Number density
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In [[physics]], [[astronomy]], and [[chemistry]], '''number density''' is an [[intensive quantity]] used to describe the degree of concentration of [[countable]] objects ([[Elementary particle|particle]]s, [[molecules]], [[phonon]]s, [[galaxy|galaxies]], copies of [[Physics World]], etc.) in the [[three-dimensional]] physical [[space]].
==Definition==
'''number density''', usually denoted by ''n'', is the number of specified objects per unit [[volume]]:<ref>{{Citation
| author = Mark G. Kuzyk
| author-link = http://www.physics.wsu.edu/Personnel/faculty/kuzyk.html
| editor-last = Kuzyk
| editor-first = Mark G.
| editor2-last = Dirk
| editor2-first = Carl William
| contribution = Relationship Between the Molecular and Bulk Response
| title = Characterization techniques and tabulations for organic nonlinear optical materials
| year = 1998
| page = 163
| place =
| publisher = [[CRC Press]]
| url =
| doi =
| id =
| isbn = 0-8247-9968-2}}</ref>
: <math>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 = Martin Sommerfeld
| author3 = Yutaka Tsuji
| title = Multiphase flows with droplets and particles: allelochemical interactions
| year = 1998
| page = 18
| place =
| publisher = [[CRC Press]]
| url =
| doi =
| id =
| isbn = 0-8493-9469-4}}</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''.
==Units==
In [[SI]] system of units, number density is measured in [[Metre|m]]<sup>−3</sup>, although [[Centimetre|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 0[[Centigrade|° C]] and 1 [[Atmosphere (unit)|atm]] as a [[yardstick]], 1 [[amagat]] = [[Loschmidt constant|2.6867774×10<sup>25</sup>]] m<sup>−3</sup> is often introduced as a unit of number density, for any substances at any contidions, not necessarily limited to an ideal gas at 0° C and 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 = 0-8911-6641-6}}</ref>
==Usage==
Using the number density as a [[function]] of [[Coordinate system|spatial coordinates]], the total number of objects ''N'' in the entire volume ''V'' can be calculated as
: <math>N=\iiint_V n(x,y,z)\;dV </math> ,
where
:<math>dV=dx\,dy\,dz</math> 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>m=\iiint_V m_0\, n(x,y,z)\;dV </math> .
Similar expressions are valid for [[electric charge]] or any other [[extensive quantity]] associated with countable objects. For example, replacing <math>m\rightarrow q</math> (total charge) and <math>m_0\rightarrow q_0</math> (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''' ''n'' (in units of m<sup>−3</sup>) can be expressed in terms of its [[molar concentration]] ''C'' (in units of [[Mole (unit)|mole]]/m<sup>3</sup>) as:
:<math>n=N_A\,C</math> ,
where ''N''<sub>A</sub> ≈ 6.022×10<sup>23</sup> mol<sup>-1</sup> 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 units of cm<sup>−3</sup> and ''C'' is in units of [[Mole (unit)|mole]]/cm<sup>3</sup>, or if ''n'' is in units of [[Litre|L]]<sup>−1</sup> and ''C'' is in units of [[Mole (unit)|mole]]/[[Litre|L]], etc.
===Mass density===
For [[atom]]s or [[molecule]]s of a well-defined [[molecular mass]] ''M''<sub>0</sub> (in units of [[Kilogram|kg]]/[[Mole (unit)|mole]]), the '''number density''' can be expressed in terms of the mass [[density]] of a substance <math>\rho</math> (in units of kg/m<sup>3</sup>) as
:<math>n=\frac{N_A}{M_0}\rho</math> .
Note that the ratio ''M''<sub>0</sub>/''N''<sub>A</sub> is ''m''<sub>0</sub>, the mass of a single atom or molecule in units of [[Kilogram|kg]].
==Examples==
The following table lists common examples of number densities at 1 [[atmosphere (unit)|atm]] and 20 [[Celsius|°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
! Material !!colspan="2" | Number density (''n'') !! [[Molar concentration]] (''C'') !![[Density]] (<math>\rho</math>) !! [[Molar mass]] (''M''<sub>0</sub>)
|-
|align="right" | [[Unit]]s ||align="center" |(10<sup>27</sup> m<sup>-3</sup>) or<br>(10<sup>21</sup> cm<sup>-3</sup>)||align="center" |([[amagat]])||align="center" |(10<sup>3</sup> [[Mole (unit)|mol]]/m<sup>3</sup>) or ([[Mole (unit)|mol]]/[[Litre|L]])||(10<sup>3</sup> [[kg]]/[[cubic meter|m<sup>3</sup>]]) or ([[gram|g]]/[[cubic centimeter|cm<sup>3</sup>]])||(10<sup>-3</sup> [[kg]]/[[Mole (unit)|mol]]) or ([[gram|g]]/[[Mole (unit)|mol]])
|-
|align="left"| [[ideal gas]] ||0.02504||0.932||0.04158||41.58×10<sup>-6</sup>×''M''||''M''
|-
|align="left"| [[Density of air|dry air]] ||0.02504||0.932||0.04158||1.2041×10<sup>-3</sup>||28.9644
|-
|align="left"| [[Water (data page)|water]] ||33.3679||1241.93||55.4086||0.99820||18.01524
|-
|align="left"| [[Carbon|diamond]] ||176.2||6556||292.5||3.513||12.01
|}
==References and notes==
{{Reflist}}
[[Category:Physical quantity]]
[[Category:Density]]
[[Category:Chemical properties]]
[[de:Teilchendichte]]
[[fr:Densité de nombre]]
[[ja:数密度]]
[[pl:Koncentracja (fizyka)]]
[[ru:Концентрация частиц]]