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. Some intensive properties, such as viscosity, are statistical in nature and not relevant to extremely small quantities.

Contents 1 Distinguishing between intensive and extensive properties 2 Intensive quantity 2.1 Combined intensive quantities 2.2 Joining systems 2.3 Examples 3 Extensive quantity 3.1 Combined extensive quantities 3.2 Examples 4 Distinction from perceptions 5 See also 6 References

[edit] Distinguishing between intensive and extensive properties

Consider the joining of two identical systems: extensive properties (such as mass and volume) will change, but intensive properties (such as temperature and the density of the system) will not.^[1]

[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 an object 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 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 intrinsic and extrinsic.

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

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 m_a and another piece of substance b of amount m_b 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 V be an intensive variable. The value of variable V corresponding to the first substance is V_a, and the value of V corresponding to the second substance is V_b. If the two pieces a and b are put together, forming a piece of substance "a+b" of amount m_a+b = m_a+m_b, then the value of their intensive variable V is:

which is a weighted mean. Further, if V_a = V_b then V_{a + b} = V_a = V_b, 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.

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 = 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

where N_i is the number of particles in gas i, and T_i is the corresponding temperature.

[edit] Examples

Examples of intensive properties include:

• color
• temperature
• pressure
• chemical potential
• density
• viscosity
• electric field
• electrical resistivity
• melting point
• boiling point
• spectral absorption maxima (in solution)
• flammability
• specific energy

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

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

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

[edit] Examples

Examples of extensive properties include:

• mass
• length
• volume
• entropy
• energy
• electrical resistance
• texture
• heat

[edit] 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:

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