# The determination of Avogadro's number

## The Brownian motion ( Robert Brown, Albert Einstein )

At the beginning of the 19th century, Robert Brown observed strange movements of pollen grains in water under his microscope. He had discovered the Brownian motion:

At the beginning of the 20th century, Albert Einstein was interested in Brownian motion He explained this phenomenon by impacts of water molecules against the pollen grains: It is observed that the grains travel along a zig-zag path : Let be n the number of elementary paths (here n=10): Then the mean distance $\overline{x^2}$ of such an elementary path is given by the formula: $\overline{x^2}$ $=$ $\frac{x_1^2+x_2^2+x_3^2+...x_n^2+}{n}$ Einstein found that $\overline{x^2}$ depends on Avogadro's number $\mathscr{N}$ by the formula: $\frac{\overline{x^2}}{t}$ $=$ $\frac{RT}{\mathscr{N}}\cdot \frac{1}{4\pi r^3 \eta}$: Searching for the measurable parmeters ($R$: ideal gas constant, $T$ absolute temperature, $t$ the amount of time required for an elementary path , $\eta$ viscosity of the medium, $r$ radius of the pollen grains) it appeared theoretically possible to calculate Avogadro's number by this formula!

Jean Perrin had the idea to centrifuge the pollen grains in order to obtain grains of the same radius $r$ which he examined under his microscope. He measured $T$, $\eta$, $\overline{x^2}$ and $t$ and was then able to calculate $\mathscr{N}$. He found: $\mathscr{N}$ = $6,8\cdot 10^{23}$ which is a very good result: The best known value we know today is : $\mathscr{N}$ = $6,023\cdot 10^{23}$