Stoichiometric Coefficients and Mole Ratios
Stoichiometry is the calculation of the quantities of reactants and products in chemical reactions, and its key tool is the mole ratio, a conversion factor taken from the coefficients of a balanced equation. In brief: in N2 + 3H2 → 2NH3 the coefficients 1 : 3 : 2 say that 1 mol of nitrogen reacts with 3 mol of hydrogen to give 2 mol of ammonia, so 3.00 mol of H2 yields 3.00 × (2 ÷ 3) = 2.00 mol of NH3.
The word stoichiometry comes from the Greek stoicheion (element) and metron (measure). It allows chemists to predict how much product a reaction will give and how much of each reactant is needed. This article explains what the coefficients mean, how to write mole ratios, how to use them as conversion factors, shows a table of ratios for several reactions, works through examples with different species, explains why equations must be balanced and describes the history of the idea.
What do the coefficients mean?
The coefficients in a balanced chemical equation give the relative numbers of particles (molecules, formula units or atoms) that react or form, and so also the relative numbers of moles. In 2H2 + O2 → 2H2O, two molecules of hydrogen react with one molecule of oxygen to form two molecules of water. Scaling up by a factor of Avogadro’s number, two moles of H2 react with one mole of O2 to form two moles of H2O. The coefficients are not mass ratios: the masses are 4.03 g of H2, 32.00 g of O2 and 36.03 g of H2O.
How do you write a mole ratio?
A mole ratio compares the coefficients of any two substances in the equation. For the reaction 2H2 + O2 → 2H2O, the possible ratios include:
- 2 mol H2 : 1 mol O2, written as the factor (2 mol H2 ÷ 1 mol O2) or its inverse (1 mol O2 ÷ 2 mol H2).
- 2 mol H2 : 2 mol H2O (which reduces to 1 : 1).
- 1 mol O2 : 2 mol H2O.
Always put the quantity you want in the numerator and the quantity you have in the denominator, so that the unit of the given substance cancels.
Mole ratios for selected reactions
| Balanced equation | Ratio of reactants | Ratio of reactant to product |
|---|---|---|
| N2 + 3H2 → 2NH3 | N2 : H2 = 1 : 3 | H2 : NH3 = 3 : 2 |
| 2Al + 3Cl2 → 2AlCl3 | Al : Cl2 = 2 : 3 | Al : AlCl3 = 1 : 1 |
| C3H8 + 5O2 → 3CO2 + 4H2O | C3H8 : O2 = 1 : 5 | C3H8 : CO2 = 1 : 3 |
| Fe2O3 + 3CO → 2Fe + 3CO2 | Fe2O3 : CO = 1 : 3 | Fe2O3 : Fe = 1 : 2 |
| 2KClO3 → 2KCl + 3O2 | (single reactant) | KClO3 : O2 = 2 : 3 |
How do you use a mole ratio?
- Write a balanced equation.
- Identify the substance you are given and the substance you want.
- Write the mole ratio so that the given substance is in the denominator.
- Multiply the given amount (in moles) by the ratio to get the wanted amount (in moles).
The general form is: moles wanted = moles given × (coefficient of wanted ÷ coefficient of given).
Worked examples
- Ammonia from hydrogen. How many moles of NH3 form from 3.00 mol of H2 in N2 + 3H2 → 2NH3? 3.00 mol H2 × (2 mol NH3 ÷ 3 mol H2) = 2.00 mol NH3.
- Oxygen needed. How many moles of O2 are needed to burn 2.00 mol of propane, C3H8 + 5O2 → 3CO2 + 4H2O? 2.00 × (5 ÷ 1) = 10.0 mol O2.
- Carbon dioxide formed. From 0.500 mol of propane: 0.500 × (3 ÷ 1) = 1.50 mol CO2. The water formed is 0.500 × (4 ÷ 1) = 2.00 mol.
- Iron from its oxide. How many moles of iron are formed from 2.50 mol of Fe2O3 in Fe2O3 + 3CO → 2Fe + 3CO2? 2.50 × (2 ÷ 1) = 5.00 mol Fe. The CO needed is 2.50 × 3 = 7.50 mol.
- Reactant ratio. How much chlorine reacts with 4.0 mol of aluminum, 2Al + 3Cl2 → 2AlCl3? 4.0 × (3 ÷ 2) = 6.0 mol Cl2, and 4.0 mol of AlCl3 forms.
- Fractions. How many moles of O2 are made by 0.250 mol of KClO3 in 2KClO3 → 2KCl + 3O2? 0.250 × (3 ÷ 2) = 0.375 mol O2.
- Scaling the recipe. Making a bicycle needs 2 wheels, 1 frame and 1 handlebar: 2W + F + H → bicycle. With 100 wheels you can build 50 bicycles using 50 frames. In the same way, if 100 molecules of H2 react in 2H2 + O2 → 2H2O, 50 molecules of O2 are needed and 100 molecules of H2O form.
Why must the equation be balanced?
An unbalanced equation gives incorrect mole ratios. The unbalanced equation H2 + O2 → H2O suggests that 1 mol of H2 reacts with 1 mol of O2 to give 1 mol of H2O, which would create or destroy oxygen atoms. Using the balanced equation 2H2 + O2 → 2H2O gives the correct ratios. If an equation can be balanced with different sets of coefficients (which are multiples of each other), such as 4H2 + 2O2 → 4H2O, the ratios remain the same.
The stoichiometry road map
| Step | Action | Conversion factor |
|---|---|---|
| 1 | Convert given quantity to moles | Molar mass, molar volume of gas or molarity |
| 2 | Convert moles of given to moles of wanted | Mole ratio from coefficients |
| 3 | Convert moles of wanted to the required unit | Molar mass, molar volume of gas or molarity |
The mole ratio in step 2 is the only step that uses the balanced equation, and it is the bridge between the two substances.
Who started stoichiometry?
The German chemist Jeremias Benjamin Richter introduced the term “stoichiometry” in 1792, after noticing that acids and bases neutralize each other in fixed mass proportions. Joseph Proust’s law of definite proportions (1799 to 1806) and John Dalton’s law of multiple proportions (1803) added further regularities, and Dalton’s atomic theory explained them. By the middle of the 19th century, with reliable atomic masses and the concept of the mole developed by Wilhelm Ostwald and others, stoichiometry became a standard tool.
Which mistakes are common?
Mistake 1: Using the coefficients as mass ratios
Coefficients relate moles. Convert grams to moles before applying them.
Mistake 2: Inverting the mole ratio
Put the wanted substance in the numerator. If the units do not cancel, flip the ratio.
Mistake 3: Using an unbalanced equation
Balance first. Wrong coefficients give wrong answers.
Mistake 4: Using subscripts instead of coefficients
The subscripts describe the formula of a substance. Only coefficients give the ratios between substances.