Gas Laws and Kinetic Theory

Gas Laws and Kinetic Theory

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

A gas exerts pressure because its molecules collide with the walls of the container; the rate and force of those collisions depend on how many molecules are present, how fast they move, and how hot the gas is. The four core relationships — Boyle’s law (PV = k), Charles’s law (V/T = k), the pressure law (P/T = k) and the ideal gas equation (PV = nRT) — connect pressure (P), volume (V), temperature (T) and amount (n). Kinetic theory derives these laws by treating a gas as identical, perfectly elastic point particles moving randomly, giving the average molecular kinetic energy as E_k = (3/2)kT and the root-mean-square speed as c_rms = √(3RT/M). For NECO SSCE: always convert temperature to Kelvin (K = °C + 273), state which variable is held constant before picking a law, and remember that diffusion of one gas into another obeys Graham’s law: rate ratio = √(M₂/M₁).

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

The Four Experimental Gas Laws

Boyle’s law states that at constant temperature, the volume of a fixed mass of gas is inversely proportional to its pressure: P₁V₁ = P₂V₂. A P–V graph at fixed T is a rectangular hyperbola.

Charles’s law holds at constant pressure: the volume of a fixed mass of gas is directly proportional to its absolute temperature, V₁/T₁ = V₂/T₂. A V–T plot is a straight line that extrapolates back to absolute zero (−273 °C or 0 K), the theoretical temperature at which an ideal gas would have zero volume.

The pressure law (Gay-Lussac’s) keeps volume fixed: pressure is directly proportional to absolute temperature, P₁/T₁ = P₂/T₂.

These three combine into the combined gas equation: P₁V₁/T₁ = P₂V₂/T₂, which is the workhorse for NECO “a gas is heated from …” calculation items.

The Ideal Gas Equation and Kinetic Theory

Avogadro’s law says equal volumes of gases at the same T and P contain equal numbers of molecules, leading to PV = nRT, where R = 8.314 J mol⁻¹ K⁻¹ and n is the number of moles. Kinetic theory models gas molecules as identical, negligibly small spheres in ceaseless random motion that collide elastically with each other and the walls. From Newton’s second law applied to wall collisions, P = (1/3)ρ<c²>, which rearranges to PV = (1/3)Nm<c²> and shows the average translational kinetic energy per molecule is E_k = (3/2)kT, where k = R/N_A is the Boltzmann constant (1.38 × 10⁻²³ J K⁻¹). Therefore a molecule’s root-mean-square speed is c_rms = √(3RT/M), with M as the molar mass in kg mol⁻¹ — heavier molecules move more slowly, which is why hydrogen diffuses faster than oxygen.

Mixtures and Diffusion

In a gas mixture, Dalton’s law of partial pressures gives P_total = P₁ + P₂ + P₃ + …, useful for collecting gases over water where water vapour contributes its own pressure. Graham’s law compares diffusion rates: r₁/r₂ = √(M₂/M₁).

Typical NECO Question Patterns

• A 2-mark item asking which variable must be kept constant before applying a named law.
• A 3–4 mark calculation using the combined gas equation with T in K.
• A short structured question linking c_rms to molar mass and asking which of two gases diffuses faster.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Edge Cases and Real-Gas Behaviour

The ideal-gas model assumes (i) molecules have negligible volume compared with the container, and (ii) no intermolecular forces except during instantaneous elastic collisions. Real gases approximate this well at low pressure and high temperature, but deviate near condensation. The van der Waals correction adds terms a (attraction) and b (excluded volume) to account for this — beyond NECO scope, but useful as background. A practical consequence: a gas compressed isothermally below a certain volume no longer obeys Boyle’s law, because intermolecular repulsion becomes significant.

Connections to Adjacent Topics

Kinetic theory bridges straight into heat and internal energy: the internal energy of a monatomic ideal gas is U = (3/2)nRT, so heating at constant volume raises temperature proportionally, while at constant pressure the extra flow of heat does work pΔV. The same c_rms = √(3RT/M) formula also explains why escape velocity from a planet depends on atmospheric molar mass — hydrogen escapes Earth easily, while heavier CO₂ does not.

Common Mistakes Examiners Exploit

1. Unit slip on temperature — using 27 °C instead of 300 K inflates Charles’s-law answers by a factor of ~1.09.
2. Holding the wrong variable constant — applying Boyle’s law when T actually changes (e.g. a gas being compressed rapidly heats up).
3. Confusing Graham’s law direction — the lighter gas diffuses faster, so the rate ratio has the heavier molar mass on top inside the square root.
4. Ignoring water-vapour pressure when a gas is collected over water in a pneumatic trough.

Worked Micro-Example

A 2.0 L flask contains 0.50 mol of N₂ at 27 °C. Find (a) the pressure and (b) the rms speed of N₂ molecules (M = 0.028 kg mol⁻¹). (a) T = 300 K, so P = nRT/V = (0.50)(8.314)(300)/(2.0 × 10⁻³) = 6.24 × 10⁵ Pa. (b) c_rms = √(3RT/M) = √[3 × 8.314 × 300 / 0.028] ≈ √(2.67 × 10⁵) ≈ 517 m s⁻¹.

Practice Prompts

1. A sealed gas at 2.0 × 10⁵ Pa and 300 K is heated at constant volume to 450 K. Find the new pressure and identify which law applies.
2. Compare the rms speeds of H₂ and O₂ at the same temperature, and predict which gas effuses faster through a porous plug, citing Graham’s law.

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