Particle Model and Pressure

In a gas, the inter-particle forces are very weak (compared with solids and liquids). Particles move:

• At high speed — typical ~500 m/s for air molecules at room T.
• In random straight lines between collisions.
• In all directions equally (no preferred direction).

When two particles collide, they bounce off each other (elastic collisions — no energy lost). When a particle hits the container wall, it bounces back, exerting a tiny force on the wall. The cumulative effect of trillions of these collisions per second is gas pressure (covered in 4.3.3.2).

Brownian motion is direct evidence of random gas-particle motion: pollen grains in water (or smoke particles in air) jiggle randomly because they are bombarded by invisible molecules from all sides at slightly unequal rates.

Temperature is a measure of the AVERAGE kinetic energy of the particles in a substance.

• Not all particles move at the same speed — there's a distribution of speeds.
• The AVERAGE kinetic energy rises in direct proportion to absolute temperature (in K).
• Doubling absolute T doubles the average particle kinetic energy.
• Doubling kinetic energy means speed rises by $\sqrt{2}$ (since $E_k \propto v^2$).

Worked example. Air molecules at 20 °C have a certain average kinetic energy. If you heat air to roughly twice the absolute temperature (293 K → 586 K = 313 °C), the average kinetic energy doubles and average speed rises by about 41 %.

Gas particles move randomly in all directions. When they hit a container wall, each collision exerts a small force on the wall (Newton's third law).

The total force per unit area from all these collisions is the gas pressure:

$P = \frac{F}{A}$

Unit: pascal (Pa). 1 Pa = 1 N/m². Atmospheric pressure ≈ 100 kPa (or 10⁵ Pa).

Pressure acts in ALL directions because particles move in all directions. A container's walls all experience the same gas pressure.

If you heat a gas held at constant volume:

1. Average particle kinetic energy increases.
2. Particles move FASTER.
3. They hit the walls more OFTEN (less time between collisions).
4. Each collision delivers MORE momentum (faster particles).
5. Both effects raise the force per unit area → pressure rises.

This is why:

• A car tyre's pressure rises after driving (tyre walls heat up).
• An aerosol can explodes if heated in a fire.
• Gas pressure rises proportionally to absolute T (Pressure Law, $P/T$ = constant at constant V; not explicitly assessed at GCSE).

Boyle's law: for a fixed amount of gas at constant temperature,

$p \, V = \text{constant}$

Equivalently $p_1 V_1 = p_2 V_2$.

• p in Pa.
• V in m³.

Worked example. A gas occupies 0.50 m³ at 100 kPa. It is compressed to 0.20 m³ at the same temperature. Find the new pressure.

$p_2 = p_1 V_1 / V_2 = (100{,}000 \times 0.50) / 0.20 = 250{,}000\,\text{Pa} = 250\,\text{kPa}$

Why? Compressing the gas into a smaller volume means particles hit the walls MORE often per second (less distance to travel). Same particle speeds (same T) but higher collision frequency → higher P.

When you do work on a gas (e.g. push down a bicycle pump piston), you transfer energy mechanically to the gas particles. They speed up, increasing internal energy. Temperature rises.

This is why:

• A bicycle pump becomes hot during inflation.
• A car engine's combustion chamber gets very hot during the compression stroke.
• Adiabatic compression in industrial gas processing produces large T rises.

If T also rises, then pV is NOT constant — Boyle's law no longer applies. You have to consider both effects simultaneously.

For Boyle's law to apply, the compression must be SLOW enough that the gas stays at the same temperature (isothermal process) — i.e. enough time for heat to leave and maintain equilibrium with surroundings.