Units and orders of magnitude

A human hair is about 70 micrometres thick; a hydrogen atom is about 0.1 nanometre across. Written side by side, it is hard to feel how far apart those numbers are — yet chemistry lives in both worlds at once: we weigh grams on a balance while thinking about individual atoms. Crossing that gap takes two tools: a notation that makes the order of magnitude obvious, and a way of converting units that cannot go wrong.

The International System of Units (SI) has only seven base units: the second (s), metre (m), kilogram (kg), ampere (A), kelvin (K), mole (mol) and candela (cd). Every other unit is built from them — density in $\text{kg}\,\text{m}^{-3}$, energy in joules, $\text{J} = \text{kg}\,\text{m}^2\,\text{s}^{-2}$.

Since 2019 all seven are defined by constants of nature with exactly fixed values: the speed of light $c$ defines the metre, the Planck constant $h$ the kilogram, the Avogadro constant $N_\mathrm{A}$ the mole. The platinum–iridium International Prototype Kilogram in Paris has retired. Note that the kilogram is the only base unit with a prefix (“kilo”) built into its name.

Scientific notation writes any number as $a \times 10^{n}$ with $1 \le a < 10$. The exponent $n$ gives the order of magnitude at a glance: the exponents of $1.54\times10^{-10}\ \text{m}$ (a C–C bond) and $7.5\times10^{-6}\ \text{m}$ (a red blood cell) differ by 4, and the mantissas add a factor of about 5, so the two differ about $5\times10^{4}$-fold. It has a second virtue: the number of digits in $a$ states how many significant figures were measured — $1.20\times10^{3}$ and $1.2\times10^{3}$ are different measurements (more in the next lesson).

To multiply or divide, combine the mantissas and add or subtract the exponents: $(3.0\times10^{8})\times(2.0\times10^{-9}) = 6.0\times10^{-1}$.

Because $1\ \text{km} = 1000\ \text{m}$, the fraction $\dfrac{1000\ \text{m}}{1\ \text{km}}$ equals exactly one. Multiplying by one leaves the quantity unchanged but changes its unit. The factor-label method multiplies by such ones, cancelling identical units in numerator and denominator as if they were numbers, until only the wanted unit remains. Which way up to use ($\tfrac{1000\ \text{m}}{1\ \text{km}}$ or $\tfrac{1\ \text{km}}{1000\ \text{m}}$) is decided by what you need to cancel.

On the bench below every card is a “one” you can apply either way up; a struck-through unit has cancelled. If the unit at the end is wrong, the answer is wrong — which is exactly why the method never fails silently.