# Intensive and extensive properties

In the physical sciences, an **intensive property** (also called a **bulk property**), is a physical property of a system that does *not* depend on the system size or the amount of material in the system. (see: examples)
By contrast, an **extensive property** of a system *does* depend on the system size or the amount of material in the system. (see: examples) Some intensive properties, such as viscosity, are empirical macroscopic quantities and are not relevant to extremely small systems.

## Intensive quantity

An **intensive quantity** (also **intensive variable**) is a physical quantity whose value does *not* depend on the amount of the substance for which it is measured. It is the counterpart of an extensive quantity. For instance, the mass of an object is not an intensive quantity, because it depends on the amount of that substance being measured. Density, on the other hand, is a bulk property of the substance. The riddle "what weighs more, a pound of feathers or a pound of lead?" is an example where it is easy to confuse the intensive and extensive quantities.

### Combined intensive quantities

At least two functions are needed to describe any thermodynamic system, an intensive one and an extensive one.

If a set of parameters, <math>\{a_i\}</math>, are intensive quantities and another set, <math>\{A_j\}</math>, are extensive quantities, then the function <math>F(\{a_i\},\{A_j\})</math> is an intensive quantity if for all <math>\alpha</math>,

- <math>F(\{a_i\},\{\alpha A_j\}) = F(\{a_i\},\{A_j\}).\,</math>

It follows, for example, that the ratio of two extensive quantities is an intensive quantity - density (intensive) is equal to mass (extensive) divided by volume (extensive).

### Joining systems

Let there be a system or piece of substance *a* of amount *m _{a}* and another piece of substance

*b*of amount

*m*which can be combined without interaction. [For example, lead and tin combine without interaction, but common salt dissolves in water and the properties of the resulting solution are not a simple combination of the properties of its constituents.] Let

_{b}*V*be an intensive variable. The value of variable

*V*corresponding to the first substance is

*V*, and the value of

_{a}*V*corresponding to the second substance is

*V*. If the two pieces

_{b}*a*and

*b*are put together, forming a piece of substance "a+b" of amount

*m*, then the value of their intensive variable

_{a+b}= m_{a}+m_{b}*V*is:

- <math> V = \frac{m_a V_a + m_b V_b}{m_a + m_b},</math>

which is a weighted mean. Further, if *V _{a}* =

*V*then

_{b}*V*=

_{a + b}*V*=

_{a}*V*, i.e. the intensive variable is independent of the amount. Note that this property holds only as long as other variables on which the intensive variable depends stay constant.

_{b}As an example, 60kg of lead, of density 11.34 g·cm−3 and 40kg of tin, of density 6.99 g·cm−3 will combine to form 60 + 40 = 100kg of 60/40 solder of density <math> \frac{60 \times 11.34 + 40 \times 6.99}{60 + 40}</math> = 9.60 g·cm−3

In a thermodynamic system composed of two monatomic ideal gases, *a* and *b*, if the two gases are mixed, the final temperature *T* is

- <math>T = \frac{N_aT_a+N_bT_b}{N_a+N_b},</math>

a weighted mean where <math>N_i</math> is the number of particles in gas *i*, and <math>T_i</math> is the corresponding temperature.

### Examples

Examples of intensive properties include:

- temperature
- pressure
- chemical potential
- density
- viscosity
- velocity
- electric field
- electrical resistivity
- spectral absorption maxima (in solution)
- specific energy

## Extensive quantity

An **extensive quantity** (also **extensive variable** or **extensive parameter**) is a physical quantity, whose value is proportional to the size of the system it describes. Such a property can be expressed as the sum of the quantities for the separate subsystems that compose the entire system.

Extensive quantities are the counterparts of intensive quantities, which are intrinsic to a particular subsystem and remain constant regardless of size. Dividing one type of extensive quantity by a different type of extensive quantity will in general give an intensive quantity (mass divided by volume gives density).

### Combined extensive quantities

If a set of parameters <math>\{a_i\}</math> are intensive quantities and another set <math>\{A_j\}</math> are extensive quantities, then the function <math>F(\{a_i\},\{A_j\})</math> is an extensive quantity if for all <math>\alpha</math>,

- <math>F(\{a_i\},\{\alpha A_j\})=\alpha F(\{a_i\},\{A_j\}).\,</math>

Thus, extensive quantities are homogeneous functions (of degree 1) with respect to <math>\{A_j\}</math>. It follows from Euler's homogeneous function theorem that

- <math>F(\{a_i\},\{A_i\})=\sum_j A_j \left(\frac{\partial F}{\partial A_j}\right),</math>

where the partial derivative is taken with all parameters constant except <math>A_j</math>. The converse is also true - any function which obeys the above relationship will be extensive.

### Examples

Examples of extensive properties include:

## Distinction from perceptions

Certain perceptions are often described (or even "measured") as if they are intensive or extensive physical properties, but in fact perceptions are fundamentally different from physical properties. For example, the colour of a solution is not a physical property. A solution of potassium permanganate may appear pink, various shades of purple, or black, depending upon the concentration of the solution and the length of the optical path through it. The colour of a given sample as perceived by an observer (ie, the degree of 'pinkness' or 'purpleness') cannot be measured, only ranked in comparison with other coloured solutions by a panel of observers. Attempts to quantify a perception always involve an observer response, and biological variability is an intrinsic part of the process for many perceived properties. A given volume of permanganate solution of a given concentration has physical properties related to the colour: the optical absorption spectrum is an * extensive* property, and the positions of the absorption maxima (which are relatively independent of concentration) are *intensive* properties. A given absorption spectrum, for a certain observer, will always be perceived as the same colour; but there may be several different absorption spectra which are perceived as the same colour: there is no precise one-to-one correspondence between absorption spectrum and colour even for the same observer.

The confusion between perception and physical properties is increased by the existence of numeric scales for many perceived qualities. However, this is not 'measurement' in the same sense as in physics and chemistry. A numerical value for a perception is, directly or indirectly, the *expected* response of a group of observers when perceiving the specified physical event.

Examples of perceptions related to an intensive physical property:

- Temperature: in this case all observers will agree which is the hotter of two objects.
- Loudness of sound; the related physical property is sound pressure level. Observers may disagree about the relative loudness of sounds with different acoustic spectra.
- Hue of a solution; the related physical property is the position of the spectral absorption maximum (or maxima).

Examples of perceptions related to an extensive physical property:

## See also

## References

- Callen, Herbert B. (1985).
*Thermodynamics and an Introduction to Themostatistics*(2nd Ed. ed.). New York: John Wiley & Sons. ISBN 0-471-86256-8. - Lewis, G.N. (1961).
*Thermodynamics*(2nd Edition ed.). New York: McGraw-Hill Book Company. Unknown parameter`|coauthors=`

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