The Idiot’s Guide to Making Atoms

**Avagadro’s Number and Moles**

Writing
this chapter has reminded me of the opening of a story by a
well-known science fiction author (whose name, needless to say, I
can’t recall): “This is a warning, the only one you’ll get so
don’t take it lightly.”

*Alice in Wonderland*or “We’re not in Kansas anymore” also pop into mind. What I mean by this is that I could find no way of writing it without requiring the reader to put his thinking (and imagining) cap on. So: be prepared.
A
few things about science in general before I plunge headlong into the
subject I’m going to cover. I have already mentioned the way
science is a step-by-step, often even torturous, process of
discovering facts, running experiments, making observations, thinking
about them, and so on; a slow but steady accumulation of knowledge
and theory which gradually reveals to us the way nature works, as
well as why. But there is more to science than this. This more has
to do with the concept, or hope I might say, of trying to understand
things like the universe as a whole, or things as tiny as atoms, or
geological time, or events that happen over exceedingly short times
scales, like billionths of a second. I say hope because in dealing
with such things, we are extremely removed from reality as we deal
with it every day, in the normal course of our lives.

The
problem is that, when dealing with such extremes, we find that most
of our normal ideas and expectations – our intuitive, “common
sense”, feeling grasp of reality – all too frequently starts to
break down. There is of course good reason why this should be, and
is, so. Our intuitions and common sense reasoning have been sculpted
by our evolution – I will resist the temptation to say designed,
although that often feels to be the case, for, ironically, the same
reasons – to grasp and deal with ordinary events over ordinary
scales of time and space. Our minds are not well endowed with the
ability to intuitively understand nature’s extremes, which is why
these extremes so often seem counter-intuitive and even absurd to us.

Take, as one of the best examples I know of this, biological
evolution,

*a lá*Darwin. As the English biologist and author Richard Dawkins has noted several times in his books, one of the reasons so many people have a hard time accepting Darwinian evolution is the extremely long time scale over which it occurs, time scales in the millions of years and more. None of us can intuitively grasp a million years; we can’t even grasp, for that matter, a thousand years, which is one-thousandth of a million. As a result, the claim that something like a mouse can evolve into something like an elephant*feels*“obviously” false. But that feeling is precisely what we should ignore in evaluating the possibility of such events, because we cannot have any such feeling for the exceedingly long time span it would take. Rather, we have to evaluate the likelihood using evidence and hard logic; commonsense can seriously mislead us.
The
same is true for nature on the scale of the extremely small. When we
start poking around in this territory, around with things like atoms
and sub-atomic particles, we find ourselves in a world which bears
little resemblance to the one we are used to. I am going to try
various ways of giving you a sense of how the ultra-tiny works, but I
know in advance that no matter what I do I am still going to be
presenting concepts and ideas that seem, if anything, more outlandish
than Darwinian evolution; ideas and concepts that might, no, probably
will, leave your head spinning. If it is any comfort, they often
leave my mind spinning as well. And again, the only reason to accept
them is that they pass the scientific tests of requiring evidence and
passing the muster of logic and reason; but they will often seem
preposterous, nevertheless.

First,
however, let’s try to grab hold of just how tiny the world we are
about to enter is. Remember Avogadro’s number, the number of a
mole of anything, from the last chapter? The reason we need such an
enormous number when dealing with atoms is that they are so
mind-overwhelmingly small. When I say mind-overwhelmingly, I really
mean it. A good illustration of just how small that I enjoy is to
compare the number of atoms in a glass of water to the number of
glasses of water in all the oceans on our planet. As incredible as
it sounds, the ratio of the former to the latter is around 10,000

**to****1. This means that if you fill a glass with water, walk down to the seashore, pour the water into the ocean and wait long enough for it to disperse evenly throughout all the oceans (if anyone has managed to calculate how long this would take, please let me know), then dip your now empty glass into the sea and re-fill it, you will have scooped up some ten thousand of the original atoms that it contained. Another good way of stressing the smallness of atoms is to note that every time you breathe in you are inhaling some of the atoms that some historical figure – say Benjamin Franklin or Muhammad – breathed in his lifetime. Or maybe just in one of their breaths; I can’t remember which – that’s how hard to grasp just how small they are.**
One
reason all this matters is that nature in general does not
demonstrate the property that physicists and mathematicians call
“scale invariance.” Scale invariance simply means that, if you
take an object or a system of objects, you can increase its size up
to as large as you want, or decrease it down, and its various
properties and behaviors will not change. Some interesting systems
that do possess scale invariance are found among the mathematical
entities called fractals: no matter how much you enlarge or shrink
these fractals, their patterns repeat themselves over and over

*ad infinitum*without change. A good example of this is the Koch snowflake:
which
is just a set of repeating triangles, to as much depth as you want.
There are a number of physical systems that have scale invariance as
well, but, as I just said, in general this is not true. For example,
going back to the mouse and the elephant, you could not scale the
former up to the size of the latter and let it out to frolic in the
African savannah with the other animals; our supermouse’s
proportionately tiny legs, for one thing, would not be strong enough
to lift it from the ground. Making flies human sized, or vice-versa,
run into similar kinds of problems (a fly can walk on walls and
ceilings because it is so small that electrostatic forces dominate
its behavior far more than gravity).

**Scale Invariance – Why it Matters**

One natural phenomenon
that we know lacks scale invariance, we met in the last chapter is
matter itself. We know now that you cannot take a piece of matter, a
nugget of gold for example, and keep cutting it into smaller and
smaller pieces, and so on until the end of time. Eventually we reach
the scale of individual gold atoms, and then even smaller, into the
electrons, protons, and neutrons that comprise the atoms, all of
which are much different things than the nugget we started out with.
I hardly need to say that all elements, and all their varied
combinations, up to stars and galaxies and larger, including even the
entire universe, suffer the same fate. I should add, for the sake of
completeness, that we cannot go in the opposite direction either; as
we move toward increasingly more massive objects, their behavior is
more and more dominated by the field equations of Einstein’s
general relativity, which alters the space and time around and inside
them to a more and more significant degree.

Why
do I take the time to mention all this? Because we are

*en route*to explaining how atoms, electrons and all, are built up and how they behave, and we need to understand that what goes on in nature at these scales is very different than what we are accustomed to, and that if we cannot adopt our thinking to these different behaviors we are going to find it very tough, actually impossible, sledding, indeed.
In
my previous book,

*Wondering About*, I out of necessity gave a very rough picture of the world of atoms and electrons, and how that picture helped explained the various chemical and biological behaviors that a number of atoms (mostly carbon) displayed. I say “of necessity” because I didn’t, in that book, want to mire the reader in a morass of details and physics and equations which weren’t needed to explain the things I was trying to explain in a chapter or two. But here, in a book largely dedicated to chemistry, I think the sledding is worth it, even necessary, even if we do still have to make some dashes around trees and skirt the edges of ponds and creeks, and so forth.
Actually,
it seems to me that there are two approaches to this field, the field
of quantum mechanics, the world we are about to enter, and how it
applies to chemistry. One is to simply present the details, as if
out of a cook book: so we are presented our various dishes of,
first, classical mechanics, then the LaGrangian equation of motion
and Hamiltonium operators and so forth, followed by Schrödinger’s
various equations and Heisenberg’s matrix approach, with
eigenvectors and eigenvalues, and all sorts of stuff that one can
bury one’s head into and never come up for air. Incidentally, if
you do want to summon your courage and take the plunge, a very good
book to start with is Melvin Hanna’s

*Quantum Mechanics in Chemistry*, of which I possess the third edition, and go perusing through from time to time when I am in the mood for such fodder.
The problem with this
approach is that, although it cuts straight to the chase, it leaves
out the historical development of quantum mechanics, which, I
believe, is needed if we are to understand why and how physicists
came to present us with such a peculiar view of reality. They had
very good reasons for doing so, and yet the development of modern
quantum mechanical theory is something that took several decades to
mature and is still in some respects an unfinished body of work.
Again, this is largely because some it its premises and findings are
at odds with what we would intuitively expect about the world
(another is that the math can be

*very*difficult). These are premises and findings such as the quantitization of energy and other properties to discrete values in very small systems such as atoms. Then there is Heisenberg’s famous though still largely misunderstood uncertainly principle (and how the latter leads to the former).**Talking About Light and its Nature**

A
good way of launching this discussion is to begin with light, or,
more precisely,

*electromagnetic radiation*. What do I mean by these polysyllabic words? Sticking with the historical approach, the phenomena of electricity and magnetism had been intensely studied in the 1800s by people like Faraday and Gauss and Ørsted, among others. The culmination of all this brilliant theoretical and experimental work was summarized by the Scottish physicist James Clerk Maxwell, who in 1865 published a set of eight equations describing the relationships between the two phenomena and all that had been discovered about them. These equations were then further condensed down into four and placed in one of their modern forms in 1884 by Oliver Heaviside. One version of these equations is (if you are a fan of partial differential equations):
Don’t
worry if you don’t understand this symbolism (most of it I don’t).
The important part here is that the equations predict the existence
of electromagnetic waves propagating through free space at the speed
of light; waves rather like water waves on the open ocean albeit
different in important respects. Maxwell at once realized that light
must be just such a wave, but, more importantly, that there must be a
theoretically infinite number of such waves, each with different
wavelengths ranging from the very longest, what we now call radio
waves, to the shortest, or gamma rays. An example of such a wave is
illustrated below:

To
assist you in understanding this wave, look at just one component of
it, the oscillating electric field, or the part that is going up and
down. For those not familiar with the idea of an electric (or
magnetic) field, simply take a bar magnet, set it on a piece of
paper, and sprinkle iron filings around it. You will discover, to
your pleasure I’m certain, that the filings quickly align
themselves according to the following pattern:

The
pattern literally traces out the, in this case, magnetic field of the
bar magnet, but we could have used an electrically charged source to
produce a somewhat different pattern. The point is, the field makes
the iron filings move into their respective positions; furthermore,
if we were to move the magnet back and forth or side to side the
filings would continuously move with it to assume their desired
places. This happens because the outermost electrons in the filings
(which, in addition to carrying an electric charge, also behave as
very tiny magnets) are basically free to orient themselves anyway
they want, so they respond to the bar’s field with gusto, in the
same way a compass needle responds to Earth’s magnetic field. If
we were using an electric dipole it would be the electric properties
of the filings’ electrons performing the trick, but the two
phenomena are highly interrelated.

Go
back to the previous figure, of the electromagnetic wave. The wave
is a combination of oscillating electric and magnetic fields, at
right angles (90°) to each other, propagating through space. Now,
imagine this wave passing through a wire made of copper or any other
metal. Hopefully you can perceive by now that, if the wave is within
a certain frequency range, it will cause the electrons in the wire’s
atoms to start spinning around and gyrating in order to accommodate
the changing electric and magnetic fields, just as you saw with the
iron filings and the bar magnet. Not only would they do that, but
the resulting electron motions could be picked up by the right kinds
of electronic gizmos, transistors and capacitators and resistors and
the like – here, we have just explained the basic working principle
of radio transmission and receiving, assuming the wire is the
antenna. Not bad for a few paragraphs of reading.

This sounds all very
nice and neat, yet it is but our first foot into the door of what
leads to modern quantum theory. The reason for this is that this
pat, pretty perception of light as a wave just didn’t jibe with
some other phenomena scientists were trying to explain at the end of
the nineteenth century

**/**beginning of the twentieth century. The main such phenomena along these lines which quantum thinking solved were the puzzles of the so-called “blackbody” radiation spectrum and the photo-electric effect.**Blackbody Radiation and the Photo-electric Effect**

If you take an object,
say, the tungsten filament of the familiar incandescent light bulb,
and start pumping energy into it, not only will its temperature rise
but at some point it will begin to emit visible light: first a dull
red, then brighter red, then orange, then yellow – the filament
eventually glows with a brilliant white light, meaning all of the
colors of the visible spectrum are present in more or less equal
amounts, illuminating the room in which we switched the light on.
Even before it starts to visibly glow, the filament emits infrared
radiation, which consist of longer wavelengths than visible red, and
is outside our range of vision. It does so in progressively greater
and greater amounts and shorter and shorter wavelengths, until the
red light region and above is finally reached. At not much higher
temperatures the filament melts, or at least breaks at one of its
ends (which is why it is made from tungsten, the metal with the
highest melting point), breaking the electric current and causing us
to replace the bulb.

The
filament is a blackbody in the sense that, to a first approximation,
it completely absorbs all radiation poured onto it, and so its
electromagnetic spectrum depends only on its temperature and not any
on properties of its physical or chemical composition. Other such
objects which are blackbodies include the sun and stars, and even our
own bodies – if you could see into right region of the infrared
range of radiation, we would all be glowing. A set of five blackbody
electromagnetic spectra are illustrated below:

Examine
these spectra, the colored curves, carefully. They all start out at
zero on the left which is the shortest end of the temperature, or
wavelength (λ, a Greek letter which is
pronounced lambda) scale; the height of the curves then quickly rises
to a maximum λ at a certain temperature,
followed by a gradual decline at progressively lower temperatures
until they are basically back at zero again. What is pertinent to
the discussion here is that, if we were living around 1900, all these
spectra would be experimental; it was not possible then, using the
physical laws and equations known at the end of the 1800s, to explain
or predict them theoretically. Instead, from the laws of physics as
known then, the predicted spectra would simply keep increasing as λ
grew shorter

**/**temperature grew higher, resulting it what was called “the ultraviolet catastrophe.”
Another,
seemingly altogether different, phenomenon that could not be
explained using classical physics principles was the so-called
photoelectric effect. The general idea is simple enough: if you
shine enough light of the right wavelength or shorter onto certain
metals – the alkali metals, including sodium and potassium, show
this effect the strongest – electrons will be ejected from the
metal, which can then be easily detected:

This
illustration not only shows the effect but also the problem 19’th
century physicists had explaining it. There are three different
light rays shown striking the potassium plate: red at a wavelength
of 700 nanometers or nm (an nm is a billionth of a meter), green at
550 nm, and purple at 400 nm. Note that the red light fails to eject
any electrons at all, while the green and purple rays eject only one
electron, with the purple electron escaping with a higher velocity,
meaning higher energy, than the green.

The
reason this is so difficult to explain with the physics of the 1800’s
is that physics then defined the energy of all waves using both the
wave’s amplitude, which is the distance from crest or highest point
to trough or lowest point, in combination with the wavelength (the
shorter the wavelength the more waves can strike within a given
time). This is something you can easily appreciate by walking into
the ocean until the water is up to your chest; both the higher the
waves are and the faster they hit you, the harder it is to stay on
your feet.

Why
don’t the electrons in the potassium plate above react in the same
way? If light behaved as a classical wave it should not only be the
wavelength but the intensity or brightness (assuming this is the
equivalent of amplitude) that determines how many electrons are
ejected and with what velocity. But this is not what we see: e.g.,
no matter how much red light, of what intensity, we shine on the
plate no electrons are emitted at all, while for green and purple
light only the shortening of the wavelength in and of itself
increases the energy of the ejected electrons, once again, regardless
of intensity. In fact, increasing the intensity only increases the
number of escaping electrons, assuming any escape at all, not their
velocity. All in all, a very strange situation, which, as I said,
had physicists scratching their heads all over at the end of the
1800s.

The
answers to these puzzles, and several others, comes back to the point
I made earlier about nature not being scale invariant. These
conundrums were simply insolvable until scientists began to think of
things like atoms and electrons and light waves as being quite unlike
anything they were used to on the larger scale of human beings and
the world as we perceive it. Using such an approach, the two men who
cracked the blackbody spectrum problem and the photoelectric effect,
Max Planck and Albert Einstein, did so by discarding the concept of
light being a classical wave and instead, as Newton had insisted two
hundred years earlier, thought of it as a particle, a particle which
came to be called a

*photon*. But they also did not allude to the photon as a classical particle either but as a particle with a wavelength; furthermore, that the energy*E*of this particle was described, or*quantized*, by the equation
in which

*c*was the speed of light,*λ*the photon’s wavelength, and*h*was Planck’s constant, the latter of which is equal to 6.626 × 10^{-34}joules seconds – please note the extremely small value of this number. In contrast to our earlier, classical description of waves, the amplitude is to be found nowhere in the equation; only the wavelength, or frequency, of the photon determines its energy.
If
you are starting to feel a little dizzy at this point in the story,
don’t worry; you are in good company. A particle with a
wavelength? Or, conversely, a wave that acts like a particle even if
only under certain circumstances? A

*wavicle*? Trying to wrap your mind around such a concept is like awakening from a strange dream in which bizarre things, only vaguely remembered, happened. And the only justification of this dream world is that it made sense of what was being seen in the laboratories of those who studied these phenomena. Max Planck, for example, was able, using this definition, to develop an equation which correctly predicted the shapes of blackbody spectra at all possible temperature ranges. And Einstein elegantly showed how it solved the mystery of the photoelectric effect: it took a minimum energy to eject an electron from a metal atom, an energy dictated by the wavelength of the incoming photon; the velocity or kinetic energy of the emitted electron came solely from the residual energy of the photon after the ejection. The number of electrons freed this way was simply equal to the number of the photons that showered down on the metal, or the light’s intensity. It all fit perfectly. The world of the quantum had made its first secure foot prints in the field of physics.
There was much, much
more to come.

**The Quantum and the Atom**

Another phenomena that
scientists couldn’t explain until the concept of the quantum came
along around 1900-1905 was the atom itself. Part of the reason for
this is that, as I have said, atoms were not widely accepted as real,
physical entities until electrons and radioactivity were discovered
by people like the Curies and J. J. Thompson, Rutherford performed
his experiments with alpha particles, and Einstein did his work on
Brownian motion and the photo-electric effect (the results of which
he published in 1905, the same year he published his papers on
special relativity and the E = mc

^{2}equivalence of mass and energy in the same year, all at the tender age of twenty-six!). Another part is that, even if accepted, physics through the end of the 1800s simply could not explain how atoms could be stable entities.
The
problem with the atomic structure became apparent in 1911, when
Rutherford published his “solar system” model, in which a tiny,
positively charged nucleus (again, neutrons were not discovered until
1932 so at the time physicists only knew about the atomic masses of
elements) was surrounded by orbiting electrons, in much the same way
as the planets orbit the sun. The snag with this rather intuitive
model involved – here we go both with not trusting intuition and
nature not being scale invariant again – something physicists had
known for some time about charged particles.

When
a charged particle changes direction, it will emit electromagnetic
radiation and thereby lose energy. Orbiting electrons are electrons
which are constantly changing direction and so, theoretically, should
lose their energy and fall into the nucleus in a tiny fraction of a
second (the same is true with planets orbiting a sun, but it takes
many trillions of years for it to happen). It appeared that the
Rutherford model, although still commonly evoked today, suffered from
a lethal flaw.

And
yet this model was compelling enough that there ought to be some
means of rescuing it from its fate. That means was published two
years later, in 1913, by Niels Bohr, possibly behind Einstein the
most influential physicist of the twentieth century. Bohr’s
insight was to take Planck’s and Einstein’s idea of the
quantitization of light and apply it to the electrons’ orbits. It
was a magnificent synthesis of scientific thinking; I cannot resist
inserting here Jacob Bronowski’s description of Bohr’s idea, from
his book

*The Ascent of Man*:
Now in a sense, of course, Bohr’s
task was easy. He had the Rutherford atom in one hand, he had the
quantum in the other. What was there so wonderful about a young man
of twenty-seven in 1913 putting the two together and making the
modern image of the atom? Nothing but the wonderful, visible
thought-process: nothing but the effort of synthesis. And the idea
of seeking support for it in the one place where it could be found:
the fingerprint of the atom, namely the spectrum in which its
behavior becomes visible to us, looking at it from outside.

Reading this reminds me of another feature of atoms I have yet to
mention. Just as blackbodies emit a spectrum of radiation, one based
purely on their temperature, so did the different atoms have their
own spectra. But the latter had the twist that, instead of being
continuous, they consisted of a series a sharp lines and were not
temperature dependent but were invoked usually by electric discharges
into a mass of the atoms. The best known of these spectra, and the
one shown below, is that of atomic hydrogen (atomic because hydrogen
usually exists as diatomic molecules, H

_{2}, but the electric discharge also dissociates the molecules into discrete atoms):
This is the visible part of the hydrogen atom spectrum, or so-called
Balmer series, in which there are four distinct lines: from right to
left, the red one at 656 nanometers (nm), the blue-green at 486 nm,
the blue-violet at 434 nm, and the violet at 410 nm.

Bohr’s
dual challenge was explain both why the atom, in this case hydrogen,
the simplest of atoms, didn’t wind down like a spinning top as
classical physics predicted, and why its spectrum consisted of these
sharp lines instead of being continuous as the energy is lost. As
said, he accomplished both tasks by invoking quantum ideas. His
reasoning was more or less as this: the planets in their paths
around the sun can potentially occupy any orbit, in the same
continuous fashion we have learned to expect from the world at large.
As we now might begin to suspect however, this is not true for the
electrons “orbiting” (I put this in quotes because we shall see
that this is not actually the case) the nucleus. Indeed, this is the
key concept which solves the puzzle of atomic structure, and which
allowed scientists and other people to finally breathe freely while
they accepted the reality of atoms.

Bohr
kept the basic solar system model, but modified it by saying that
there was not a continuous series of orbits the electrons could
occupy but instead a set of discrete ones, in-between which there was
a kind of no man’s land where electrons could never enter. Without
going into details you can see how, at one stroke, this solved the
riddle of the line spectra of atoms: each spectral line represented
the transition of an electron from a higher orbit (more energy) to a
lower one (less energy). For example, the 656 nm red line in the
Balmer spectrum of hydrogen is caused by an electron dropping from
orbit level three to orbit level two:

Here
again we see the magical formula

*hυ*, the energy of the emitted photon, in this case being equal to E, the difference in energy between the two orbits. Incidentally, if the electron falls further inward, from orbit level two to orbit level one – this is what is known as the Lyman series, in this case accompanied by a photon emission of 122 nm, well into the ultraviolet and invisible to our visual systems. Likewise, falls to level three from above, the so-called Paschen series, occur in the equally invisible infrared spectrum. There is also a level four, five, six … potentially out to infinity. It was the discovery of these and other series which confirmed Bohr’s model and in part earned him the Nobel Prize in physics in 1932.
This is fundamentally
the way science works. Inexplicable features of reality are solved,
step by step, sweat drop by tear drop , and blood drop by drop, by
the application of known physical laws; or, when needed, new laws and
new ideas are summoned forth to explain them. Corks are popped, the
bubbly flows, and awards are apportioned among the minds that made
the breakthroughs. But then, as always, when the party is over and
the guests start working off their hangovers, we realize that
although, yes, progress has been made, there is still more territory
to cover. Ironically, sometimes the new territory is a direct
consequence of the conquests themselves.

Bohr’s
triumph over atomic structure is perhaps the best known entré in
this genre of the story of scientific progress. There were two
problems, one empirical and one theoretical, which arose from it in
particular, problems which sobered up the scientific community. The
empirical problem was that Bohr’s atomic model, while it perfectly
explained the behavior of atomic hydrogen, could not be successfully
applied to any other atom or molecule, not even seemingly simple
helium or molecular hydrogen (H

_{2}), the former of which is just after hydrogen in the periodic table. The theoretical problem was that the quantitization of orbits was purely done on an*ad hoc*basis, without any meaningful physical insight as to*why*it should be true.
And
so the great minds returned to their offices and chalkboards,
determined to answer these new questions.

**Key Ideas in the Development of Quantum Mechanics**

The
key idea which came out of trying to solve these problems was that,
if that which had been thought of as a wave, light, could also
possess particle properties, then perhaps the reverse was also true:
that which had been thought of as having a particle nature, such as
the electron, could also have the characteristics of waves. Louis de
Broglie, in his 1924 model of the hydrogen atom, introduced this,
what was to become called the

*wave-particle duality*concept, explaining the discrete orbits concept of Bohr by recasting them as distances from the nuclei where standing electron waves could exist only in whole numbers, as the mathematical theory behind waves demanded:
De Broglie’s model was supported in the latter 1920’s by
experiments which showed that electrons did indeed show wave
features, at least under the right conditions. Yet, though a
critical step forward in the formulation of the quantum mechanical
description of atoms, de Broglie still fell short. For one thing,
like Bohr, he could only predict the properties of the simplest atom,
hydrogen. Second, and more importantly, he still gave no fundamental
insight as to how or why particles could behave as waves and

**/**or vice-versa. Although I have said that reality on such small scales should not be expected to behave in the same matter as the scales we are used to, there still has to be some kind of underlying theory, an intellectual glue, that allows us to make at least some sense of what is really going on. And scientists in the early 1920’s still did not possess that glue.
That
glue was first provided by people like Werner Heisenberg and Max
Born, who, only a few years after de Broglie’s publication, created
a revelation, or perhaps I should say revolution, of one of
scientific – no, philosophic – history’s most astonishing
ideas. In 1925 Heisenberg, working with Born, introduced the
technique of matrix mechanics, one of the modern ways of formulating
quantum mechanical systems. Crucial to the technique was the concept
that at the smallest levels of nature, such as with electrons in an
atom, neither the positions nor motions of particles could be defined
exactly. Rather, these properties were “smeared out” in a way
that left the particles with a defined uncertainty. This led, within
two years, to Heisenberg’s famous Uncertainty Principle, which
declared that certain pairs of properties of a particle in any system
could not be simultaneously known with perfect precision, but only
within a region of uncertainty. One formulation of this principle
is, as I have used before:

*x*× *s*≤ *h*/ (2π ×*m*)
which states that the product of the uncertainty of a particle’s
position (

*x*) and its speed (*s*) is always less than or equal to Planck’s (*h*) constant divided by 2π times the object’s mass (*m*). Now, there is something I must say upfront. It is critical to understand that this uncertainty is*not*due to deficiencies in our measuring instruments, but is built directly into nature, at a fundamental level. When I say fundamental I mean just that. One could say that, if God or Mother Nature really exists, even He Himself (or Herself, or Itself) does not and cannot know these properties with zero uncertainty. They simply do not*have*a certainty to reveal to any observer, not even to a supernatural one, should such an observer exist.
Yes, this is what I am
saying. Yes, nature is this strange.

**The Uncertainty Principle and Schrödinger’s Breakthrough**

Another, more precise
way of putting this idea is that you can specify the exact position
of an object at a certain time, but then you can say nothing about
its speed (or direction of motion); or the reverse, that speed /
direction can be perfectly specified but then the position is a
complete unknown. A critical point here is that the reason we do not
notice this bizarre behavior in our ordinary lives – and so, never
suspected it until the 20th century – is that the product of these
two uncertainties is

*inversely*proportional to the object’s mass (that is, proportional to*1/m*) as well as directly proportional to the tiny size of Planck’s constant*h*. The result of this is that large objects, such as grains of sand, are simply much too massive to make this infinitetesimally small uncertainty product measurable by any known or even imaginable technique.
Whew.
I know. And just what does all this talk about uncertainty have to
do with waves? Mainly it is that trigonometric wave functions, like
sine and cosine, are closely related to probability functions, such
as the well-known Gaussian, or bell-shaped, curve. Let’s start
with the latter. This function starts off near (but never at) zero
at very large negative x, rises to a maximum y = f(x) value at a
certain point, say x = 0, and then, as though reflected through a
mirror, trails off again at large positive x. A simple example
should help make it clear. Take a large group of people. It could
be the entire planet’s human’s population, though in practice
that would make this exercise difficult. Record the heights of all
these people, rounding the numbers off to a convenient unit, say,
centimeters or cm. Now make sub-groups of these people, each
sub-group consisting of all individuals of a certain height in cm.
If you make a plot of the number of people within each sub-group, or
the y value, versus the height of that sub-group, the x value, you
will get a graph looking rather (but not exactly) like this:

Here, the y or f(x) value is called dnorm(x). Value x = 0 represents
the average height of the population, and each x point (which have
been connected together in a continuous line) the greater or lesser
height on either side of x = 0. You see the bell shape of this
curve, hence its common name.

What
about those trigonometric functions? As another example, a sine
function, which is the typical shape of a wave, looks like this:

The resemblances, I assume, are obvious; this function looks a lot
like a bunch of bell shaped curves (both upright and upside-down),
all strung together. In fact the relationship is so significant that
a probability curve such as the Gaussian can be modeled using a
series of sine (and cosine) curves in what mathematicians call a

*Fourier**transformation*. So obvious that Erwin Schrödinger, following up de Broglie’s work, in 1926 produced what is now known as the*Schrödinger wave equation*, or equations rather, which described the various properties of physical systems via one or more differential equations (if you know any calculus, these are equations with relate a function to one or more of its derivatives; if you don’t, don’t worry about it), whose solutions were a series of complex wave functions (a complex function or number is one that includes the imaginary number i, or square root of negative one), given the formal symbolic designation*ψ*. In addition to his work with Heisenberg, Max Born almost immediately followed Schrödinger‘s discovery with the description of the so-called complex square of*ψ,*or*ψ** *ψ*^{ }, being the probability distribution of the object, in this case, the electron in the atom.
It
is possible to set up Schrödinger’s equation for any physical
system, including any atom. Alas, for all atoms except hydrogen, the
equation is unsolvable due to a stone wall in mathematical physics
known as the

*three-body problem*; any system with more than two interacting components, say the two electrons plus nucleus of helium, simply cannot be solved by any closed algorithm. Fortunately, for hydrogen, where there is only a single proton and a single electron, the proper form of the equation can be devised and then solved, albeit with some horrendous looking mathematics, to yield a set of*ψ*, or wave functions. The complex squares of these functions as described above, or solutions I should say as there are an infinite number of them, describe the probability distributions and other properties of the hydrogen atom’s electron.
The nut had at last
been (almost) cracked.

**Solving Other Atoms**

So all of this
brilliance and sweat and blood, from Planck to Born, came down to the
bottom line of, find the set of wave functions, or

*ψ*s, that solve the Schrödinger equation for hydrogen and you have solved the riddle of how electrons behave in atoms.
Scientists,
thanks to Robert Mullikan in 1932, even went so far as to propose a
name for the squared functions, or probability distribution
functions, a term I dislike because it still invokes the image of
electrons orbiting the nucleus: the atomic

*orbital*.
Despite
what I just said, actually, we haven’t completely solved the
riddle. As I said, the Schrödinger equation cannot be directly
solved for any other atom besides hydrogen. But nature can be kind
sometimes as well as capricious, and thus allows us to find side door
entrances into her secret realms. In the case of orbitals, it turns
out that their basic pattern holds for almost all the atoms, with a
little tweaking here, and some further (often computer intensive)
calculations there. For our purposes here, it is the basic pattern
that matters in cooking up atoms.

Orbitals.
Despite the name, again, the electrons do not circle the nucleus
(although most of them do have what is called

*angular momentum*, which is the physicists’ fancy term for moving in a curved path). I’ve thought and thought about this, and decided that the only way to begin describing them is to present the general solution (a wave function, remember) to the Schrödinger equation for the hydrogen atom in all its brain-overloading detail:
Don’t
panic: we are not going to muddle through all the symbols and
mathematics involved here. What I want you to do is focus on three
especially interesting symbols in the equation:

*n*, ℓ, and*m*. Each appears in the*ψ*function in one or more places (search carefully), and their numeric values determine the exact form of the*ψ*we are referring to. Excuse me, I mean the exact form of the*ψ** *ψ*, or squared wave function, or orbital, that is.
The
importance of

*n*, ℓ, and*m*lies in the fact that they are not free to take on any values, and that the values they can have are interrelated. Collectively, they are called*quantum numbers*, and since*n*is dubbed the*principle quantum number*, we will start with it. It is also the easiest to understand: its potential values are all the positive integers (whole numbers), from one on up. Historically, it roughly corresponds to the orbit numbers in Bohr’s 1913 orbiting model of the hydrogen atom. Note that one is its lowest possible value; it cannot be zero, meaning that the electron cannot collapse into the nucleus.*Also sprach Zarathustra*!
The
next entry in the quantum number menagerie is ℓ,
the

*angular momentum quantum number*. As with*n*it is also restricted to integer values, but with the additional caveat that for every*n*it can only have values from zero to*n*-one. So, for example, if*n*is one, then ℓ can only equal one value, that of zero, while if*n*is two, then ℓ can be either zero or one, and so on. Another way of thinking about ℓ is that it describes the*kind*of orbital we are dealing with: a value of zero refers to what is called an s orbital, while a value of one means a so-called p orbital.
What
about

*m*,*the magnetic moment quantum number*? This can range in value from – ℓ to ℓ, and represents the number of orbitals of a given type, as designated by ℓ. Again, for an*n*of one, ℓ has just the one value of zero; furthermore, for ℓ equals zero*m*can only be zero (so there is only one s orbital), while for ℓ equals one*m*can be one of three integers: minus one, zero, and one. Seems complicated? Play around with this system for a while and you will get the hang of it. See? College chemistry isn’t so bad after all.
* * *

Let’s
summarize before moving on. I have mentioned two kinds of orbitals,
or electron probability distribution functions, so far: s
and p. When ℓ equals zero
we are dealing only with an s orbital, while for
ℓ equals one the orbital is type p.
Furthermore, when ℓ equals one

*m*can be either minus one, zero, or one, meaning that at each level (as determined by n) there are always three p orbitals, and only one s orbital.
What
about when

*n*equals two? Following our scheme, for this value of*n*there are three orbital types, as ℓ can go from zero to one to two. The orbital designation when ℓ equals two is d; and as*m*can now vary from minus two to plus two (-2, -1, 0, 1, 2), there are five of these d type orbitals. I could press onward to ever increasing*n*s and their orbital types (f, g, etc.), but once again nature is cooperative, and for all known elements we rarely get past f orbitals, at least at the ground energy level (even though*n*reaches seven in the most massive atoms, as we shall see).**Explaining the Periodic Table. Atomic Orbitals.**

It
hopefully is now beginning to make some sense. Each horizontal row,
or period, in the table represents a specific primary quantum number,
or

*n*, starting with one at hydrogen (H) and going up to seven at francium (Fr) in the seventh row. As we move from left to right across a period, we are filling the elements in said period, by which I mean their various orbitals, with electrons. For*n*equals one there is only ℓ equals zero, thus*m*equaling zero, meaning we only have an s orbital to fill in hydrogen and helium – each orbital can only hold a maximum of two electrons, for reasons we will get to. For the period just below hydrogen and helium, where*n*equals two, ℓ can equal either zero or one, meaning we have one s orbital and three p orbitals to fill, the latter with*m*values equaling minus one, zero, and one; this gives us a total of one + three = four orbitals at this level, each orbital containing a maximum of two electrons to give us the eight elements in this row / period, Li through Ne.
The
title of this book,

*The Third Row*, refers to the period beginning with sodium (Na) and ending with argon (Ar). The first two columns, or groups, known as the alkali metals and the alkaline earths, represent s orbitals being filled, while the last six groups – of which the last two are called the halogens and the noble gasses – involve p orbital filling. The central, sunken region, or transition metals, are d orbitals being occupied, while the two offset rows at the bottoms are f orbitals being filled. We will get to the reasons why the d and f orbital periods are sunken**/**disconnected later. The first question you should ask is, how many electrons does it take to fill an orbital? From what I’ve said and you’ve just seen, the answer is two, but we can do a little better than that and explain why. It turns out I have been holding out on you.
Well,
no, I haven’t really. We are following the historical development
of quantum mechanics, and now is the time to include some important
concepts I have been ignoring so far. It turns out that there is a

*fourth*quantum number, known as*s*or spin (do not confuse this with s orbitals!), which comes about when quantum mechanics is reformulated using Einststein’s special relativity. This turns out to be necessary because the electrons in an atom move at a significant fraction of the speed of light (they can’t move as fast or faster than light speed, as relativity also says) and so relativistic effects cannot be ignored. This new quantum number*s*is of course also quantized, and so can have only one of two values: +½*ħ*or -½*ħ*, where*ħ*or h-bar as it is called, is equal to*h*/2π. This work was done by a number of individuals, some of whom we have already met, but the main new name to enter here is Wolfgang Pauli. Using the resulting relativistic quantum field theory devised from special relativity and quantum mechanics, Pauli was able to show in 1925 that no two electrons in an atom can have the same quantum numbers*n*, ℓ,*m*, and*s*. More broadly, he showed that fundamental particles known then (and now) could be divided into two camps: fermions, which obey the*Pauli Exclusion Principle*, and bosons, which do not. Later it came to be realized that fermions are particles which constitute the main mass of matter, such as electrons, protons, and neutrons, while bosons are force carrying particles, or the glue if you like, holding the whole menagerie of particles together. Photons are a good example of bosons: they carry the electromagnetic force. And there are others, as we learned from the last chapter but did not elaborate much on, such as nuclear forces.
If
an orbital can hold only two electrons (same

*n*, ℓ,*m*, but with different*s’*s) then we can see how the atoms can be built up, step by step, filling in the lower level orbitals and then expanding out to increasingly higher, or more energetic, level ones. The entire periodic table finally snaps into focus, and we find, to our astonishment, that we can grasp its rationale, or at least some of it. And yet, yes, I still haven’t covered one of the most important topics of all. What do these orbitals look like and how do they behave? And why should we care?# The Shapes and Behaviors of the Atomic Orbitals

What
I am about to show you can be misleading, or at least confusing. I
will talk about the shapes and sizes of orbitals, and even show
pictures of them. The misleading part is in thinking that orbitals,
in and of themselves, are actual, concrete things, filling space like
any other material object. This image, or illusion I should say,
though easily fallen for, is something we have to resist if we are
truly to understand orbitals, their meanings, and the functions they
serve.

A
good place to start is with the s orbital where

*n*equal to one, i.e., the 1s orbital, first because it is the simplest one in shape and also because there is an s orbital for every value of*n*– that is, every row in the periodic table. Let me begin by reminding you of the Gaussian distribution function, shown several pages earlier:
I do this because this function is essentially the shape of the 1s
orbital. The only difference is that this is a two dimensional
figure while of course orbitals are three dimensional entities. We
should redraw the 1s orbital more appropriately
as something like this:

Can you see how this smeared out sphere is the 3D equivalent to the
Gaussian curve? It is densest

**/**highest at the center, and then exponentially drops off from there; you get the picture if you will follow the density of the dots, ever closing in but never quite reaching zero as you go out from the center (the dots do*not*represent electrons themselves, but the probability of them being found at a certain place). Incidentally, the same picture fundamentally applies to all s orbitals, not just those where*n*equals one; for successively higher values of*n*(2, 3, 4, etc.) these still have the highest density in the center but the exponential decay decreases progressively more slowly, and circular nodes of zero density start to form rings in the distribution.
What
about p orbitals? Remembering that there are
three of them, the best description of them is as dumbbell shapes,
one dumbbell for each 3D axis (x, y, z), having a node of zero at the
center of the atomic nucleus:

Displayed here is a p

_{x}orbital, more specifically the 2p_{x}as we only start having these orbitals when the principle quantum number is two or higher. There are two other such orbitals, the 2p_{y}and the 2p_{z}, which have the same shapes but are oriented along their respective axes. And again, as*n*goes up, these orbitals become larger and more spread out, and develop nodes.
Technically
what comes next are the d orbitals, but again a
reminder before we proceed. These orbitals are not material entities
in any definition of the word; they are more akin to the various
states describing the electrons in an atom. If we were to propose a
(admittedly strained) analogy with our own world of space in time,
saying that an electron occupies a given orbital is rather like
saying a car is going so many miles per hour down a highway. In this
analogy, talking about an empty orbital is akin to talking about the
state of going so many mph, although no vehicle may actually be in
that state. I insert this caution precisely because we will at times
speak of orbitals as though they really are physical, even solid,
manifestations, for example when we combine them to make new
orbitals; but this is just a convenient way of talking about them –
don’t lose sight of what they really are, just the complex squares
of wave functions found by solving the Schrödinger equation for the
hydrogen atom.

With
this warning in mind, on to d orbitals. These
exist only for

*n*equals three or higher, so they don’t exist at all until we reach the third row of the periodic table, which as I have said, runs from sodium (Na) to argon (Ar). Here is also where we find ourselves faced with the puzzle of why the transition, or d-block, elements don’t begin here but only with the fourth period (*n*= 4). There are five such orbitals for every period that has them, but unfortunately they cannot easily be described by a few words so the only thing to do is show them in their full splendor:
Here, because it makes them easier to visualize, for instead of the
fuzzy pictures made from dots I used for s and p
orbitals I am using solid shapes (bearing in mind yet again my
warning that orbitals themselves are not solid, material things)
which enclose approximately ninety percent of each orbital’s
probability distribution. This is as good a place as any to solve
the mystery of the sunken d-block elements in the
table, not to mention the offset f-block
(lanthanides and actinides) as well.

In the beginning was the solution to the Schrödinger equation for
the hydrogen atom:

in which the energies of the orbitals was dependent solely on

*n*, the principle quantum number. Recall, however, that some tweaks and calculations are needed as we moved upward through the elements, because they have multiple electrons and so we can’t solve the equation for them directly. One of those changes is that, as soon as we start filling the orbitals with electrons, the kinds of orbitals at each level, s, p, or d, begin to diverge in energy, with the higher ℓ orbitals increasing over their lower siblings. Even by the time we get to*n*equals two in the table the p orbitals have higher energies than the s, and for*n*equals three the d orbitals are higher still.
This
is what accounts for the sunken transition elements. By the time

*n*equals three the 3d orbitals now lie at higher energies than the 4s orbitals which we naively expected should lie below them. Thus, we must wait for the 4s orbitals to be filled (which they are in the elements potassium or K, and calcium or Ca, through argon, Ar) before filling the 3d orbitals in the first row of transition metals, which goes from scandium (Sc) to zinc (Zn); only then can we move on to the 4p elements, from potassium (K) to krypton (Kr). A similar, even greater disparity in energy accounts for the f-block elements, the lanthanides and actinides, which is why they are set off below the main body of the table. It is a good thing nature doesn’t go as far as g orbitals, or our pretty table would become horrifically complicated!
Let
us recapitulate before moving on. We began this chapter by noting
that, in general, reality is not scale invariant, meaning that the
appearances and behavior of objects, and even the underlying physical
laws for them, appear to change as we move either to the world of the
immensely large or infinitesimally small. For the latter, we
discovered that nature at this level obeys the laws of quantum
mechanics, a system of physics that was mainly developed between 1900
and the 1930s. Electrons are so tiny that they fall well within the
range of this new system of physics; for example, they can not move
in simple orbits about the atomic nucleus as planets do around the
sun, but rather, their behavior is determined by wave functions
derived by solving the Schrödinger equation for the hydrogen atoms
(and then adding some extra tweaks and calculations). All of this
has been a considerable trek to understanding the whys and hows of
the periodic table of elements. And so, take a breather, and we
shall see where this will take us.