslash \})=\backslash alpha\; F(\backslash \{a\_i\backslash \},\backslash \{A\_j\backslash \})\backslash ,$.

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

$F(\backslash \{a\_i\backslash \},\backslash \{A\_i\backslash \})=\backslash sum\_j\; A\_j\; \backslash left(\backslash frac\{\backslash partial\; F\}\{\backslash partial\; A\_j\}\backslash right)$,

where the partial derivative is taken with all parameters constant except $A\_j$. The converse is also true - any function which obeys the above relationship will be extensive.

Examples of extensive quantities

• Electric charge
• Energy
• Enthalpy and any other thermodynamic potentials
• Entropy
• Mass
• Momentum
• Volume

See also

• Intensive quantity
• Physical quantity

Physical quantity

Extensive Größe | Extensieve grootheid | Zmienna ekstensywna | Ekstenzivna količina