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# 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 mass density then applying the relation

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

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 = 8.314 472(15) J mol–1 K–1, that is a relative standard uncertainty of 1.7×10–6, according to the 2006 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

## 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

High quality single crystals of ultrapure silicon are 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),[5] 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][6]

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.[7]

## References

1. ^ a b International Union of Pure and Applied Chemistry (1993). Quantities, Units and Symbols in Physical Chemistry (2nd Edn). Oxford: Blackwell Science. ISBN 0-632-03583-8. p. 41. Electronic version.
2. ^ CODATA value: molar gas constant. NIST. Retrieved on 2007-10-14.
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). "{{{title}}}". Phys. Rev. Lett. 33: 463–66.
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: 1713–1852. doi:0047-2689/99/28(6)/1713/140.
5. ^ CODATA value: Avogadro constant. NIST. Retrieved on 2007-10-14.
6. ^ CODATA value: Planck constant. NIST. Retrieved on 2007-10-14.
7. ^ CODATA value: molar volume of silicon. NIST. Retrieved on 2007-10-14.