The Gas Laws

What causes gas pressure. Gas molecules move quickly and randomly. They collide with the walls of the container, and each collision changes the molecule's momentum, exerting a small force on the wall. The total force from all these collisions, divided by the area of the wall, is the pressure.

Two things therefore raise the pressure: more frequent collisions, and harder collisions.

Boyle's law — constant temperature:

p₁ V₁ = p₂ V₂

At constant temperature, halving the volume doubles the pressure. Why: the molecules are moving at the same average speed, but in a smaller space they reach the walls more often, so collisions become more frequent. The speed of each collision is unchanged; only the rate changes.

The pressure law — constant volume:

p₁ / T₁ = p₂ / T₂

with T in kelvin. At constant volume, doubling the absolute temperature doubles the pressure. Why: raising the temperature raises the average kinetic energy, so the molecules move faster. They hit the walls both more often and harder, and both effects raise the pressure.

The kelvin rule is absolute. Boyle's law needs no temperature at all, but the pressure law will give nonsense unless temperatures are converted to kelvin first. Convert before you do anything else.

Working a gas problem:

1. Write down what you have and what you want.
2. Convert every temperature to kelvin.
3. Check which quantity is being held constant, which tells you which law to use.
4. Substitute and solve.

Example: a gas occupies 0.060 m³ at 2.0 × 10⁵ Pa. Compressed at constant temperature to 0.020 m³, the new pressure is p₂ = p₁V₁ ÷ V₂ = (2.0 × 10⁵ × 0.060) ÷ 0.020 = 6.0 × 10⁵ Pa. The volume fell to a third, so the pressure tripled.

A real-world case worth knowing: an aerosol can warns against heating because the volume is fixed. Raising the temperature raises the pressure in proportion to the absolute temperature, and the can eventually bursts.