Number density

Key takeaways

Number density belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Number density to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Number density from memory before moving on to harder problems.

Reference excerpt

The number density (symbol: n or ρN) is an intensive quantity used to describe the degree of concentration of countable objects (particles, molecules, phonons, cells, galaxies, 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 N) is sometimes used in chemistry for the same quantity, particularly when comparing with other concentrations.

Definition Volume number density is the number of specified objects per unit volume:

n = N V , {\displaystyle n={\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 error, but V is chosen to be small enough that the resulting n does not depend much on the size 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 ′ = N A , {\displaystyle n'={\frac {N}{A}},}

Similarly, linear number density is the number of specified objects per unit length, L:

n ″ = N 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:

n c ′ = ∫ n d s . {\displaystyle n'_{c}=\int n\,\mathrm {d} s.}

It's related to column mass density, with the volumetric number density replaced by the volume mass density.

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 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: n0 = 1 amg = 2.6867774 × 1025 m−3 is often introduced to define a relative number density (a dimensionless quantity), for any substances at any conditions (not necessarily limited to an ideal gas at 0 °C and 1 atm).

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 = ∭ V n ( x , y , z ) d V , {\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

m = ∭ V m 0 n ( x , y , z ) d V . {\displaystyle m=\iiint _{V}m_{0}n(x,\,y,\,z)\,\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).

Relation to other quantities

Molar concentration For any substance, the number density can be expressed in terms of its amount concentration c (in mol/m3) as

n = N A c {\displaystyle n=N_{\rm {A}}c}

where NA 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 L−1 and c is in mol/L, etc.

Mass density For atoms or molecules of a well-defined molar mass M (in kg/mol), the number density can sometimes be expressed in terms of their mass density ρm (in kg/m3) as

n = N A M ρ m . {\displaystyle n={\frac {N_{\rm {A}}}{M}}\rho _{\mathrm {m} }.}

Note that the ratio M/NA 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.

References and notes

Illustrations

Worked examples

Example 1 — a first encounter with Number density

Start with the simplest possible case. Write down what Number density claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Number density before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Number density ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Number density

In research: Number density appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Number density in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Number density is common in secondary-school and first-year university syllabi. It links to neighbouring topics Concentration, Density, Plasma parameters, so understanding it makes those chapters shorter.
In everyday life: Look for Number density outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Number density in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Number density means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Number density out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Number density in simple terms?

The number density (symbol: n or ρN) is an intensive quantity used to describe the degree of concentration of countable objects (particles, molecules, phonons, cells, galaxies, etc.) in physical space: three-dimensional volumetric number density, two-dimensional areal number density, or one-dimensi…

Why does Number density matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Number density?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Number density.