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Molecular kinetic theory
The gas laws were measured; kinetic theory explains them, deriving pressure from nothing but molecules bouncing off walls. It is one of the few derivations AQA examines in full, and it ends by telling you what temperature actually is.
Builds on Ideal gases and the gas laws and Momentum and impulse.
IN THIS TOPIC
- Contrast the empirical gas laws with the kinetic theory model, and cite Brownian motion as evidence for atoms.
- Reproduce the derivation of pV = ⅓Nm(crms)2 from the assumptions.
- Use the average molecular kinetic energy ½m(crms)2 = 3kT/2 = 3RT/2NA.
WHAT YOU PROBABLY THINK
Air pressure comes from air's weight pressing down.
A theory, and its evidence
The spec draws the philosophical line itself: the gas laws are empirical, generalisations from measurement, while kinetic theory arises from theory, deriving the same relationships from a molecular model. When the derivation reproduces Boyle's hyperbola from first principles, the model earns its keep.
The model's foundational evidence is Brownian motion: pollen grains in water, or smoke in air, jitter along ceaseless random walks. A grain vastly heavier than any molecule visibly trembles because the molecular bombardment is random and, instant by instant, uneven. Einstein's 1905 analysis of exactly this motion is what finally convinced the sceptics that atoms exist.
The assumptions
The ideal-gas model rests on assumptions worth listing, since papers ask for them: the gas contains a very large number of identical molecules in random motion; the molecules' own volume is negligible beside the container's; collisions, with walls and each other, are elastic and of negligible duration; and no forces act between molecules except during collisions. Each assumption is a small honest lie about real gases, which is why real gases follow the ideal equation best when hot and sparse.
The required derivation
AQA examines this derivation in full, so here it is as a story in five moves, with momentum and impulse from Year 12 doing the work. One: a molecule of mass m travels at velocity component vx toward a wall of a box of side L; each elastic bounce reverses that component, a momentum change of 2mvx. Two: it returns after a round trip of 2L, so hits arrive every 2L/vx seconds, giving an average force on the wall of 2mvx/(2L/vx) = mvx2/L.
Three: sum over all N molecules, replacing each vx2 by its average. Four: random motion shares speed equally among three directions, so the mean of vx2 is one third of the mean square speed. Five: pressure is that total force over the wall's area L2, and with volume V = L3,
where crms, the root mean square speed, is the square root of the mean of the squared speeds, the honest average for quantities that matter through their squares.
What temperature is
Set the derived equation beside the empirical pV = NkT and the meaning of temperature falls out:
The average kinetic energy per molecule is proportional to absolute temperature, and to nothing else: not the gas, not the pressure, just T. Temperature, at the molecular level, is average kinetic energy. And since an ideal gas has no intermolecular forces, it has no potential-energy account at all: an ideal gas's internal energy is entirely kinetic. As for the lie: air pressure is not the atmosphere's weight leaning on you; it is molecular bombardment, striking floor, walls and ceiling alike, which is why the pressure pushes in every direction at once.
THE EXAM BIT
- Empirical versus theoretical is quotable spec language: the gas laws come from experiment; kinetic theory derives them from a model. One sentence, one mark.
- Brownian motion's inference chain: visible random jitter of grains → uneven bombardment → by particles far too small to see → atoms exist. Keep the arrows in order.
- The derivation is examined stepwise: momentum change 2mvx, hit interval 2L/vx, force mvx2/L, sum and average, the factor ⅓ from three dimensions. Practise until the five moves are automatic.
- crms is root-mean-square: square, average, then root. Averaging the speeds first is the classic error, and the two differ.
- Doubling the kelvin temperature doubles the mean KE but multiplies crms by √2: the square root sits between energy and speed.
CHECK YOURSELF
Find the root mean square speed of nitrogen molecules (molar mass 0.028 kg mol−1) at 300 K. (R = 8.31 J K−1 mol−1.)
Show a hint
Write the average-KE equation with molar quantities, then solve for crms.
Show the answer
Per mole: ½M(crms)2 = 3RT/2, so (crms)2 = 3RT/M.
(crms)2 = 3 × 8.31 × 300 / 0.028 = 2.67 × 105 m2 s−2, giving crms = 520 m s−1.
Faster than the speed of sound in air, which is no coincidence: sound is carried by exactly these molecules, and cannot outrun its messengers.
Pressure is bombardment, derived from momentum.
Temperature is average kinetic energy, in kelvin.
No animated video for this topic yet; these notes stand alone. InkPhysics on YouTube.