Stoichiometry and the limiting reagent

What a balanced equation says

LPG, the gas in a kitchen cylinder, is mostly propane. It burns like this:

The same equation can be read in four ways.

• Molecules. One molecule of propane reacts with five molecules of oxygen. They give three molecules of carbon dioxide and four of water.
• Moles. $1\ \text{mol} + 5\ \text{mol} \rightarrow 3\ \text{mol} + 4\ \text{mol}$.
• Masses. $44\ \text{g} + 160\ \text{g} \rightarrow 132\ \text{g} + 72\ \text{g}$. Both sides add up to $204\ \text{g}$, so mass is conserved.
• Gas volumes. At the same temperature and pressure, equal moles of gases fill equal volumes. At STP ($273.15\ \text{K}$ and $1\ \text{bar}$), one mole of any gas fills $22.7\ \text{L}$.

The coefficients give the mole ratio. Working out amounts from this ratio is called stoichiometry.

Mass to mass in three steps

A balanced equation links moles, never grams. So every problem goes through moles.

1. Mass to moles. Divide the mass by the molar mass.
2. Moles to moles. Multiply by the mole ratio from the equation.
3. Moles to mass. Multiply by the molar mass of the substance you want, or by $22.7\ \text{L}$ for a gas at STP.

Example. How much carbon dioxide forms when $22\ \text{g}$ of propane burns, and how much oxygen does it use?

1. Propane: $\dfrac{22}{44} = 0.5\ \text{mol}$.
2. Carbon dioxide: $0.5 \times 3 = 1.5\ \text{mol}$, which is $1.5 \times 44 = 66\ \text{g}$, or $1.5 \times 22.7 = 34.05\ \text{L}$ at STP.
3. Oxygen: $0.5 \times 5 = 2.5\ \text{mol}$, which is $2.5 \times 32 = 80\ \text{g}$.

The limiting reagent

A sandwich needs $2$ slices of bread and $1$ slice of cheese. With $10$ slices of bread and $4$ of cheese, you can make only $4$ sandwiches, and $2$ slices of bread are left. The cheese limits the number of sandwiches.

In a reaction, the reactant that is used up first is the limiting reagent. It decides how much product forms. The other reactant is in excess, and some of it is left over.

Example. $56\ \text{g}$ of nitrogen and $9\ \text{g}$ of hydrogen react: $\text{N}_2 + 3\text{H}_2 \rightarrow 2\text{NH}_3$.

1. Moles. Nitrogen: $\dfrac{56}{28} = 2\ \text{mol}$. Hydrogen: $\dfrac{9}{2} = 4.5\ \text{mol}$.
2. Divide each by its coefficient. Nitrogen: $\dfrac{2}{1} = 2$. Hydrogen: $\dfrac{4.5}{3} = 1.5$. The smaller share belongs to hydrogen, so hydrogen is the limiting reagent.
3. Product. Ammonia: $\dfrac{2}{3} \times 4.5 = 3\ \text{mol}$, which is $3 \times 17 = 51\ \text{g}$.
4. Excess. Nitrogen used: $1.5\ \text{mol}$. Left over: $0.5\ \text{mol}$, which is $14\ \text{g}$.
5. Check. At the start, $56 + 9 = 65\ \text{g}$. At the end, $51 + 14 = 65\ \text{g}$.

Practise with new numbers

Try it in Python

Aluminium burns in chlorine: $2\text{Al} + 3\text{Cl}_2 \rightarrow 2\text{AlCl}_3$. The program finds the limiting reagent, the product and what is left over.

Common mistakes

• Using an equation that is not balanced. The mole ratio comes from the coefficients, so balance first.
• Using a mass ratio. The equation gives moles. $1\ \text{g}$ of nitrogen does not react with $3\ \text{g}$ of hydrogen.
• Choosing the limiting reagent by its mass, or by its moles alone. Divide the moles by the coefficient, then compare.
• Forgetting the excess. The reactant in excess is not all used. Work out what is left.

Remember these

• The coefficients of a balanced equation give the mole ratio.
• Mass problems go mass to moles, moles to moles, moles to mass.
• At STP, one mole of a gas fills $22.7\ \text{L}$. Equal moles of gases fill equal volumes.
• The limiting reagent has the smallest value of moles divided by its coefficient. It fixes the amount of product.

Check yourself

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

1. In the burning of propane, $\text{C}_3\text{H}_8 + 5\text{O}_2 \rightarrow 3\text{CO}_2 + 4\text{H}_2\text{O}$, how many moles of oxygen does $2$ mol of propane need?

2. Magnesium burns: $2\text{Mg} + \text{O}_2 \rightarrow 2\text{MgO}$. What mass of magnesium oxide comes from $12\ \text{g}$ of magnesium? Use $\text{Mg} = 24$ and $\text{O} = 16$.

3. Riya has $6$ slices of bread and $5$ slices of cheese. A sandwich needs $2$ slices of bread and $1$ of cheese. How many sandwiches can she make, and what is left?

4. $1$ mol of nitrogen and $1$ mol of hydrogen are mixed to make ammonia. Which is the limiting reagent?

5. What volume does $0.5$ mol of oxygen gas fill at STP?