Second Example (Brownian Movement, Diffusion)

Fill the lower part of a closed glass vessel with fog of minute droplets.

The upper boundary of the fog gradually sinks with a speed determined by:

• the viscosity of the air
• the size of a sinking droplet
• the specific gravity of the droplets

Under the microscope, the droplets individually do not permanently sink with constant speed.

Instead it has a very irregular ‘Brownian’ movement.

• This matches a regular sinking only on the average.

These droplets are:

• small and light
• affected by other molecules that hammer their surface

This knocks them around.

They follow the influence of gravity only on the average.

This example shows what funny and disorderly experience we should have if our senses were susceptible to the impact of a few molecules only. There are bacteria and other organisms so small that they are strongly affected by this phenomenon.

Their movements are determined by the thermic whims of the surrounding medium.

They have no choice. If they had some locomotion of their own they might nevertheless succeed in getting from one place to another - but with some difficulty, since the heat motion tosses them like a small boat in a rough sea.

Diffusion is akin to Brownian movement.

Water when mixed with some coloured substance dissolved in it, say potassium permanganate, not in uniform concentration, but rather as in Fig. 4, where the dots indicate the molecules of the dissolved substance (permanganate) and the concentration diminishes from left to right.

If you leave this system alone a very slow process of ‘diffusion’ sets in, the permanganate spreading in the direction from left to right, that is, from the places of higher concentration towards the places of lower concentration, until it is equally distributed through the water.

This process is in caused by any force driving the permanganate molecules away from the crowded region to the less crowded one.

Each these molecules behaves independently of all the others. They all get knocked around.

Their motion is like a blindfolded person walking on a large surface, changing his line continuously.

All the permanganate molecules, has this random walk.

the same for all of them, should yet produce a regular flow towards the smaller concentration and ultimately make for uniformity of distribution, is at first sight perplexing - but only at first sight.

If you contemplate in Fig. 4 thin slices of approximately constant concentration, the permanganate molecules which in a given moment are contained in a particular slice will, by their random walk, it is true, be carried with equal probability to the right or to the left.

But precisely in consequence of this, a plane separating two neighbouring slices will be crossed by more molecules coming from the left than in the opposite direction, simply because to the left there are more molecules engaged in random walk than there are to the right.

As long as that is so the balance will show up as a regular flow from left to right, until a uniform distribution is reached.

When these considerations are translated into mathematical language the exact law of diffusion is reached in the form of a partial differential equation

ap= DV2 p at '

which I shall not trouble the reader by explaining, though its meaning in ordinary language is again simple enough.

I The reason for mentioning the stern ‘mathematically exact’ law here, is to emphasize that its physical exactitude must nevertheless be challenged in every particular application.

It is based on pure chance. Its validity is only approximate.

If it is, as a rule, a very good approximation, that is only due to the enormous number of molecules that co-operate in the phenomenon.

The smaller their number, the larger the quite haphazard deviations we must expect - and they can be observed under favourable circumstances.