Intensive and extensive properties

An intensive property is a physical quantity whose value does not depend on the amount of substance which was measured. The most obvious intensive quantities are ratios of extensive quantities. In a homogeneous system divided into two halves, all its extensive properties, in particular its volume and its mass, are divided into two halves. All its intensive properties, such as the mass per volume (mass density) or volume per mass (specific volume), must remain the same in each half.

The temperature of a system in thermal equilibrium is the same as the temperature of any part of it, so temperature is an intensive quantity. If the system is divided by a wall that is permeable to heat or to matter, the temperature of each subsystem is identical. Additionally, the boiling temperature of a substance is an intensive property. For example, the boiling temperature of water is 100 °C at a pressure of one atmosphere, regardless of the quantity of water remaining as liquid.

Examples of intensive properties include:^[5][2][1]

See List of materials properties for a more exhaustive list specifically pertaining to materials.

An extensive property is a physical quantity whose value is proportional to the size of the system it describes,^[8] 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).^[9]

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 of extensive properties include:^[5][2][1]

In thermodynamics, some extensive quantities measure amounts that are conserved in a thermodynamic process of transfer. They are transferred across a wall between two thermodynamic systems or subsystems. For example, species of matter may be transferred through a semipermeable membrane. Likewise, volume may be thought of as transferred in a process in which there is a motion of the wall between two systems, increasing the volume of one and decreasing that of the other by equal amounts.

On the other hand, some extensive quantities measure amounts that are not conserved in a thermodynamic process of transfer between a system and its surroundings. In a thermodynamic process in which a quantity of energy is transferred from the surroundings into or out of a system as heat, a corresponding quantity of entropy in the system respectively increases or decreases, but, in general, not in the same amount as in the surroundings. Likewise, a change in the amount of electric polarization in a system is not necessarily matched by a corresponding change in electric polarization in the surroundings.

In a thermodynamic system, transfers of extensive quantities are associated with changes in respective specific intensive quantities. For example, a volume transfer is associated with a change in pressure. An entropy change is associated with a temperature change. A change in the amount of electric polarization is associated with an electric field change. The transferred extensive quantities and their associated respective intensive quantities have dimensions that multiply to give the dimensions of energy. The two members of such respective specific pairs are mutually conjugate. Either one, but not both, of a conjugate pair may be set up as an independent state variable of a thermodynamic system. Conjugate setups are associated by Legendre transformations.

The ratio of two extensive properties of the same object or system is an intensive property. For example, the ratio of an object's mass and volume, which are two extensive properties, is density, which is an intensive property.^[10]

More generally properties can be combined to give new properties, which may be called derived or composite properties. For example, the base quantities^[11] mass and volume can be combined to give the derived quantity^[12] density. These composite properties can sometimes also be classified as intensive or extensive. Suppose a composite property ${\displaystyle F}$ is a function of a set of intensive properties ${\displaystyle \{a_{i}\}}$ and a set of extensive properties ${\displaystyle \{A_{j}\}}$, which can be shown as ${\displaystyle F(\{a_{i}\},\{A_{j}\})}$. If the size of the system is changed by some scaling factor, ${\displaystyle \lambda }$, 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 ${\displaystyle F(\{a_{i}\},\{\lambda A_{j}\})}$.

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, ${\displaystyle \lambda }$,

${\displaystyle F(\{a_{i}\},\{\lambda A_{j}\})=F(\{a_{i}\},\{A_{j}\}).\,}$

(This is equivalent to saying that intensive composite properties are homogeneous functions of degree 0 with respect to ${\displaystyle \{A_{j}\}}$.)

It follows, for example, that the ratio of two extensive properties is an intensive property. To illustrate, consider a system having a certain mass, ${\displaystyle m}$, and volume, ${\displaystyle V}$. The density, ${\displaystyle \rho }$ is equal to mass (extensive) divided by volume (extensive): ${\displaystyle \rho ={\frac {m}{V}}}$. If the system is scaled by the factor ${\displaystyle \lambda }$, then the mass and volume become ${\displaystyle \lambda m}$ and ${\displaystyle \lambda V}$, and the density becomes ${\displaystyle \rho ={\frac {\lambda m}{\lambda V}}}$; the two ${\displaystyle \lambda }$s cancel, so this could be written mathematically as ${\displaystyle \rho (\lambda m,\lambda V)=\rho (m,V)}$, which is analogous to the equation for ${\displaystyle F}$ above.

The property ${\displaystyle F}$ is an extensive property if for all ${\displaystyle \lambda }$,

${\displaystyle F(\{a_{i}\},\{\lambda A_{j}\})=\lambda F(\{a_{i}\},\{A_{j}\}).\,}$

(This is equivalent to saying that extensive composite properties are homogeneous functions of degree 1 with respect to ${\displaystyle \{A_{j}\}}$.) It follows from Euler's homogeneous function theorem that

${\displaystyle F(\{a_{i}\},\{A_{j}\})=\sum _{j}A_{j}\left({\frac {\partial F}{\partial A_{j}}}\right),}$

where the partial derivative is taken with all parameters constant except ${\displaystyle A_{j}}$.^[13] This last equation can be used to derive thermodynamic relations.

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, ${\displaystyle C_{p}}$, by the mass of the system gives the specific heat capacity, ${\displaystyle c_{p}}$, 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.^[5]

If the amount of substance in moles can be determined, then each of these thermodynamic properties may be expressed on a molar basis, and their name may be qualified with the adjective molar, yielding terms such as molar volume, molar internal energy, molar enthalpy, and molar entropy. The symbol for molar quantities may be indicated by adding a subscript "m" to the corresponding extensive property. For example, molar enthalpy is ${\displaystyle H_{\mathrm {m} }}$.^[5] Molar Gibbs free energy is commonly referred to as chemical potential, symbolized by ${\displaystyle \mu }$, particularly when discussing a partial molar Gibbs free energy ${\displaystyle \mu _{i}}$ for a component ${\displaystyle i}$ in a mixture.

For the characterization of substances or reactions, tables usually report the molar properties referred to a standard state. In that case a superscript ${\displaystyle ^{\circ }}$ is added to the symbol. Examples:

• ${\displaystyle V_{\mathrm {m} }^{\circ }}$ = 22.4L/mol is the molar volume of an ideal gas at standard conditions of 1atm (101.325kPa) and 0°C (273.15K).^[14]
• ${\displaystyle C_{P,\mathrm {m} }^{\circ }}$ is the standard molar heat capacity of a substance at constant pressure.
• ${\displaystyle \mathrm {\Delta } _{\mathrm {r} }H_{\mathrm {m} }^{\circ }}$ is the standard enthalpy variation of a reaction (with subcases: formation enthalpy, combustion enthalpy...).
• ${\displaystyle E^{\circ }}$ is the standard reduction potential of a redox couple, i.e. Gibbs energy over charge, which is measured in volt = J/C.

The general validity of the division of physical properties into extensive and intensive kinds has been addressed in the course of science.^[15] Redlich noted that, although physical properties and especially thermodynamic properties are most conveniently defined as either intensive or extensive, these two categories are not all-inclusive and some well-defined concepts like the square-root of a volume conform to neither definition.^[1]

Other systems, for which standard definitions do not provide a simple answer, are systems in which the subsystems interact when combined. Redlich pointed out that the assignment of some properties as intensive or extensive may depend on the way subsystems are arranged. For example, if two identical galvanic cells are connected in parallel, the voltage of the system is equal to the voltage of each cell, while the electric charge transferred (or the electric current) is extensive. However, if the same cells are connected in series, the charge becomes intensive and the voltage extensive.^[1] The IUPAC definitions do not consider such cases.^[5]

Some intensive properties do not apply at very small sizes. For example, viscosity is a macroscopic quantity and is not relevant for extremely small systems. Likewise, at a very small scale color is not independent of size, as shown by quantum dots, whose color depends on the size of the "dot".