Dimensional Analysis

Dimensional Analysis

When you work through numerical problems in chemistry, you often need to convert a quantity expressed in one set of units into the equivalent quantity in another set of units. The method that makes these conversions clean and nearly foolproof is called dimensional analysis. It is also known as the factor label method or the unit factor method.

The core idea is simple: you multiply the quantity you start with by one or more carefully chosen fractions that are each equal to 1. Because multiplying by 1 does not change the value of a quantity, the numerical magnitude may change but the actual physical amount stays the same. The trick is to choose the fractions so that the unwanted units cancel and the desired units remain.

The Unit Factor

A unit factor is a fraction constructed from an equivalence between two units. For example, the equivalence

gives two unit factors:

Both fractions equal 1 because the numerator and denominator represent the same physical length. You choose the one whose denominator contains the unit you want to cancel and whose numerator contains the unit you want to end up with.

Example: inches to centimetres

A piece of metal is 3 inches long. What is its length in centimetres?

We know $1 \text{ in} = 2.54 \text{ cm}$. The unit factor that cancels inches and leaves centimetres is $\frac{2.54 \text{ cm}}{1 \text{ in}}$. Multiply:

Notice that the unit "in" appears in the numerator of the starting quantity and in the denominator of the unit factor, so it cancels just like a number would. The result is in centimetres.

Handling Units Like Numbers

In dimensional analysis, units obey the same algebraic rules as numbers. They can be cancelled, multiplied, divided, squared, or cubed. This is a powerful feature: if you set up the conversion correctly, the units will automatically simplify to the ones you want, and you can check your work by seeing whether the units come out right.

Example: litres to cubic metres

A jug contains 2 L of milk. Calculate the volume in m$^3$.

We need two equivalences:

From the second equivalence we get the unit factor $\frac{1 \text{ m}}{100 \text{ cm}} = 1$. To convert cubic centimetres to cubic metres, we need to cube this unit factor:

Now convert 2 L to cm$^3$ first:

Then multiply by the cubed unit factor:

Chaining Multiple Conversions in One Step

When a conversion requires several steps, you can string the unit factors together in a single calculation. Each factor cancels one unit and introduces the next, until only the desired unit remains.

Example: days to seconds

How many seconds are there in 2 days?

We need the chain:

Write the unit factors so that each unwanted unit cancels: …