D3.4 Explain quantitative relationships in balanced equations

Ontario Grade 11 Chemistry

Quantities in Chemical Reactions

How coefficients describe particle, mole, and mass relationships

When substances react, the amounts used and formed are connected. A balanced chemical equation shows this connection with coefficients, the numbers placed in front of chemical formulas. These numbers compare particles and amounts in moles. They do not directly compare masses. To find masses, use the equation’s mole relationship together with molar mass.

What you will learn

Explain what the coefficients in a balanced chemical equation tell us.
Use coefficient ratios to relate amounts of substances in moles.
Distinguish mole relationships from mass relationships.
Check that calculations use the correct ratio and units.

1. Review: what balancing tells us

A chemical reaction changes substances into different substances. In a closed reaction, atoms are rearranged; they are not created or destroyed. A chemical formula shows which atoms make up a substance. A coefficient is the number written in front of a formula. It tells how many units of that substance are represented.

To balance an equation, adjust coefficients until each kind of atom has the same total on both sides. Never change a formula’s subscripts to balance an equation. A subscript is part of the substance’s identity. Changing it would describe a different substance.

For hydrogen and oxygen forming water, the balanced equation is shown below. The coefficients are the smallest whole-number values that balance the atoms. There are four hydrogen atoms and two oxygen atoms on each side.

\mathrm{2H_2(g) + O_2(g) \rightarrow 2H_2O(l)}

Balance atoms by changing coefficients, not subscripts.
A balanced equation has the same number of each kind of atom on both sides.

2. Coefficients are quantitative relationships

A particle is a small unit of a substance, such as a molecule. In the particle model, two molecules of hydrogen react with one molecule of oxygen to form two molecules of water. This is the particle relationship represented by the coefficients in the balanced equation.

The same coefficient ratio applies when comparing amounts in moles. A mole is a counting unit used for a very large number of particles. It lets chemists compare substance amounts without counting individual particles.

For the water equation, the mole relationship is two moles of hydrogen to one mole of oxygen to two moles of water. A coefficient of one is understood even when it is not written. The amounts can be scaled up or down, but the ratio stays the same.

The equation does not mean that two grams of hydrogen react with one gram of oxygen. Coefficients compare particle counts or mole amounts, not masses. To find masses, first use the coefficients to relate amounts in moles. Then use molar mass to convert each amount to mass.

2\ \mathrm{mol\ H_2}:1\ \mathrm{mol\ O_2}:2\ \mathrm{mol\ H_2O}

Coefficients give ratios of particles and ratios of moles.
Coefficients are not direct mass ratios.
Molar mass connects an amount in moles to a mass.

3. Use a coefficient ratio to relate amounts

A mole ratio is a ratio between the mole amounts of substances in a balanced equation. Choose the two substances named in the question. Use their coefficients in the same order as the requested conversion. This links the known amount to the amount you need.

For example, if two moles of one substance react with one mole of another, the second amount is half the first. If one mole of a substance produces two moles of another, the product amount is twice the reacting amount. The coefficients set the relationship; the given amount sets its scale.

Arrange a conversion ratio so the starting unit cancels and the wanted unit remains. To convert an amount of substance A to an amount of substance B, multiply by the coefficient of B over the coefficient of A. The symbols in this relationship stand for the substances’ coefficients, not their chemical formulas.

If a question gives mass instead of moles, first divide the mass by molar mass to find the amount in moles. Apply the coefficient ratio next. If the question asks for a mass, multiply the final amount in moles by that substance’s molar mass.

n_B=n_A\times\frac{c_B}{c_A}

Select coefficients for the two substances involved.
Arrange the ratio so the starting unit cancels.
Convert between grams and moles with molar mass when needed.

4. From mole relationships to mass relationships

Molar mass is the mass of one mole of a substance. It is usually measured in grams per mole. For a compound, calculate molar mass using its formula and the atomic masses from the periodic table.

To find mass from an amount in moles, multiply by molar mass. To find an amount in moles from a mass, divide by molar mass. In these calculations, units show whether the operation leaves grams or moles.

For the amounts represented by a balanced equation, the total mass of reactants equals the total mass of products. This is consistent with atoms being rearranged rather than created or destroyed. Small differences may appear when calculated values are rounded.

A state symbol identifies a substance as a solid, liquid, gas, or aqueous solution. Aqueous means dissolved in water. State symbols add information about the reaction, but do not change the coefficient ratio.

m=nM

Use molar mass to connect moles and grams.
A balanced equation represents equal total mass on both sides, allowing for rounding.
State symbols do not alter coefficient ratios.

Worked example

Find amounts and compare masses

Use the balanced equation for forming water. If 3.00 mol of hydrogen reacts completely, find the amount of oxygen used and the amount of water formed. Then find the masses of all three substances. Use molar masses of 2.02 g/mol for hydrogen, 32.00 g/mol for oxygen, and 18.02 g/mol for water.

Read the balanced relationship
The coefficients show that two moles of hydrogen react with one mole of oxygen and form two moles of water. Match each coefficient to its substance before setting up a ratio.
$\mathrm{2H_2(g) + O_2(g) \rightarrow 2H_2O(l)}$
Find the oxygen amount
Use the oxygen-to-hydrogen coefficient ratio. The hydrogen units cancel, leaving moles of oxygen.
$3.00\ \mathrm{mol\ H_2}\times\frac{1\ \mathrm{mol\ O_2}}{2\ \mathrm{mol\ H_2}}=1.50\ \mathrm{mol\ O_2}$
Find the water amount
Hydrogen and water have equal coefficients. Their mole amounts are therefore equal for this reaction ratio.
$3.00\ \mathrm{mol\ H_2}\times\frac{2\ \mathrm{mol\ H_2O}}{2\ \mathrm{mol\ H_2}}=3.00\ \mathrm{mol\ H_2O}$
Convert amounts to masses
Multiply each amount by its molar mass. The mole units cancel, leaving grams. The starting amount has three significant figures, so report masses to three significant figures.
$m(\mathrm{H_2})=6.06\ \mathrm{g},\quad m(\mathrm{O_2})=48.0\ \mathrm{g},\quad m(\mathrm{H_2O})=54.1\ \mathrm{g}$

Answer: The reaction uses 1.50 mol of oxygen and forms 3.00 mol of water. The corresponding masses are 6.06 g of hydrogen, 48.0 g of oxygen, and 54.1 g of water.

Check: Before rounding, the reactant masses total 54.06 g, and the product mass is 54.06 g. The amounts follow the coefficient ratio, and the masses agree.

Common mistakes and how to avoid them

Treating coefficients as masses, such as saying a coefficient ratio of two to one means two grams to one gram.

Correction: Coefficients compare particle counts and mole amounts. Use molar masses to calculate masses.

Changing a subscript to balance an equation.

Correction: Change coefficients only. Subscripts are part of a chemical formula.

Using the right coefficients in the wrong order for a conversion.

Correction: Place the coefficient for the wanted substance in the numerator and the coefficient for the known substance in the denominator, so the known unit cancels.

Leaving units out of a calculation.

Correction: Write units with each amount and conversion factor. Confirm that the final unit answers the question.

Lesson summary

A balanced equation has the same number of each kind of atom on both sides.
Coefficients give ratios of particles and moles.
Use coefficient ratios to relate amounts in moles.
Use molar mass to convert between moles and grams.
Keep units through calculations and round answers appropriately.

Check your understanding

Question 1

In the balanced equation for making water, what amount of oxygen reacts with 4.0 mol of hydrogen?

1.0 mol O₂
2.0 mol O₂
4.0 mol O₂
8.0 mol O₂

Show answer and explanation

2.0 mol O₂

The hydrogen-to-oxygen coefficient ratio is two to one. Half of 4.0 mol of hydrogen is 2.0 mol of oxygen.

Question 2

What does a coefficient ratio directly compare?

Masses in grams only
Particle counts or amounts in moles
Temperatures of reactants
The sizes of atoms

Show answer and explanation

Particle counts or amounts in moles

Coefficients give relative particle counts and mole amounts. Molar mass is needed to calculate masses.

Key terms

Balanced equation: A chemical equation with the same number of each kind of atom on both sides.
Coefficient: A number in front of a chemical formula that tells how many units of that substance are represented.
Mole: A counting unit used for a very large number of particles.
Mole ratio: A ratio between the mole amounts of substances, based on coefficients in a balanced equation.
Molar mass: The mass of one mole of a substance, usually measured in grams per mole.

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D3.4 · Explain quantitative relationships in balanced equations