# Intensive and extensive properties

(Redirected from Extensive quantity)

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] This reflects the corresponding mathematical ideas of mean and measure, respectively.

An intensive property is a bulk property, meaning that it is a local physical property of a system that does not depend on the system size or the amount of material in the system. Examples of intensive properties include temperature, T; refractive index, n; density, ρ; and hardness of an object, η. In hardness, when a diamond is cut, the pieces maintain their intrinsic hardness, so hardness is independent of the size of the system while the size, number or total area of the diamond pieces is not.

By contrast, an extensive property is additive for subsystems.[3] This means the system could be divided into any number of subsystems, and the extensive property measured for each subsystem; the value of the property for the system would be the sum of the property for each subsystem. For example, both the mass, m, and the volume, V, of a diamond are directly proportional to the amount that is left after cutting it from the raw mineral.

The terms intensive and extensive quantities were introduced by Richard C. Tolman in 1917.[4]

Among others, Otto Redlich has noted that these two categories are not exhaustive, since some physical properties are neither intensive nor extensive.[5] For example, the electrical impedance of two subsystems is additive when — and only when — they are combined in series; whilst if they are combined in parallel, the resulting impedance is less than that of either subsystem.

## 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, the temperature of each subsystem is identical. The same applies to the density of a homogeneous system; if the system is divided in half, the mass and the volume are both divided in half and the density remains unchanged. 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, according to the state postulate: "The state of a simple compressible system is completely specified by two independent, intensive properties". Other intensive properties are derived from those two variables.

### Examples

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

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 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][4][5]

## 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 also be classified as intensive or extensive.[dubious ] 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, ${\displaystyle \alpha }$, 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 ${\displaystyle F(\{a_{i}\},\{\alpha A_{j}\})}$.

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, ${\displaystyle \alpha }$,

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

(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 ${\displaystyle \alpha }$, then the mass and volume become ${\displaystyle \alpha m}$ and ${\displaystyle \alpha V}$, and the density becomes ${\displaystyle \rho ={\frac {\alpha m}{\alpha V}}}$; the two ${\displaystyle \alpha }$s cancel, so this could be written mathematically as ${\displaystyle \rho (\alpha m,\alpha V)=\rho (m,V)}$, which is analogous to the equation for ${\displaystyle F}$ above.

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

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

(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, Cp, by the mass of the system gives the specific heat capacity, cp, 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 Hm.[3] Molar Gibbs free energy is commonly referred to as chemical potential, symbolized by μ, particularly when discussing a partial molar Gibbs free energy μi for a component 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 ° is added to the symbol. Examples:

## Potential sources of confusion

The use of the term intensive is potentially confusing. The meaning here is not the colloquial one, meaning "more" or "more aggressively". Instead it refers to "something within the area, length, or size of something", and often constrained by it, as opposed to "extensive", "something without the area, more than that".

## 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.[5] 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.[5]

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.[5] 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. ^ a b c Tolman, Richard C. (1917). "The Measurable Quantities of Physics". Phys. Rev. 9 (3): 237–253.
5. Redlich, O. (1970). "Intensive and Extensive Properties". J. Chem. Educ. 47 (2): 154–156. Bibcode:1970JChEd..47..154R. doi:10.1021/ed047p154.2.
6. ^ Chang, R.; Goldsby, K. (2015). Chemistry (12 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.
12. ^ George N. Hatsopoulos, G. N.; Keenan, J. H. (1965). Principles of General Thermodynamics. John Wiley and Sons. pp. 19–20. ISBN 9780471359999.CS1 maint: Multiple names: authors list (link)