Number density

Units

In SI units, number density is measured in m⁻³, although cm⁻³ is often used. However, these units are not quite practical when dealing with atoms or molecules of gases, liquids or solids at room temperature and atmospheric pressure, because the resulting numbers are extremely large (on the order of 10²⁰). Using the number density of an ideal gas at 0 °C and 1 atm as a yardstick: n₀ = 1 amg = 2.6867774 × 10²⁵ m⁻³ is often introduced to define a relative number density (a dimensionless quantity), for any substances at any conditions (not necessarily limited to an ideal gas at 0 °C and 1 atm).

Usage

Using the number density as a function of spatial coordinates, the total number of objects N in the entire volume V can be calculated as $${\displaystyle N={\iiint }_{V}n(x,y,z)\mathrm{d}V,}$$ where dV = dx dy dz is a volume element. If each object possesses the same mass m₀, the total mass m of all the objects in the volume V can be expressed as $${\displaystyle m={\iiint }_V{m}_{0}n(x,y,z)\mathrm{d}V.}$$

Similar expressions are valid for electric charge or any other extensive quantity associated with countable objects. For example, replacing m with q (total charge) and m₀ with q₀ (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 moles per unit volume (and thus called molar concentration).

Relation to other quantities

Molar concentration

For any substance, the number density can be expressed in terms of its amount concentration c (in mol/m³) as $${\displaystyle n=N_{\mathrm{A}}c}$$ where N_A 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 cm⁻³ and c is in mol/cm³, or if n is in L⁻¹ and c is in mol/L, etc.

Mass density

For atoms or molecules of a well-defined molar mass M (in kg/mol), the number density can sometimes be expressed in terms of their mass density ρ_m (in kg/m³) as $${\displaystyle n=\frac{N_{\mathrm{A}}}{M}{\rho }_{\mathrm{m}}.}$$ Note that the ratio M/N_A is the mass of a single atom or molecule in kg.

Examples

The following table lists common examples of number densities at 1 atm and 20 °C, unless otherwise noted.