Intensive and extensive properties

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In physics and chemistry an intensive property (also called a bulk property) of a system is a physical property of the system that does not depend on the system size or the amount of material in the system. By contrast, an extensive property of a system does depend on the system size or the amount of material in the system. However, some of the intensive properties are statistical in nature (e.g. viscosity) and are relevant only in aggregate scales.

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[edit] Distinction from perceptions

Certain perceptions are often described (or even "measured") as if they are an intensive or extensive physical property. However, they are fundamentally different. For example, the colour of a solution is not a physical property. A dilute solution of potassium permanganate is pink, a more concentrated solution deep purple, and a larger volume of a strong solution is black. The colour (ie, the degree of 'pinkness' or 'purpleness'), is a perception and 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. The same volume of permanganate solution 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. Although the colour is related to the absorption spectrum, they are not synonymous. The truth of this becomes apparent when comparing two objects that have the same colour despite very different spectra.

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:

  • loudness of sound:    the related physical property is sound pressure level
  • 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:

  • color of a solution:    the related physical property is the transmission or absorption spectrum

[edit] 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 a substance is not a bulk property, because it depends on the amount of that substance being measured. Density on the other hand, is a bulk property.

[edit] 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, {ai}, are intensive quantities and another set, {Aj}, are extensive quantities, then the function F({ai},{Aj}) is an intensive quantity if for all α,

F(\{a_i\},\{\alpha A_j\}) = F(\{a_i\},\{A_j\}).\,

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).

[edit] Joining systems

Let there be a system or piece of substance a of amount ma and another piece of substance b of amount mb. Let V be an intensive variable. The value of variable V corresponding to the first substance is Va, and the value of V corresponding to the second substance is Vb. If the two pieces a and b are put together, forming a piece of substance "a+b" of amount ma+b = ma+mb, then the value of their intensive variable V is:

V_{a+b} = \frac{m_a V_a + m_b V_b}{m_a + m_b},

which is a weighted mean. Further, if Va = Vb then Va + b = Va = Vb, 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.

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

T = \frac{N_aT_a+N_bT_b}{N_a+N_b},

where Ni is the number of particles in gas i, and Ti is the corresponding temperature.

[edit] Examples

Examples of intensive properties include:

[edit] 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).

[edit] Combined extensive quantities

If a set of parameters {ai} are intensive quantities and another set {Aj} are extensive quantities, then the function F({ai},{Aj}) is an extensive quantity if for all α,

F(\{a_i\},\{\alpha A_j\})=\alpha F(\{a_i\},\{A_j\}).\,

Thus, extensive quantities are homogeneous functions (of degree 1) with respect to {Aj}. It follows from Euler's homogeneous function theorem that

F(\{a_i\},\{A_i\})=\sum_j A_j \left(\frac{\partial F}{\partial A_j}\right),

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

[edit] Examples

[edit] See also

[edit] References

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