Intensive and extensive properties

Physical properties of materials and systems can often be categorized as being either intensive or extensive quantities, according to how the property changes when the size (or extent) of the system changes. According to IUPAC, an intensive property is one whose magnitude is independent of the size of the system. An extensive property is one whose magnitude is additive for subsystems.^[1]

An intensive property is a bulk property, meaning that it is a 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 (IUPAC symbols^[1] are used throughout this article). When a diamond is cut, the pieces maintain their intrinsic hardness (until their size reaches a few atoms thick), so hardness is independent of the size of the system.

By contrast, an extensive property is additive for subsystems.^[2] 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. Mass and volume are extensive properties, but hardness is intensive.

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.^[3]

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

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 change in the identical ratio 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, a sufficiently simple system consisting of a single substance requires only two independent intensive variables to fully specify the system's entire state. Other intensive properties are derived from those two variables.

Examples

Examples of intensive properties include:^[2][4][5]

• chemical potential, μ
• color^[6]
• concentration, c
• density, ρ (or specific gravity)
• magnetic permeability, μ
• melting point and boiling point^[7]
• molality, m or b
• pressure, p
• specific heat capacity, c_p
• specific volume, v
• standard reduction potential,^[7] E°
• temperature, T

Extensive properties

The IUPAC Gold Book defines an extensive property as a physical quantity whose magnitude is additive for subsystems.^[1] The value of such an additive property is proportional to the size of the system it describes, or to the quantity of matter in the system. For example, the amount of heat required to melt ice at constant temperature and pressure is an extensive property, known as the enthalpy of fusion. The amount of heat required to melt one ice cube would be much less than the amount of heat required to melt an iceberg, so it is dependent on the quantity.

Extensive properties are not just dependent on the amount of material in a system; the relation must be additive. If, say, a property depended on the square of the mass, it would not be an extensive property. (Consider a system consisting of two 1 gram weights. The total mass is 2 g, squaring that gives 4 g². Squaring and summing the individual masses gives 2 g². This property is not additive for the two subsystems.)

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:^[2][4][5]

• amount of substance, mol
• energy, E
• enthalpy, H
• entropy, S
• Gibbs energy, G
• heat capacity, C_p
• Helmholtz energy, A or F
• internal energy, U
• mass, m
• volume, V

Composite properties

Properties can be combined to give new properties, which may be called derived,^[1] or composite properties. For example, mass and volume can be combined to give density. These composite properties can also be classified as intensive or extensive. Suppose a composite property, $F$, is a function of a set of intensive properties, $\{a_{i}\}$, and a set of extensive properties, $\{A_{j}\}$, which can be shown as $F(\{a_{i}\},\{A_{j}\})$. If the size of the system is changed by some scaling factor, $\alpha$, 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, $\alpha$,

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

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

It follows, for example, that the ratio of two extensive properties is an intensive property. To illustrate, consider a system having a certain mass, $m$, and volume, $V$. The density, $\rho$ is equal to mass (extensive) divided by volume (extensive): ${\displaystyle \rho ={\frac {m}{V}}}$. If the system is scaled by the factor $\alpha$, 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 $\alpha$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 $F$ above.

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

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

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

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

where the partial derivative is taken with all parameters constant except $A_{j}$.^[8] This last equation can be used to derive thermodynamic relations.

"Specific" properties

A specific property is obtained by dividing an extensive property of a system by the mass of the system. For example, heat capacity is an extensive property of a system. Dividing heat capacity, C_p, by the mass of the system gives the specific heat capacity, 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.^[2]

Specific properties derived from extensive properties

Extensive property	Symbol	SI units	Intensive property	Symbol	SI units
Volume	V	m^3 or L	Specific volume*	v	m^3/kg or L/kg
Internal energy	U	J	Specific internal energy	u	J/kg
Entropy	S	J/K	Specific entropy	s	J/(kg·K)
Enthalpy	H	J	Specific enthalpy	h	J/kg
Gibbs free energy	G	J	Specific Gibbs free energy	g	J/kg
Heat capacity at constant volume	CV	J/K	Specific heat capacity at constant volume	cv	J/(kg·K)
Heat capacity at constant pressure	CP	J/K	Specific heat capacity at constant pressure	cP	J/(kg·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 H_m.^[2] A well known molar volume, V_m, is that of an ideal gas at standard conditions for temperature and pressure, with the value 22.41L/mol. Molar Gibbs free energy is commonly referred to as chemical potential, symbolized by μ, particularly when discussing a partial molar Gibbs free energy μ_i for a component i in a mixture.

Limitations

The general validity of the division of physical properties into extensive and intensive kinds has been addressed in the course of science.^[9] 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 system, 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.^[2]

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. McNaught, A. D.; Wilkinson, A.; Nic, M.; Jirat, J.; Kosata, B.; Jenkins, A. (2014). IUPAC. Compendium of Chemical Terminology, 2nd ed. (the "Gold Book"). 2.3.3. Oxford: Blackwell Scientific Publications. doi:10.1351/goldbook.E02281. ISBN 0-9678550-9-8.
2. 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.
3. Canagaratna, Sebastian G. (1992). "Intensive and Extensive: Underused Concepts". J. Chem. Educ. 69 (12): 957–963. Bibcode:1992JChEd..69..957C. doi:10.1021/ed069p957.
4. 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. 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. Alberty, R. A. (2001). "Use of Legendre transforms in chemical thermodynamics" (PDF). Pure Appl.Chem. 73 (8): 1349–1380. doi:10.1351/pac200173081349.
9. George N. Hatsopoulos, G. N.; Keenan, J. H. (1965). Principles of General Thermodynamics. John Wiley and Sons. pp. 19–20. ISBN 9780471359999.