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 = mc2
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, H2, 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 entrée 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 (H2), 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 if you prefer, 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 20’th 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 ψs, 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 ns
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).
