Foundation lesson

Quantities and units

A physical quantity is a number times a unit. Only quantities of the same kind can be added or compared, so every term of a correct equation carries the same units, and checking that catches many slips.

Written for this edition, not translated from Einstein. Editorial review pending.

Can a number of metres be compared with a number of seconds?

"The particle moved 6" says nothing until the unit is added: 6 micrometres, or 6 seconds? A physical quantity is a number times a unit, and the number alone changes when the unit does. Six micrometres is 0.006 millimetres: the same length, written with another unit.

Quantities of different kinds cannot be added or compared: 3 metres plus 2 seconds has no meaning. So in a correct equation every term, on both sides, carries the same units. Checking that is the quickest test a formula can pass or fail.

Take the Brownian diffusion coefficient, $D = k_BT/(6\pi\eta a)$ . Boltzmann's constant is in joules per kelvin, T in kelvins, the viscosity η in pascal-seconds and the radius a in metres. The units work out to square metres per second, which is what a diffusion coefficient must be. The worked example below goes through it.

Units also tell apart quantities that share a letter. The light paper's $\rho_\nu$ is an energy per unit volume per unit of frequency, joules per cubic metre per hertz, not per unit of wavelength: two different quantities with different units. And its L is the speed of light in §§1 and 2 but an absorbed energy in §9. A unit check starts from what the quantity is, not from its letter.

Worked example: Checking that D comes out in square metres per second

$k_BT$ : joules per kelvin times kelvins is joules, which are newton-metres.
$6\pi\eta a$ : 6π has no units, and pascal-seconds times metres is (newtons per square metre) × seconds × metres, which is newton-seconds per metre.
Divide: newton-metres divided by newton-seconds per metre is square metres per second.
With numbers: 1.380649 × 10⁻²³ × 290.15 ÷ (6π × 1.35 × 10⁻³ × 5 × 10⁻⁷) = 3.15 × 10⁻¹³ m²/s. That is the modern Boltzmann constant (set modern-si-2019) with the paper's water at 17 °C and its particle.

JavaScript is off, so the controls below cannot change anything. The construction shows its first setting, and its closing paragraph, “What it shows, in words”, describes the rest.

Try it: cancelling units

Choose a formula. Each factor is written in the base units kilogram, metre, second and kelvin, the powers are added, and the sum is compared with the units the result must have. Two of the formulas are wrong on purpose.

Formula

D = k_BT/(6πηa)

k_B, Boltzmann's constant: J/K, which is kg m² s⁻² K⁻¹, on top.
T, the temperature: K, on top.
6π, six times pi, is a pure number with no units.
η, the viscosity: Pa s, which is kg m⁻¹ s⁻¹, underneath.
a, the particle's radius: m, underneath.

Adding the powers gives m² s⁻¹. A diffusion coefficient must be m² s⁻¹: the units agree.

What it shows, in words

A formula's units can be checked without any numbers: write every factor in kilograms, metres, seconds and kelvins, add the powers of the factors on top, subtract those underneath, and halve those under a square root. The Stokes-Einstein diffusion coefficient comes out in square metres per second. Leave out the radius and a metre too many survives; leave off the square root in the spread and the result is an area, not a length. The check catches both slips, but it cannot catch a wrong pure number such as 6π.

Where this lesson stops

This lesson stops at checking units. Converting the papers' own units into SI units is a separate step, with its own conventions.

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