IB Physics SLTopic B.3 — The Behaviour of GasesPaper 1 & 2Average KE · Internal Energy · U = ³⁄₂nRT~7 min read
Average Molecular Kinetic Energy
Temperature isn’t just a number on a thermometer — for an ideal gas, it’s a direct measure of how fast the molecules are moving, on average. This page turns that idea into a formula you can actually calculate with.
📘 What you need to know
An ideal gas’s molecules only exert forces on each other during a collision — so between collisions, they store no potential energy.
That means the internal energy of an ideal, monatomic gas is entirely kinetic energy.
The average kinetic energy of one molecule is Ek = ½mv² = 3⁄2kBT.
A hotter gas means a genuinely higher average kinetic energy per molecule — temperature and molecular motion are directly linked.
This simple relationship only applies to monatomic gases (helium, neon, argon) — diatomic molecules store energy in rotation too.
Why an Ideal Gas Has No Potential Energy
One of the core assumptions of kinetic theory is that gas molecules only interact through brief, elastic collisions — outside of that instant, no forces act between them at all. No force between molecules means no potential energy stored between them, since potential energy comes from the work done against a force. Strip away potential energy entirely, and whatever energy an ideal gas holds has to be kinetic energy — the energy of its molecules’ own random motion.
Average Kinetic Energy of One Molecule
Because collisions conserve both momentum and total kinetic energy, and because there’s no potential energy to account for, the average kinetic energy of a single gas molecule connects directly to temperature:
Average kinetic energy of one moleculeEk = ½mv² = 3⁄2kBT
where Ek is the average kinetic energy of one molecule (J), v² is the mean square speed (m² s⁻²), m is the mass of one molecule (kg), kB is the Boltzmann constant, and T is the temperature of the gas (K). Double the thermodynamic temperature and you double the average kinetic energy of every molecule — this equation is why raising temperature always means faster-moving molecules.
Total Internal Energy of a Gas
Since the total internal energy of an ideal, monatomic gas is just the total kinetic energy of all its molecules added together, scaling the single-molecule equation up by the number of molecules gives the internal energy of the whole gas:
Internal energy — particle formU = 3⁄2NkBT
Internal energy — mole formU = 3⁄2nRT
where U is the internal energy (J), N is the number of molecules, and n is the number of moles. Since U ∝ T, transferring heat energy into a gas held at constant volume raises its internal energy — and because that internal energy is entirely kinetic, the temperature has to rise right along with it.
Monatomic vs Diatomic Molecules
These internal energy equations only hold for monatomic gases — helium, neon and argon are the classic examples. A single-atom molecule can only move from place to place, so translational kinetic energy is the whole story. A diatomic molecule, with two atoms bonded together, can do that too — but it can also tumble and spin, storing additional rotational kinetic energy on top. That extra energy store means U = 3⁄2nRT underestimates the true internal energy of a diatomic gas.
A monatomic molecule only has translational kinetic energy. A diatomic molecule has translational kinetic energy too, but can also spin — giving it rotational kinetic energy that the simple U = ³⁄₂nRT formula doesn’t capture.
Quick recap: for an ideal gas, internal energy is entirely kinetic energy. Eₖ = ³⁄₂k_BT for one molecule, and U = ³⁄₂nRT for the whole gas — but only for monatomic gases, since diatomic molecules also store rotational kinetic energy.
WE 1
350 J of thermal energy is transferred to 2.5 g of neon gas held at a constant volume. The molar mass of neon is 20.2 g mol⁻¹. Calculate the resulting change in temperature of the gas.
Step 1 — Find the number of molesn = m ÷ M_r = 2.5 ÷ 20.2 ≈ 0.1238 molStep 2 — Rearrange the internal energy equationU = 3⁄2nRT → ΔT = 2U ÷ (3nR)
Step 3 — SubstituteΔT = (2 × 350) ÷ (3 × 0.1238 × 8.31)ΔT ≈ 227 K (3 s.f.)Neon is monatomic, so U = ³⁄₂nRT applies directly — this wouldn’t work this simply for a diatomic gas like nitrogen.
WE 2
Calculate the average kinetic energy of a single gas molecule at a temperature of 310 K.
Step 1 — Write the equationEk = 3⁄2kBTStep 2 — SubstituteEₖ = 1.5 × (1.38 × 10⁻²³) × 310Eₖ ≈ 6.42 × 10⁻²¹ J (3 s.f.)A single molecule’s kinetic energy is tiny — this equation only becomes useful once scaled up to a huge number of molecules.
💡 Top tips
If you’re using Ek = 3⁄2kBT, remember it’s for one molecule — multiply by N to get the total internal energy of a whole gas.
Spot which form of the internal energy equation fits the question: given a number of moles, use nRT; given a number of molecules, use NkBT.
Check whether a gas is monatomic before applying U = 3⁄2nRT directly — it’s only valid when rotational kinetic energy isn’t part of the picture.
⚠ Common mistakes
Applying U = 3⁄2nRT to a diatomic gas without accounting for its additional rotational kinetic energy.
Forgetting that Ek = 3⁄2kBT gives the energy of one molecule, not the whole gas.
Leaving temperature in °C — this relationship, like every other gas law equation, requires thermodynamic temperature in kelvin.
That covers the core physics of kinetic theory — from pressure at the molecular level through to what temperature actually measures. Amount of Substance is still waiting whenever you’d like to loop back to it, and after that this topic is complete.
Want this to actually stick before the exam?
Book a free session and we’ll work through internal energy and kinetic theory problems until they’re second nature.
