View source for Number density

{{short description|Degree of concentration of countable objects}}
{{distinguish|Natural density}}
The '''number density''' (symbol: ''n'' or ''ρ''<sub>N</sub>) is an [[intensive quantity]] used to describe the degree of [[concentration]] of [[countable]] objects ([[Elementary particle|particle]]s, [[molecules]], [[phonon]]s, [[Cell (biology)|cell]]s, [[galaxy|galaxies]], etc.) in physical space: [[Three-dimensional space|three-dimensional]] '''volumetric number density''', [[Plane (mathematics)|two-dimensional]] '''areal number density''', or [[One-dimensional space|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]] ''[[letter N|N]]'') is sometimes used in chemistry for the same quantity, particularly when comparing with other [[concentration]]s.

==Definition==
Volume number density is the number of specified objects per unit [[volume]]:<ref>{{GoldBookRef | file = N04260 | title = number concentration}}</ref>
<math display="block">n = \frac{N}{V},</math>
where ''N'' is the total number of objects in a volume ''V''. 

Here it is assumed<ref>{{Citation
  | author1 = Clayton T. Crowe
  | author2 = John D. Schwarzkopf
  | author3 = Martin Sommerfeld
  | author4 = Yutaka Tsuji
  | title = Multiphase flows with droplets and particles: allelochemical interactions
  | year = 2011
  | page = 18
  | place = 
  | publisher = [[CRC Press]]
  | url = https://www.taylorfrancis.com/books/mono/10.1201/b11103/multiphase-flows-droplets-particles-clayton-crowe-john-schwarzkopf-martin-sommerfeld-yutaka-tsuji
  | doi = 10.1201/b11103
  | id = 
  | isbn = 9780429106392
  }}</ref> that ''N'' is large enough that [[rounding]] of the count to the nearest [[integer]] does not introduce much of an [[Random error|error]], however ''V'' is chosen to be small enough that the resulting ''n'' does not depend much on the [[Volume|size]] or [[shape]] of the volume ''V'' because of large-scale features.

Area number density is the number of specified objects per unit [[area]], ''A'':
<math display="block">n' = \frac{N}{A},</math>
Similarly, linear number density is the number of specified objects per unit [[length]], ''L'':
<math display="block">n'' = \frac{N}{L},</math>

''Column number density'' is a kind of areal density, the number or count of a substance per unit area, obtained integrating volumetric number density along a vertical path:
<math display="block">n'_c = \int n \, \mathrm{d}s.</math>
It's related to [[column mass density]], with the volumetric number density replaced by the volume mass density.

==Units==
In [[International System of Units|SI]] units, number density is measured in m<sup>−3</sup>, although cm<sup>−3</sup> is often used. However, these units are not quite practical when dealing with atoms or molecules of [[gas]]es, [[liquid]]s or [[solid]]s at [[room temperature]] and [[atmospheric pressure]], because the resulting numbers are extremely large (on the order of 10<sup>20</sup>). Using the number density of an [[ideal gas]] at {{nowrap|0 [[Celsius|°C]]}} and {{nowrap|1 [[Atmosphere (unit)|atm]]}} as a [[yardstick]]: {{nowrap|1=[[Loschmidt constant|''n''<sub>0</sub>]] = 1 [[Amagat|amg]] = {{val|2.6867774}} × 10<sup>25</sup> m<sup>−3</sup>}} is often introduced as a unit of number density, for any substances at any conditions (not necessarily limited to an ideal gas at {{nowrap|0 °C}} and {{nowrap|1 atm}}).<ref>{{Citation
  | author = Joseph Kestin
  | title = A Course in Thermodynamics
  | volume = 2 
  | year = 1979
  | page = 230
  | place = 
  | publisher = Taylor & Francis
  | url = 
  | doi = 
  | id = 
  | isbn = 0-89116-641-6
}}</ref>

==Usage==
Using the number density as a [[Function (mathematics)|function]] of [[Coordinate system|spatial coordinates]], the total number of objects ''N'' in the entire volume ''V'' can be calculated as
<math display="block">N = \iiint_V n(x,\,y,\,z)\,\mathrm{d}V,</math>
where d''V'' = d''x'' d''y'' d''z'' is a volume element. If each object possesses the same [[mass]] ''m''<sub>0</sub>, the total mass ''m'' of all the objects in the volume ''V'' can be expressed as
<math display="block">m = \iiint_V m_0 n(x,\,y,\,z)\,\mathrm{d}V.</math>

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 ''m''<sub>0</sub> with ''q''<sub>0</sub> (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 [[Mole (unit)|moles]] per unit volume (and thus called [[molar concentration]]).

==Relation to other quantities==
===Molar concentration===
For any substance, the number density can be expressed in terms of its [[amount concentration]] ''c'' (in [[Mole (unit)|mol]]/m<sup>3</sup>) as
<math display="block">n = N_{\rm A}c</math>
where {{nobreak|''N''<sub>A</sub>}} 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<sup>−3</sup> and ''c'' is in mol/cm<sup>3</sup>, or if ''n'' is in [[Litre|L]]<sup>−1</sup> and ''c'' is in mol/L, etc.

===Mass density===
For [[atom]]s or [[molecule]]s of a well-defined [[molar mass]] ''M'' (in [[Kilogram|kg]]/mol), the number density can sometimes be expressed in terms of their [[mass density]] ''ρ''<sub>m</sub> (in kg/m<sup>3</sup>) as
<math display="block">n = \frac{N_{\rm A}}{M} \rho_\mathrm{m}.</math>
Note that the ratio ''M''/''N''<sub>A</sub> is the mass of a single atom or molecule in kg.

==Examples==
The following table lists common examples of number densities at {{nobreak|1 atm}} and {{nobreak|20 °C}}, unless otherwise noted.

{| class="wikitable" align="center" style="text-align:center"
|+ Molecular<ref name=mol>For [[Chemical element|elemental]] substances, atomic densities/concentrations are used</ref> number density and related parameters of some materials{{cn|date=May 2015}}
|-
! rowspan="2" | Material
! colspan="2" | Number density, ''n''
! [[Amount concentration]], ''c''
! [[Mass density]], ''ρ''<sub>''m''</sub>
! [[Molar mass]], ''M''
|-
! (10<sup>27</sup> m<sup>−3</sup> = 10<sup>21</sup> cm<sup>−3</sup>)
! ([[Amagat|amg]])
! (10<sup>3</sup> [[mole (unit)|mol]]/m<sup>3</sup> = [[mole (unit)|mol]]/[[litre|L]])
! (10<sup>3</sup> [[kilogram|kg]]/m<sup>3</sup> = [[gram|g]]/cm<sup>3</sup>)
! (10<sup>−3</sup> [[kilogram|kg]]/[[mole (unit)|mol]] = [[Gram|g]]/[[mole (unit)|mol]])
|-
| [[Ideal gas]] || 0.02504 || 0.932 || 0.04158 || 41.58 × 10<sup>−6</sup> ''M'' || ''M''
|-
| [[Density of air|Dry air]] || 0.02504 || 0.932 || 0.04158 || 1.2041 × 10<sup>−3</sup> || 28.9644
|-
| [[Water (data page)|Water]] || 33.3679 || 1,241.93 || 55.4086 || 0.99820 || 18.01524
|-
| [[Carbon|Diamond]] || 176.2 || 6,556 || 292.5 || 3.513 || 12.01
|}

==See also==
*[[Columnar number density]]

==References and notes==
{{reflist}}

{{Chemical solutions}}

[[Category:Density]]
[[Category:Population geography]]
[[Category:Scalar physical quantities]]
[[Category:Plasma parameters]]
[[Category:Concentration]]