Editing Molar volume

{{Short description|Volume occupied by a given amount of particles of a substance}}
{{Infobox physical quantity
| bgcolour = {default}
| name = Molar volume
| unit = m<sup>3</sup>/mol
| otherunits = dm<sup>3</sup>/mol, cm<sup>3</sup>/mol
| symbols = ''V''<sub>m</sub>, <math>\tilde V</math>
| dimension = '''L'''<sup>3</sup> '''N'''<sup>−1</sup>
}}

In [[chemistry]] and related fields, the '''molar volume''', symbol ''V''<sub>m</sub>,<ref name="GreenBook">{{GreenBookRef2nd|page=41}}</ref> or <math>\tilde V</math> of a substance is the ratio of the [[volume]] (''V'') occupied by a substance to the [[amount of substance]] (''n''), usually at a given [[temperature]] and [[pressure]]. It is also equal to the [[molar mass]] (''M'') divided by the [[mass density]] (''ρ''):
<math display="block">V_{\text{m}} = \frac{V}{n} = \frac{M}{\rho}</math>

The molar volume has the [[SI unit]] of [[cubic metre]]s per mole (m<sup>3</sup>/mol),<ref name="GreenBook"/> although it is more typical to use the units [[cubic decimetre]]s per mole (dm<sup>3</sup>/mol) for [[gas]]es, and [[cubic centimetre]]s per mole (cm<sup>3</sup>/mol) for [[liquid]]s and [[solid]]s.

== Definition ==
[[File:Excess Volume Mixture of Ethanol and Water-int.svg|thumb|Change in volume with increasing ethanol fraction.]]
The molar volume of a substance ''i'' is defined as its [[molar mass]] divided by its density ''ρ''<sub>''i''</sub><sup>0</sup>:
<math display="block">V_{\rm m,i} = {M_i\over\rho_i^0}</math>
For an [[ideal mixture]] containing ''N'' components, the molar volume of the mixture is the [[weighted sum]] of the molar volumes of its individual components. For a real mixture the molar volume cannot be calculated without knowing the density:
<math display="block">V_{\rm m} = \frac{\displaystyle\sum_{i=1}^{N} x_i M_i}{\rho_{\mathrm{mixture}}}</math>
There are many liquid–liquid mixtures, for instance mixing pure [[ethanol]] and pure [[water]], which may experience contraction or expansion upon mixing. This effect is represented by the quantity '''excess volume''' of the mixture, an example of [[excess property]].

=== Relation to specific volume ===
Molar volume is related to [[specific volume]] by the product with [[molar mass]]. This follows from above where the specific volume is the [[reciproc]]al of the density of a substance:
<math display="block">V_{\rm m,i} = {M_i \over \rho_i^0} = M_i v_i</math>

== Ideal gases ==
For [[ideal gas]]es, the molar volume is given by the [[ideal gas equation]]; this is a good approximation for many common gases at [[standard temperature and pressure]].
The ideal gas equation can be rearranged to give an expression for the molar volume of an ideal gas:
<math display="block">V_{\rm m} = \frac{V}{n} = \frac{RT}{P}</math>
Hence, for a given temperature and pressure, the molar volume is the same for all ideal gases and is based on the [[gas constant]]: ''R''&nbsp;= {{val|8.31446261815324|u=m<sup>3</sup>⋅Pa⋅K<sup>−1</sup>⋅mol<sup>−1</sup>}}, or about {{val|8.20573660809596e-5|u=m<sup>3</sup>⋅atm⋅K<sup>−1</sup>⋅mol<sup>−1</sup>}}.

The molar volume of an ideal gas at 100&nbsp;[[Pascal (unit)|kPa]] (1&nbsp;[[bar (unit)|bar]]) is
: {{val|0.022710954641485|end=...|u=m3/mol}} at 0&nbsp;°C,
: {{val|0.024789570296023|end=...|u=m3/mol}} at 25&nbsp;°C.
The molar volume of an ideal gas at 1 atmosphere of pressure is
: {{val|0.022413969545014|end=...|u=m3/mol}} at 0&nbsp;°C,
: {{val|0.024465403697038|end=...|u=m3/mol}} at 25&nbsp;°C.

== Crystalline solids ==
For [[Crystal|crystalline solids]], the molar volume can be measured by [[X-ray crystallography]].
The [[unit cell]] volume (''V''<sub>cell</sub>) may be calculated from the [[unit cell]] parameters, whose determination is the first step in an X-ray crystallography experiment (the calculation is performed automatically by the structure determination software). This is related to the molar volume by
<math display="block">V_{\rm m} = {{N_{\rm A}V_{\rm cell}}\over{Z}}</math>
where ''N''<sub>A</sub> is the [[Avogadro constant]] and ''Z'' is the number of formula units in the unit cell. The result is normally reported as the "crystallographic density".

=== Molar volume of silicon ===
{{See also|Avogadro project}}
Ultra-pure [[silicon]] is routinely made for the [[electronics industry]], and the measurement of the molar volume of silicon, both by X-ray crystallography and by the ratio of molar mass to mass density, has attracted much attention since the pioneering work at [[NIST]] in 1974.<ref>{{cite journal | last=Deslattes | first=R. D. |author2=Henins, A. |author3=Bowman, H. A. |author4=Schoonover, R. M. |author5=Carroll, C. L. |author6=Barnes, I. L. |author7=Machlan, L. A. |author8=Moore, L. J. |author9= Shields, W. R. | year=1974 | journal=[[Physical Review Letters|Phys. Rev. Lett.]] | volume=33 | issue=8 | pages=463–66 | doi = 10.1103/PhysRevLett.33.463
 | title = Determination of the Avogadro Constant | bibcode=1974PhRvL..33..463D}}</ref> The interest stems from that accurate measurements of the unit cell volume, [[atomic weight]] and mass density of a pure crystalline solid provide a direct determination of the Avogadro constant.<ref name="CODATA98">{{cite journal |author-first1=Peter J. |author-last1=Mohr |author-first2=Barry N. |author-last2=Taylor |title=CODATA recommended values of the fundamental physical constants: 1998 |journal=[[Journal of Physical and Chemical Reference Data]] |volume=28 |issue=6 |pages=1713–1852 |bibcode=1999JPCRD..28.1713M |doi=10.1063/1.556049 |year=1999 |url=https://www.nist.gov/pml/div684/fcdc/upload/rmp1998-2.pdf |archive-url=https://web.archive.org/web/20171001122752/https://www.nist.gov/sites/default/files/documents/pml/div684/fcdc/rmp1998-2.pdf|archive-date=2017-10-01}}</ref>

The CODATA recommended value for the molar volume of silicon is {{physconst|VmSi|ref=no}}, with a relative standard uncertainty of {{physconst|VmSi|runc=yes|after=.}}

== See also ==
* [[Specific volume]]
* [[Standard temperature and pressure#Molar volume of a gas]]

== References ==
{{reflist}}

== External links ==
* [http://www.webelements.com/periodicity/molar_volume/ Interactive table of molar volumes]

{{Mole concepts}}
{{Authority control}}

[[Category:Volume]]
[[Category:Molar quantities]]
[[Category:Chemical quantities]]