# 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.

## Calculation

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

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

If the sample is a mixture containing N components, the molar volume is complex. It can be simply the sum of the individual components, and calculated using:

${\displaystyle V_{\rm {m}}={\frac {\displaystyle \sum _{i=1}^{N}x_{i}M_{i}}{\rho _{\mathrm {mixture} }}}}$.

But this is violated by many liquid-liquid mixtures. For instance mixing pure ethanol into pure water causes a decrease in the volume calculated by this formula. This effect is called "excess volume".

## Ideal gases

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

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

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 = 0.082 057 338(47) L atm K−1 mol−1, that is a relative standard uncertainty of 5.7×10−7, according to the 2014 CODATA recommended value[2]. The molar volume of an ideal gas at 100 kPa (1 bar) is

0.022 710 980(38) m3/mol at 0 °C,
0.024 789 598(42) m3/mol at 25 °C.

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

0.022 414 m3/mol at 0 °C,
0.024 465 m3/mol at 25 °C.

## Crystalline solids

For crystalline solids, the molar volume can be measured by X-ray crystallography. 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

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

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]