Theme B · Particulate Nature of Matter · SL + HL · 6 hours

B.3 Gas Laws

How are the macroscopic properties of a gas related to molecular behaviour?

An ideal gas is a beautifully tractable fiction: point particles that undergo perfectly elastic collisions and have no intermolecular forces except during collision. Real gases approximate this well at low pressure and high temperature. From this model, three empirical laws — Boyle's (constant T), Charles's (constant P), and Gay-Lussac's (constant V) — collapse into one equation: PV = nRT.

The kinetic theory derivation of pressure is a triumph of atomic-scale thinking. Each molecule exerts a tiny impulsive force on the container wall; the average of billions of these impacts per second gives the measurable macroscopic pressure. The result connects pressure directly to average molecular speed: P = (1/3)ρv̄². This is not a postulate — it is derived from Newton's laws applied at the molecular scale.

A gas in a sealed container has pressure 1.5 × 10⁵ Pa at 300 K. The container is heated until the pressure is 2.0 × 10⁵ Pa. What is the new temperature? What assumptions are you making, and when would they break down?

Key equations

Pressure: P = F/A

Amount of substance: n = N / N_A

Ideal gas law: PV = Nk_BT = nRT

Pressure from kinetic theory: P = (1/3)ρv̄²

Internal energy (monatomic ideal gas): U = (3/2)Nk_BT = (3/2)nRT

What students must understand

Ideal gas: point masses, elastic collisions, no intermolecular forces except during collision
The three empirical gas laws: Boyle's (PV = const at constant T), Charles's (V/T = const at constant P), Gay-Lussac's (P/T = const at constant V)
The ideal gas law PV = nRT (or PV = Nk_BT) as a unification of the three laws
Pressure from kinetic theory: molecular collisions with walls produce macroscopic pressure P = (1/3)ρv̄²
Internal energy of a monatomic ideal gas: U = (3/2)nRT
Differences between ideal and real gases; conditions under which the ideal approximation holds
P–V diagrams to represent changes of state