Number density

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In physics, astronomy, and chemistry, number density 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.

1 Definition
2 Units
3 Usage
4 Relation to other quantities
- 4.1 Molar concentration
- 4.2 Mass density
5 Examples
6 References and notes

[edit] Definition

number density, usually denoted by n, is the number of specified objects per unit volume:^[1]

$n = \frac{N}{V}$ ,

where

N is the total number of objects in a volume V.

Here it is assumed^[2] 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.

[edit] 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.^[3]

[edit] 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

$N=\iiint_V n(x,y,z)\;dV$ ,

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

$m=\iiint_V m_0\, n(x,y,z)\;dV$ .

Similar expressions are valid for electric charge or any other extensive quantity associated with countable objects. For example, replacing $m\rightarrow q$ (total charge) and $m_0\rightarrow q_0$ (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).

[edit] Relation to other quantities

[edit] 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:

$n=N_A\,C$ ,

where N_A ≈ 6.022×10²³ mol^-1 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 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.

[edit] Mass density

For atoms or molecules of a well-defined molecular mass M₀ (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

$n=\frac{N_A}{M_0}\rho$ .

Note that the ratio M₀/N_A is m₀, the mass of a single atom or molecule in units of kg.

[edit] 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
Material	Number density (n)		Molar concentration (C)	Density ( $ρ$ )	Molar mass (M₀)
Units	(10²⁷ m^-3) or (10²¹ cm^-3)	(amagat)	(10³ mol/m³) or (mol/L)	(10³ kg/m³) or (g/cm³)	(10^-3 kg/mol) or (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	1241.93	55.4086	0.99820	18.01524
diamond	176.2	6556	292.5	3.513	12.01

[edit] References and notes

^ [|Mark G. Kuzyk] (1998), "Relationship Between the Molecular and Bulk Response", in Kuzyk, Mark G.; Dirk, Carl William, Characterization techniques and tabulations for organic nonlinear optical materials, CRC Press, p. 163, ISBN 0-8247-9968-2
^ Clayton T. Crowe; Martin Sommerfeld; Yutaka Tsuji (1998), Multiphase flows with droplets and particles: allelochemical interactions, CRC Press, p. 18, ISBN 0-8493-9469-4
^ Joseph Kestin (1979), A Course in Thermodynamics, 2, Taylor & Francis, p. 230, ISBN 0-8911-6641-6
^ For elemental substances, atomic densities/concentrations are used

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Categories: Physical quantities | Density | Amount of substance