Three experiments, three proportionalities
Long before anyone believed in molecules, careful experimenters found simple rules. In 1662 Robert Boyle trapped air and squeezed it: at fixed temperature, halving the volume doubled the pressure. Pressure and volume are inversely proportional — that is Boyle's law.
Boyle's law: at fixed temperature and amount, pressure times volume is constant. Squeeze a gas and its pressure rises in exact inverse proportion.
A century later Jacques Charles found that at fixed pressure a gas's volume grows in direct proportion to its temperature (Charles's law), while Gay-Lussac found that at fixed volume its pressure grows in proportion to temperature (Gay-Lussac's law). Heat a sealed gas and it pushes harder; let it expand freely and it swells.
Charles's law (left, constant P) and Gay-Lussac's law (right, constant V). In both, T must be the absolute temperature — read the tip below.
Counting molecules: the mole
Amedeo Avogadro added the last piece: at the same temperature and pressure, equal volumes of any gas contain equal numbers of molecules, so volume is proportional to the amount n. But molecules are so numerous that counting them in ordinary units is hopeless — so chemists bundle them into a mole, a fixed pack of Avogadro's number of particles.
Avogadro's number: one mole contains 6.022 × 10²³ particles. A mole of any ideal gas fills 22.4 litres at 0 °C and 1 atm.
One equation to rule them all
Now stitch the four proportionalities together — V \propto 1/P, V \propto T, V \propto n — and one equation of state falls out, the ideal gas law. It is the single most useful equation in all of thermal physics.
The ideal gas law. R is the universal gas constant, the same for every gas — its universality is itself a clue that gases share a common microscopic story.
Chemists count in moles, but a physicist thinking about individual molecules prefers to count molecules directly, N = nN_A. Rewriting gives an equivalent form in which the Boltzmann constant k_B plays the role of the gas constant per molecule.
The molecular form of the gas law. k_B is the bridge between the macroscopic constant R and the microscopic world — it will reappear the moment we link temperature to molecular energy.
Worked example: moles in a cylinder
Problem. A 20.0-litre steel cylinder holds oxygen at a temperature of 300 K and a pressure of 1.5 × 10⁷ Pa (about 150 atmospheres). How many moles of oxygen are inside, and what is their mass? (Molar mass of O₂ ≈ 32 g/mol.)
- Convert to SI units so R = 8.314 works cleanly: V = 20.0 L = 0.0200 m³, T = 300 K, P = 1.5 × 10⁷ Pa. (Temperature is already absolute — good.)
- Rearrange the ideal gas law for the unknown amount: n = PV / (RT).
- Substitute: n = (1.5 × 10⁷ Pa)(0.0200 m³) / [(8.314)(300)] = 3.0 × 10⁵ / 2494 ≈ 120 mol.
- Find the mass: m = n × (molar mass) = 120 mol × 32 g/mol ≈ 3840 g ≈ 3.8 kg of oxygen.
- Sanity check: at ordinary pressure 120 mol would fill 120 × 22.4 L ≈ 2700 L, yet it fits in 20 L. That factor of ~135 is exactly the ~150-fold compression the high pressure implies — the numbers hang together.