## Section 4.3: Brownian Motion

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In 1827 botanist Robert Brown noticed that grains of pollen suspended in a liquid moved erratically. This motion is called Brownian motion. Restart. Click on the Brown animation to see his observations. The graph shows the path of the pollen grain. In 1905 Einstein explained this motion by creating a mental picture or thought (Gedanken) experiment that involved the invisible molecules in the liquid colliding into the larger pollen molecules. The Einstein animation shows the *hidden* molecules. Einstein's explanation allowed for a calculation of the number of atoms in a mole (Avogadro's number) and put *atomicity* on solid footing.

In the Einstein animation, the smaller (and invisible in optical microscopes) molecules have a kinetic energy that is proportional to their temperature (temperature is a measure of average kinetic energy). These particles collide into the larger particle and cause it to move. Einstein's relationship for the average displacement, *D*_{rms}, of a spherical particle of radius *r* is given by

*D*_{rms} = (*k*_{B}*Tt*/πη*r*)^{1/2},

where *k*_{B} is the Boltzmann constant, *T* is the temperature, *t* is the time, and η is the viscosity of the fluid in which the particle is suspended. All of the quantities could be measured except *k*_{B}, the Boltzmann constant. Determining *k*_{B}, however, gave a value for *N*_{A}, Avogadro's number since *N*_{A} = *R*/*k*_{B} where *R* is the gas constant (as in the ideal gas law: *pV* = *nRT* = *Nk*_{B}*T*). Jean-Baptiste Perrin used this method in conjunction with two other methods (one of which was based on the rotation of suspended particles exhibiting Brownian motion, which was also explained by Einstein) to determine Avogadro's number. Perrin won the Nobel Prize in physics in 1926 for this series of measurements that firmly established the idea of atoms as fundamental constituents of nature.

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