Gas Behaviour from Kinetic Theory
Imagine a football stadium packed with tiny, invisible billiard balls flying in every direction at hundreds of metres per second, constantly bouncing off the walls and each other. That mental picture is essentially what kinetic theory says a gas is. Kinetic theory bridges the microscopic world of individual molecules and the macroscopic world of temperature, pressure, and volume that we measure in the laboratory. It turns out that the 'feelable' properties of a gas are simply averages over the chaotic motion of an enormous number of particles. Pressure, for example, is not a mysterious push that a gas exerts all at once — it is the collective result of billions of molecular collisions against a surface every second. Each collision transfers a tiny amount of momentum to the wall; summed over all particles, this produces the steady force per unit area we call pressure. Similarly, temperature is a measure of the average kinetic energy of the particles, not some independent quantity. This microscopic picture explains, from first principles, why heating a gas increases its pressure or volume, and why compressing a gas raises its temperature. The great success of kinetic theory is that it does not just offer a vague story — it produces exact, testable mathematical relationships. From a small set of simple assumptions (particles are point masses, collisions are perfectly elastic, the particles exert no forces on each other except during collisions), one can derive the ideal gas law and connect it to Newton's laws of motion. This article walks through that derivation and shows how each of the empirical gas laws emerges naturally from the microscopic picture.
- The Model: What We Assume
- One Molecule Hitting a Wall
- Summing Over All Molecules
- Connecting to Temperature
- The Gas Laws Fall Out Automatically
The Model: What We Assume
We model a gas as N identical molecules, each of mass m, moving randomly inside a cubic container of side length L (volume V = L³). Key assumptions: (1) molecules are tiny compared to the space between them; (2) all collisions — molecule–molecule and molecule–wall — are perfectly elastic, so kinetic energy is conserved; (3) there are no intermolecular forces between collisions; (4) the distribution of velocities is random and isotropic (no preferred direction). These assumptions define the ideal gas.
One Molecule Hitting a Wall
Consider a single molecule moving with velocity component cₓ in the x-direction toward the right wall. On collision, the x-component reverses: Δ(momentum) = m(cₓ) − m(−cₓ) = 2mcₓ. The molecule must travel to the opposite wall and back — a distance 2L — before hitting the same wall again, so the time between successive hits is Δt = 2L/cₓ. By Newton's second law, the average force from this one molecule on that wall is F = Δp/Δt = 2mcₓ/(2L/cₓ) = mcₓ²/L.
Summing Over All Molecules
Adding contributions from all N molecules and dividing by the wall area L², the pressure on that wall becomes P = (1/L³) × m × Σcₓ². Replacing Σcₓ² with N⟨cₓ²⟩ (where ⟨cₓ²⟩ is the mean square of the x-component), and using the isotropy condition ⟨cₓ²⟩ = ⟨cᵧ²⟩ = ⟨c_z²⟩ = ⟨c²⟩/3, we get PV = (1/3)Nm⟨c²⟩. This is the fundamental result of kinetic theory.
Connecting to Temperature
The ideal gas law (from experiment) states PV = NkT, where k is Boltzmann's constant and T is absolute temperature. Setting the two expressions equal: (1/3)Nm⟨c²⟩ = NkT, which gives (1/2)m⟨c²⟩ = (3/2)kT. The average translational kinetic energy of one molecule equals (3/2)kT. Temperature is literally the average kinetic energy of the molecules, scaled by Boltzmann's constant. Hotter gas → faster molecules → higher pressure. This is no longer just an analogy; it is a derived result.
The Gas Laws Fall Out Automatically
From PV = (1/3)Nm⟨c²⟩, all three classic gas laws follow immediately. Boyle's Law (P ∝ 1/V at constant T): if T is fixed, ⟨c²⟩ is fixed, so PV = constant. Charles's Law (V ∝ T at constant P): at constant P, V must grow linearly with T to keep PV = NkT. Gay-Lussac's Law (P ∝ T at constant V): if V is fixed, P must increase with T as more energetic collisions occur more forcefully. Each law is a special case of the same microscopic equation.
The Key Equation
- P — Pressure (Pa): Force per unit area exerted by gas on container walls
- V — Volume (m³): Volume of the gas container
- N — Number of molecules (dimensionless): Total count of gas molecules in the sample
- m — Molecular mass (kg): Mass of one gas molecule
- \langle c^2 \rangle — Mean square speed (m²s⁻²): Average of the square of all molecular speeds
- k_B — Boltzmann's constant (J K⁻¹): k_B = 1.38 × 10⁻²³ J K⁻¹; links temperature to energy
- T — Absolute temperature (K): Temperature measured from absolute zero
Worked Example
Problem. A sealed container holds 3.0 × 10²⁴ molecules of nitrogen gas (each with mass 4.65 × 10⁻²⁶ kg) at a temperature of 300 K. (a) Find the average kinetic energy per molecule. (b) Find the root-mean-square (rms) speed of the nitrogen molecules. (c) If the volume is 0.050 m³, find the pressure.
Answer. Average KE ≈ 6.2 × 10⁻²¹ J; rms speed ≈ 517 m/s; Pressure ≈ 2.5 × 10⁵ Pa
Origins · 1 minJames Clerk Maxwell and how this was discovered
Maxwell derived the statistical distribution of molecular speeds in a gas (the Maxwell–Boltzmann distribution) in 1860, showing that molecules do not all move at the same speed but follow a characteristic bell-shaped curve that depends on temperature. He also formalised the kinetic theory of gases mathematically, deriving expressions for viscosity, thermal conductivity, and diffusion — all from molecular motion alone. Maxwell published his first scientific paper at age 14, on the geometry of oval curves, presented to the Royal Society of Edinburgh — which thought it too sophisticated to have been written by a teenager.
Related Concepts
- The Ideal Gas Law — PV = nRT is the macroscopic law that kinetic theory derives from first principles; understanding both levels deepens insight into each.
- Maxwell–Boltzmann Speed Distribution — Kinetic theory shows that ⟨c²⟩ exists; the Maxwell–Boltzmann distribution describes the full spread of molecular speeds around that average.
- Internal Energy and Heat — The internal energy of an ideal gas is purely kinetic energy of its molecules — a direct consequence of kinetic theory linking U = (3/2)NkT.
- Momentum and Impulse — The derivation of pressure relies on calculating the impulse (change in momentum) delivered to the container wall by each molecular collision.