Thenumber density (symbol:n orρ_N) is anintensive quantity used to describe the degree ofconcentration ofcountable objects (particles,molecules,phonons,cells,galaxies, etc.) in physical space:three-dimensionalvolumetric number density,two-dimensionalareal number density, orone-dimensionallinear number density.Population density is an example of areal number density. The termnumber concentration (symbol:lowercasen, orC, to avoid confusion withamount of substance indicated byuppercaseN) is sometimes used in chemistry for the same quantity, particularly when comparing with otherconcentrations.

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

Here it is assumed^[2] thatN is large enough thatrounding of the count to the nearestinteger does not introduce much of anerror, howeverV is chosen to be small enough that the resultingn does not depend much on thesize orshape of the volumeV because of large-scale features.

Area number density is the number of specified objects per unitarea,A:$${\displaystyle n^{\prime }=\frac{N}{A},}$$Similarly, linear number density is the number of specified objects per unitlength,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:$${\displaystyle n_c^{\prime }=\int n\mathrm{d}s.}$$It's related tocolumn mass density, with the volumetric number density replaced by the volume mass density.

InSI 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 ofgases,liquids orsolids atroom temperature andatmospheric pressure, because the resulting numbers are extremely large (on the order of 10²⁰). Using the number density of anideal gas at0°C and1atm as ayardstick:n₀ = 1amg =2.6867774 × 10²⁵ m⁻³ is often introduced to define arelative number density (adimensionless quantity), for any substances at any conditions (not necessarily limited to an ideal gas at0 °C and1 atm).^[3]

Using the number density as afunction ofspatial coordinates, the total number of objectsN in the entire volumeV 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 samemassm₀, the total massm of all the objects in the volumeV can be expressed as$${\displaystyle m={\iiint }_V{m}_{0}n(x,y,z)\mathrm{d}V.}$$

Similar expressions are valid forelectric charge or any otherextensive quantity associated with countable objects. For example, replacingm withq (total charge) andm₀ withq₀ (charge of each object) in the above equation will lead to a correct expression for charge.

The number density ofsolute molecules in asolvent is sometimes calledconcentration, although usually concentration is expressed as a number ofmoles per unit volume (and thus calledmolar concentration).

For any substance, the number density can be expressed in terms of itsamount concentrationc (inmol/m³) as$${\displaystyle n=N_{\mathrm{A}}c}$$whereN_A is theAvogadro constant. This is still true if thespatial dimension unit, metre, in bothn andc is consistently replaced by any other spatial dimension unit, e.g. ifn is in cm⁻³ andc is in mol/cm³, or ifn is inL⁻¹ andc is in mol/L, etc.

Foratoms ormolecules of a well-definedmolar massM (inkg/mol), the number density can sometimes be expressed in terms of theirmass densityρ_m (in kg/m³) as$${\displaystyle n=\frac{N_{\mathrm{A}}}{M}{\rho }_{\mathrm{m}}.}$$Note that the ratioM/N_A is the mass of a single atom or molecule in kg.

The following table lists common examples of number densities at1 atm and20 °C, unless otherwise noted.

• Density
• Population geography
• Scalar physical quantities
• Plasma parameters
• Concentration

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