Balancing equations using moles

A balanced equation shows the ratio in which substances react and form products. To find that ratio from masses, convert each mass to an amount in moles, then compare the mole amounts.

A mole is a fixed number of particles, so mole amounts are directly proportional to numbers of particles. The simplest whole-number ratio of the mole amounts therefore gives the balancing coefficients in the equation.

Converting masses to moles

For each substance, use its mass and molar mass:

The molar mass has the same numerical value as the substance’s relative formula mass, but is expressed in $$\mathrm{g,mol^{-1}}$$. Calculate the relative formula mass by adding the relative atomic masses of the atoms in the formula.

Finding coefficients from mass data

Suppose $$4\ \mathrm{g}$$ of hydrogen reacts with $$32\ \mathrm{g}$$ of oxygen to form $$36\ \mathrm{g}$$ of water. The unbalanced equation is $$\ce{H2 + O2 -> H2O}$$.

The molar masses are $$2\ \mathrm{g,mol^{-1}}$$ for $$\ce{H2}$$, $$32\ \mathrm{g,mol^{-1}}$$ for $$\ce{O2}$$ and $$18\ \mathrm{g,mol^{-1}}$$ for $$\ce{H2O}$$. Converting each mass to moles gives:

• Hydrogen: $$4 \div 2 = 2\ \mathrm{mol}$$
• Oxygen: $$32 \div 32 = 1\ \mathrm{mol}$$
• Water: $$36 \div 18 = 2\ \mathrm{mol}$$

The mole ratio $$\ce{H2} : \ce{O2} : \ce{H2O}$$ is therefore $$2 : 1 : 2$$. This is already the smallest whole-number ratio, so the coefficients are $$2$$, $$1$$ and $$2$$:

The equation shows that two moles of hydrogen react with one mole of oxygen to form two moles of water. A coefficient of $$1$$ is not normally written.

When the mole amounts are not already in their smallest whole-number ratio, divide them all by the smallest amount. If the resulting ratio contains fractions, multiply every part by the same whole number to obtain whole numbers. Place these numbers before the formulae; do not change the subscripts within the formulae.

Rearranging equations

Changing the subject of an equation means isolating a chosen quantity on one side, so that it is expressed in terms of the other quantities. For example, if an equation gives $$y$$ in terms of $$x$$, it can be rearranged to give $$x$$ in terms of $$y$$.

The two sides of an equation are equal. To preserve that equality, apply the same valid operation to both sides. The rearranged equation expresses the same relationship as the original.

Undoing operations

To isolate a quantity, undo the operations acting on it, generally in reverse order. Subtraction undoes addition, and division undoes multiplication.

For example, to make $$x$$ the subject of the following equation:

First subtract $$2$$ from both sides. This undoes the final addition:

Then divide both sides by $$3$$ to isolate $$x$$:

Rearranging when the subject is in a denominator

Suppose concentration $$c$$ is given by mass $$m$$ divided by volume $$V$$:

To make $$V$$ the subject, first multiply both sides by $$V$$. This removes it from the denominator:

Then divide both sides by $$c$$:

Division by zero is not valid, so this final form requires $$c \ne 0$$. The original equation also requires $$V \ne 0$$.