The Gas Laws from Kinetic Theory
Five experimental laws, all following from one equation.
Lesson at a glanceShow the key points
- Boyle: pV constant at fixed T. Charles: V ∝ T at fixed p; p ∝ T at fixed V.
- Avogadro: equal volumes at equal p and T contain equal numbers of molecules (equal average KE).
- Graham: diffusion rate ∝ v_rms ∝ 1/√ρ at equal pressure.
- Dalton: total pressure = sum of the pressures each gas would exert alone.
Boyle, Charles, Avogadro, Graham and Dalton found their gas laws by experiment. Kinetic theory derives every one of them from pV = ⅓Nmv²_rms and v²_rms ∝ T.
Boyle’s law
At a fixed temperature, v²_rms is fixed; for a given mass of gas, N and m are fixed. So pV = ⅓Nmv²_rms is constant:
Charles’s laws
At fixed pressure, V ∝ v²_rms ∝ T. At fixed volume, p ∝ v²_rms ∝ T (which is really the definition of the absolute scale).
A balloon holds 2.0 L of air at 17°C. At constant pressure, what is its volume at 47°C?
Avogadro’s law
Take equal volumes of two gases at the same pressure and temperature. From pV = ⅓Nmv² for each,
Same temperature means equal average kinetic energies, ½m₁v₁² = ½m₂v₂². Dividing:
Graham’s law of diffusion
When two gases at the same pressure and temperature diffuse into each other, it is reasonable that each diffuses at a rate proportional to its rms speed. Since v_rms = √(3p/ρ), at equal pressures:
How much faster does hydrogen (2 g/mol) diffuse than oxygen (32 g/mol) at the same pressure and temperature?
Dalton’s law of partial pressures
Pressure comes from molecules hitting the walls, and the total force is just the sum of the forces from each gas’s molecules. Each part, F₁/A, F₂/A…, is the pressure that gas would exert if it filled the container alone:
A 3.0 L vessel at 300 K contains 0.10 mol of nitrogen and 0.05 mol of oxygen. Find the total pressure.
- 1Boyle: pV constant at fixed T. Charles: V ∝ T at fixed p; p ∝ T at fixed V.
- 2Avogadro: equal volumes at the same p, T hold equal numbers of molecules.
- 3Graham: diffusion rate ∝ 1/√ρ.
- 4Dalton: total pressure = sum of partial pressures.
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