In physics, astronomy, chemistry, biology and geography number density (symbol: n) is an intensive quantity used to describe the degree of concentration of countable objects (particles, molecules, phonons, cells, galaxies, etc.) in physical space: threedimensional volume number density, twodimensional area number density, or onedimensional line number density. Population density is an example of areal number density. The term number concentration (symbol: C) is sometimes used in chemistry for the same quantity, particularly when comparing with other concentrations.
Definition
Volume number density is the number of specified objects per unit volume:^{[1]}

n = \frac{N}{V} ,
where

N is the total number of objects in a volume V.
Here it is assumed^{[2]} 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.
Units
In SI system of 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 10^{20}). Using the number density of an ideal gas at 0 °C and 1 atm as a yardstick: 1 amagat = 2.6867774×10^{25} 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).^{[3]}
Usage
Using the number density as a function of spatial coordinates, the total number of objects N in the entire volume V can be calculated as

N=\iiint_V n(x,y,z)\;dV ,
where

dV=dx\,dy\,dz is a volume element. If each object possesses the same mass m_{0}, the total mass m of all the objects in the volume V can be expressed as

m=\iiint_V m_0\, n(x,y,z)\;dV .
Similar expressions are valid for electric charge or any other extensive quantity associated with countable objects. For example, replacing m\rightarrow q (total charge) and m_0\rightarrow q_0 (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 quantities
Molar concentration
For any substance, the number density n (in units of m^{−3}) can be expressed in terms of its molar concentration c (in units of mole/m^{3}) as:

n=N_{\rm A}\,c ,
where N_{A} is the Avogadro constant ≈ 6.022×10^{23} mol^{−1}. 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 units of cm^{−3} and c is in units of mole/cm^{3}, or if n is in units of L^{−1} and c is in units of mole/L, etc.
Mass density
For atoms or molecules of a welldefined molar mass M (in units of kg/mole), the number density can be expressed in terms of the mass density of a substance ρ (in units of kg/m^{3}) as

n=\frac{N_{\rm A}}{M}\rho .
Note that the ratio M/N_{A} is the mass of a single atom or molecule in units of kg.
Examples
The following table lists common examples of number densities at 1 atm and 20 °C, unless otherwise noted.
Molecular^{[4]} number density and related parameters of some materials
Material

Number density (n)

Molar concentration (c)

Density (\rho)

Molar mass (M)

Units

(10^{27} m^{−3}) or
(10^{21} cm^{−3})

(amagat)

(10^{3} mol/m^{3}) or (mol/L)

(10^{3} kg/m^{3}) or (g/cm^{3})

(10^{−3} kg/mol) or (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

water

33.3679

1241.93

55.4086

0.99820

18.01524

diamond

176.2

6556

292.5

3.513

12.01

See also
References and notes

^ IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "number concentration".

^ Clayton T. Crowe; Martin Sommerfeld; Yutaka Tsuji (1998), Multiphase flows with droplets and particles: allelochemical interactions,

^ Joseph Kestin (1979), A Course in Thermodynamics 2, Taylor & Francis, p. 230,

^ For elemental substances, atomic densities/concentrations are used
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