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If your Christmas festivities are anything like mine, several cups of
well-sweetened black coffee will feature in the proceedings. As you gaze
into the murky depths of the healing brew, ponder on the fact that Albert
Einstein, shortly before he became known to most other scientists, used
a similar concoction to cure a headache that had been troubling scientists
for some time. He worked out how big molecules are.

As Einstein’s biographer Abraham Pais points out, among the scientific
papers published before 1912 and cited in papers published between 1961
and 1975, three of the ‘top ten’ are by Einstein. And of these three the
most frequently cited is his 1904 doctoral thesis. A sequel to the thesis
is the next most frequently cited of Einstein’s early works. The third is
one of the three famous papers he published in 1905, on Brownian motion.
But Einstein’s paper announcing the special theory of relativity is nowhere
to be found in the pre-1912 top ten.

Down to Earth

One reason for this curious absence is that special relativity became
such a standard feature of physics that by the 1960s hardly anyone was bothering
to read Einstein’s original 1905 paper. But Einstein’s thesis is clearly
something special. It has been so widely quoted in recent decades because
it deals with the properties of particles suspended in a fluid, a topic
which has rather more everyday application than the special theory of relativity.
It has found uses in calculations relating to such diverse phenomena as
the way sand particles get stirred up in cement mixers, the properties of
cow’s milk, and the way fine particles of dust and droplets of liquid (aerosols)
are suspended in clouds.

But Einstein was not concerned with any of these things when he set
about his investigation of the way particles are suspended in fluids. As
he later told his friend and scientific sparring partner Max Born, ‘my main
purpose for doing this was to find facts which would attest to the existence
of atoms of definite size’. Amazing as it might seem today, at the beginning
of this century many scientists still doubted the reality of atoms. In
searching for evidence for their existence, Einstein was following a well-established
tradition, which went back almost a hundred years. In his thesis he came
to the very brink of a proof that would persuade remaining doubters of the
reality of molecules and atoms; the final proof came in his Brownian motion
paper, which was also an extension of the thesis.

Einstein’s work devel-oped an idea that goes back to the mid-1860s,
when a German chemist, Johann Joseph Loschmidt made a neat attempt at estimating
the sizes of molecules. Earlier, an Italian called Amedeo Avogadro, had
put forward the hypothesis that a box of a certain volume, filled with gas
at a certain temperature and pressure, must always contain the same number
of atoms or molecules, no matter what the chemical composition of the gas
and the weight of each molecule. The nub of Loschmidt’s approach was to
use two sets of equations to determine simultaneously two properties of
molecules – their sizes, and Avogadro’s number, which is now defined as
the number of carbon-12 atoms in 12 grams of carbon.

If you want to find one unknown quantity, you can do so by solving a
single equation in which that quantity appears. If you have two unknowns,
you need two equations in which both appear. Both Loschmidt and Einstein
stuck, for this particular calculation, with such a pair of simultaneous
equations, solving them for two unknown quantities. Loschmidt’s calculations
involved the average distance a molecule travels between collisions in a
gas – the ‘mean free path’ – and the fraction of the volume of the gas occupied
by the molecules alone. He assumed that in a liquid all the molecules are
touching each other, which gave him a handle on the volume occupied by all
the molecules in a given quantity of liquid When the same liquid is heated
to become a gas, he knew that the volume of the gas molecules alone must
be the same as the volume of the original liquid. The rest of the volume
of the gas is simply the empty space that the molecules whiz through.

He carried out his calculations for air, and had to use estimates of
the densities of liquid nitrogen and liquid oxygen which were not as accurate
as modern measurements. Yet he still came up with answers that stand up
well today. Loschmidt said that the diameter of a typical molecule of air
must be measured in millionths of a millimetre, and he gave, in 1866, a
value for Avogadro’s number of 0.5 x 10 ²³.

Using modern data, the mean free path of molecules of air turns out
to be just 13 millionths of a metre at 0 °C, and an oxygen molecule
in air at that temperature will be travelling at just over 461 metres per
second. So it undergoes more than 3.5 billion collisions every second. The
modern value for Avogadro’s number is 6.02 x 10 ²³.

Einstein used a similar form of mathematical reasoning, solving two
simultaneous equations with the same two unknown quantities in them. But
he app-lied his reasoning to solutions, in which molecules of one compound
(the solute) are spread more or less evenly through a liquid (the solvent)
made up of molecules of another compound. The solutions Einstein based his
calculations on were simply sugar in water: his calculations would apply
accurately to the molecular behaviour of a cup of sweet tea or coffee.

The starting point for this work was the discovery, made in the 1880s,
that the molecules in a solution behave in some respects like the molecules
of a gas. One example of this behaviour is osmosis, an effect which relies
on the use of a ‘semipermeable’ membrane – one which will allow solvent
molecules to pass, but not the solute. Just as gas spreads out (diffuses)
from one side of a box to fill the entire box when a partition in the middle
of the box is removed, so solvent molecules in a weak solution (water, in
Einstein’s equation) will tend to pass through a semipermeable barrier separating
it from a stronger solution, until both solutions are at similar concentrations.

To meet the second law of thermodynamics – that is, for information
to be lost and entropy to increase – the strong solution must somehow be
made weaker, more like the weak solution, so the overall result is that
solvent passes from the weak to the strong solution. This causes the level
of solution to rise in the side of the container that contains the sugar,
and to fall in the side containing only water. The process stops when the
extra pressure of the stronger solution, caused by the weight of the extra
height of liquid in that side of the container, called osmotic pressure,
is strong enough to stop the flow of solvent through the membrane. The osmotic
pressure depends on the number of molecules of solute in the solution –
the more concentrated the solution is, the stronger the pressure. And, once
again, the sizes of the molecules comes into the calculation in terms of
the fraction of volume of solution that is actually occupied by those molecules.

The second equation used by Einstein involved the mean free path of
the molecules of the solute, which he related to the speed with which liquid
molecules diffused through the membrane. He also determined other properties,
such as the relation between this diffusion and the viscosity of a liquid,
which have proved so interesting to engineers investigating cement, milk
and all the rest.

Blue sky project

In the thesis, Einstein calculated a value for Avogadro’s number of
2.1 x 10 ²³, with estimates for molecular sizes in the now familiar
range of a few hundredths of a micrometre. In an updated version published
in 1906, he was able to improve the calculation, with the help of new data
from more accurate measurements of the behaviour of sugar solutions. This
gave him a value of 4.15 x 10 ²³. In 1911, Einstein presented
a paper in which he gave Avogadro’s number as 6.6 x 10 ²³. By
then, this crucial number had been det-ermined reasonably accurately by
others in a dozen ways, each giving similar values. There was no longer
any doubt that atoms and molecules were real.

Einstein had one more trick up his sleeve. In a paper written in October
1910, he considered the way in which the blue colour of the sky is produced.
As far back as 1869, the British physicist John Tyndall had explained that
the blueness might be due to small dust particles or droplets of liquid
in the air bouncing short-wavelength blue light around, scattering it to
all parts of the sky, while longer-wavelength red and orange passed through
relatively unaffected. Other scientists had realised that the scattering
must be caused by the molecules of air themselves, but it was Einstein who
put the numbers in. He thereby proved that the blueness of the sky was connected
with the existence of molecules and, in the process, derived the value of
Avo-gadro’s number in yet another way. It was the ultimate piece of blue-sky
research.

Further reading Einstein: A life in science by John Gribbin and Michael
White (Simon & Schuster); Subtle is the Lord by Abraham Pais (Oxford
University Press).