The number density (symbol: n or ρN) is an intensive quantity used to describe the degree of concentration of countable objects (particles, molecules, phonons, cells, galaxies, etc.) in physical space: three-dimensional volumetric number density, two-dimensional areal number density, or one-dimensional linear number density. Population density is an example of areal number density. The term number concentration (symbol: lowercase n, or C, to avoid confusion with amount of substance indicated by uppercase N) is sometimes used in chemistry for the same quantity, particularly when comparing with other concentrations.
where N is the total number of objects in a volume V.
Here it is assumed that N is large enough that rounding of the count to the nearest integer does not introduce much of an error, however V is chosen to be small enough that the resulting n does not depend much on the size or shape of the volume V because of large-scale features.
In SI units, number density is measured in m−3, although cm−3 is often used. However, these units are not quite practical when dealing with atoms or molecules of gases, liquids or solids at room temperature and atmospheric pressure, because the resulting numbers are extremely large (on the order of 1020). Using the number density of an ideal gas at 0 °C and 1 atm as a yardstick: n0 = 1 amg = 2.6867774 × 1025 m−3 is often introduced as a unit of number density, for any substances at any conditions (not necessarily limited to an ideal gas at 0 °C and 1 atm).
where dV = dx dy dz is a volume element. If each object possesses the same mass m0, the total mass m of all the objects in the volume V can be expressed as
Similar expressions are valid for electric charge or any other extensive quantity associated with countable objects. For example, replacing m with q (total charge) and m0 with q0 (charge of each object) in the above equation will lead to a correct expression for charge.
The number density of solute molecules in a solvent is sometimes called concentration, although usually concentration is expressed as a number of moles per unit volume (and thus called molar concentration).
Relation to other quantitiesEdit
where NA is the Avogadro constant. This is still true if the spatial dimension unit, metre, in both n and c is consistently replaced by any other spatial dimension unit, e.g. if n is in cm−3 and c is in mol/cm3, or if n is in L−1 and c is in mol/L, etc.
Note that the ratio M/NA is the mass of a single atom or molecule in kg.
The following table lists common examples of number densities at 1 atm and 20 °C, unless otherwise noted.
|Material||Number density, n||Amount concentration, c||Mass density, ρm||Molar mass, M|
|(1027 m−3 = 1021 cm−3)||(amg)||(103 mol/m3 = mol/L)||(103 kg/m3 = g/cm3)||(10−3 kg/mol = g/mol)|
|Ideal gas||0.02504||0.932||0.04158||41.58 × 10−6 M||M|
|Dry air||0.02504||0.932||0.04158||1.2041 × 10−3||28.9644|
References and notesEdit
- IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "number concentration". doi:10.1351/goldbook.N04260
- Clayton T. Crowe; John D. Schwarzkopf; Martin Sommerfeld; Yutaka Tsuji (2011), Multiphase flows with droplets and particles: allelochemical interactions, CRC Press, p. 18, doi:10.1201/b11103, ISBN 9780429106392
- Joseph Kestin (1979), A Course in Thermodynamics, 2, Taylor & Francis, p. 230, ISBN 0-89116-641-6
- For elemental substances, atomic densities/concentrations are used