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## Einstein's Analysis of the Brownian Motion

Walter Pfeifer
Switzerland
2004, revised 2008

### 1 Einstein's formula

At the beginning of the 19th century the fact was well-known that microscopically small particles suspended in a liquid move continuously in an irregular fashion. The botanist Robert Brown showed in 1828 for the first time that this random motion exists everywhere and is not caused by organisms moving of their own effect. Subsequently, several scientists could prove that

a) the motion of agitation does not depend on time,
b) the magnitude of the motion is large if the particle observed is small,
c) the agitation increases with the temperature,
d) the magnitude of the motion is large if the viscosity (see below) of the medium is small.

On the basis of these facts the following kinetic model of this Brownian motion was formed. Every suspended particle is constantly subject to the bombardment of the molecules of the surrounding medium. Because this particle is relatively small the collision forces don't sum up to zero at each instant of time. The accidentally distributed impacts push the particle to and fro.
In the framework of this kinetic model Einstein developed a formula in 1905 (today this year is named his Annus Mirabilis), which connects the measured quantities and a material property of the participating particles and of the medium.
We proceed here in such a way to present and explain this formula and the quantities which it contains. Then we will show that it meets the facts a) up to d) qualitatively. After these expositions held as readily comprehensible as possible, in sections 2 and 3 a physical derivation of Einstein's formula will follow.

(1.1)

The quantity  is the absolute temperature. It depends on the Celsius temperature as follows

absolute temperature = Celsius temperature + 273.15 Grades.

Although the unit of the absolute temperature agrees with the unit of the Celsius scale one denotes it by 1 Kelvin, abbreviated by 1 K.

The constant  plays a role in the gas laws (perfect-gas equation). It is named Boltzmann constant and has the value .

We assume that the suspended particles are spheres with the radius .

The quantity  is the viscosity of the medium. It reports how viscous the surroundings of the particle are (the viscosity of gases is much smaller). If an object is pulled by a force through a medium, a corresponding velocity results. It is the higher, the smaller the viscosity is.

The quantity  is most costly to determine. In figure 1.1 the wobbly path of a suspended particle is outlined. The positions of the particle at equal time steps  are marked in by points. The distance in x-direction (here to the right or to the left) between points coming after one another is named displacement . The mean of the squares of all observed displacements has to be formed, i.e.

(1.2)

The more displacements are taken into account the more meaningful is the mean-square value of . The formula (1.1) says that this value is proportional to the absolute temperature  and to the length of the chosen time interval . Of course,    is the greater the smaller the radius  of the particle and the viscosity  are. Therefore, formula (1.1) corresponds to the facts b) up to d). Because the growing time (not the time interval ) does not appear in the formula the property a) is also satisfied.

At the end of section 3 the quantity  will be calculated for an example.

### 2 The diffusion coefficient of suspended spherical particles

In this and in the next section physical and mathematical knowledge is required, which students have at their disposal in about the 4th semester.

In the two first sections of his publication in 1905 about the motion of suspended particles Einstein derived an expression for the diffusion coefficient of such particles using properties of the osmotic pressure. However, many scientists felt this treatise as difficult, for which reason in 1908 Einstein published an "elementare Theorie der Brown'schen Bewegung". In the section on hand we follow this work in detail.

First, we deal with the osmotic pressure of suspended particles in a liquid medium.

A vessel may contain a liquid with the pressure . It may be subdivided by a semipermeable …

This is only an excerpt. The whole publication can be ordered from the category «order for free» in this site.