1905 was the year in which 25-year-old Einstein wrote three very different physics papers in three diverse fields, each of which was easily a Nobel prize on its own. It turns out there was a fourth paper, just as brilliantly original — his doctoral dissertation.
There were at the time powerful abstract arguments for belief in atoms, but no one knew how big they were. How many hydrogen atoms are in one gram of hydrogen? The answer had been known for almost a hundred years as Avogadro’s number, but no one knew what the number was.
Here’s Einstein’s idea for calculating it. Adding sugar to water makes the water flow more slowly. Think “slow as molasses”. How slowly it flows can be measured in a number called viscosity. Lab scientists had a pretty good idea how many grams of sugar would produce how much viscosity.
For his dissertation, Einstein came up with a theory about how fluid flow would change if you added tiny beads to the fluid. (Today, we could manufacture uniform plastic beads, but I don’t think there was any equivalent experiment that could be done in 1905, so it was all a matter of theory.) He calculated the way the viscosity would change with the size and also the number of beads. By thinking of the dissolved sugar as Avogadro’s number of beads, and assigning a size for each bead based on the known volume of sugar divided by the same Avogadro’s number, Einstein was able to come up with an equation that he solved for Avogadro’s number.
Four years later, in 1909, the Millkin Oil Drop experiment resolved the question via a different route. The answer was substantially more accurate than Einstein’s, but Einstein’s calculation was in the right ballpark.
I just heard this story from Areeba Merriam’s account