Units and dimensional analysis

A unit is part of the number

"The distance is 5." Five what? Five metres or five kilometres? A number from a measurement means nothing without its unit. In chemistry we always carry the units through a calculation, and treat them like numbers. They can be multiplied, divided and cancelled.

Conversion factors

We know that $1\ \text{in} = 2.54\ \text{cm}$. Divide both sides by $1\ \text{in}$:

A fraction like this, whose top and bottom are equal, is a conversion factor, or a unit factor. Multiplying by $1$ does not change a quantity, so we can multiply by a unit factor whenever we like. We choose it so that the unwanted unit cancels.

How many centimetres are in $4.00\ \text{in}$?

The inches cancel, and centimetres are left. If we had used the factor upside down, the units would be $\text{in}^2/\text{cm}$, which is clearly wrong.

Chains of conversions

Often one factor is not enough. Then we use a chain, and each step cancels one unit. How many seconds are there in $2$ days?

Squared and cubed units need the factor squared or cubed. Since $1\ \text{m} = 100\ \text{cm}$,

and $1\ \text{m}^3 = 1000\ \text{L}$.

Density and other derived units

A derived unit, such as $\text{g/mL}$, can also act as a factor. Ethanol has a density of $0.789\ \text{g/mL}$. What is the mass of $250\ \text{mL}$ of it?

To go the other way, from mass to volume, turn the factor upside down: $\dfrac{1\ \text{mL}}{0.789\ \text{g}}$.

Checking an answer by its units

The units of an answer are a free check. If you want a mass and your units come out as $\text{mL}^2/\text{g}$, the setup is wrong, even before you look at the number. This method is called dimensional analysis. Use it on every problem.

Practise with new numbers

Try it in Python

The program multiplies a quantity by a chain of unit factors and prints each step, as you would write it on paper.

Common mistakes

• A factor upside down. Check that the unit you want to remove is on the bottom of the factor.
• Forgetting to cube. $1\ \text{m}^3$ is $10^{6}\ \text{cm}^3$, not $100\ \text{cm}^3$.
• Mixing litres and millilitres. Write every unit at every step, and the mistake shows itself.
• Rounding in the middle of a chain. Round only the final answer.

Remember these

• A conversion factor is a fraction equal to one, such as $\dfrac{1000\ \text{mL}}{1\ \text{L}}$.
• Units multiply, divide and cancel like numbers.
• For squared or cubed units, square or cube the factor.
• If the units of the answer are wrong, the setup is wrong.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. How many litres are in $2.5\ \text{m}^3$?

2. Change $0.45\ \text{kg}$ into milligrams.

3. A cheetah runs at $30\ \text{m/s}$. What is this in km/h?

4. Which factor turns grams into kilograms: $\dfrac{1000\ \text{g}}{1\ \text{kg}}$ or $\dfrac{1\ \text{kg}}{1000\ \text{g}}$?

5. Ravi divides a volume in mL by a density in g/mL to find a mass, and his units come out as $\text{mL}^2/\text{g}$. What went wrong?