Derived Units

Introduction

The seven SI base units cannot describe everything on their own. How fast is a car travelling? How much space does a flask hold? How concentrated is a solution? None of these is a single base quantity. Instead, scientists combine base units by multiplying and dividing them, producing derived units . Once you understand how this works, you can work out the unit of almost any quantity simply from the equation that defines it.

Core explanation

What is a derived unit? A derived unit is any unit formed by combining base units through multiplication or division, with no extra numerical factor. Because the SI is coherent, a derived unit built from base units needs no conversion factor: 1 m ÷ 1 s is exactly 1 m s⁻¹.

Units follow the equation. The rule is simple: whatever you do to the quantities, you do to their units. If a quantity is defined as one quantity divided by another, its unit is one unit divided by the other.

- Speed = distance ÷ time, so its unit is metres ÷ seconds = m/s . - Area = length × length, so its unit is m × m = m² . - Volume = length × length × length, so its unit is m³ . - Density = mass ÷ volume, so its unit is kg ÷ m³ = kg/m³ . - Concentration = amount of substance ÷ volume, so its unit is mol/m³ (chemists usually use mol/dm³). - Acceleration = change in speed ÷ time = (m/s) ÷ s = m/s² .

Two ways of writing "per". The unit "metres per second" can be written as m/s or as m s⁻¹ . A negative index means "divide by", so s⁻¹ is "per second" and m⁻³ is "per cubic metre". Density in kg/m³ becomes kg m⁻³ , and concentration in mol/dm³ becomes mol dm⁻³ . Advanced courses and data books prefer negative indices because they avoid ambiguity when several units are divided, such as J mol⁻¹ K⁻¹.

Spacing and order. In compound units a space (or a raised dot) separates the parts: N m , not Nm, and kg m⁻³ . This matters: "ms" means millisecond, while "m s" means metre second.

Common derived units in chemistry.

Quantity Defining equation SI unit --- --- --- Area length × length m² Volume length³ m³ Density mass ÷ volume kg m⁻³ Speed distance ÷ time m s⁻¹ Concentration amount ÷ volume mol m⁻³ Rate of reaction change in concentration ÷ time mol m⁻³ s⁻¹ Molar mass mass ÷ amount kg mol⁻¹

Some derived units are used so often that they are given special names — the newton, pascal, joule and watt among them. Those are covered separately, but each is still just a combination of base units.

Step-by-step reasoning

To find the unit of any quantity:

1. Write the defining equation in words. 2. Replace each quantity with its SI unit. 3. Multiply or divide the units exactly as the equation does. 4. Simplify, combining repeated units as powers, and write "per" as a negative index if required.

Visual explanation

Picture base units as building blocks of different colours: a metre block, a kilogram block, a second block. A derived unit is a small model made by stacking blocks on top (multiplying) or placing them underneath a fraction line (dividing). Density is a kilogram block on top of three metre blocks.

Real-world analogy

Base units are like the letters of an alphabet and derived units are like words. With only 26 letters you can spell thousands of words; with only seven base units you can build a unit for every measurable quantity in science.

Real-world example

Car speedometers show kilometres per hour, and fuel economy is often quoted in litres per 100 kilometres. Both are derived units combining length, time and volume. Engineers convert them to SI units such as m s⁻¹ when doing calculations.

Why?

Why keep only seven base units rather than defining a new one for every quantity? Fewer independent units means fewer standards to maintain, and every derived unit automatically fits with the others. Calculations then work without extra conversion factors.

Common misconception

"The unit of a quantity has to be memorised." You rarely need to memorise one: if you know the defining equation, the unit follows automatically by treating units algebraically.

Worked example

Question: The molar mass of a substance is its mass divided by its amount in moles. Deduce the SI unit of molar mass and write it in negative-index form.

Reasoning: Molar mass = mass ÷ amount. The unit of mass is kg and of amount is mol, so the unit is kg ÷ mol.

Answer: kg/mol, written kg mol⁻¹. (Chemists usually quote molar masses in g mol⁻¹.)

Quick check

1. What is the derived SI unit of speed, written with a negative index? Answer: m s⁻¹ (metres per second).

Exam focus

Examiners ask you to "deduce the units" of a quantity from an equation. Show the substitution of units clearly and simplify carefully. Know both m/s and m s⁻¹ forms, and never mix them in a way that creates a double slash such as mol/dm³/s — write mol dm⁻³ s⁻¹ instead.

Advanced insight

Dimensional analysis treats each base quantity as a dimension: length L, mass M, time T. Speed has dimensions L T⁻¹ and density M L⁻³. Both sides of a valid physical equation must have identical dimensions, which gives a powerful check on formulae before any numbers are used.

Summary

Derived units are formed by multiplying and dividing the seven SI base units. The unit of a quantity follows directly from its defining equation. "Per" can be written with a slash or a negative index, for example kg/m³ or kg m⁻³. Derived units in chemistry include m³ for volume, kg m⁻³ for density and mol m⁻³ for concentration.

Practice questions

1. Define a derived unit. Answer: A unit formed by multiplying or dividing SI base units, such as m s⁻¹ or kg m⁻³. 2. Deduce the SI unit of density and write it in two ways. Answer: Mass ÷ volume gives kg/m³, also written kg m⁻³. 3. Rate of reaction can be measured as change in concentration (mol dm⁻³) divided by time (s). Give the unit. Answer: mol dm⁻³ s⁻¹. 4. Why is it better to write "m s" rather than "ms" for metre second? Answer: "ms" means millisecond, so a space is needed to show that metre and second are multiplied.