Molecules and the ideal gas

All matter is made of molecules. In a solid or a liquid they touch one another; in a gas at ordinary conditions they are about 10 times farther apart than their own size (about 2 Å across). That is why a gas is easy to squeeze, and why its molecules can be treated as tiny points.

Three old gas laws, each with one quantity held fixed:

• Boyle's law (T fixed): PV = constant, so P₁V₁ = P₂V₂. Plot P against V and you get a falling curve; plot P against 1/V and you get a straight line through the origin.
• Charles' law (P fixed): V ∝ T, with T in kelvin.
• Gay-Lussac's law (V fixed): P ∝ T.

Put together, PV/T stays constant, and for μ moles this is the ideal gas equation: PV = μRT = Nk_BT.

• R = 8.31 J mol⁻¹ K⁻¹ (about 2 cal mol⁻¹ K⁻¹). N is the number of molecules, and k_B = R/N_A = 1.38 × 10⁻²³ J K⁻¹ (Boltzmann's constant) is R for a single molecule.
• At STP (1 atm = 76 cm of mercury = 1.013 × 10⁵ Pa, and 273 K), one mole of any gas fills 22.4 L.
• Most mistakes here are units: T in kelvin, P in Pa, V in m³ (1 L = 10⁻³ m³).

Example: a 2 L bottle of air at 27 °C (300 K) and 1 atm holds μ = PV/RT = 1.013 × 10⁵ × 0.002/(8.31 × 300) ≈ 0.081 mol, which is about 0.081 × 6.02 × 10²³ ≈ 4.9 × 10²² molecules.

An ideal gas obeys PV = μRT exactly. It would need molecules of no size and no forces between them, so its energy would be all kinetic. Real gases come close at low pressure and high temperature, where the molecules are far apart and fast.

Boyle's law at work: an air bubble rises from the bottom of a lake and its volume triples by the surface (temperature unchanged). Then the pressure at the bottom was 3 atm: the water above adds 2 atm, and since 1 atm equals about 10.3 m of water, the lake is about 20.7 m deep.

Dalton's law: in a mixture of gases that do not react, the total pressure is the sum of the partial pressures, each being the pressure that gas would exert alone in the whole vessel. A partial pressure is proportional to the number of moles: 2 mol of nitrogen and 1 mol of oxygen at a total of 3 atm give 2 atm and 1 atm.