Why Do Tiny Particles Floating in Water Never Stop Trembling?
― The Scientists Who Measured the Jiggle and Counted Invisible Atoms
Dilute some milk and look at it under a microscope, and you will see white particles jiggling endlessly. There is no wind and no current. This trembling is called "Brownian motion", and it became firm proof that water is a crowd of invisible molecules. Just over a century ago, many people still doubted that atoms were real.
In science class, you put one drop of milk diluted with water on a glass slide and look at it under a microscope. As you raise the magnification, many small round particles come into view.
Look closely and none of them keeps still. A little to the right, a little up, then left again. They tremble at random, almost as if they were alive.
The particles in milk are not alive. So what is shaking them? It took about 80 years to answer that question.
There are only two reasons for the endless trembling
Water is made of countless tiny molecules flying about. The particle is hit all the time, and whenever the hits from left and right are slightly uneven, the particle is pushed.
The direction of each push is random, so the particle keeps going back and forth. Wait four times as long, and it ends up only twice as far from where it started. This property became the key to counting atoms.
The first is "why it trembles", and the second is "how it trembles". Let's look at each in turn, following the history of the discovery.
A botanist finds a "lifelike" motion
In 1827, the British botanist Robert Brown was studying pollen soaked in water under a microscope. Even smaller particles came out of the pollen, and they trembled without stopping.
At first, Brown suspected the motion had something to do with life. So he tried specimens of plants that had been dried for many years. The particles still moved. He then tried powders made by crushing glass and rock finely. They are said to have moved too.
In this way, Brown established that the trembling had nothing to do with living things. But he could not tell why the particles moved. Over the next 80 years or so, many explanations came and went, among them uneven temperature in the water, currents, and the heat of light.
Many scientists of the time thought that atoms and molecules were merely "convenient fictions" for doing calculations. Nobody had ever seen a single molecule.
Einstein's prediction and Perrin's patient measurements
In 1905, Einstein, then a 26-year-old still working at the patent office, published a paper. Its message was: "If water is made of molecules, a floating particle must tremble. And we can calculate how far the trembling spreads."
The key point was that he said to measure not the particle's "speed" but the "spread of its drift". Earlier researchers had tried to measure the speed of the particles and are said to have been frustrated, because they got a different value every time.
The particle keeps changing direction in tiny steps, so the answer for its speed changes just by changing the interval at which you look. Look at Figure 1. The left shows an example of the path a particle follows, and the right shows how the size of the drift relates to time.
When you change the waiting time with the slider, the dotted ring on the left (a guide to the drift) grows. But even if you make the time ten times longer, the ring becomes only about three times larger.
The person who tested this prediction in the lab was the French physicist Jean Perrin. Perrin made small spheres of uniform size from a resin called gamboge. He is said to have matched their sizes only after repeating centrifugation many times.
Then, under the microscope, he kept recording the positions of the particles at fixed intervals. The spread of the drift grew with the square root of time, just as Einstein's formula said. What is more, from the size of that spread he could work backward to the number of molecules in a given amount of water.
The value Perrin obtained agreed well with values estimated by completely different methods. After this, even leading chemists who had doubted atoms are said to have accepted that they are real. Perrin received the Nobel Prize in Physics in 1926.
Why don't big things tremble?
A water molecule is said to be about three hundred-millionths of a centimetre across. A milk particle is about 1 micrometre (one thousandth of a millimetre) in diameter, thousands of times larger than a molecule. Even so, it is still "small".
The more collisions there are, the more the pushes from left and right average out and cancel. A bigger object is hit by many more molecules at once, so the imbalance hardly shows and the motion disappears. That is why your finger or a marble does not tremble.
Put the other way round, the trembling is stronger when the particle is smaller, when the water is hotter, and when the water is thinner and runnier. In "Check it with a formula" below, you can see these three points gathered into a single equation.
In a beam of light from a window, you can see dust drifting and swirling. It is often given as an example of Brownian motion, but dust you can see is far too big. The main cause of that motion is thought to be the slow movement of air in the room.
Small molecules inside cells are shaken constantly by the same mechanism. Oxygen and nutrients reach every corner of a cell thanks to this non-stop "random shaking".
Summary
The trembling of tiny particles floating in water is proof that unseen molecules strike from every side. The drift spreads only as the square root of time, and from that spreading, Einstein and Perrin showed that atoms could be counted.
Atoms could be counted even though no one could see them.
The trembling of tiny particles revealed their "shadow".
How water and fat separate, as in the particles of milk, is covered in Why Don't Oil and Water Mix?, and how thin a layer of atoms can become is covered in How Thin Can Gold Be Beaten?.
- Dilute milk with about ten times as much water, put one drop on a glass slide, and cover it with a cover glass.
- Under a microscope at about 400× magnification, you can see small particles trembling finely. Sometimes even a children's microscope will show them.
- If all the particles are drifting the same way, that is a current in the water. Wait a little until the flow settles, then look for the trembling in which each particle goes in its own direction.
Also compare them: smaller particles tremble more vigorously, and larger ones more calmly.
For those who want to know more ― terms, formulas and links to textbooksEach part is labelled with its level, from junior high science to university specialist courses
- Junior highCovered in junior high school science
- High schoolCovered in high school physics and chemistry
- High school+Advanced high school material, or treated as a textbook sidebar
- UniversityNot taught in high school; university specialist courses (statistical mechanics, physical chemistry)
- ResearchNot yet taught as settled fact even at university; researchers are still investigating it
Junior highTerms: this phenomenon has names
- Brownian motion: the constant, irregular movement of small particles floating in a liquid or gas, caused by collisions with the surrounding molecules. It is named after its discoverer, Robert Brown.
- Diffusion: the spreading of molecules or particles from where they are crowded to where they are sparse, as a result of random motion. Brownian motion is diffusion watched one particle at a time.
- Avogadro constant: the number of particles in one mole of a substance, about 6.02×1023. Perrin estimated this number from his measurements of Brownian motion.
Junior highHigh schoolCheck it with a formula: how far does a milk particle drift in one minute?
We calculate how far, on average, a particle 1 micrometre across drifts in one chosen direction in water at 20 °C.
| In symbols | x ＝ √( 2 × D × t ) ， D ＝ kT ÷ ( 6π × η × r ) |
| In words | Drift ＝ the square root of "2 × spreading ease × time". Spreading ease ＝ strength of the thermal shaking ÷ resistance of the water |
| Where it comes from | The Einstein relation, derived from the link between the strength with which molecules shake the particle (set by temperature) and the drag the water puts on a moving particle (Stokes' formula) |
| Symbol | Meaning and unit |
| x | Average drift in one direction (micrometres) |
| D | Diffusion coefficient = spreading ease (square micrometres per second) |
| t | Waiting time (seconds) |
| kT | The Boltzmann constant times the absolute temperature. The size of the thermal shaking per molecule |
| η | Stickiness of water (viscosity, pascal-seconds) |
| r | Radius of the particle (micrometres) |
| kT at 20 °C (in units of pascals and cubic micrometres) | 0.00404 |
| Viscosity η of water at 20 °C (pascal-seconds) | 0.0010 |
| Particle radius r (micrometres) | 0.5 |
| Pi, π | 3.14 |
| Work out 6π | 6 × 3.14 ＝ 18.84 |
| Multiply by the viscosity | 18.84 × 0.0010 ＝ 0.01884 |
| Multiply by the radius to get the size of the water's resistance | 0.01884 × 0.5 ＝ 0.00942 |
| Spreading ease D (square micrometres per second) | 0.00404 ÷ 0.00942 ≒ 0.429 |
| "2 × D" per second | 2 × 0.429 ＝ 0.858 |
| Drift in 1 second | The square root of 0.858 is about 0.93 micrometres |
| "2 × D × t" for 60 seconds (1 minute) | 0.858 × 60 ＝ 51.48 |
| Drift in 1 minute | The square root of 51.48 is about 7.2 micrometres |
| If it kept going straight for 1 minute at the 1-second pace | 0.93 × 60 ＝ 55.8 |
Even after waiting a full minute, the particle has drifted only about 7 micrometres. That is about a tenth of the width of a human hair (about 70 micrometres). Had it kept going at its first-second pace, it would have reached 56 micrometres, but with all the back-and-forth it gets only about an eighth of that. Perrin used this calculation in reverse: from the measured drift he found kT, and from that the Avogadro constant.
High schoolHigh school+Temperature and the motion of molecules
High schoolIn high school physics, "the motion of gas molecules" teaches that temperature represents the average kinetic energy per molecule. A particle undergoing Brownian motion is like a member of the same temperature group as the surrounding molecules, and on average it has the same kinetic energy. It is heavier, so it simply moves more slowly.
High school+That the drift grows as the square root of time is a property shared by all random walks. If you take random steps, one pace forward or back at a time, after 100 steps you will on average be only about 10 paces from where you began.
UniversityFluctuation and resistance spring from the same root
The formula above, D ＝ kT ÷ ( 6π × η × r ), is called the Stokes–Einstein relation. Its denominator comes from Stokes' law, which gives the drag a slowly moving sphere feels from a liquid. The motion of the particle can be written with the Langevin equation, which includes both a random force and a drag force. The idea that the force that shakes the particle and the drag that stops it arise from the same collisions was generalised as the fluctuation–dissipation theorem.
📖 Derivations and further reading: Brownian motion (Wikipedia, in Japanese) ／ Einstein's original 1905 paper (in German)
ResearchWhat is still not well understood
- Motion over extremely short times. The particle's "instantaneous speed" between collisions was long thought impossible to measure. Since around 2010, experiments have been measuring it with particles held by light.
- Jiggling inside crowded cells. A cell is packed with molecules, and living things also shake it with energy they spend themselves. Motions that depart from the simple square-root law have been found, and understanding them is still work in progress.
- Molecular machines that use the jiggling. How far proteins such as those in muscle make use of random jiggling to move in one direction is still debated.
In other words, this article too is "an explanation of what is known so far". Almost 200 years after its discovery, random trembling still keeps raising new questions.
Links to textbooks (by level)
| Level | Subject / unit | Where in this article |
|---|---|---|
| Junior high | Science, year 1: "States of matter" and the particle model | Water is made of tiny particles; diffusion |
| High school | Physics: "Motion of gas molecules"; Chemistry: "Amount of substance" | Temperature and kinetic energy; the Avogadro constant |
| High school+ | Mathematics: advanced "Probability" | Random walks and the square-root law |
| University | Statistical mechanics, physical chemistry, fluid mechanics | The Stokes–Einstein relation; the Langevin equation |
| Research | Biophysics, soft matter physics | Jiggling inside cells; molecular machines |
| ― | Links to daily life | Why milk and paint do not settle easily; how smells spread through a room |
- Einstein, "On the Movement of Small Particles Suspended in Stationary Liquids Required by the Molecular-Kinetic Theory of Heat" (German physics journal, 1905)
- Jean Perrin, Les Atomes, Japanese translation by Bun'ichi Tamamushi (Iwanami Bunko, 玉蟲文一 訳 / 岩波文庫)
- Robert Brown, "A brief account of microscopical observations on the particles contained in the pollen of plants" (1828)
- Wikipedia, "Brownian motion" (in Japanese, ブラウン運動)
- Wikipedia, "Avogadro constant" (in Japanese, アボガドロ定数)
* This article is a popular-science explanation for general readers. The figures given are rough guides to help you understand how things work. Details of historical events are open to differing accounts.