# Intensive and extensive properties

Physical 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. According to IUPAC, an intensive quantity is one whose magnitude is independent of the size of the system[1] whereas an extensive quantity is one whose magnitude is additive for subsystems.[2]

An intensive property does not depend on the system size or the amount of material in the system. It 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 of an object, η.

By contrast, extensive properties such as the mass, volume and entropy of systems are additive for subsystems.[3]

Though it is very often convenient to define physical quantities to make them intensive or extensive, they do not necessarily fall under those classifications. For example, the square root of the mass is neither intensive nor extensive.[4]

The terms intensive and extensive quantities were introduced into physics by German writer Georg Helm in 1898, and by American physicist and chemist Richard C. Tolman in 1917.[4][5]

## Intensive properties

An intensive property is a physical quantity whose value does not depend on the amount of the substance for which it is measured. For example, the temperature of a system in thermal equilibrium is the same as the temperature of any part of it. If the system is divided by a wall that is permeable to heat or to matter, the temperature of each subsystem is identical; if a system divided by a wall that is impermeable to heat and to matter, then the subsystems can have different temperatures. Likewise for the density of a homogeneous system; if the system is divided in half, the extensive properties, such as the mass and the volume, are each divided in half, and the intensive property, the density, remains the same in each subsystem. Additionally, the boiling point of a substance is another example of an intensive property. For example, the boiling point of water is 100 °C at a pressure of one atmosphere, which remains true regardless of quantity.

The distinction between intensive and extensive properties has some theoretical uses. For example, in thermodynamics, the state of a simple compressible system is completely specified by two independent, intensive properties, along with one extensive property, such as mass. Other intensive properties are derived from those two intensive variables.

### Examples

Examples of intensive properties include:[3][5][4]

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 generally gives an intensive value—for example: mass (extensive) divided by volume (extensive) gives density (intensive).

### Examples

Examples of extensive properties include:[3][5][4]

## Conjugate quantities

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 move 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 of 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 of 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.

## Composite properties

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.[8]

More generally properties can be combined to give new properties, which may be called derived or composite properties. For example, the base quantities[9] mass and volume can be combined to give the derived quantity[10] 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 ${\displaystyle A_{j}}$.[11] 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, ${\displaystyle C_{p}}$, 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.[3]

Specific properties derived from extensive properties
Extensive
property
Symbol SI units Intensive (specific)
property
Symbol SI units Intensive (molar)
property
Symbol SI units
Volume V m3 or L Specific volume* v m3/kg or L/kg Molar volume Vm m3/mol or L/mol
Internal energy U J Specific internal energy u J/kg Molar internal energy Um J/mol
Enthalpy H J Specific enthalpy h J/kg Molar enthalpy Hm J/mol
Gibbs free energy G J Specific Gibbs free energy g J/kg Chemical potential Gm or µ J/mol
Entropy S J/K Specific entropy s J/(kg·K) Molar entropy Sm J/(mol·K)
Heat capacity
at constant volume
CV J/K Specific heat capacity
at constant volume
cV J/(kg·K) Molar heat capacity
at constant volume
CV,m J/(mol·K)
Heat capacity
at constant pressure
CP J/K Specific heat capacity
at constant pressure
cP J/(kg·K) Molar heat capacity
at constant pressure
CP,m J/(mol·K)
*Specific volume is the reciprocal of density.

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 ${\displaystyle H_{\mathrm {m} }}$.[3] 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 an additional superscript ${\displaystyle ^{\circ }}$ is added to the symbol. Examples:

• ${\displaystyle V_{\mathrm {m} }^{\circ }}$ = 22.41L/mol is the molar volume of an ideal gas at standard conditions for temperature and pressure.
• ${\displaystyle C_{P,\mathrm {m} }^{\circ }}$ 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.

## Limitations

The general validity of the division of physical properties into extensive and intensive kinds has been addressed in the course of science.[12] 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 physical properties conform to neither definition.[4] Redlich also provides examples of mathematical functions that alter the strict additivity relationship for extensive systems, such as the square or square root of volume, which may occur in some contexts, albeit rarely used.[4]

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.[4] The IUPAC definitions do not consider such cases.[3]

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".

## References

1. ^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version:  (2006–) "Intensive quantity". doi:10.1351/goldbook.I03074
2. ^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version:  (2006–) "Extensive quantity". doi:10.1351/goldbook.E02281
3. Cohen, E. R.; et al. (2007). IUPAC Green Book (PDF) (3rd ed.). Cambridge: IUPAC and RSC Publishing. pp. 6 (20 of 250 in PDF file). ISBN 978-0-85404-433-7.
4. Redlich, O. (1970). "Intensive and Extensive Properties" (PDF). J. Chem. Educ. 47 (2): 154–156. Bibcode:1970JChEd..47..154R. doi:10.1021/ed047p154.2.
5. ^ a b c Tolman, Richard C. (1917). "The Measurable Quantities of Physics". Phys. Rev. 9 (3): 237–253.[1]
6. ^ Chang, R.; Goldsby, K. (2015). Chemistry (12th ed.). McGraw-Hill Education. p. 312. ISBN 978-0078021510.
7. ^ a b Brown, T. E.; LeMay, H. E.; Bursten, B. E.; Murphy, C.; Woodward; P.; Stoltzfus, M. E. (2014). Chemistry: The Central Science (13th ed.). Prentice Hall. ISBN 978-0321910417.
8. ^ Canagaratna, Sebastian G. (1992). "Intensive and Extensive: Underused Concepts". J. Chem. Educ. 69 (12): 957–963. Bibcode:1992JChEd..69..957C. doi:10.1021/ed069p957.
9. ^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version:  (2006–) "Base quantity". doi:10.1351/goldbook.B00609
10. ^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version:  (2006–) "Derived quantity". doi:10.1351/goldbook.D01614
11. ^ Alberty, R. A. (2001). "Use of Legendre transforms in chemical thermodynamics" (PDF). Pure Appl. Chem. 73 (8): 1349–1380. doi:10.1351/pac200173081349. S2CID 98264934.
12. ^ George N. Hatsopoulos, G. N.; Keenan, J. H. (1965). Principles of General Thermodynamics. John Wiley and Sons. pp. 19–20. ISBN 9780471359999.