Big picture
Air can support a tyre because its molecules continually strike the walls. Kinetic theory connects these microscopic collisions with measurable pressure and temperature. The gas may appear stationary as a whole while individual molecules move rapidly in random directions.
The theory uses an idealised model rather than tracking every molecule. Its success depends on knowing its assumptions and limits. For example, increasing temperature raises average molecular kinetic energy, but it does not mean every molecule acquires the same speed. Keep microscopic quantities and bulk variables clearly separated throughout calculations.
Learning goals
State ideal-gas assumptions and explain pressure microscopically.
Apply the ideal-gas equation using moles or molecules.
Relate temperature to translational kinetic energy and rms speed.
Use equipartition to determine idealised heat capacities.
Explain mean free path and model limitations.
Core ideas
1. Ideal-gas model
An ideal gas contains a very large number of molecules treated as point particles whose total volume is negligible compared with container volume. Molecules move randomly, obey Newtonian mechanics and interact mainly through brief elastic collisions. Intermolecular forces are neglected between collisions, and the walls are taken as rigid during the pressure derivation.
These assumptions work best at low density and sufficiently high temperature, away from condensation. Real gases depart from the model when molecular size and attractions become important. Dilute air at ordinary conditions can often be approximated as ideal, whereas a dense vapour near liquefaction requires more care. Elastic collisions conserve total kinetic energy but may redistribute it between individual molecules.
2. Gas equation and constants
The ideal-gas equation is PV = nRT = NkBT, where P is absolute pressure in pascals, V volume in m³, n amount in moles, N number of molecules and T absolute temperature in kelvin. R ≈ 8.31 J mol⁻¹ K⁻¹ and kB ≈ 1.38 × 10⁻²³ J K⁻¹, with R = NₐkB.
For 0.50 mol at 300 K in 0.012 m³, P = nRT/V = 0.50 × 8.31 × 300 ÷ 0.012 = 103,875 Pa, about 1.04 × 10⁵ Pa. Use kelvin, not Celsius, and convert litres by 1 L = 10⁻³ m³. For fixed amount, constant-temperature compression gives PV constant; at constant volume P/T is constant. These relations require specifying what remains fixed.
3. Deriving pressure
Consider molecules in a cube of side L. A molecule of mass m with x-component vₓ reverses that component during an elastic wall collision, transferring momentum 2mvₓ. The time between successive collisions with the same wall is 2L/|vₓ|. Its average force contribution is therefore mvₓ²/L.
Summing over molecules and dividing by wall area gives P = (m/V)Σvₓ². Random isotropic motion implies average vₓ² = average vᵧ² = average v𝓏² = average v²/3. Hence P = (1/3)(Nm/V)〈v²〉 = (1/3)ρvᵣₘₛ². The mean vector velocity is zero for a gas at rest, but the mean square speed is positive. Replacing the latter with the former would incorrectly predict zero pressure.
For density 1.2 kg m⁻³ and rms speed 500 m s⁻¹, P = (1/3) × 1.2 × 500² = 100,000 Pa. The dimensions kg m⁻³ × m² s⁻² reduce to pressure units, providing a useful check.
4. Temperature and molecular energy
Combining kinetic pressure with PV = NkBT gives ½m〈v²〉 = (3/2)kBT. Absolute temperature is proportional to mean translational kinetic energy per molecule. At the same temperature, different ideal gases have the same mean translational kinetic energy, not the same rms speed.
For one mole, translational kinetic energy is (3/2)RT. At 400 K it is 1.5 × 8.31 × 400 = 4,986 J mol⁻¹. For 2 mol of a monatomic ideal gas at 300 K, internal energy U = (3/2)nRT = 1.5 × 2 × 8.31 × 300 = 7,479 J. This monatomic expression does not automatically include rotational or vibrational contributions in a molecular gas.
Temperature is a collective statistical quantity. A single molecule can have an instantaneous kinetic energy above or below the mean, so assigning it an exact thermodynamic temperature from one observed speed is not generally meaningful.
5. RMS speed and speed comparisons
The root-mean-square speed is vᵣₘₛ = √〈v²〉 = √(3kBT/m) = √(3RT/M), where M is molar mass in kg mol⁻¹. It differs from mean speed and most probable speed in the Maxwell distribution. At a given temperature, vᵣₘₛ is the largest of these three characteristic speeds.
For M = 0.028 kg mol⁻¹ at 300 K, vᵣₘₛ = √(3 × 8.31 × 300/0.028) = about 516.8 m s⁻¹. At fixed gas composition, raising T from 300 K to 1,200 K multiplies rms speed by √(1,200/300) = 2, not four. At equal temperature, comparing helium with molar mass 4 g mol⁻¹ and oxygen with 32 g mol⁻¹ gives vHe/vO₂ = √(32/4) = √8 ≈ 2.83.
6. Degrees of freedom and equipartition
A degree of freedom is an independent mode contributing to the energy description. The equipartition theorem assigns an average energy ½kBT to each active quadratic energy term. Translational motion has three terms, so every ideal-gas molecule has mean translational energy 3kBT/2.
A monatomic gas has three active translational degrees of freedom in the elementary model. A rigid diatomic molecule at ordinary temperatures commonly has three translational and two rotational degrees, giving f = 5. Vibration can become active at higher temperatures; each vibrational mode contributes kinetic and potential quadratic terms. A nonlinear rigid polyatomic molecule has three rotational degrees rather than two. Quantum restrictions mean some modes are effectively frozen at low temperatures, so f is temperature-dependent in real gases.
7. Internal energy and heat capacities
For f active quadratic degrees per molecule, U = (f/2)nRT and molar Cᵥ = (f/2)R. For an ideal gas, Cₚ − Cᵥ = R, so Cₚ = ((f + 2)/2)R and γ = Cₚ/Cᵥ = (f + 2)/f. The difference arises because heating at constant pressure also supplies expansion work.
Idealised gas | f | Cᵥ | Cₚ | γ |
|---|---|---|---|---|
Monatomic | 3 | 3R/2 | 5R/2 | 5/3 |
Rigid diatomic | 5 | 5R/2 | 7R/2 | 7/5 |
Nonlinear rigid polyatomic | 6 | 3R | 4R | 4/3 |
For a rigid diatomic gas, Cᵥ = 2.5 × 8.31 = 20.775 J mol⁻¹ K⁻¹. Heating 0.40 mol by 25 K at constant volume requires Q = nCᵥΔT = 0.40 × 20.775 × 25 = 207.75 J. At constant pressure, use Cₚ instead; do not substitute total internal-energy change for the entire heat supplied.
8. Mean free path
The mean free path, λ, is the average distance travelled between collisions. In a simple hard-sphere gas, λ = 1/(√2 πd²nN), where d is molecular collision diameter and nN = N/V is number density in m⁻³. At fixed temperature, using nN = P/(kBT), λ = kBT/(√2 πd²P).
For d = 3.0 × 10⁻¹⁰ m and nN = 2.0 × 10²⁵ m⁻³, λ = 1/[√2 × π × (3.0 × 10⁻¹⁰)² × 2.0 × 10²⁵] = about 1.25 × 10⁻⁷ m. Doubling number density halves λ. The molecule travels rapidly yet changes direction frequently, explaining why molecular transport across a room is much slower than dividing room length by rms speed.
9. Experiments and interpretation
A gas in a sealed rigid vessel can be warmed while pressure is measured after thermal equilibrium. A plot of absolute pressure against kelvin temperature is approximately linear for a dilute gas, consistent with P/T constant. Keep the amount and volume fixed; leakage or flexible walls would invalidate the intended test.
In a conceptual piston experiment, compressing gas at constant temperature increases collision frequency per unit area and raises pressure without increasing mean translational energy. If compression is rapid and not isothermal, temperature can rise, so the same explanation needs modification. Always distinguish a model prediction from an experimentally controlled condition.
A reliable approach to questions
List whether amount is given in molecules, moles or mass.
Convert temperature, volume and molar mass to SI units.
Choose the gas equation, pressure relation, energy formula or mean-free-path relation.
State active degrees of freedom before using heat capacities.
Check square-root scaling and distinguish mean velocity from rms speed.
Connections that matter
Mechanics supplies momentum conservation and elastic-collision reasoning.
Thermodynamics links microscopic energy with heat, work and internal energy.
Waves connects gas compressibility and γ with sound speed.
Common mistakes
Using Celsius in PV = nRT breaks absolute-temperature proportionality.
Using g mol⁻¹ in the SI speed formula introduces a factor-of-1,000 mass error.
Equating zero mean velocity with zero molecular motion predicts false zero pressure.
Giving every gas f = 3 ignores active rotational and vibrational contributions.
Revision checklist
I can derive the pressure relation from collisions.
I can calculate energy and rms speed with correct units.
I can compare gas speeds using ratios.
I can determine idealised heat capacities from degrees of freedom.
I can explain mean free path and the assumptions behind it.