Extensive quantity

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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 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 <math>\{a_i\}</math> are intensive quantities and another set <math>\{A_j\}</math> are extensive quantities, then the function <math>F(\{a_i\},\{A_j\})</math> is an extensive quantity if for all <math>\alpha</math>,

<math>F(\{a_i\},\{\alpha A_j\})=\alpha F(\{a_i\},\{A_j\})</math>.

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

<math>F(\{a_i\},\{A_i\})=\sum_j A_j \left(\frac{\partial F}{\partial A_j}\right)</math>,

where the partial derivative is taken with all parameters constant except <math>A_j</math>. 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 quantitynl:Extensieve grootheid

pl:Zmienna ekstensywna sl:Ekstenzivna količina