Einstein's Masterwork

by John Gribbin

  Einstein actually completed the paper that became his dissertation, prosaically titled ‘A New Determination of Molecular Dimensions’, at the end of April, but he didn’t submit it to the University of Zurich until 20 July 1905. The delay may have been because he was busy working on his other ideas in the spring and didn’t make a final decision about which paper to submit until then. Although the title of the dissertation focuses on the sizes of molecules, the technique Einstein describes actually also gives a measurement of the number of molecules (or atoms) present, in this case in a solution. This is typical of the kinds of methods used to estimate the numbers and sizes of atoms and molecules in the 19th and early 20th centuries, and shows Einstein building on what has gone before, rather than leaping off in a new direction.

  The problem was that in mathematical descriptions of the behaviour of atoms and molecules, both the sizes and the numbers of these particles appear in the equations. As we all learned in school, if you have one equation involving one unknown quantity (usually denoted by x), you can solve the equation to find a value for the unknown quantity. But if you have a single equation involving two unknowns (x and y), you cannot find out what the values of the unknowns are. To do that, you need two different equations each involving x and y. So all the ‘classical’ methods for determining the sizes of molecules and the numbers of molecules in a certain amount of matter depended on using two equations to work out the two unknowns.

  The number that comes into these calculations was called Avogadro’s number, after the Italian who came up with the idea in 1811.b It is the number of particles (atoms or molecules) contained in an amount of material whose weight in grams is numerically equal to the atomic (or molecular) weight of the substance. The atomic weight of carbon, for example, is twelve; so twelve grams of carbon contains Avogadro’s number of atoms. The atomic weight of hydrogen is one, but each hydrogen molecule contains two atoms, so its molecular weight is two. So two grams of hydrogen gas also contains Avogadro’s number of molecules – and so on.

  One early attempt to work out the value of this number, and simultaneously the sizes of molecules, was made by the German Johann Loschmidt in the mid-1860s. His calculations involved the average distance travelled by particles in a gas between collisions with one another (called the ‘mean free path’) and the fraction of the volume of the gas actually occupied by the molecules themselves. He reasoned that in a liquid all the molecules must be touching each other with no gaps in between; so measuring the density of the liquid, which depends on the number and size of the molecules present, would tell you the volume occupied by Avogadro’s number of molecules. When the liquid is heated to become a gas, the actual molecules must still occupy the same volume as the original liquid, but now with lots of empty space between them. It’s only in the gas that the mean free path comes into the calculations.

  Loschmidt carried out his calculations for air, which is almost completely a mixture of oxygen and nitrogen, and had to use estimates for the density of liquid nitrogen and liquid oxygen which were not as accurate as modern measurements. He combined these with calculations of the mean free path, which also depends on the number and size of the molecules present, based on measurements of the way the pressure exerted by air changes when it is squeezed into a smaller volume. He found that a typical molecule of air must be a few millionths of a millimetre across, and estimated Avogadro’s number to be 0.5 × 1023 – which means a 5 followed by 22 zeroes, or 50 thousand billion billion.

  Einstein’s approach to the problem was in the same spirit of solving two different equations simultaneously, but used a very different kind of physical system. He realised that the sizes of molecules (and Avogadro’s number) could be inferred from measurements that had already been carried out on the behaviour of solutions of sugar in water. But, as we have said, he didn’t do the experiments. What was new about Einstein’s work (and what justified the award of his PhD) was the mathematical way in which he calculated how molecules of sugar would behave in such a solution, and how this would affect the measurable properties of the solution. What was particularly clever about the work wasn’t that it gave a value for Avogadro’s number or the size of molecules – the techniques based on the kinetic theory of gases, such as Loschmidt’s method, had already done that. Einstein’s special contribution was to find a way to get results as good as those obtained from the kinetic theory of gases using liquids alone. Previously, estimates based on studies of liquids had been very rough and ready. Along the way, as we shall see, he developed techniques with widespread applications for industry wherever suspensions of particles in liquids are used.

  The technique depended on the fact that sugar molecules are very much larger than molecules of water. In fact, as Einstein realised, because some water molecules actually attach themselves to the sugar molecules in the solution, the effective size of the sugar molecules is even bigger, which makes the assumptions used in his calculations even more accurate.

  It is easy to describe the thinking behind those calculations. When something is dissolved in water, the viscosity of the solution – its stickiness – increases. By assuming that each sugar molecule is a large sphere embedded in a sea of much smaller water molecules, Einstein was able to work out an equation which related this change in viscosity (which can be measured) to the total volume of the fluid occupied by the sugar (which depends on two unknown quantities, the size of each sugar molecule and the number of sugar molecules present). Because experimenters always measure the weight of sugar (or other stuff) being added to the solution, the number of molecules present can always be expressed in terms of Avogadro’s number.

  Then, Einstein looked at the way sugar diffuses through water, and calculated the force acting on a single sugar molecule as it moves through the sea of water molecules. This could be related to another measurable property of the solution, called its ‘osmotic pressure’, through another equation which itself depended on both the number of sugar molecules present and their size. So Einstein derived two equations, each of which included the two unknown quantities, Avogadro’s number and the size of a sugar molecule, and each of which was directly related to measurable properties of the solution, its viscosity and the osmotic pressure.

  Once he had that pair of equations, it was a simple matter to plug in the numbers for viscosity and osmotic pressure that were already well known and had been published in standard tables listing the properties of such solutions. For the record, the ‘answers’ that came out of the equations were that sugar molecules (which are much bigger than molecules of nitrogen or oxygen) have a radius of about 9.9 × 10–8 cm (9.9 hundred-millionths of a centimetre) and Avogadro’s number is 2.1 × 1023 (210 thousand billion billion). This pretty much agrees with Loschmidt’s estimate, but the impressive part of the dissertation is the bit we can’t put into words, the sophisticated mathematical techniques that Einstein used to deduce the relevant equations.c

  The best way to appreciate just how good the maths was is to look at what the professors who examined the work said in their official report to the University of Zurich. Alfred Kleiner commented that: ‘the arguments and calculations to be carried out are among the most difficult in hydrodynamics and could be approached only by someone who possesses understanding and talent for the treatment of mathematical and physical problems … Herr Einstein has provided evidence that he is capable of occupying himself successfully with scientific problems’. His colleague Heinrich Burkhardt said that: ‘the mode of treatment demonstrates fundamental mastery of the relevant mathematical methods.’3 Einstein himself later told his biographer Carl Seelig that the only official comment he had received on the dissertation was that it was too short, and that in response he had added a single sentence, whereupon it was accepted. Even though the dissertation itself was officially accepted by the University of Zurich early in August 1905, it took until 15 January 1906 to complete all the various formalities required by the University for the degree to be conferred. So all the paper
s he completed during the annus mirabilis were the work of simple ‘Herr Einstein’, not ‘Herr Doktor Einstein’.

  Just after the thesis was accepted, Einstein submitted a slightly revised version to the Annalen der Physik, but publication was delayed until 1906 because the editor of the journal, Paul Drude, knew of some more accurate and upto-date measurements of the properties of sugar solutions than the ones Einstein had used. When he asked Einstein to take account of these data, the result was a slight change in the numbers he came up with – in the right direction, we now know. Even that wasn’t the end of the story, because Einstein’s paper eventually encouraged other experimenters to measure the relevant properties of these and other solutions even more accurately. It also turned out that Einstein had made a minor error in his calculations, which had not been spotted by either of his examiners or by Drude. The final, definitive version of Einstein’s method for calculating Avogadro’s number from the properties of sugar solutions only appeared in the Annalen der Physik in 1911, and gave a value of 6.56 × 1023, which is very close to the accepted modern value, 6.02 × 1023.

  The saga of Einstein’s doctoral dissertation did not end there, however. In scientific terms, this is the most mundane of the papers he wrote during the annus mirabilis. But it has one curious distinction. It became far more widely quoted than any of his truly revolutionary papers from the same year.

  One way in which scientists measure the value of scientific papers is to record how often they are referred to in other scientific papers. This is by no means a perfect system, as witnessed by the fact that Einstein’s original paper on the Special Theory of Relativity has been very seldom referred to. The reason for that, of course, is that the content of the paper quickly became part of the established fabric of science, something taught from textbooks that ‘everybody knows’, so that few scientists have even read the paper, let alone cited it.

  By contrast, the paper based on the doctoral dissertation has been very widely cited. Just how widely was brought home in 1979, as part of the celebrations to mark the centenary of Einstein’s birth. Two researchers carried out a survey of the citations received not just by Einstein’s papers but by all the papers in science (what they called the ‘exact’ sciences, like physics and chemistry) published before 1912.4 The twist was that they only looked at citations in papers that had themselves been published between 1961 and 1975; so they came up with a list of all the papers that were still important enough to be quoted at least 50 years after they had originally been published. Out of the top eleven ‘most cited’ papers in this survey, four were by Einstein. (No other scientist had more than one paper in the top eleven.) And top of the four papers by Einstein came his doctoral dissertation.

  Why was this seemingly mundane paper cited so often between 1961 and 1975? Simply because it is mundane. It deals with practical things important in the everyday world – the behaviour of fluids with particles suspended in them. The equations Einstein derived are relevant to (among other places) the dairy industry, where it is important to understand and predict the behaviour of milk during the process of making cheese; the study of pollution and the way tiny particles called aerosols get spread through the atmosphere; and problems involving the behaviour of cement being transported in liquid form, and the design of the lorries to carry the cement. The work Einstein did for his doctorate in 1905 turned out to be of widespread importance in many practical applications in the second half of the 20th century, and is still relevant today, more than a hundred years after the dissertation was written.

  Jiggling atoms

  For the same reasons, the second most cited of Einstein’s papers in that 1979 survey did not concern the Special Theory of Relativity or quantum physics (indeed, neither of those papers even made the top four), but the phenomenon known as Brownian motion. Appropriately, it was the paper on Brownian motion that Einstein returned to as soon as he had finished drafting what would become his doctoral dissertation, at the end of April 1905. On 11 May, Paul Drude received a paper with the splendid title ‘On the Motion of Small Particles Suspended in Liquids at Rest Required by the Molecular-Kinetic Theory of Heat’.d He had no hesitation in accepting it for publication.

  Brownian motion got its name from the Scottish botanist Robert Brown, who first studied the phenomenon in detail in 1827. Intriguingly, though, Einstein wasn’t trying to explain Brownian motion in this paper; indeed, in a sense he was predicting it, on the basis of his statistical approach to the kinetic theory, honed in the series of three papers mentioned earlier. That’s why the term ‘Brownian motion’ doesn’t appear in the title of the paper. In the first paragraph of the paper, Einstein says:

  It is possible that the motions to be discussed here are identical with so-called Brownian molecular motion; however, the data available to me on the latter are so imprecise that I could not form a judgement on the question.

  But since we now know enough to form that judgement, it makes sense to introduce this aspect of Einstein’s work by looking at just what it was that Robert Brown discovered.

  Even before Brown’s time, people had noticed the way tiny grains of material, notably pollen, seem to dance about in a jittery kind of motion, something like running on the spot, when they are suspended in a liquid such as water and observed through the microscope. So Brown didn’t discover Brownian motion. Before Brown’s work, however, the obvious explanation for this motion seemed to be that the particles were alive – after all, pollen grains are a kind of plant equivalent to the sperm cells in animals, and if sperm can move under their own steam, why shouldn’t pollen? When Brown began his detailed studies in the summer of 1827 (the results were published in 1828), he thought that this was the most probable explanation. But then he made the next logical step. He took a series of clearly inanimate materials, such as ground up fragments of glass and granite, and suspended them in water. He found exactly the same behaviour for these definitely non-living materials, proving that the motion of a particle in suspension has nothing to do with any mysterious life force. ‘These motions,’ he wrote in that 1828 paper, ‘were such as to satisfy me, after frequently repeated observation, that they arose neither from currents in the fluid nor from its gradual evaporation, but belonged to the particle itself.’5 It was this discovery, a result of the truly scientific way he went about his work, that meant his name would be forever associated with the phenomenon.

  But if a life force wasn’t causing the motion, what was? Over the next few decades, people considered the possibility that convection currents might be involved (in spite of Brown’s comments), or electrical effects, or the same force that caused capillary action, and other more or less wild ideas. The key experimental discoveries were that the speed of this jiggling increased if the temperature of the water increased, and was less for bigger particles. Combining this with the ideas of the kinetic theory gave rise to the suggestion that the particles were being bombarded by the molecules in the water, and were being jerked about in response to the kicks they received from individual molecules. But in order for a single molecule to produce a visible shift in a pollen grain or a speck of granite dust, the molecule would either have to be impossibly big, or travelling impossibly fast.

  This was more or less where the puzzle of Brownian motion stood at the beginning of the 20th century. It is clear from his writings, though, that Einstein had not read up on all of these developments, and was not up-to-date on the subject. He was aware of the phenomenon of Brownian motion, but his theoretical studies of how particles suspended in liquids ought to move were not specifically intended to explain that phenomenon. Rather, they were a logical development from the work in his doctoral dissertation.

  As Einstein has told us, what he was really interested in at that time was proving the reality of atoms and molecules. He was completely convinced of the validity of the kinetic theory of heat, and saw in this extension of his PhD work a way to convince others as well. For this purpose, the distinction between atoms and molecules is
of no significance. Atoms are the fundamental component of elements, such as hydrogen and oxygen, and molecules are the basic components of compound substances, such as water (where two hydrogen atoms combine with one oxygen atom in each molecule of water).

  At its simplest, the kinetic theory says that everything is made of tiny particles (atoms or molecules) which can be regarded as little, hard spheres. In a solid, the little spheres are packed closely together and do not move past one another. In a liquid, the little spheres buffet each other and slide past one another like people moving through a dense crowd, but they are still essentially touching all their neighbours. In a gas, the little spheres fly freely through empty space, bouncing off each other and the walls of any container they are in. The hotter a substance is, the faster the spheres move, which explains the transition from solid to liquid to gas as a substance is heated, and from gas back to liquid and then solid when it cools.

  In the paper that became his doctoral dissertation, Einstein had already used the idea that molecules of sugar dissolved in water are being bombarded by water molecules from all sides, and that the way the sugar molecules move through the sea of water molecules affects measurable properties of the solution: its viscosity and its osmotic pressure. The success of Einstein’s results from that paper already provided powerful circumstantial evidence in favour of the kinetic theory, but even that was not direct proof that atoms and molecules exist. To obtain that, the effects of the bombardment by water molecules had to be scaled up somehow, to become visible, at least under the microscope. A pollen grain, tiny though it is by any human standard (about a thousandth of a millimetre across), is enormously much bigger than a water molecule (measured in millionths of a millimetre), or even a molecule of sugar. But Einstein made the huge mental leap of realising that as far as the behaviour of particles suspended in liquids was concerned, this was the only difference that mattered between a pollen grain (or a fragment of granite) and a sugar molecule. In what I shall refer to as the Brownian motion paper, he said: