Percentage composition and formulas

What a formula tells you

A chemical formula tells you which elements a compound holds, and in what ratio. There are two kinds of formula.

• The molecular formula gives the actual number of atoms of each element in one molecule. Glucose is $\text{C}_6\text{H}_{12}\text{O}_6$.
• The empirical formula gives the simplest whole-number ratio of the atoms. For glucose, divide every number by $6$: $\text{CH}_2\text{O}$.

Sometimes the two are the same, as for water. An analysis in the laboratory gives the empirical formula first. The molar mass then tells us how many empirical units make one molecule.

Mass percent of an element

The mass percent of an element is the share of the compound's mass that comes from that element:

• $\text{mass percent} = \dfrac{\text{its mass in 1 mol}}{M} \times 100$, where $M$ is the molar mass of the compound.

Example: calcium carbonate, $\text{CaCO}_3$, the compound of marble and chalk. Use $\text{Ca} = 40$, $\text{C} = 12$ and $\text{O} = 16$.

1. Molar mass. $40 + 12 + 3 \times 16 = 100\ \text{g mol}^{-1}$.
2. Calcium. $\dfrac{40}{100} \times 100 = 40\%$.
3. Carbon and oxygen. Carbon is $12\%$, and oxygen is $\dfrac{48}{100} \times 100 = 48\%$.
4. Check. $40 + 12 + 48 = 100$. The percentages of all the elements always add up to $100$.

From percentages to an empirical formula

An analysis gives the percentages. Four steps turn them into a formula.

1. Take $100\ \text{g}$ of the compound. Each percent becomes a mass in grams.
2. Change each mass to moles. Divide it by the atomic mass of the element.
3. Divide by the smallest number of moles. This gives the ratio of the atoms.
4. Make whole numbers. A number that ends near $.5$ needs every number multiplied by $2$. One that ends near $.33$ or $.67$ needs $3$, and one near $.25$ or $.75$ needs $4$.

Example. A red oxide of iron is $69.94\%$ iron and $30.06\%$ oxygen. Find its empirical formula. Use $\text{Fe} = 56$ and $\text{O} = 16$.

The empirical formula is $\text{Fe}_2\text{O}_3$. Rounding $1.50$ up to $2$ would give the wrong formula, $\text{FeO}_2$. Never round a ratio that is far from a whole number. Multiply instead.

From the empirical formula to the molecular formula

Find the empirical formula mass, the mass of one empirical unit. Then divide the molar mass by it:

• $n = \dfrac{\text{molar mass}}{\text{empirical formula mass}}$, and the molecular formula is $n$ times the empirical formula.

Example. A liquid has the empirical formula $\text{CH}$ and a molar mass of $78\ \text{g mol}^{-1}$. One unit of $\text{CH}$ has a mass of $13\ \text{g mol}^{-1}$. So $n = \dfrac{78}{13} = 6$, and the molecular formula is $\text{C}_6\text{H}_6$, benzene.

Practise with new numbers

Try it in Python

This program follows the four steps for a hydrocarbon gas, and then uses its molar mass to find the molecular formula.

Common mistakes

• Using the percents as moles. Divide each mass by its atomic mass first.
• Rounding too early. A ratio of $1.5$ is not $2$. Multiply every number by $2$ instead.
• Rounding a third away. A ratio such as $2.33$ needs every number multiplied by $3$.
• Mixing up the two formulas. The empirical formula is only a ratio. Only the molar mass gives the molecular formula.
• Using the atomic number. Carbon's atomic number is $6$, but its atomic mass is $12$.

Remember these

• The molecular formula counts the atoms in a molecule. The empirical formula gives their simplest whole-number ratio.
• Mass percent of an element $=$ its mass in one mole of the compound, divided by the molar mass, times $100$.
• Empirical formula: grams in $100\ \text{g}$, then moles, then divide by the smallest, then whole numbers.
• Molecular formula: $n = \dfrac{\text{molar mass}}{\text{empirical formula mass}}$.

Check yourself

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

1. What is the empirical formula of octane, $\text{C}_8\text{H}_{18}$?

2. What percent of the mass of a water molecule is hydrogen? Use $\text{H} = 1$ and $\text{O} = 16$.

3. After dividing by the smallest, a ratio comes out as $\text{C} : \text{H} = 1 : 2.5$. What do you do next, and what is the formula?

4. Methanal ($30\ \text{g mol}^{-1}$) and lactic acid ($90\ \text{g mol}^{-1}$) both have the empirical formula $\text{CH}_2\text{O}$. Write the molecular formula of each.

5. Why do you need the molar mass to find a molecular formula?