How Big is Big and How Small is Small

by Smith, Timothy Paul

  When you look at different parts of an oily puddle you are looking at different angles and also at different thicknesses. The angle and the thickness will select the wavelength and so the color of light you see. Measure the color and angles and you can figure out the thickness of the oil. For some oils that thickness may be the size of a single molecule. Oil is typically 1–2 × 10−9 m thick.

  Actually Benjamin Franklin first observed the way oil spreads on water and Lord Rayleigh measured the thickness by seeing how far a single drop of oil would spread. Since he could measure his oil drop as a sphere and the surface of a flat oil slick with a meter stick, Lord Rayleigh’s method gives us the size of the molecule directly. But en route we have touched on interference, something we will continue to use.

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  The world smaller than cells yet larger than atoms is the world of chemistry. This is a complex place where atoms can combine in a plethora of ways to build innumerable types of molecules (although actually, since the number of atoms in the observed universe is finite, the number of combinations is also finite and even calculable.) So I will not discuss particular molecules like benzene rings or carbon chains. What I will talk about is why they stay together and how that affects their shape and size.

  Why is it that molecules do not just fall apart? Atoms are bound together into molecules by chemical forces and chemical forces are derived from electromagnetic forces. Most atoms are electrically neutral yet they still bind, a problem alluded to in Chapter 7.

  The most important point of electromagnetism is something we all learned with bar magnets in elementary school: opposites attract and likes repel and neutrals do neither. So when an electron passes by an atom, it sees the atom as neutral and so it is neither attracted nor repelled. In fact, it is attracted to the atom’s proton and repelled by its electron, effects which tend to balance out. However, if it is a close encounter things change. The electron’s trajectory could take it closer to the atom’s proton than the atom’s electron and so it would be slightly more attracted. It might even push on the atom’s electrons, distorting their orbit. Think of the atom’s electrons in a dance around the proton. An electron from the neighboring atom might be close enough to ‘cut-in’ and join the dance.

  When two or more atoms are near each other we can picture them as a group of dancers. Dance partners are primarily attracted to their partner but they are not blind to the opposite sex in other dancing pairs. Are they only attracted, or do they sometime cut-in and switch partners? When nature has a choice she usually chooses both. Sometimes atoms are mildly attracted from a distance, sometimes partners are shared, and sometimes dance partners are stolen outright, leaving the abandoned partner orbiting the central waltz. All of chemistry is just understanding the attraction, repulsion and dynamics of a dance floor. The different solutions—shared, borrowed, or stolen partners—are analogous to the various types of chemical bonds. What this tells us about size is that chemical bonds are short, no more than the size of an atom. If the atoms were well separated, they would look neutral to each other and not interact. If they were too close, the atoms would get mixed together. So bond length should be about an angstrom, 10−10 m.

  As an aside, the angstrom is a measurement unit often used in atomic physics and chemistry. It is defined as 10−10 meters. It has the symbol Å and was named after a Swedish physicist Anders Ångström who was studying the spectrum of the Sun. The symbol, Å is part of the Swedish alphabet and, in English, is called an A-ring. The angstrom is close to the size of atoms and bond lengths. The diameter of a hydrogen atom is about 1.12 Å and if two hydrogen atoms are bound together they are separated by 0.74 Å. Most chemical bonds are between 1 and 3 Å.

  One of the consequences of short bond lengths is that atoms tend to bind only to their neighbors. If two carbon atoms in a benzene ring or a hydrocarbon chain have another carbon atom between them, they do not really affect each other, except in that they may modify that intermediary atom. Chemistry is essentially about atoms binding to neighbors and so the models people build of molecules out of balls and rods are pretty good pictures. Molecules have structure. They are not just amorphous collections of atoms.

  To picture a large molecule we can again return to our dance analogy. Imagine a large room full of square dancers, where each square represents an atom with a number of electrons. In this dance members of one square will occasionally switch with members of an adjacent square. This means that squares (atoms) need to stay near each other in order for dancers (electrons) to be shared or swapped in a short time. The molecule can be quite complex, with squares of different size representing different elements. An individual electron dancer may migrate across the whole floor, but the squares need to stay put for the dancers to find their place; in other words molecules are stable.

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  The smallest molecule will have the least number of atoms (two), the smallest atoms (hydrogen or helium) and the shortest bond length. Since helium does not chemically bind, the smallest molecule is H2.

  The largest molecule is essentially without bounds. An atom can be bound to two other atoms and form a chain, or more than two and create a complex web-like structure. There are molecules with millions of atoms. A single strand of DNA can have several billion atoms (1010), but only occupy a space about 1000 Å across. This brings us back to the dimensions of a virus, a measurement that we looked at in Chapter 2, which is not surprising since a virus is primarily a strand of DNA (or RNA).

  There are molecules even bigger than those found in biology. Plastics can be made from long, chain-like molecules and carbon nanotubes and fibers are in some sense mega-molecules. Larger than these are crystals, which are atoms held together with chemical bonds. Actually people are hesitant to calls these true molecules, because they have repeating patterns without limit and they do not really follow the law of definite proportions, which Dalton (Chapter 2) saw as critical. Still, they are very interesting.

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  In the world of mathematics crystals are cool: they have lots of symmetries. This is because crystals are made up of a grid of atoms, each atom linked to its neighbor in a regular pattern that is repeated over and over again, much like tiles on a floor. If the tiles are square or hexagons or triangles, that will dictate the pattern of the whole floor. If the first few atoms are arranged at right angles to each other, in a cubic structure, the macroscopic crystal will also be made with right angles. NaCl, common salt, is a go example of this. If the original was a tetrahedron, that will also show up in the macroscopic crystal. When you look at a quartz crystal you can see the tetrahedral angles, the angles between the bonds of silicon and oxygen.

  Perhaps the most classic crystal is a diamond. As hard as hard can be, diamonds are the definition of 10 on the Mohs scale. They are made up of carbon atoms and carbon can bind to four adjacent atoms. When you try to link up carbon in a regular, three-dimensional grid you find that all the angles are the angles that show up in an octahedron. An octahedron is a three-dimensional shape with eight faces and each face is a triangle. It looks like two Egyptian pyramids with their bases joined together. This regular shape, starting at the atomic scale, is repeated again and again, trillions of times, until the crystal is big enough to see. Put together 1022 carbon atoms and you have a one carat diamond. This octahedral shape is the natural shape of a diamond.

  A diamond like the Great Star of Africa, among the Crown Jewels in the Tower of London, or the Hope Diamond in the Smithsonian Institute are not octahedrons. They are cut diamonds. In fact the classic shape of a diamond that we associate with engagement rings is a round brilliant cut. This is one of many ways of cutting a diamond. The way you cut a diamond is actually dictated by the underlying atomic structure. There are preferred planes, or facets, along which you can cleave the crystal.

  One way to think of a facet is to picture an orchard where the trees have been planted in perfect columns and rows (see Figure 10.4). Here a facet is where I could draw a long line through the
orchard and not encounter a tree trunk. If I look between two rows I can see to the other side of the orchard, so that is a facet. I can also sight between columns. I can also look at 45°, on the diagonal, and see a facet. This facet is like the way a bishop moves on a chess board. There is even a facet at about 26°, what you might call the knight’s facet, over two columns and up a row. But you cannot just look at any angle. Trees do block the view in some directions.

  Figure 10.4 Trees in an orchard demonstrate the facets in a crystal. Where the trees are in perfect columns and rows there are special angles where you can see all the way through the orchard. These sight lines are analogous to facets in a crystal.

  Crystal facets can be used to cut gems, and the angles of the cuts can tell us about the underlying atomic grid. Actually, the facets of a diamond are more complex than that of an orchard, because the atomic grids are three-dimensional, but the reasoning is the same.

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  One last curiosity of crystals is that their regular pattern of planes and facets can also interact with light to form interference patterns, much like our examples of lasers and hair, or sunlight on oily water. We cannot use visible light because the light waves are much to big compared to the distance between facets. But if we shine X-rays with a wavelength similar to the atomic spacing we can produce beautiful interference patterns (see Figure 10.5). These patterns remind me of snowflakes, a kaleidoscope of bright spots in all directions, which have encoded in them the facet angles and spacing within the crystal. In fact it was using X-rays on DNA and looking at the resulting interference patterns that first pointed the way towards their double-helix structure.

  Figure 10.5 The interference pattern of X-rays from a crystal and DNA. (Left) silicon crystal, courtesy of C.C. Jones, Union College, Schenectady, NY. (Right) DNA, an image created by Rosalind Franklin and R. G. Gosling and which represents some of the first evidence of the double-helix structure. Reprinted by permission of Macmillan Publishers Ltd, Nature April 25, 1953.

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  Our first vision of an atom is of a nucleus in the middle, made up of protons and neutrons and electrons zipping around. Let us correct that image right away. The nucleus is about 100,000 times smaller than an atom. If you drew an atom on a computer screen the nucleus would occupy a fraction of a pixel. Seeing the nucleus inside an atom is like looking for a single sheep in the center of Wales. Our image really should be electrons zipping around a minuscule point in the middle. The size and shape of the atom is the size and shape of the electron orbits.

  What we know about orbits is based on our experience with planets and satellites. If you toss a satellite into space with enough speed or energy it will settle down into a nice orbit. With too little energy it will come back to Earth, and with a bit more energy a wider, larger orbit. However, if you give it too much energy, a speed that is greater than the escape velocity you will lose your satellite, or perhaps you have just launched a deep-space probe. So orbits happen when objects have the right speed or kinetic energy. The Moon, or other satellites around the Earth not only curve through space, but they come back to their starting point, repeatedly. This is the vision Niels Bohr (1885–1962) first gave us for the structure of the atom: electrons in nice ring-like orbits about the nucleus; a planetary system. But this is not the way nature acts at this scale. At the size of the atom and smaller the world is governed by the rules of quantum mechanics.

  Quantum mechanics has two unique features that are required to describe nature at these scales. The first is embedded in the name of this theory. The world is quantized, which means that it happens at certain values and not other values. When Bohr drew his atom he took this to mean that for some unknown reason electrons could orbit with certain allowed energies and with certain well-defined circular trajectories. Also it meant that there were energies and orbits where electrons were not allowed. This is different than for a satellite. If you add a little energy to a satellite it will go slightly faster and occupy an orbit that is slightly larger. The amount of energy you could add is continuous, not quantized.

  The second unique feature of quantum mechanics is that matter, including electrons, has a wave-like nature. In fact this particle–wave nature explains quantization and a lot more. But before we look at quantum waves I want to step back and look at a guitar string and the wave it produces.

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  The lowest note on a standard guitar is made when plucking the sixth string, the thickest string, usually on the bottom. If unfretted it will produce an E 2 note, which means that it vibrates at 82.41 Hz. The string is of the right material and tension such that after you pluck it, it will produce that note for a while. That is not the frequency with which you plucked it. In fact, plucking or strumming a guitar is a violent act. At the point where it was plucked, the string is vibrating with all sorts of frequencies. Consider one frequency and the wave associated with it. It will travel up to the nut and down to the saddle, where the waves will bounce and propagate back to the middle of the string. If the frequency is on tune, when the two waves meet they will support each other, add constructively and continue to propagate up and down the string. If they do not match, the waves will be out of phase (one on the rise and one falling) and they will cancel each other and that frequency will quickly die out. When you tune a guitar you are changing the tension, which changes the propagation speed, and so selects a certain frequency. Electron orbits in atoms are selected the same way.

  Electrons have waves associated with them, so when we picture an orbit we should think of a wave flowing around the nucleus of an atom. If the electron has the wrong wavelength when it wraps around the atom it will not be in tune—it will be out of phase—which is not a good orbit. The stable orbits of an atom are the in-tune solutions and in quantum mechanics these solutions are called wavefunctions. The wave-fuction describes a region around the nucleus where the electron will be found. It is not a thin line, a planetary orbit. It is more like a cloud or a halo.

  Back to the guitar once more. The E 2 string vibrates at 82.41 Hz. It also vibrates at 164.82 Hz, 247.230 Hz, 329.64 Hz and other frequencies. These are waves that have gone through two, three, or more oscillations when they meet on the string and merge again. So these too are stable solutions to the guitar string problem. Not only can they happen, they happen all the time in addition to the 82.41 Hz solution. These are the harmonics or overtones of this string. In fact the strength of various harmonics is how you can tell the difference between instruments. A guitar, piano and horn can all play the same note, but your ear can tell them apart because the harmonics have different strengths.

  The string of a guitar also have nodes, places that do not vibrate. The first harmonic has half the wavelength of the fundamental frequency, so half way down the string it has a node. If you pluck the E 2 string and then put your finger at the mid-point and fret it, you will kill the fundamental (82.41 Hz) frequency, but the first harmonic will continue.

  Atoms also have harmonics or overtones. There are multiple wave-function solutions to an atom. These are other orbits that are available to the electron. Much as the guitar string will vibrate with distinc harmonics, and not just any frequency, the atom also has distinct, quantized solutions or wavefunctions.

  This is where things get fun. The most common wavefunction is called the ground state. It is a spherical region surrounding the nucleus. This is sometimes described as a shell, but that conjures up an image of a well-defined, rigid object. The wavefunctions do not have hard edges, they just fade at large distances, like a morning fog. Where a wave-function is large is where an electron is likely to be, where it is small the electron rarely goes. So a picture of the wavefunction is like a long time-exposure photograph of the atom.

  An orbit with greater energy produces a wavefunction or shell with larger radius, with smaller shells nested inside of larger ones. Like the guitar string, the higher harmonics also have nodes in them, spherical regions where the electron will not be found (see Figure 10.6).


  But it gets even stranger. Because the atom is a fully three-dimensional object, its modes are more complex than the vibrational mode of a string. In fact there are wavefunctions that look like lobes and rings. The lobes and rings are tantalizing, but hard to see in nature because if you add the lobes and rings together you end up with a sphere.

  Figure 10.6 Wavefunctions of the hydrogen atom. Each image is 10 Å (1 × 10−9 m) wide. (A) ground state, (B–D) first excited state, (E–J) second excited state; spherical, lobes, ring, ring and lobes, double ring, ring. The rings in D, G, I and J are really two wavefunctions combined.

  Most of the time the hydrogen atom is in its ground state (Figure 10.6A). But the world is not made of just hydrogen. If you have an atom with a lot of electrons, only two can go in each wavefunction and so to build an atom is to add wavefunctions on top of wavefunctions. Now we will find that the rings of one wavefunction nicely mesh with the lobes of another wavefunction, such that when you build up the atom to elements such as neon, argon and krypton, they are nicely spherical. These atoms without bumps or corners are the non-interacting, inert or noble gasses.

  In non-noble gasses—the rightmost column of the periodic table in Figure 10.7—those bumps and lobes become important. That is what defines the angles of chemical bonds and the geometry of molecules.

  Figure 10.7 The periodic table of the elements. As electrons are added to an atom, you move across a row. Elements with completely filled shells, a smooth wavefunction, are the non-interacting noble gases, in the right-hand column.