Boyle's Law Calculations

Boyle’s law states that for a fixed mass of gas at constant temperature:

$p_1 V_1 = p_2 V_2$

Pressure is inversely proportional to volume: $p \propto \frac{1}{V}$.

Graph shapes

• $p$ vs $V$: a curve (rectangular hyperbola).
• $p$ vs $\frac{1}{V}$: a straight line through the origin.

Worked example

A gas has a volume of 500 cm$^3$ at 100 kPa. The gas is compressed to 200 cm$^3$ at constant temperature. Find the new pressure.

$p_1 V_1 = p_2 V_2$ $100 \times 500 = p_2 \times 200$ $p_2 = \frac{100 \times 500}{200} = 250 \text{ kPa}$

Why Boyle’s law works (kinetic theory)

Decreasing the volume means particles hit the walls more often (they have less distance to travel). The force per unit area increases, so pressure increases. Temperature is constant, so the speed of particles does not change.

Conditions

• Fixed mass of gas (no gas added or removed).
• Constant temperature.
• Units of $p$ and $V$ can be anything, as long as they are the same on both sides.

Common errors and how to correct them

• Using this law when temperature changes. If temperature changes, use the combined gas law instead.
• Getting the rearrangement wrong. If volume decreases, pressure must increase (and vice versa). Check your answer makes physical sense.

How to apply this in an exam

Write $p_1 V_1 = p_2 V_2$. Substitute the three known values. Solve for the unknown. Check: if $V$ decreased, $p$ should have increased.