Number density
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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.
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[edit] Definition
Number density is the number of specified objects per unit volume:[1]
,
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
,
where
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
.
Similar expressions are valid for electric charge or any other extensive quantity associated with countable objects. For example, replacing 
(total charge) and 
(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:
,
where NA 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.
[edit] 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/NA is 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.
| Material | Number density (n) | Molar concentration (c) | Density (ρ) | Molar mass (M) | |
|---|---|---|---|---|---|
| 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
