Physical properties of materials and systems are often described as intensive and extensive
properties. This classification relates to the dependency of the properties upon the
size or extent of the system or object in question.
The distinction is based on the concept that smaller, non-interacting identical subdivisions
of the system may be identified so that the property of interest does or does not change
when the system is divided or combined. An intensive property is a bulk property,
meaning that it is a physical property of a system that does not depend on the system
size or the amount of material in the system. Examples of intensive properties include temperature,
refractive index, density, and hardness of an object. No matter how small a diamond is
cut, it maintains its intrinsic hardness. By contrast, an extensive property is one
that is additive for independent, noninteracting subsystems. The property is proportional to
the amount of material in the system. For example, both the mass and the volume of a
diamond are directly proportional to the amount that is left after cutting it from the raw
mineral. Mass and volume are extensive properties, but hardness is intensive.
The ratio of two extensive properties is scale-invariant, and is therefore an intensive property. For
example, when gravity effects can be neglected, the ratio of the extensive properties mass
and volume, the density, is an intensive property. This terminology of intensive and extensive
properties was introduced by Richard C. Tolman in 1917.
Intensive 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 change in the identical ratio and the density remains unchanged. Additionally,
the boiling point of a substance is another example of an intensive property. For example,
the boiling point for water is 100°C at a pressure of one atmosphere, a fact which remains
true regardless of quantity. According to the state postulate, for a sufficiently
simple thermodynamic system, only two independent intensive variables are needed to fully specify
the entire state of a system. Other intensive properties can be derived from the two known
values. Some intensive properties, such as viscosity,
are empirical macroscopic quantities and are not relevant to extremely small systems.
Combined intensive properties There are four properties in any thermodynamic
system, two are intensive and two are extensive. If the set of parameters, , are intensive
properties and another set, , are extensive properties, then the function is an intensive
property if for all ,
It follows, for example, that the ratio of two extensive properties is an intensive property
- density is equal to mass divided by volume. Examples
Examples of intensive properties include:
Extensive properties An extensive property is defined by the IUPAC
Green Book as a physical quantity which is the sum of the properties of separate noninteracting
subsystems that compose the entire system. The value of such an additive property is
proportional to the size of the system it describes, or to the quantity of matter in
the system. Taking on the example of melting ice, the amount of heat required to melt ice
is an extensive property. The amount of heat required to melt one ice cube would be much
less than the amount of heat required to melt an iceberg, so it is dependent on the quantity.
Extensive properties are the counterparts of intensive properties, which are intrinsic
to a particular subsystem. Dividing one type of extensive property by a different type
of extensive property will in general give an intensive value. For example, mass divided
by volume gives density. Combined extensive properties
If a set of parameters are intensive properties and another set are extensive properties,
then the function is an extensive property if for all ,
Thus, extensive properties are homogeneous functions with respect to . It follows from
Euler's homogeneous function theorem that
where the partial derivative is taken with all parameters constant except . The converse
is also true - any function which obeys the above relationship will be extensive.
Examples Examples of extensive properties include:
Related extensive and intensive properties
Although not true for all physical properties, some properties have corresponding extensive
and intensive analogs, many of which are thermodynamic properties. Examples of such extensive thermodynamic
properties, that are dependent on the size of the thermodynamic system in question, include
volume, internal energy, enthalpy, entropy, Gibbs free energy, Helmholtz free energy,
and heat capacity. The symbols of these extensive thermodynamic properties shown here are capital
letters. For homogeneous substances, these extensive
thermodynamic properties each have corresponding intensive thermodynamic properties, which
are expressed on a per mass or volume basis. The name is usually prefixed with the adjective
specific to indicate that they are bulk properties, valid at any location in a thermodynamic system.
They may be dependent on other conditions at any 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. Specific volume is volume per mass, the reciprocal
of density which equals mass per volume. * 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 amount of substance can be determined, then each of these thermodynamic
properties may be expressed on a molar basis, and their name may be qualified with the adjective
molar, yielding terms such as molar volume, molar internal energy, molar enthalpy, molar
entropy. Standards for the symbols of molar quantities do not exist. A well known molar
volume is that of an ideal gas at standard conditions for temperature and pressure, with
the value 22.41liters/mol. Molar Gibbs free energy is commonly referred to as chemical
potential, symbolized by μ, particularly when discussing a partial molar Gibbs free
energy μi for a component i in a mixture. Generality of classification
The general validity of the division of physical properties into extensive and intensive kinds
has been addressed in the course of science. Redlich noted that physical properties and
especially thermodynamic properties are most conveniently defined as either intensive or
extensive, however, these two categories are not all-inclusive and some well-defined physical
properties conform to neither definition. Redlich also provides examples of mathematical
functions that alter the strict additivity relationship for extensive system, such as
the square or square root of volume, which occur in some contexts, altbeit rarely used.
Other systems, for which the 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 in which 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 is extensive. However if the same cells are connected in series, the charge
becomes intensive and the voltage extensive. The IUPAC definitions do not consider such
cases. References
Callen, Herbert B.. Thermodynamics and an Introduction to Themostatistics. New York:
John Wiley & Sons. ISBN 0-471-86256-8. Lewis, G.N.; Randall, M.. Thermodynamics.
New York: McGraw-Hill Book Company. Linder, Bruno.. Thermodynamics and Introductory
Statistical Mechanics. New York: John Wiley & Sons. ISBN 0-471-47459-2.