Intensive and extensive 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.

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).

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. 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.

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".

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".

Ilya Prigogine’s [13] groundbreaking 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 the 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, the frequency is intensive. It appears that as one passes 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 the 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 formulas 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 numerator of pressure and voltage and the denominator is still temperature. This means lower than the level of molecules there are no definite stable units.