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Intensive and extensive properties: Wikis


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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: it is scale invariant. 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.

For example, density is an intensive quantity (it does not depend on the quantity), while mass and volume are extensive quantities. Note that the ratio of two extensive quantities that scale in the same way is scale-invariant, and hence an intensive quantity.


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 an extensive quantity, because it depends on the amount of that substance being measured. Density, on the other hand, is an intensive property of the substance.


Combined intensive quantities

There are four quantities in any thermodynamic system, two intensive ones and two extensive ones.

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).

Joining systems

Let there be a system or piece of substance a of amount ma and another piece of substance b of amount mb 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 extensive 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 extensive variable V is:

 V = \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},

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

Note that you have to measure the amounts in the same unit that was used to calculate the intensive quantity from the extensive quantity. So when you interpolate density, you have to measure the quantities in volume, as density is mass per volume. The formula makes no sense when you measure the quantities in mass (kg).


Examples of intensive properties include:

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.[citation needed]

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. For example, mass (extensive) divided by volume (extensive) gives density (intensive).

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.[citation needed]


Examples of extensive properties include[citation needed]:

Related extensive and intensive properties

Although not true for all physical properties, there are a number of properties which have corresponding extensive and intensive analogs, many of which are thermodynamic properties. Examples of such extensive thermodynamic properties, which are dependent on the size of the thermodynamic system in question, include volume (V), internal energy (U), enthalpy (H), entropy (S), Gibbs free energy (G), Helmholtz free energy (A), and heat capacities (Cv and Cp) (in the sense of thermal mass). Note that the main symbols of these extensive thermodynamic properties shown here are capital letters. Except for volume (V), these extensive properties are dependent on the amount of material (substance) in the thermodynamic system in question.

For homogeneous substances, these extensive thermodynamic properties each have analogous intensive thermodynamic properties, which can be expressed on a per mass basis, and the corresponding intensive property symbols would be the lower case letters of the corresponding extensive property. Examples of intensive thermodynamic properties, which are independent on the size of the thermodynamic system in question and are analogous to the extensive ones mentioned above, include specific volume (v), specific internal energy (u), specific enthalpy (h), specific entropy (s), specific Gibbs free energy (g), specific Helmholtz free energy (a), and specific heat capacities (cv and cp, sometimes simply called specific heats). These intensive thermodynamic properties are effectively material properties which are valid at a point in a thermodynamic system or at a point in space at a certain time. These intensive properties are dependent on the conditions at that point such as temperature, pressure, and material composition, but are not considered dependent on the size of a thermodynamic system or on the amount of material in the system. See the table below. Specific volume is volume per mass, the reciprocal of density which equals mass per volume.

Corresponding extensive and intensive thermodynamic properties
Symbol SI units Intensive
Symbol SI units
m3 or l*
Specific volume***
m3/kg or l*/kg
Internal energy
Specific internal energy
Specific entropy
Specific enthalpy
Gibbs free energy
Specific Gibbs free energy
Heat capacity
at constant volume
Specific heat capacity
at constant volume
Heat capacity
at constant pressure
Specific heat capacity
at constant pressure
* l = liter, J = joule
** specific properties, expressed on a per mass basis
*** Specific volume is the reciprocal of density.

If a molecular weight can be assigned for the substance, or the number of moles in the system can be determined, then each of these thermodynamic properties can be expressed on a per mole basis. These intensive properties could be named after the analogous extensive properties but with the word "molar" preceding them; thus molar volume, molar internal energy, molar enthalpy, molar entropy, etc. Although the same small letters can be used as in the analogous specific properties indicating they are intensive, sometimes the corresponding capital letters have been used (and understood to be on a per mole basis), and there seems to be no universally agreed upon symbol convention for these molar properties. A well known molar volume is that of an ideal gas at STP (Standard Temperature and Pressure); this molar volume = 22.41 liters per mole.

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

See also


  • 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.; Randall, M. (1961). Thermodynamics (2nd Edition ed.). New York: McGraw-Hill Book Company. 


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