IB Physics HLThe Behaviour of GasesPaper 1 & 2KE & Temperature~9 min read
Average Kinetic Energy of a Molecule
Here’s the payoff for the whole chapter. Our derivation showed that pressure depends on molecular speed squared — which is just kinetic energy in disguise. Now we make the link explicit: temperature is a measure of the average kinetic energy of the molecules. That single idea ties motion, temperature and a gas’s internal energy neatly together.
📘 What you need to know
The average kinetic energy of one molecule: Ek = ½mv̄2 = ³⁄₂kBT
Temperature is a direct measure of average molecular kinetic energy — hotter means faster
An ideal gas has no intermolecular forces, so no potential energy: its internal energy is all kinetic
Internal energy of an ideal (monatomic) gas: U = ³⁄₂NkBT = ³⁄₂nRT
This applies to monatomic gases (He, Ne, Ar), whose molecules have only translational motion
Diatomic molecules can also rotate, so they store energy in more ways
Temperature measures kinetic energy
Put two of our results side by side. The kinetic theory equation can be written PV = ⅓Nmv̄2, and the ideal gas equation says PV = NkBT. Setting them equal, the N‘s cancel and a little rearranging gives the average kinetic energy of a single molecule:
Average KE of one moleculeEk = ½mv̄2 = ³⁄₂kBT
Read the right-hand side carefully: the average kinetic energy depends on nothing but the temperature. Raise T and every molecule, on average, moves faster. This is what temperature really is — a measure of how energetically the particles jiggle about.
Warm a gas and its molecules speed up. Since Ek = ³⁄₂kBT, the average kinetic energy rises in step with the temperature.
WE 1
Find the average kinetic energy of a gas molecule at a temperature of 300 K. (kB = 1.38 × 10−23 J K−1.)
Step 1 — use Ek = ³⁄₂ kB TEk = 1.5 × (1.38×10⁻²³) × 300Ek = 6.2 × 10⁻²¹ JTiny for one molecule — but multiply by the ~10²³ molecules in a mole and it adds up fast.
Internal energy of an ideal gas
An ideal gas has no forces between its molecules (except in collisions), so the molecules have no potential energy. That makes its total internal energy simply the sum of all the molecular kinetic energies. Multiply the average KE by the number of molecules N:
Internal energy of an ideal gasU = N × ³⁄₂kBT = ³⁄₂NkBTU = ³⁄₂nRT
The two versions are the same thing counted differently (recall NkB = nR). Either way, U ∝ T: heat a fixed amount of gas and its internal energy — and temperature — rise together.
WE 2
500 J of heat is transferred to 4.0 g of helium gas held at constant volume. The molar mass of helium is 4.0 g mol−1. Find the rise in temperature.
Step 1 — find the moles: n = m / mrn = 4.0 ÷ 4.0 = 1.0 molStep 2 — rearrange U = ³⁄₂ nRT for T
ΔT = 2U / (3nR)
ΔT = (2 × 500) ÷ (3 × 1.0 × 8.31)ΔT ≈ 40 KAll the heat goes into kinetic energy (constant volume, no work done), so it all shows up as a temperature rise.
Monatomic vs diatomic molecules
One catch: the tidy U = ³⁄₂nRT applies to monatomic gases — single atoms like helium, neon and argon. A single atom can only move through space (translational motion), and that’s the only place it can store kinetic energy.
A monatomic atom stores energy only in translation. A diatomic molecule can also rotate, giving it extra ways to hold energy — which is why the simple ³⁄₂nRT result is stated for monatomic gases.
One molecule ³⁄₂ kBT
× N molecules
Whole gas U = ³⁄₂ NkBT
NkB = nR
U = ³⁄₂ nRT
💡 Top tips
³⁄₂kBT is for one molecule — multiply by N for the whole gas.
Temperature must be in kelvin in every one of these equations.
Ideal gas ⇒ internal energy is all kinetic (no intermolecular PE).
Spot the combos:NkB and nR are the same — that’s how the two U forms connect.
rms speed: if asked for a speed, use ½mv̄2 = ³⁄₂kBT, then square-root v̄2.
⚠ Common mistakes
Using ³⁄₂kBT for the whole gas — it’s per molecule
Leaving temperature in °C instead of kelvin
Applying U = ³⁄₂nRT to a diatomic gas (it has extra rotational energy)
Mixing up N (molecules) and n (moles) in the two internal-energy forms
Thinking an ideal gas has potential energy — it has none between collisions
Quick recap: Temperature is average molecular kinetic energy: Ek = ½mv̄2 = ³⁄₂kBT. With no potential energy, an ideal gas’s internal energy is all kinetic: U = ³⁄₂NkBT = ³⁄₂nRT, so U ∝ T. This holds for monatomic gases; diatomic molecules also rotate.
And that completes The Behaviour of Gases! Look how far you’ve come: from “pressure is force per area” all the way to a molecular explanation of temperature and internal energy. You can now move fluently between pressure, volume, temperature, moles, molecular speeds and energy — the full toolkit. Revisit any page that still feels shaky, and you’ll have this whole topic firmly in hand for the exam.
Want this to actually click before the exam?
Book a free meeting and let’s work through the tricky bits together.
