Number density

 ( Density ) 

The number density (symbol: n or ρ_N) is an intensive quantity used to describe the degree of concentration of countable objects (particles, molecules, , cells, galaxy, etc.) in physical space: three-dimensional volumetric number density, two-dimensional areal number density, or one-dimensional linear number density. Population density is an example of areal number density. The term number concentration (symbol: lowercase n, or C, to avoid confusion with amount of substance indicated by uppercase letter N) is sometimes used in chemistry for the same quantity, particularly when comparing with other .

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Definition Volume number density is the number of specified objects per unit volume: $$n\; =\; \backslash frac\{N\}\{V\},$$ where N is the total number of objects in a volume V.

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

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

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: $$n\text{'}\_c\; =\; \backslash int\; n\; \backslash ,\; \backslash mathrm\{d\}s.$$ It's related to column mass density, with the volumetric number density replaced by the volume mass density.

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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 , or 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 and as a yardstick: is often introduced to define a relative density (a dimensionless quantity), for any substances at any conditions (not necessarily limited to an ideal gas at and ).

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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 $$N\; =\; \backslash iiint\_V\; n(x,\backslash ,y,\backslash ,z)\backslash ,\backslash mathrm\{d\}V,$$ where d V = d x d y d z 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 $$m\; =\; \backslash iiint\_V\; m\_0\; n(x,\backslash ,y,\backslash ,z)\backslash ,\backslash mathrm\{d\}V.$$

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).

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Relation to other quantities

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Molar concentration For any substance, the number density can be expressed in terms of its amount concentration c (in mol/m³) as $$n\; =\; N\_\{\backslash rm\; A\}c$$ where 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 Litre⁻¹ and c is in mol/L, etc.

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

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Examples The following table lists common examples of number densities at and , unless otherwise noted.

+ MolecularFor Chemical element substances, atomic densities/concentrations are used number density and related parameters of some materials

M

28.9644

18.01524

12.01

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References and notes

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