Intensive and extensive properties

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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 whereas an extensive quantity is one whose magnitude is additive for subsystems. 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.

By contrast, extensive properties such as the mass, volume and entropy of systems are additive for subsystems because they increase and decrease as they grow larger and smaller, respectively.

These two categories are not exhaustive, since some physical properties are neither exclusively intensive nor extensive. 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.

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

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:

• chemical potential, μ
• color
• concentration, c
• density, ρ (or specific gravity)
• magnetic permeability, μ
• melting point and boiling point
• molality, m or b
• pressure, p
• Specific conductance (or electrical conductivity)
• specific heat capacity, c_p
• specific internal energy, u
• specific rotation, [α]
• specific volume, v
• standard reduction potential, E°
• surface tension
• temperature, T
• thermal conductivity
• viscosity

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:

• amount of substance, n
• 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

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.

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 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, ${\displaystyle \alpha }$, 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 ${\displaystyle A_{j}}$. 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, 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.

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 H_m. 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.

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:

• V_m° = 22.41L/mol is the molar volume of an ideal gas at standard conditions for temperature and pressure.
• C_P,m° is the standard molar heat capacity of a substance at constant pressure.
• Δ_rH_m° is the standard enthalpy variation of a reaction (with subcases: formation enthalpy, combustion enthalpy...).
• E° is the standard reduction potential of a redox couple, i.e. Gibbs energy over charge, which is measured in volt = J/C.

Potential sources of confusion

The use of the term intensive is potentially confusing. The meaning here is "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. 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. 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.

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. The IUPAC definitions do not consider such cases.

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". 

Complex systems and entropy production

Ilya Prigogine’s  ground breaking work shows that every form of energy is made up of an intensive variable and an extensive variable. Measuring these two factors and taking the product of these two variables gives us an amount for that particular form of energy. If we take the energy of expansion the intensive variable is pressure (P) and the extensive variable is volume (V) we get PxV this is then the energy of expansion. Likewise one can do this for density/mass movement where density and velocity (intensive) and volume (extensive) essentially describe the energy of the movement of mass.

Other energy forms can be derived from this relationship also such as electrical, thermal, sound, springs. Within the quantum realm it appears that energy is made up of intensive factors mainly. For example frequency is intensive. It appears that as one pass to the subatomic realms the intensive factor is more dominant. The example is the quantum dot where color (intensive variable) is dictated by size, size is normally an extensive variable. There appears to be integration of these variables. This then appears as the basis of the quantum effect.

The key insight to all this is that the difference in the intensive variable gives us the entropic force and the change in the extensive variable gives us the entropic flux for a particular form of energy. A series of entropy production formula can be derived.

∆S _heat= [(1/T)_a-(1/T)_b] x ∆ thermal energy

∆S _expansion= [(pressure/T)_a-(pressure/T)_b] x ∆ volume

∆S _electric = [(voltage/T)_a-(voltage/T)_b] x ∆ current

These equations have the form

∆S_s = [(intensive)_a -(intensive)_b] x ∆ extensive
where the a and b are two different regions.

This is the long version of Prigogine’s equation

∆S_s = X_sJ_s
where X_s is the entropic force and J_s is the entropic flux.

It is possible to derive a number of different energy forms from Prigogine’s equation. 

Note that in thermal energy in the entropy production equation the intensive factor’s numerator is 1. Whilst the other equations we have a numerators of pressure and voltage and the denominator is still temperature. This means lower than the level of molecules there are no definite stable units.