Monday, September 05, 2016
Atoms Old and New, 2: From Newton to Einstein
by Paul Braterman
Part 1 of this series, "Atoms Old and New: Atoms in Antiquity" can be read here.
The transition to modern thinking
"It seems probable to me, that God in the beginning formed matter in solid, massy, hard, impenetrable, movable particles… even so very hard, as never to wear or break in pieces; no ordinary power being able to divide what God Himself made one in the first creation." So wrote Sir Isaac Newton in his 1704 work, Opticks. Apart from the reference to God, there is nothing here that Democritus would have disagreed with. There is, however, very little that the present-day scientist would fully accept. In this and later posts, I discuss how atoms reemerged as fundamental particles, only to be exposed, in their turn, as less than fundamental.
The scientific revolution and the revival of corpuscular theory - 1543–1687
In 1543, on his death-bed, Nicholas Copernicus received a copy of the first edition of his book, On the Revolutions of the Heavenly Bodies, in which he argued that the Sun, not the Earth, was thecentre of what we now call the Solar System. In 1687, Isaac Newton published his Mathematical Principles of Natural Philosophy, commonly known as the "Principia". With hindsight, we can identify the period between these events as a watershed in the way that educated people in the West thought about the world, and number the political revolutions in America and France, and the economic revolutions in agriculture and industry, among its consequences.
Before this scientific revolution, European thinking about nature still followed that of Aristotle. The Earth lay at the centre of the Universe. Objects on Earth moved according to their nature; light bodies, for instance, containe, air or fire in their makeup, and these had a natural tendency to rise. Earth was corrupt and changeable, while the heavens were perfect and immutable, and the heavenly bodies rode around the centre on spheres within spheres because the sphere was the most perfect shape. By its end, Earth was one of several planets moving round the Sun in elliptical orbits, the movements of objects were the result of forces acting on them, the laws of Nature were the same in the heavens as they were on Earth, and all objects tended to move in straight lines unless some force deflected them from this path. The Universe ran, quite literally, like clockwork. This mechanical world-view was to last in its essentials until the early 20th century, and still remains, for better or worse, what many non-scientists think of as the "scientific" viewpoint.
Left: manuscript where Galileo records his observations of the motion of the moons of Jupiter, dethroning Earth from its special position as centre of celestial motion. Below right, Gallileo demonstrates the telescope to the Doge of Venice, fresco by Bertini. Click to enlarge
In 1611, Galileo turned the newly-invented telescope on the heavens, discovered sunspots, and moons round Jupiter, and realised that the belief in a perfect and unchanging1 celestial realm was no longer sustainable. Earlier, he had studied the motion of falling bodies. In work that he started in 1666, Newton showed how the laws of falling bodies on Earth, and the movement of heavenly bodies in a Copernican solar system, could be combined into a single theory. To use present-day language, the Moon is in free fall around the Earth, pulled towards it by the same force of gravity as a falling apple. This force gets weaker as we move away from Earth, according to the famous inverse square law, which says that if we double the distance, the force falls to a quarter of its value. Then with a certain amount of intellectual effort (involving, for example, the invention of calculus), Newton was able to work out, from the acceleration of falling bodies on Earth, and from the Earth-Moon distance, just how long it should take the Moon to go round the Earth, and came up with the right answer. He was also able to work out just how long it would take satellites at different distances to go through one complete orbit. Of course, at that time, Earth only had one satellite (the Moon), but six were known for the Sun (Mercury, Venus, Earth, Mars, Jupiter, Saturn), and his theory correctly predicted how the length of the year of these different planets would vary with their distance from the Sun (the answer is a 2/3 power law; an eight-fold increase in distance gives a fourfold increase in time). Celestial and terrestrial mechanics were united.
It was around this time that a Dutchman, Anthony van Leeuwenhoek, began an extensive series of microscope studies, using single lens instruments of his own devising. Among the first to observe spermatozoa, he also described bacteria, yeast, the anatomy of the flea, and the stem structure of plants. He communicated his results to the Royal Society in London. Formally established around 1660, under the patronage of Charles II, this was and remains among the most prestigeful of learned societies. Here they caught the attention of Robert Boyle (of Boyle's Law for gases). Boyle tried to explain such properties of matter as heat, and the pressure of gases, in terms of the mechanics of small particles, or "corpuscles", and hoped that the other aspects of matter could be explained in the same kind of way. This was, after all, simply an extension downwards of the mechanical system that Newton had so successfully extended upwards. It is instructive to consider how far this hope was fulfilled. Atoms and molecules are in some ways similar in their behavior to small objects obeying the everyday laws of mechanics, but in others they are very different, and it is these differences that must be invoked if we are to understand the forces involved in the chemical bonding.
Early modern theory - 1780-1840
Between 1780 and 1840, chemistry underwent a revolution, that transformed it into the kind of science that we would recognise today. It is no accident that this was the same period as the beginning of the industrial revolution in Europe. Materials were being mined, and iron and steel produced and worked, on a larger scale than ever before. By the end of the period, mineral fertilisers were already in large scale use to feed the growing population. Demand for machinery led to improvements in engineering, and this in turn made possible improvements in the precision of scientific instruments. Much of the new interest in chemistry grew out of mining, mineralogy, and metallurgy, while improvements in manufacture and glass-blowing led to the precision balance, and to new apparatus for handling gases.
Here I will summarise some of the most important discoveries, as seen from our present point of view, and using today's language. This means running the risk of creating a misleading impression of smoothness and inevitability. Inevitability, perhaps yes; the world really is what it is, and once certain questions had been asked, it was inevitable that we would eventually find the right answers. Smoothness, no; the very concept of atoms, let alone bonding between atoms, remained controversial in some circles way into the 20th century. Outsiders sometimes criticise scientists for taking their theories too seriously, but more often they are reluctant to take them seriously enough.
Overall, mass is conserved; the mass of the products of a reaction is always the same as the mass of the reactants. This is because atoms are not created or destroyed in a chemical reaction.2 Single substances can be elements or compounds, and the enormous number of known compounds can be formed by assembling together the atoms of a much smaller number of different elements. We owe our distinction between elements and compounds to Lavoisier ("The banker who lost his head"). Boyle had come close a hundred years earlier, but was so taken with the transformations of matter that he rejected the notion that its fundamental constituents were immutable.3
The combustion of carbon (its reaction with oxygen) gives a gas, the same gas as is formed when limestone is heated. But there is no chemical process that gives carbon on its own, or oxygen on its own, by reaction between two other substances. So we regard carbon and oxygen as elements, whereas the gas formed by burning carbon (what we now call carbon dioxide) is a compound of these two elements. The production of this same gas, together with a solid residue, by the heating of limestone, shows that limestone is a compound containing carbon, oxygen, and some other element.4 To us, using today's knowledge, limestone is calcium carbonate, and decomposes on heating to give carbon dioxide and calcium oxide. In Lavoisier's time, there was no way of breaking down calcium oxide into simpler substances, so he considered it to be an element.
A short philosophical digression (and every scientist has a working philosophy, whether they realise it or not): Lavoisier could make as much progress as he did because he had introduced an operational definition of an element, referring not to some inner essence but to observationally defined properties. And implicit in this was the principle of fallibilism; conclusions are always in principle revisable in the light of further observation, as the example of calcium oxide shows.
Air is a mixture, and burning means reacting with one of its components, which we call oxygen. Metals in general become heavier when they burn in air. This is because they are removing oxygen from the air, and the weight (more strictly speaking, the mass) of the compound formed is equal to that of the original metal plus the weight of oxygen. (Mass is an amount of matter; weight is the force of gravity acting on that matter. Atoms are weightless when moving freely in outer space, but not massless.)
Different elements combine with different amounts of oxygen; these relative amounts are a matter of experiment. In modern language, when some typical metals (magnesium, aluminium, titanium, none of which were known when Lavoisier was developing his system) react with oxygen, they form oxides with the formulas MgO, Al2O3, TiO2.
About one fifth of the air is oxygen, and if we burn anything in a restricted supply of air, the fire will go out when the oxygen has been used up. Nothing can burn in (or stay alive by breathing) the remaining air. Some materials, like wood and coal, appear to lose weight when they burn, but this is because they are in large measure converted to carbon dioxide and water vapour, which are gases, and we need to take the weight of these gases into account.
It was also shown during this period that the relative amounts of each element in a compound are fixed (Law of Definite Proportions). For instance, water always contains 8 grams of oxygen for each gram of hydrogen. Moreover, when the same elements form more than one different compound, there is always a simple relationship between the amounts in these different compounds (Law of Multiple Proportions). Thus hydrogen peroxide, also a compound of hydrogen and oxygen, contains 16 grams of oxygen for each gram of hydrogen. Similarly, the gas (carbon dioxide to us) formed by burning carbon in an ample supply of oxygen contains carbon and oxygen in the weight ratio 3:8, but when the supply of oxygen is restricted, another gas (carbon monoxide) is formed, in which the ratio is 3:4. Carbon monoxide is intermediate in composition between between carbon and carbon dioxide, but it is not intermediate in its properties. For a start, it is very poisonous; it sticks to the oxygen-carrying molecules in the blood even more strongly than oxygen itself, thus putting them out of action. It is formed when any carbon-containing fuel, not just carbon itself, burns in an inadequate supply of air, That is why car exhaust fumes are poisonous, and why it is so important to make sure that gas-burning appliances are properly vented. It is also one of the components of cigarette smoke, which helps explain why cigarettes cause heart disease and reduce fitness.
Left: Dalton's table of the elements, with relative weights, based on H = 1. The correct value for oxygen is 16. Dalton's value is based on an assumed formula HO for water, together with experimental error; likewise for other elements. Click to enlarge
All these facts can be explained if the elements are combined in molecules that are made out of atoms, the atoms of each element all have the same mass,5 and each compound has a constant composition in terms of its elements. For instance, each molecule of water contains two atoms of hydrogen and one of oxygen (hence the formula H2O); hydrogen peroxide is H2O2; carbon dioxide is CO2; carbon monoxide is CO; and the masses of atoms of hydrogen, oxygen, and carbon are in the ratio 1:16:12. Using these same ratios, we can also explain the relative amounts of the elements in more complicated molecules, such as those present in octane (a component of gasoline), C8H18, and sucrose (table sugar), C12H22O11. Why C8H18 and not C4H9, which would have the same atomic ratio? This can be inferred from the density of the vapour, using Avogadro's hypothesis (see below).
Thus, by the early 19th century, chemists were in the process of developing consistent sets of relative atomic weights (sometimes known as relative molar masses). However, there was more than one way of doing this. For instance, John Dalton, the first to explain chemical reactions in terms of atoms, thought that water was HO and that the relative weight of hydrogen to oxygen was one to eight. This uncertainty even led some of the most perceptive to question whether atoms were real objects, or merely book-keeping devices to describe the rules of chemical combination.
Evidence from the behavior of gases (to around 1860)
A French chemist, Joseph Gay-Lussac, noticed that the volumes of combining gases and of their gaseous products, were in simple ratios to each other. In 1811, the Italian Count Amadeo Avogadro explained this by a daring hypothesis, that under the same conditions of temperature and pressure equal volumes of gases contain equal numbers of molecules. We now know this to be (very nearly) true, except at high pressures or low temperatures.
Avogadro's Hypothesis, as we still call it, gives us a way of directly comparing the relative weights of different molecules, and of inferring the relative weights of different atoms. For example, if we compare the weights of a litre of oxygen and a litre of hydrogen at the same temperature and pressure, we find that the oxygen gas weighs sixteen times as much as the hydrogen. (This is not a difficult experiment. All we need to do is to pump the air out of a one litre bulb, weigh it empty, and then re-weigh it full of each of the gases of interest in turn.) But Avogadro tells us that they contain equal number of molecules. It follows that each molecule of oxygen weighs sixteen times as much as each molecule of hydrogen.
One litre of hydrogen will react with one litre of chlorine to give two litres of the gas we call hydrogen chloride. Thus, by Avogadro's Hypothesis, one molecule of hydrogen will react with one molecule of chlorine to give two molecules of hydrogen chloride. So one molecule of hydrogen chloride contains half a molecule of hydrogen, and half a molecule of chlorine. It follows that the molecules of hydrogen and of chlorine are not fundamental entities, but are capable of being split in two. Making a distinction between atoms and molecules that is obvious to us now but caused great confusion at the time, each molecule of chlorine, must contain (at least) two separate atoms.6 By similar reasoning, since 2 litres of hydrogen react with 1 litre of oxygen to give 2 litres of steam, water must have the familiar formula H2O, and not HO as Dalton had assumed for the sake of simplicity.
Avogadro's hypothesis was put forward in 1811, but it was not until 1860 or later that his view was generally accepted. Why were chemists so slow to accept his ideas? Probably because they could not fit it into their theories of bonding. We now recognise two main kinds of bonding that hold compounds together - ionic bonding and covalent bonding. Ionic bonding takes place between atoms of very unlike elements, such as sodium and chlorine, and was at least partly understood by the early 19th century, helped by the excellent work of Davy and Faraday in studying the effect of electric currents on dissolved or molten salts. They showed that sodium chloride contained electrically charged particles, and inferred, correctly, that the bonding in sodium chloride involved transfer of electrical charge (we would now say transfer of electrons) from one atom to another. But, as we have seen, Avogadro's hypothesis implies that many gases, hydrogen and chlorine for instance, each contain two atoms of the same kind per molecule, which raises the question of what holds them together. These are examples of what we now call covalent bonding or electron sharing, a phenomenon not properly understood until the advent of wave mechanics in the 1920s.
Physicists, meanwhile, were developing the kinetic theory of gases, which treats a gas as a collection of molecules flying about at random, bouncing off each other and off the walls of their container. This theory explains the pressure exerted by a gas against the walls of its container in terms of the impact of the gas molecules, and explains temperature as a measure of the disorganised kinetic energy (energy of motion) of the molecules. The theory then considers that this energy is spread out in the most probable (random) way among large numbers of small colliding molecules. It can be shown that molecules of different masses but at the same temperature will then end up on average with the same kinetic energy, and it is this energy that at a fundamental level defines the scale of temperature. This is a statistical theory, where abandoning the attempt to follow any one specific molecule allows us to make predictions about the total assemblage.
The kinetic theory explains the laws (Boyle's law, Charles' law) describing how pressure changes with volume and temperature. Avogadro's hypothesis can also be shown to follow from this treatment. Many other physical properties of gases, such as viscosity (which is what causes air drag) and heat capacity (the amount of heat energy needed to increase temperature), are quantitatively explained by the kinetic theory, and by around 1850 the physicists at least were fully persuaded that molecules and, by implication, atoms, were real material objects.
Right; kinds of isomer. The nature of optical isomers was established by Pasteur. Simple rotamers, such as the pair shown, readily interconvert at room temperature, giving an equilibrium mixture. The other kinds shown generally do not
Chemists were on the whole harder to convince than the physicists, but were finally won over by the existence of isomers, chemical substances whose molecules contain the same number of atoms of each element, but are nonetheless different from each other, with different boiling points and chemical reactivity is. This only made sense if the atoms were joined up to each other in different ways in these different substances. So atoms were real, as were molecules, and the bonding between the atoms in a molecule controlled its properties. This is what we still think today.
Einstein and Lucretius The piece of evidence that finally convinced even the most skeptical scientists came from an unexpected direction, from botany. In 1827, a Scottish botanist called Robert Brown had been looking at some grains of pollen suspended in water under the microscope, and noticed that they were bouncing around, although there was no obvious input of energy to make them do so. This effect, which is shown by any small enough particle, is still known as Brownian motion. Brown thought that the motion arose because the pollen grains were alive, but it was later discovered that dye particles moved around in the same way. The source of the motion remained a mystery until Albert Einstein explained it in 1905. (This was the same year that he developed the theory of Special Relativity, and explained the action of light on matter in terms of photons). Any object floating in water is being hit from all sides by the water molecules. For a large object, the number of hits from different directions will average out, just as if you toss an honest penny a large number of different times the ratio of heads to tails will be very close to one. But if you toss a coin a few times only, there is a reasonable chance that heads (or tails) will predominate. and if you have a small enough particle there is a reasonable chance that it will be hit predominantly from one side rather than the other. Pollen grains are small enough to show this effect. But this is only possible if the molecules are real objects whose numbers can fluctuate; if they were just a book-keeping device for a truly continuous Universe, the effects in different directions would always exactly cancel out. And if molecules are real, then so are atoms. It is just as Lucretius said, looking at dust in the air two thousand years earlier:
So think about the particles that can be seen moving to and fro in a sunbeam, for their disordered motions are a sign of underlying invisible movements of matter.
1 In fact (see earlier post), the Arabs had already recognized the variability of the star Algol
2 We cheat. There are, of course, processes (radioactive decay, nuclear fusion) where the number of atoms of each kind is not conserved because one element is transformed into another. We simply decide to call these physical processes, so that our statement remains true by definition. Nonetheless, it is useful, because it is usually pretty obvious whether a process should be called "chemical" or "physical", on other grounds, such as whether or not it involves the formation of new bonds between atoms.
3 The Architecture of Matter, S. Toulmin and J. Goodfield, Hutchinson, 1962
4 In present-day notation,
C + O2 = CO2 and CaCO3 = CaO + CO2
5 This is not quite true. Most elements are a mixture of atoms of slightly different mass but very similar properties. The relative atomic masses of the elements as they occur in nature are an average of the masses of these chemically identical isotopes
6 So we can write the reactions as H2 + Cl2 = 2HCl and 2H2 + O2 = 2H2O
An earlier version of some of this material appeared in my From Stars to Stalagmites, World Scientific. Leeuwenhoek material via Buffalo Library. Dalton's table of elements and their symbols via Chemogenesis. Isomers image by Vladsinger via Wikipedia
Posted by Paul Braterman at 12:35 AM | Permalink