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## Einstein's Analysis of the Brownian Motion

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

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.

It reads

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.

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 4^{th} 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.