Number density

The number density (symbol: n or ρ_N) is an

.

Definition

Volume number density is the number of specified objects per unit volume:^[1] $${\displaystyle n={\frac {N}{V}},}$$ where N is the total number of objects in a volume V.

Here it is assumed

of the volume V because of large-scale features.

Area number density is the number of specified objects per unit area, A: $${\displaystyle n'={\frac {N}{A}},}$$ Similarly, linear number density is the number of specified objects per unit length, L: $${\displaystyle n''={\frac {N}{L}},}$$

Column number density is a kind of areal density, the number or count of a substance per unit area, obtained integrating volumetric number density along a vertical path: $${\displaystyle n'_{c}=\int n\,\mathrm {d} s.}$$ It's related to

, with the volumetric number density replaced by the volume mass density.

In

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

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

).

Relation to other quantities

Molar concentration

For any substance, the number density can be expressed in terms of its

/m³) as $${\displaystyle n=N_{\rm {A}}c}$$ where N_A is the ⁻¹ and c is in mol/L, etc.

Mass density

For

mass density

ρ_m (in kg/m³) as $${\displaystyle n={\frac {N_{\rm {A}}}{M}}\rho _{\mathrm {m} }.}$$ 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.

Molecular^[4] number density and related parameters of some materials^{[citation needed]}

Material	Number density, n		Amount concentration , c	Mass density , ρm	Molar mass, M
	(1027 m−3 = 1021 cm−3)	(amg)	(103 mol/m3 = mol/L)	(103 kg/m3 = g/cm3)	(10−3 kg/mol = g/mol)
Ideal gas	0.02504	0.932	0.04158	41.58 × 10−6 M	M
Dry air	0.02504	0.932	0.04158	1.2041 × 10−3	28.9644
Water	33.3679	1,241.93	55.4086	0.99820	18.01524
Diamond	176.2	6,556	292.5	3.513	12.01

See also

• Columnar number density

References and notes