The kinetic theory model and its derivation

A theory, and its evidence

The spec draws the philosophical line itself. Gas laws are empirical, generalisations from measurement, while kinetic theory arises from theory and derives the same relationships from a molecular model. The derivation reproduces Boyle's law from first principles, which is evidence that the model's assumptions describe a real gas.

Brownian motion provides indirect evidence for molecular 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 turned that jitter into a quantitative prediction, tying how far a grain wanders to the Avogadro constant, and so to the number of molecules doing the shoving. Jean Perrin spent the years that followed measuring exactly that, and got a value for the Avogadro constant out of it. Theory and experiment together moved the atom from a useful assumption to a measured object. No single experiment settled the question; the evidence accumulated as independent measurements of the Avogadro constant agreed.

The demonstration is worth being able to describe, because it is asked for. A smoke cell is a small glass box holding smoke from a smouldering waxed taper, closed and set on a microscope stage. Light it from the side, so that each smoke particle scatters light up into the objective and shows as a bright speck against a dark field, and focus on the specks. What you see is not the smoke drifting as a body. Each individual speck jerks about on its own, changing direction constantly, at random, and never settling.

Then read the inference off it. The specks are smoke particles, far too large to be single molecules and visible only because they are that large. Nothing in the sealed cell is stirring them. So the only thing left to jolt them is the air molecules, arriving from every side and, at any instant, more of them on one side than the other. Their random jitter is visible evidence of the random molecular motion the model assumes, and the smaller the particle you watch, the more violent the jitter, exactly as an uneven bombardment would predict.

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, and 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 an approximation to a real gas, and the approximations are closest, so the equation fits best, when a gas is hot and sparse.

The required derivation

Describe the drawing

Left: the derivation, acted out. One molecule ping-pongs between the walls, and every round trip of 2L/v it delivers its 2mv kick to the flashing wall; many molecules doing this at once is pressure. Right: the visible consequence. A pollen grain far too big to see molecules is jolted by their uneven bombardment into a jitter you can see, the walk that convinced everyone atoms are real.

AQA, CIE and Edexcel all examine this derivation in full, so here it is in five moves, with momentum and impulse from Year 12 as the tools. One. A molecule of mass m travels at velocity component v_x towards a wall of a box of side L, and each elastic bounce reverses that component, a momentum change of 2mv_x. Two. It returns after a round trip of 2L, so hits arrive every 2L/v_x seconds, giving an average force on the wall of 2mv_x/(2L/v_x) = mv_x²/L.

Three. Sum over all N molecules, replacing each v_x² by its average. Four. Random motion shares speed equally among the three directions, so the mean of v_x² 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 c_rms, the root mean square speed, is the square root of the mean of the squared speeds, the appropriate average for quantities that matter through their squares.