Physical or chemical properties of materials and systems can often be categorized as being either intensive or extensive, according to how the property changes when the size (or extent) of the system changes.
The terms "intensive and extensive quantities" were introduced into physics by German mathematician Georg Helm in 1898, and by American physicist and chemist Richard C. Tolman in 1917.
According to International Union of Pure and Applied Chemistry (IUPAC), an intensive property or intensive quantity is one whose magnitude (extent) is independent of the size of the system.
An intensive property is not necessarily homogeneously distributed in space; it can vary from place to place in a body of matter and radiation. Examples of intensive properties include temperature, T; refractive index, n; density, ρ; and hardness, η.
By contrast, an extensive property or extensive quantity is one whose magnitude is additive for subsystems.
Examples include mass, volume and Gibbs energy.
Not all properties of matter fall into these two categories. For example, the square root of the volume is neither intensive nor extensive.
- concentration, c
- energy density, ρ
- magnetic permeability, μ
- mass density, ρ (or specific gravity)
- melting point and boiling point E°
- surface tension
- temperature, T
- thermal conductivity
- velocity v
- viscosity
See List of materials properties for a more exhaustive list specifically pertaining to materials.
Extensive properties
An extensive property is a physical quantity whose value is proportional to the size of the system it describes, or to the quantity of matter in the system. For example, the mass of a sample is an extensive quantity; it depends on the amount of substance. The related intensive quantity is the density which is independent of the amount. The density of water is approximately 1g/mL whether you consider a drop of water or a swimming pool, but the mass is different in the two cases.
Dividing one extensive property by another extensive property gives an intensive property—for example: mass (extensive) divided by volume (extensive) gives density (intensive).
Any extensive quantity E for a sample can be divided by the sample's volume, to become the "E density" for the sample;
similarly, any extensive quantity "E" can be divided by the sample's mass, to become the sample's "specific E";
extensive quantities "E" which have been divided by the number of moles in their sample are referred to as "molar E".
Examples
Examples of extensive properties include:
More generally properties can be combined to give new properties, which may be called derived or composite properties. For example, the base quantities mass and volume can be combined to give the derived quantity density. These composite properties can sometimes also be classified as intensive or extensive. Suppose a composite property <math>F</math> is a function of a set of intensive properties <math>\{a_i\}</math> and a set of extensive properties <math>\{A_j\}</math>, which can be shown as <math>F(\{a_i\},\{A_j\})</math>. If the size of the system is changed by some scaling factor, <math>\lambda</math>, only the extensive properties will change, since intensive properties are independent of the size of the system. The scaled system, then, can be represented as <math>F(\{a_i\},\{\lambda A_j\})</math>.
Intensive properties are independent of the size of the system, so the property F is an intensive property if for all values of the scaling factor, <math>\lambda</math>,
:<math>F(\{a_i\},\{\lambda A_j\}) = F(\{a_i\},\{A_j\}).\,</math>
(This is equivalent to saying that intensive composite properties are homogeneous functions of degree 0 with respect to <math>\{A_j\}</math>.)
It follows, for example, that the ratio of two extensive properties is an intensive property. To illustrate, consider a system having a certain mass, <math>m</math>, and volume, <math>V</math>. The density, <math>\rho</math> is equal to mass (extensive) divided by volume (extensive): <math>\rho=\frac{m}{V}</math>. If the system is scaled by the factor <math>\lambda</math>, then the mass and volume become <math>\lambda m</math> and <math>\lambda V</math>, and the density becomes <math>\rho=\frac{\lambda m}{\lambda V}</math>; the two <math>\lambda</math>s cancel, so this could be written mathematically as <math>\rho (\lambda m, \lambda V) = \rho (m, V)</math>, which is analogous to the equation for <math>F</math> above.
The property <math>F</math> is an extensive property if for all <math>\lambda</math>,
:<math>F(\{a_i\},\{\lambda A_j\})=\lambda F(\{a_i\},\{A_j\}).\,</math>
(This is equivalent to saying that extensive composite properties 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_j\})=\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>. This last equation can be used to derive thermodynamic relations.
Specific properties
A specific property is the intensive property obtained by dividing an extensive property of a system by its mass. For example, heat capacity is an extensive property of a system. Dividing heat capacity, <math>C_p</math>, by the mass of the system gives the specific heat capacity, <math>c_p</math>, which is an intensive property. When the extensive property is represented by an upper-case letter, the symbol for the corresponding intensive property is usually represented by a lower-case letter. Common examples are given in the table below.
- <math>C_{P,\mathrm m}^{\circ}</math> is the standard molar heat capacity of a substance at constant pressure.
- <math>\mathrm \Delta_{\mathrm r} H_{\mathrm m}^{\circ}</math> is the standard enthalpy variation of a reaction (with subcases: formation enthalpy, combustion enthalpy...).
- <math>E^{\circ}</math> is the standard reduction potential of a redox couple, i.e. Gibbs energy over charge, which is measured in volt = J/C.
Limitations
The intensive/extensive classification is most useful in macroscopic equilibrium thermodynamics, where “scaling up” is understood as combining independent copies of a system so that extensive quantities are additive for subsystems, while intensive quantities are independent of the extent (size) of the system.
This classification is not exhaustive. One can form well-defined derived quantities (for example, <math>\sqrt{V}</math>) that are neither additive (extensive) nor size-independent (intensive). Such quantities are typically not used as independent thermodynamic state variables, but they show that “intensive vs extensive” is a convenient classification rather than a universal taxonomy.