Original paper: Einstein, Perrin, and the reality of atoms: 1905 revisited
There are many things that we “know” about the world around us. We know that the Earth revolves around the Sun, that gravity makes things fall downward, and that the apparently empty space around us is actually filled with the air that we breathe. We take for granted that these things are true. But how often do we consider whether we have seen evidence that supports these truths instead of trusting our sources of scientific knowledge?
Students in school are taught from an early age that matter is made of atoms and molecules. However, it wasn’t so long ago that this was a controversial belief. In the early 20th century, many scientists thought that atoms and molecules were just fictitious objects. It was only through the theoretical work of Einstein [1] and its experimental confirmation by Perrin [2] in the first decade of the 20th century that the question of the existence of atoms and molecules was put to rest. Today’s paper by Newburgh, Peidle, and Rueckner at Harvard University revisits these momentous developments with a holistic viewpoint that only hindsight can provide. In addition to re-examining Einstein’s theoretical analysis, the researchers also repeat Perrin’s experiments and demonstrate what an impressive feat his measurement was at that time.
In the mid-1800s, the botanist Robert Brown observed that small particles suspended in a liquid bounce around despite being inanimate objects. In an effort to explain this motion, Einstein started his 1905 paper on the motion of particles in a liquid with the assumption that liquids are, in fact, made of molecules. According to his theory, the molecules would move around at a speed determined by the temperature of the liquid: the warmer the liquid, the faster the molecules would move. And if a larger particle were suspended in the liquid, it would be bounced around by the molecules in the liquid.
Einstein knew that a particle moving through a liquid should feel the drag. Anyone who has been in a swimming pool has probably felt this; it is much harder to move through water than through air. The drag should increase with the viscosity, or thickness, of the fluid. Again, this makes sense: it is harder to move something through honey than through water. It is also harder to move a large object through a liquid than a small object, so the drag should increase with the size of the particle.
Assuming that Brownian motion was caused by collisions with molecules, and balancing it with the drag force, Einstein determined an expression for the mean square displacement of a particle suspended in a liquid. This relationship indicates how far a particle moves, on average, from its starting point in a given amount of time. He concluded that it should be given by
$latex \langle \Delta x (\tau) ^2 \rangle = \frac{RT}{3 \pi \eta N_A r} \tau$
where R is the gas constant, T is the temperature, $latex \eta$ is the viscosity of the liquid, $latex N_A$ is Avogadro’s number [3], r is the radius of the suspended particle, and $latex \tau$ is the time between measurements [4]. With this result, Einstein did not claim to have proven that the molecular theory was correct. Instead, he concluded that if someone could experimentally confirm this relationship, it would be a strong argument in favor of the atomistic viewpoint.
This is where Perrin came in. Nearly five years after Einstein’s paper was published, he successfully measured Avogadro’s number using Einstein’s equation, confirming both the relationship and the molecular theory behind it. However, with the resources available at the time, this experiment was a challenge. Perrin had to first learn how to make micron-size spherical particles that were small enough that their Brownian motion could be observed, but still large enough to see in a microscope. In order to measure the particles’ motion, he used a camera lucida attached to a microscope to see the moving particles on a surface where he could trace their outlines and measure their displacements by hand. Perrin obtained a value of $latex N_A = 7.15 \times 10^{23}$ by measuring the displacements of around 200 distinct particles in this way.
Performing this experiment in the 21st century was much simpler than it was for Perrin. Newburgh, Peidle, and Rueckner were able to purchase polystyrene microspheres of various sizes, eliminating the need to synthesize them. They also used a digital camera to record the particle positions over time instead of tracking the particles by hand. Using particles with radii of 0.50, 1.09, and 2.06 microns, they measured values of $latex 8.2 \times 10^{23}$, $latex 6.4 \times 10^{23}$, and $latex 5.7 \times 10^{23}$. Perhaps surprisingly, even with all of their modern advantages, the researchers’ results are not significantly closer to the actual value of $latex N_A = 6.02 \times 10^{23}$ than Perrin’s was a hundred years earlier.
For those of us who work in the field of soft matter, the existence of Brownian motion and the linear mean square displacement of a particle undergoing such motion are well-known scientific facts. The authors of this paper remind us that, not so long ago, even the existence of molecules was not generally accepted. And, although we often take for granted that these results are correct, first-hand observations can be useful for developing a deeper understanding and appreciation: “…one never ceases to experience surprise at this result, which seems, as it were, to come out of nowhere: prepare a set of small spheres which are nevertheless huge compared with simple molecules, use a stopwatch and a microscope, and find Avogadro’s number.” [5]
[1] A. Einstein, “On a new determination of molecular dimensions,” doctoral dissertation, University of Zürich, 1905.
[2] J. Perrin, “Brownian movement and molecular reality,” translated by F. Soddy Taylor and Francis, London, 1910. The original paper, “Le Mouvement Brownien et la Réalité Moleculaire” appeared in the Ann. Chimi. Phys. 18 8me Serie, 5–114 1909.
[3] Avogadro’s number is the number of atoms or molecules in one mole of a substance.
[4] In 1908, three years after Einstein’s paper, Langevin also obtained the same result using a Newtonian approach. (P. Langevin, “Sur la Theorie du Mouvement Brownien,” C. R. Acad. Sci. Paris 146, 530–533 1908.)
[5] A. Pais, Subtle Is the Lord (Oxford U. P., New York, 1982), pp. 88–92.
