Length: The Metre

Introduction

How long is a metre? Roughly the distance from your nose to your outstretched fingertips, or the height of a kitchen worktop. But "roughly" is not good enough for science. The metre must be defined so precisely that a laboratory in Canada and one in India produce exactly the same length. Today that definition rests on the speed of light, one of the most fundamental constants in the universe. Length appears everywhere in chemistry, from the size of glassware to the distance between atoms.

Core explanation

The modern definition. The metre (symbol m) is the SI base unit of length. It is defined as the distance travelled by light in a vacuum in exactly 1/299 792 458 of a second. Put another way, the speed of light is fixed at exactly 299 792 458 m/s. Because the second is defined very precisely by atomic clocks, fixing the speed of light fixes the metre too.

A short history. In the 1790s the metre was defined as one ten-millionth of the distance from the North Pole to the equator along the meridian through Paris. Later, a platinum–iridium bar kept near Paris became the standard. In 1960 the metre was redefined using the wavelength of light from krypton atoms, and in 1983 it gained its current definition based on the speed of light. Each change made the metre more precise and easier to reproduce anywhere.

Lengths across chemistry. Chemists deal with an enormous range of lengths:

Object Typical length --- --- Laboratory bench about 2 m Test tube about 0.15 m (15 cm) Width of a pencil lead about 0.5 mm Diameter of a human hair about 0.07 mm Diameter of a hydrogen atom about 0.1 nm

To cope with this range we use prefixes: 1 km = 1000 m, 1 cm = 0.01 m, 1 mm = 0.001 m, 1 μm = 10⁻⁶ m and 1 nm = 10⁻⁹ m.

Instruments for length. A metre rule or tape measure reads to the nearest millimetre. Vernier callipers can read to about 0.1 mm or better, and a micrometer screw gauge to about 0.01 mm. Atoms and molecules are far too small to see with any of these; their sizes are found indirectly using techniques such as X-ray diffraction.

Length builds other units. Area (m²) and volume (m³) are derived from length. The volume of liquid in a measuring cylinder, in cm³, is really a length unit cubed: 1 cm³ is the volume of a cube 1 cm along each edge.

Step-by-step reasoning

To measure the length of a small object with a ruler:

1. Place the object flat against the ruler, parallel to the scale. 2. Line up one end with the zero mark (or note the starting reading if the end of the ruler is worn). 3. Look straight down at the other end to avoid parallax error. 4. Read the scale and estimate to the nearest half millimetre if possible. 5. Subtract the starting reading, if any, and record the length with its unit.

Visual explanation

Draw a long horizontal line labelled 1 m. Divide it into ten parts labelled 10 cm each, then zoom into one part and divide it into 10 mm. Continue zooming by factors of a thousand to reach μm and then nm, showing a bacterium at the μm scale and a molecule at the nm scale.

Real-world analogy

Defining the metre by the speed of light is like defining a race track by how far a perfectly reliable runner goes in a fixed time. If the runner's speed never changes, and your stopwatch is perfect, you can mark out exactly the same distance anywhere in the world.

Real-world example

Satellite navigation systems work by timing radio signals that travel at the speed of light. A timing error of just a millionth of a second would place you about 300 m from your true position. This link between time and distance is the same one used to define the metre.

Why?

Why define the metre through light rather than keep a metal bar? A bar can expand with temperature, be scratched or be lost, and it exists in only one place. The speed of light in a vacuum is the same everywhere and never changes, so any well-equipped laboratory can reproduce the metre exactly.

Common misconception

"The centimetre is the SI base unit of length." The base unit is the metre. The centimetre is a convenient smaller unit made with the prefix centi, meaning one hundredth. Many laboratory calculations use cm, but SI answers are ultimately expressed in metres.

Worked example

Question: A strip of magnesium ribbon is 45 mm long. Express this in cm and in m.

Reasoning: There are 10 mm in 1 cm, so 45 mm = 45 ÷ 10 = 4.5 cm. There are 1000 mm in 1 m, so 45 mm = 45 ÷ 1000 = 0.045 m.

Answer: 4.5 cm and 0.045 m.

Quick check

1. What constant of nature is used to define the metre? Answer: The speed of light in a vacuum, fixed at exactly 299 792 458 m/s.

Exam focus

Be able to convert confidently between km, m, cm and mm, and to state which instrument suits a given length. Questions often ask about parallax: explain that you must read the scale with your eye level with, and perpendicular to, the mark.

Advanced insight

Bond lengths between atoms are typically 0.1–0.3 nm. Chemists sometimes use the older unit the ångström (Å), where 1 Å = 10⁻¹⁰ m = 0.1 nm, because bond lengths then come out as convenient numbers such as 1.54 Å for a carbon–carbon single bond. The ångström is not an SI unit, so SI answers use nm or pm instead (154 pm).

Summary

The metre is the SI base unit of length, defined by fixing the speed of light at exactly 299 792 458 m/s. It replaced earlier definitions based on the Earth and on a metal bar. Prefixes let us express lengths from kilometres down to nanometres. Rulers, callipers and micrometers measure everyday lengths; atomic sizes are found indirectly.

Practice questions

1. Convert 2.5 m into centimetres and millimetres. Answer: 250 cm and 2500 mm. 2. Which instrument would you use to measure the diameter of a thin wire, and why? Answer: A micrometer screw gauge, because it reads to about 0.01 mm, which is needed for such a small length. 3. How many nanometres are there in one metre? Answer: 1 000 000 000 (10⁹) nm. 4. Give one advantage of defining the metre using the speed of light rather than a metal bar. Answer: The speed of light never changes and can be used anywhere, while a bar can change, be damaged and exists only in one place.