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Molecular reality: Brown, Einstein, Perrin

Molecular reality: the contributions of Brown, Einstein and Perrin Mick Nott The story of Brownian motion and its importance in modern science

Einstein’s 1905 paper on the random movement of small particles in a fluid was original and used (the then recent) statistical methods for explaining this phenomenon. He made predictions that other scientists could independently experimentally verify but he was not the first to use these small particles to convince scientists that atoms were real.

Robert Brown and his motion Textbooks tell us that, in 1827, Robert Brown noticed that fragments of pollen grains, suspended in water, moved in a random jiggling motion. Textbooks do not tell us that he had sex on his mind. Brown, the curator of the botany collection at the British Museum and an expert microscopist, was looking for evidence of ‘vital forces’ in living material. In particular, he was trying to distinguish by their movement male parts of plants, which he thought would be active, from female parts of plants, which he thought would be passive. He found that the fragments of pollen particles did move and at first thought his hypothesis promising. He went on to test fragments of pollen grains from a range of plants that had recently died ABSTRACT This article tells a story about Brownian motion from 1827 when Robert Brown investigated it to 1926 when Jean Perrin was awarded the Nobel Prize in Physics for his work on measuring the distribution and motion of Brownian particles in fluids. The article finishes with a plea to keep Brownian motion as a part of the school science curriculum and discussion of its modern relevance and importance.

and then, drawing on the collections around him, from plants that had been dead for over 100 years. Age and vitality of the plant didn’t matter; all the fragments moved. Brown was trying to understand reproduction in gymnosperms, plants that don’t involve flowers in reproduction, and he went on to test parts of what he thought might be a female part of the plant. Again he found that the fragments moved. He thought the activity might be intrinsic to, and characteristic of, both male and female parts of plants. Then, accidentally, some fragments from a crushed leaf got into his sample and these moved too. He didn’t think these leaves would have either male or female characteristics. So, he abandoned his original hypothesis and set out to test another contemporary one: all parts of all organic bodies will contain small ‘molecules’ (Brown’s word) that are elementary and contain vital life forces. His research showed that the movement was intrinsic to small fragments of plants: it was not an indicator of sex and it was not an indicator of whether the plant was alive or dead. Brown then wondered whether the motion would indicate the products of plants so he tested fragments from a variety of vegetable products (‘particularly the gum resins’) and found they moved too. He went on to test fragments of animal tissue and these moved as well, irrespective of whether the animal had been alive or dead. His thinking led him on to test fragments from fossilised organic matter, for example coal, and he found London soot seemed to be ‘entirely composed of these molecules’. He tested fragments of petrified wood: same result. He concluded that this motion was in everything provided the particles were small

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Molecular reality: Brown, Einstein, Perrin

enough. To test this generalisation he tried glass, metals and rocks; he raided other bits of the British Museum’s collections and found the movement in a crushed piece of the nose of the Sphinx. He concluded that he did not know why this movement of the particles occurred but he did know that it was not a way to tell the living from the non-living or the neverlived. He privately circulated this paper in 1828 to a small but select audience (Brown, 1828); he was criticised and other hypotheses were suggested. Brown decided to test the criticisms that the motion may be due to some kind of attraction or repulsion between the small particles or due to evaporation of the water drop that contained the particles. He devised a new experiment. He took some water, in which active particles were suspended, and shook it up in almond oil, which has approximately the same relative density as water. Thus he produced a suspension of minute water drops in the oil and active ‘molecules’ were suspended within these water drops – in fact some of the water drops had only one active particle. The active particles still moved in their random jiggling way, even though the water could not evaporate. In the water drops that contained only one active particle the solitary particle was still seen to be moving, even though there was not another active particle present to attract or repel it (Figure 1). water drops with ‘active’ particles almond oil water drop with one ‘molecule’, still active

Figure 1 Diagram of Brown’s water drops suspended in almond oil.

He then took some almond oil and shook it up in water in which the active particles were suspended. Thus he produced a suspension in the water of minute drops of almond oil as well as the active particles. He found that the oil drops were also active provided they were small enough. Brown answered his critics by publishing these results in 1829 (Brown, 1829). He

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decided that this research programme was not going to be solvable or productive and he abandoned it.

Constant anomaly At the time, other scientists either had other interests or had no inkling of why the particles were moving, and so during the nineteenth century this random jiggling motion remained an unexplained anomaly. By the time Brown died in 1858, it was so well known that it had acquired the soubriquet of ‘Brownian motion’. By the end of the nineteenth century Maxwell had established a physical kinetic theory that explained some macroscopic phenomena about gases in terms of hard, elastic, indivisible and invisible particles, atoms or molecules, but it still had anomalies as there were some macroscopic phenomena that it could not explain. Boltzmann in Europe and Gibbs in America were developing Maxwell’s ideas into statistical mechanics. Chemists were using atoms and molecules as accounting devices for chemical reactions. Many, if not most, of these nineteenth-century scientists were content to use atoms and molecules as ways of predicting and accounting for physical and chemical phenomena in an instrumental or operational sense, that is, the theories worked, but they were not necessarily content to believe in atoms and molecules in a positive or real sense. The theories may have worked but they did not confirm that atoms and molecules are real. By the beginning of the twentieth century, the reality of atoms and molecules was the subject of animated debate amongst scientists (Box 1).

Einstein’s contribution At the beginning of the twentieth century, Einstein was determined to provide a theoretical explanation that predicted measurements that could be made and would strengthen the atomic hypothesis. He imagined that the myriad atoms in a fluid would fluctuate in their movement and that, statistically, in small intervals of time these fluctuations would be unsymmetrical: the sum of the momentum of atoms or particles moving one way would be greater than the sum of the momentum of the atoms or molecules moving the other way. This would mean that a microscopic but visible particle would experience unbalanced forces on it over a short period of time

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Molecular reality: Brown, Einstein, Perrin

Box 1 Arguments used for and against the existence of atoms and molecules in the early twentieth century Objections to the reality of atoms

Support for the reality of atoms

■ Invisible atoms can have no directly observable evidence.

■ Wave theories predict soap film depths smaller than we can directly measure but they are considered real. ■ Results from radioactivity and cathode rays suggest that atoms may not be indivisible. ■ Kinetic theory and statistical methods have explained many phenomena and the anomalies will be explained in time – they are not counterexamples. ■ Highly successful theories do not always last either – scientists must remain open to changing their ideas if the evidence warrants it.

■ Indivisible atoms cannot account for the energy changes in chemical reactions. ■ Kinetic theory has its own unexplained anomalies and may be overthrown. ■ Believing in atoms and molecules assumes that matter is discontinuous and would undermine scientists’ belief in highly successful theories, e.g. thermodynamics and wave theories, that assume that matter is continuous. ■ Too much imagination with no successful experimental corroboration will give science and scientists, who rely on public money, a bad name.

due to more collisions with atoms moving one way than the other. As a consequence, it would be buffeted one way and the other in a random fashion, and as it tried to accelerate it would be slowed down by friction with the fluid. Summing the movement due to this constant buffeting over longer periods of time shows that the small, microscopic but visible particles ‘drift’ in the fluid in a random motion, sometimes called ‘drunkard’s walk’. Using statistical methods and considering diffusion, osmosis and the friction due to the viscosity of fluid, Einstein’s 1905 paper, ‘On the movement of small particles suspended in a stationary liquid demanded by the molecular-kinetic theory of heat’ (see endnote 1), derived a formula that related the displacement of the visible microscopic particle over time to the temperature and viscosity of the suspending fluid, the radius of the particle and, most importantly, Avogadro’s number. The determination of Avogadro’s number, NA – the number of particles (6.02 × 1023 mol–1) in one mole of a substance – was akin to a search for the Holy Grail amongst atomists. If you could understand Einstein’s paper, here was something to measure. But who had Brownian particles of a precise calibrated diameter and the skills with a microscope and the research students to undertake this work?

■ Science is characterised by the production of imaginative models tested against reality.

Perrin and the determination of NA The person who had all these was the French scientist Jean Baptiste Perrin, another committed atomist. He had cut his research teeth at the end of the nineteenth century on cathode rays (readers may have a Perrin’s tube in their school cupboards somewhere). He moved on to colloid chemistry and this brought him into close contact with Brownian motion. Supervising a PhD on the diffusion of colloids gave him an idea for a research programme to support the atomic hypothesis. He started this research programme about 1905 – but Einstein’s original paper was difficult for most scientists to understand and Perrin was unaware of it for at least another two years. As a physical chemist, Perrin knew the work of Van’t Hoff and Raoult that had shown that dilute solutions of undissociated particles could be shown to obey the gas laws. Perrin speculated that if the gas laws worked for these hypothesised solute particles, then a suspension of Brownian particles, which are just visible, should also obey some of the gas laws as predicted by kinetic theory. If this were so, he reasoned that a suspension of Brownian particles should distribute themselves in an isothermal solution according to the exponential law for a gas atmosphere

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Box 2 The law of atmospheres Consider a thin slice of gas of cross-sectional area A and height ∆h in equilibrium in an isothermal atmosphere. A

∆h

The force up on the slice due to the pressure below the slice is given by Fup = ph A

The universal gas law is given by

(2)

R = kN A where k is Boltzmann’s constant and NA is Avogadro’s number. Therefore (8) can be written as

Total force due to excess pressure is given by

Fup − Fdown = ph A − ph +∆h A

(3)

Weight of the slice = nmgA∆h

(4)

where n is the number of gas particles per unit volume, m is the mass of each particle, and g is the acceleration due to gravity. At equilibrium these two forces must sum to zero (Newton I). ( ph A − ph +∆h A) + nmgA∆h = 0

(5)

( ph − ph +∆h )A = −nmgA∆h

(6)

pV = μkN AT

as ∆p and (6) becomes

(9)

N N A where N is the total number of particles in the slice of gas. Therefore (9) can be rewritten as

μ=

p=

but

Divide through by A and write ( ph − ph +∆h ) ∆p = −nmg∆h

(8)

where p, V and T are the pressure, volume and absolute temperature, R is the molar gas constant and μ is the number of moles of gas.

The force down on the slice due the pressure above the slice is given by Fdown = ph +∆h A

pV = μRT

(1)

1 N kN AT V NA

(10)

N = n therefore (10) becomes V p = nkT

(11)

(7)

Divide (7) by (11):

∆p nmg∆h =− p nkT

(12)

which, with the usual physicist’s disregard for the finer points of calculus, becomes h

h

dp mg =− dh p kT ∫0 0

(13)

ph mgh =− p0 kT

(14)

ph = p 0e −mgh / kT

(15)

∫

ln

which is the exponential law of atmospheres: comparing 11 and 14, and as it is an isothermal atmosphere, we can also write

nh = n0e −mgh / kT

(16)

The number of particles at any particular height is an exponential function dependent on height from the origin and also dependent on k which is R/NA. Perrin measured nh, T and m and calculated a value for NA. This method for determining Avogadro’s number is independent of Einstein’s work and original to Perrin.

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Molecular reality: Brown, Einstein, Perrin

Figure 2 Perrin’s diagrams of his apparatus.

(see Box 2). If his experiment showed this distribution, then the Brownian particles obeyed kinetic theory and, if this were true for particles that were just visible, then it would also be true for particles that were just invisible – and one could infer that all particles would obey kinetic theory. The first task entailed the painstaking manufacture of precisely calibrated Brownian particles. He spent months centrifuging gum resin precipitates, isolating a particular fraction, and then re-centrifuging and isolating again until he had a few tenths of a gram of uniformly sized particles (diameter approx 1 micron), which, if they had been gas particles, would have a gram molecular mass of about 100 000 tonnes! Then he made suspensions of these particles and measured the distribution of the particles through direct observation and photography at different heights in fluids (see Figures 2 and 3). He varied the density of the fluid and tried fluids that were less dense than the particles, checking that the exponential distribution was inverted – more particles at the top than the bottom. Perrin and his research students found that the Brownian particles did distribute themselves according to an exponential law. The formula for this exponential distribution contains Avogadro’s number and, by 1908, independently of Einstein, Perrin had determined a value of between 6.5 and 7.2 × 1023, close enough to values that were emerging from experiments in other fields, such as electrolysis. When Perrin was shown Einstein’s paper (and an alternative easier-to-understand derivation by Langevin) he had all the resources to verify Einstein’s predictions straight away and the results agreed with his earlier determination. Perrin was delighted to be able to do this, and his 1916 book Atoms contains many diagrams of drunkard’s walk and the distribution of Brownian particles from their starting position over time (Figure 4). At last, the atomists had two

Figure 3 Perrin’s diagram of distribution of Brownian particles with height (5 photos at equidistant levels put one on top of the other from bottom to top level).

independently derived and empirically determined, consistent values for Avogadro’s number, deduced from the data from experiments that measured the displacement of particles one could see, and they had a kinetic and statistical explanation of Brownian motion – this 80-year-old anomaly. Perrin’s own determination of Avogadro’s number, its consistency with determinations by other methods, and his verification of Einstein’s predictions proved the watershed for the atomists and their theories: by about 1910, atoms and molecules were considered to be real by the scientific establishment. Perrin received the Nobel Prize for Physics in 1926, five years after Einstein got his for his work on the photoelectric effect and 99 years after Robert Brown’s original observations, for his advocacy and proof of the atomic hypothesis and for his direct empirical determinations of Avogadro’s number.

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Molecular reality: Brown, Einstein, Perrin

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Figure 4 Diagram of position of three particles plotted every 30 s (16 divisions on the graticule equals 50 microns).

Why teach Brownian motion in schools? Avogadro’s number can be determined in other ways, but this experiment, with its directness and small number of steps of inference and induction from the just visible to the just invisible, was the experiment that swung the balance for the reality of atoms. I was never taught it at school but I had to learn to use the Whitley Bay Smoke Cell in my teacher training. Textbooks and teachers can indicate confusion about what is happening. The smoke particles jiggle randomly not because they have been hit by an air particle but because the sum of the collisions with the myriad air particles at any one point in time is not zero (1 cm3 of air at room temperature and pressure contains about 2.5 × 1019 air particles). A collision with one air particle would only cause an infinitesimal, unobservable movement of a smoke particle. Also, the smoke particles do not move because they hit each other – which, in my experience, is an error of observation frequently made by students and which has to be corrected by careful questioning on the part of the teacher. We can now do the experiment differently (see the Science note, pages 18–24) and software and imaging allows us to collect results and display them graphically very quickly.

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Around 1990, Benoit Mandelbrot’s 1982 book, The fractal geometry of nature, was becoming popularised through the use of PCs to generate the Mandelbrot set; that was when I dipped into it. One of Perrin’s diagrams (see Figure 5) is one of the few diagrams in the book because Mandelbrot was inspired by something that Perrin wrote. Perrin noted that drunkard’s walk plots the position of the particle at equal time intervals (30 s in Perrin’s case), but went on to say it would be useless to try to measure an average velocity of the particles as the ‘apparent average velocity varies crazily in magnitude and direction’. He noted that his diagrams of drunkard’s walk ‘only give a weak idea of the prodigious entanglement of the real trajectory. If [the] particles’ positions were marked down 100 times more frequently each interval would be replaced by a polygon smaller than the whole drawing but just as complicated’. In other words, the path is infinitely complex and scale invariant: if you could plot the position every second then you would have smaller displacements but a path just as complex, and so on ad infinitum. Mandelbrot notes that topologically the path has a dimension of 1 but that it is a plane-filling path, which must have in some sense a dimension of 2, and therefore it has a fractal dimension between 1 and 2.

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Molecular reality: Brown, Einstein, Perrin

Figure 5 The diagram of a particle trajectory Perrin used to indicate the discontinuity and irregularity of the motion. The path between any two plotted points would be just as irregular and discontinuous ad infinitum.

Brownian motion has implications for the second law of thermodynamics. A nineteenth-century thought experiment was to imagine light threads attached to Brownian particles, with the threads running over frictionless pulleys and ratchets attached to weights. Wouldn’t one be able to get work out of this without putting in more heat energy? This is just as relevant today, when scientists are still debating and experimenting to see if Brownian motion violates the second law in nanoscale and small time intervals. Brownian motion may have consequences in nanotechnology; it may be that Brownian motion may set a limit to how small we can make things to operate, in ordinary atmospheres, reliably and not haphazardly

(see endnote 2). The polystyrene beads that are used in the Science note (pages 18–24) are used as a substrate in microbiology experiments and the effects of Brownian motion are notable in them. Brownian motion should be studied because it is significant in science; its explanation and the derived results from its study convinced the scientific establishment that atoms and molecules were real and therefore it is culturally relevant. And Brownian motion is still contemporary: from fractals, chaos and stochastic processes to nanosystems and microbiology experiments, it still acts as a stimulus for new research or has to be taken into account.

Endnotes 1 The original English translation of Einstein’s paper is at http://lorentz.phl.jhu.edu/AnnusMirabilis/ AeReserveArticles/eins_brownian.pdf 2 See http://rsc.anu.edu.au/~gmw/newsonFT.rtf for a fascinating collection of articles and papers about Brownian motion and its consideration in nanosystems.

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Molecular reality: Brown, Einstein, Perrin

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Sources

Websites

Brown, R. (1828) A brief account of microscopical observations made in the months of June, July and August 1827 on the particles contained in the pollen of plants and on the general existence of active molecules. Privately published, Science Museum London. For an abridged version see Source book in physics, ed. Magie, W. F. (1965) Massachusetts: Harvard University Press.

All the scientists involved in the story can be ‘Googled’ and the Nobel Prize website has Perrin’s Nobel Lecture. NB The photographs of the scientists on the Nobel site can be used for educational purposes. I tried two quick Google searches: ‘latex beads’+ ‘Brownian motion’ and ‘Einstein’+ ‘Brownian Motion’. It’s a good start. Brown’s observations have been reproduced with his original microscope as a ‘test’ to settle a dispute over whether he could have seen the eponymous motion. For more details about Brown and a convincing video-clip that Brown could have seen Brownian motion with his microscope visit: http://www.brianjford.com/wbbrowna.htm (last viewed 18 April). Many thanks to Dave Sang for leading me to this URL.

Brown, R. (1829) Additional remarks on active molecules. Privately published, Science Museum London. An unabridged version is in Science before Darwin, ed. Cohen, I. B. and Jones, H. M. (1963) London: Deutsch. Brush, S. G. (1968) A history of random processes: Brownian movement from Brown to Perrin. Archives of the Exact Sciences, 5, 1–36. Nott, M. (1992) The nature of science or why teach Brownian motion? In Open chemistry, ed. Atlay, M. et al. pp. 3–16. Milton Keynes: Open University Press. Nye, M. J. (1972) Molecular reality: a perspective on the scientific works of Jean Perrin. London: Macdonald. Perrin, J. B. (1916) Atoms. London: Constable. (This is the first English translation of Perrin’s book Les Atomes, which was published in France in 1913. The translation is the fourth revised French edition.) Figures 2–5 have been taken from this book. Walton, A. J. (1983) The three phases of matter. Oxford: Clarendon Press.

Mick Nott is currently a Consultant with the Science Enhancement Programme (www.sep.org.uk).

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