The Ideal Gas Law — The Complete Physics Guide
The ideal gas law, PV = nRT, is one of the great unifying equations of classical physics and chemistry. It connects the macroscopic properties of a gas — pressure, volume, temperature, and amount — in a single elegant relationship derived from the microscopic behaviour of billions of molecules. From understanding how a bicycle pump works to designing turbines, refrigerators, and internal combustion engines, PV = nRT underpins an enormous range of applications.
The Ideal Gas Law Explained
The four variables in PV = nRT each describe a measurable property of a gas sample. P is absolute pressure in pascals (Pa), not gauge pressure. V is volume in cubic metres (m³). n is the amount of gas in moles. T is absolute temperature in kelvin (K) — never Celsius. R is the universal gas constant: R = 8.314 J/(mol·K).
The "ideal gas" approximation assumes molecules have negligible volume compared to their container, and exert no forces on each other except during brief elastic collisions. Real gases approach this behaviour at low pressures and high temperatures. At high pressures or low temperatures, intermolecular forces and molecular volume become significant — requiring the Van der Waals equation or other corrections.
The ideal gas law synthesises three historically separate empirical laws: Boyle's Law (P ∝ 1/V at constant T), Charles's Law (V ∝ T at constant P), and Gay-Lussac's Law (P ∝ T at constant V). Each was discovered independently and can be derived from PV = nRT by holding two of the four variables constant.
The Three Special Cases
Boyle's Law (isothermal — constant T): PV = constant, or P₁V₁ = P₂V₂. Doubling pressure halves volume. This describes what happens when you compress a gas at constant temperature — the molecules are forced into a smaller space and collide with the walls more frequently, increasing pressure.
Charles's Law (isobaric — constant P): V/T = constant, or V₁/T₁ = V₂/T₂. Doubling absolute temperature doubles volume. This is why hot air rises (lower density), why a balloon shrinks in cold weather, and why gases must be measured at defined temperatures for consistency.
Gay-Lussac's Law (isochoric — constant V): P/T = constant, or P₁/T₁ = P₂/T₂. This explains why an aerosol can can explode if heated — the volume is fixed, so increasing temperature directly increases pressure.
Worked Example 1 — Pressure Change
Problem: A gas at 2 atm (202,650 Pa) and 300 K occupies 0.5 m³. It is heated to 400 K at constant pressure. Find the new volume.
V₂ = V₁ × T₂/T₁ = 0.5 × 400/300 = 0.667 m³
Worked Example 2 — Combined Gas Law
Problem: A gas at 100 kPa, 27°C, 2 litres is compressed to 1 litre and cooled to -73°C. Find the new pressure.
T₁ = 300 K, T₂ = 200 K, V₁ = 2 L, V₂ = 1 L, P₁ = 100 kPa
P₂ = P₁ × (V₁/V₂) × (T₂/T₁) = 100 × 2 × (200/300) = 133 kPa
Microscopic Interpretation — Kinetic Theory
The ideal gas law can be derived from the kinetic theory of gases — the model where gas consists of many small, rapidly-moving molecules. Pressure arises from molecules colliding with the walls of the container. Each collision transfers momentum to the wall; the rate of collisions and the momentum per collision together determine the pressure.
The kinetic theory derivation yields PV = NkT, where N is the number of molecules and k is Boltzmann's constant (k = 1.381 × 10⁻²³ J/K). Since N = nN_A (where N_A = 6.022 × 10²³ is Avogadro's number), and R = kN_A, this gives PV = nRT — the ideal gas law.
This connection is profound: the macroscopic gas constant R emerges from the microscopic Boltzmann constant k scaled by the number of molecules per mole. Temperature, in this framework, is a measure of the average kinetic energy of molecules: ½m⟨v²⟩ = (3/2)kT. Absolute zero (T = 0 K) corresponds to molecules at rest — the minimum possible kinetic energy.
Real-World Applications
Internal combustion engines: The compression stroke of a petrol engine compresses an air-fuel mixture — Boyle's Law predicts the pressure increase. The power stroke involves combustion at approximately constant volume — Gay-Lussac's Law predicts the resulting pressure spike that drives the piston. Engine design is fundamentally about optimising these gas law processes.
Scuba diving: At 10 m depth, water pressure is about 2 atm total. By Boyle's Law, air in a scuba tank at 200 atm expands 100-fold when breathed at 2 atm — a 10 litre tank holds about 2,000 litres of air at surface pressure. Divers must never hold their breath while ascending — as pressure decreases, lung volume would expand, risking barotrauma.
Weather balloons: A weather balloon is partially inflated at launch and expands as it ascends into lower-pressure regions. By Boyle's Law, the volume increases as pressure decreases. Balloons are designed to burst at around 30 km altitude, where the lower pressure has expanded the gas to the point of rupturing the latex envelope.