# Molar volume

The molar volume, symbol Vm,[1] is the volume occupied by one mole of a substance (chemical element or chemical compound) at a given temperature and pressure. It is equal to the molar mass (M) divided by the mass density (ρ). It has the SI unit cubic metres per mole (m3/mol),[1] although it is more practical to use the units cubic decimetres per mole (dm3/mol) for gases and cubic centimetres per mole (cm3/mol) for liquids and solids.

The molar volume of a substance can be found by measuring its molar mass and density then applying the relation

$V_{\rm m} = {M\over\rho}$.

If the sample is a mixture containing N components, the molar volume is calculated using:

$V_{\rm m} = \frac{\displaystyle\sum_{i=1}^{N}x_{i}M_{i}}{\rho_{\mathrm{mixture}}}$.

For ideal gases, the molar volume is given by the ideal gas equation: this is a good approximation for many common gases at standard temperature and pressure. For crystalline solids, the molar volume can be measured by X-ray crystallography.

## Ideal gases

The ideal gas equation can be rearranged to give an expression for the molar volume of an ideal gas:

$V_{\rm m} = {V\over{n}} = {{RT}\over{P}}$.

Hence, for a given temperature and pressure, the molar volume is the same for all ideal gases and is known to the same precision as the gas constant: R = 0.08206 L mol−1 K−1, that is a relative standard uncertainty of 9.1×10−7, according to the 2010 CODATA recommended value.[2] The molar volume of an ideal gas at 100 kPa (1 bar) is

22.710 980(38) dm3/mol at 0 °C
24.789 598(42) dm3/mol at 25 °C

The molar volume of an ideal gas at 1 atmosphere of pressure is

22.414 dm3/mol at 0 °C
24.465 dm3/mol at 25 °C

## Crystalline solids

The unit cell volume (Vcell) 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

$V_{\rm m} = {{N_{\rm A}V_{\rm cell}}\over{Z}}$

where NA 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

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 by Deslattes et al. (1974).[3] The interest stems from the fact 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.[4] At present (2006 CODATA recommended value), the precision of the value of the Avogadro constant is limited by the uncertainty in the value of the Planck constant (relative standard uncertainty of 5×10−8).[4][5]

The 2006 CODATA recommended value for the molar volume of silicon is 12.058 8349(11)×10−6 m3/mol, with a relative standard uncertainty of 9.1×10−8.[5]

## References

1. ^ a b International Union of Pure and Applied Chemistry (1993). Quantities, Units and Symbols in Physical Chemistry, 2nd edition, Oxford: Blackwell Science. ISBN 0-632-03583-8. p. 41. Electronic version.
2. ^ "CODATA value: molar gas constant". NIST. Retrieved 2011-10-26.
3. ^ Deslattes, R. D.; Henins, A.; Bowman, H. A.; Schoonover, R. M.; Carroll, C. L.; Barnes, I. L.; Machlan, L. A.; Moore, L. J.; Shields, W. R. (1974). "Determination of the Avogadro Constant". Phys. Rev. Lett. 33 (8): 463–66. Bibcode:1974PhRvL..33..463D. doi:10.1103/PhysRevLett.33.463.
4. ^ a b Mohr, Peter J.; Taylor, Barry N. (1999). "CODATA recommended values of the fundamental physical constants: 1998". J. Phys. Chem. Ref. Data 28 (6): 1713–1852. doi:10.1103/RevModPhys.72.351.
5. ^ a b Mohr, Peter J.; Taylor, Barry N.; Newell, David B. (2008). "CODATA Recommended Values of the Fundamental Physical Constants: 2006". Rev. Mod. Phys. 80 (2): 633–730. arXiv:0801.0028. Bibcode:2008RvMP...80..633M. doi:10.1103/RevModPhys.80.633.