In physics, astronomy, and chemistry, number density (symbol: n) is an intensive quantity used to describe the degree of concentration of countable objects (particles, molecules, phonons, galaxies, copies of Physics World, etc.) in the three-dimensional physical space. Area number density (number of entities per unit surface area) and linear number density (number of entities per unit length) are defined analogously. The term number concentration (symbol: C) is sometimes used in chemistry for the same quantity, particularly when comparing with other concentrations.

Definition

Number density is the number of specified objects per unit volume:

where

Here it is assumed that N is large enough that rounding of the count to the nearest integer does not introduce much of an error, however V is chosen to be small enough that the resulting n does not depend much on the size or shape of the volume V.

Units

In SI system of 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: 1 amagat = 2.6867774×10²⁵ m⁻³ is often introduced as a unit of number density, 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

where


Similar expressions are valid for electric charge or any other extensive quantity associated with countable objects. For example, replacing $m\backslash rightarrow\; q$ (total charge) and $m\_0\backslash rightarrow\; q\_0$ (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 n (in units of m⁻³) can be expressed in terms of its molar concentration c (in units of mole/m³) as:

where N_A is the Avogadro constant ≈ 6.022×10²³ mol⁻¹. 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 units of cm⁻³ and c is in units of mole/cm³, or if n is in units of L⁻¹ and c is in units of mole/L, etc.

Mass density

For atoms or molecules of a well-defined molecular mass M (in units of kg/mole), the number density can be expressed in terms of the mass density of a substance ρ (in units of kg/m³) as

Note that the ratio M/N_A is the mass of a single atom or molecule in units of kg.

Examples

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

References and notes

Category:Physical quantitiesCategory:DensityCategory:Amount of substance
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