Brownian Motion

Brown's 1827 experiment of a pollen grain moving on the surface of water.

Teaching notes

Suggested use

Press Run with the default settings. All that's visible at first is a single large orange disc drifting around, leaving a dotted trail behind it. That is the pollen grain, and it is already being tracked. Watch the trail for a while before touching anything else. It jitters, doubles back, stalls, then lurches off somewhere else. None of this has been scripted to look random; it is what a real pollen grain does under a microscope, exactly as Robert Brown first saw in 1827.

Ask what's making it move like that before revealing the answer. Then click Show water molecules. The box fills with small, fast blue dots. This is what was invisible the whole time, and it is what strikes the pollen grain constantly from every direction. No single collision does anything dramatic to something so much bigger and heavier. But with hundreds of molecules arriving every second, there is always a small random imbalance in how many are hitting from which side at any instant, and that imbalance is what produces the jiggle. Turn the water back off and the pollen grain keeps moving exactly as before; hiding the water changes what is drawn, not the physics.

Try the Pollen size slider. Shrink it towards the water molecules' own size and the "pollen" starts darting around rapidly, behaving just like an ordinary water molecule. Grow it and the grain becomes slow and heavily damped, barely drifting at all even with the same bombardment going on. This is the size range Brown himself was working in: pollen grains happen to be big enough to see clearly under a microscope, yet light enough that molecular collisions still visibly move them. Much smaller and you couldn't see the particle at all; much larger and it wouldn't visibly jiggle.

Number of water molecules and Water molecule speed both change how vigorously the pollen grain gets jostled. More molecules, or faster ones, mean more frequent and more energetic collisions, and a busier-looking trail.

What is happening

Every particle in this simulation, water molecule or pollen grain alike, obeys the same physics: straight-line motion between collisions, and a fully elastic collision conserving both momentum and kinetic energy whenever two particles touch or one hits a wall. Nothing about the pollen grain's motion is scripted. Its jitter is the calculated outcome of many individual elastic collisions with the water molecules around it.

What makes the pollen grain behave so differently from a water molecule is mass. At a given moment, every particle in the box shares, on average, the same kinetic energy. This is the equipartition of energy, and it is why a much heavier pollen grain ends up moving much more slowly than the water molecules jostling it: with $\frac{1}{2}mv^2$ roughly equal for both, a heavier particle needs a smaller $v$. A collision that sends a light water molecule bouncing back at full speed barely nudges a much heavier pollen grain, but with a great many water molecules arriving from every direction, those nudges don't always cancel out. At any instant there is a small net imbalance, and that imbalance, repeated relentlessly and randomly, produces the visible jiggle.

Brown himself, in 1827, was originally trying to work out whether the constant movement he saw in pollen grains suspended in water meant they were somehow alive. He ruled that out by showing dust and ground-up rock did the same thing. What he had no explanation for was the cause, and neither did anyone else for nearly 80 years. Einstein's 1905 paper finally explained it exactly as this simulation shows it: unseen, constantly colliding water molecules buffeting a much larger suspended particle from random directions.

What came next matters as much as the explanation itself. Because Einstein's theory made quantitative predictions about how far a grain should wander in a given time, those predictions could be tested. Jean Perrin did exactly that between 1908 and 1913, using real pollen-sized grains under a microscope. The measurements matched Einstein's theory closely enough to yield one of the first solid experimental values for Avogadro's constant, which turned atoms from a convenient calculating device into something whose existence and size had been measured.

Key relationship

The parameter that controls everything here is the mass ratio between the pollen grain and a single water molecule. Equipartition of energy links the two particles' typical speeds by

$$ \frac{v_{\text{pollen}}}{v_{\text{water}}} = \sqrt{\frac{m_{\text{water}}}{m_{\text{pollen}}}} $$

A pollen grain 100 times heavier than a water molecule moves, on average, only a tenth as fast. A tenth as fast is still very much in motion. The jiggle you see with the water hidden is the leftover signature of that motion after millions of individual, momentum-conserving collisions: no single collision matters, but their statistical imbalance, moment to moment, is real and measurable. That is the insight Einstein and Perrin turned into hard evidence for the existence of atoms.

Suggested questions

Q1. The pollen size slider is set so a pollen grain is 80 times heavier than a single water molecule. Using equipartition, roughly how many times slower does the pollen grain move than a water molecule, on average?

Show answer

$$ \frac{v_{\text{pollen}}}{v_{\text{water}}} = \frac{1}{\sqrt{80}} \approx \frac{1}{8.9} $$

So the pollen grain moves roughly 9 times slower than a water molecule, on average, despite being struck by them constantly. This is why the pollen grain drifts sedately across the box while the hidden water molecules would be a blur.

Q2. A student watches the simulation with water molecules hidden and says "the pollen grain's motion looks completely random, so nothing about what's happening is predictable or governed by a law." Using what you can see by revealing the water molecules, explain what's wrong with this claim.

Show answer

Every individual collision between the pollen grain and a water molecule obeys exactly the same predictable law: conservation of momentum and kinetic energy, given each particle's mass and velocity at that instant. Nothing about any single collision is random or unpredictable, since Newtonian mechanics determines the outcome exactly. What looks random is only the net effect of enormous numbers of these individually deterministic collisions, arriving from unpredictable directions at unpredictable times. The randomness is a statistical property of the whole system rather than a breakdown of the physical law governing each collision.

Q3. Explain, using the idea of equipartition of energy, why Brownian motion is only visible for particles within a fairly narrow range of sizes, and cannot be seen for individual molecules or for large visible objects like a grain of sand.

Show answer

An individual molecule is far too small to see directly under any ordinary microscope, whatever its motion; the effect is real at that scale but not observable. At the other extreme, equipartition means a much larger, heavier particle such as a grain of sand moves so slowly on average, and is so hard to shift by individual molecular collisions, that any residual jiggle becomes too small to detect against its own size. It is effectively damped by its own inertia. Pollen grains sit in the size range where both conditions are satisfied at once: large enough to resolve under a microscope, yet light enough that the statistical imbalance in molecular bombardment still produces a visible, measurable displacement.