Chapter Five -- All Molecules Great and Small

All Molecules Great and Small

I used the word bond(s) quite frequently in the last chapter, and by now you should have some grasp of what they are:  when two atoms approach closely enough, the most stable state of their outermost, or valence, electrons usually centers between the atoms.  Whenever two or more atoms can bond this way, the resulting structure is called a molecule; i.e., two hydrogen atoms and an oxygen atom will bond together to make the stable water molecule, of H₂O.  We were also introduced to the idea that different atoms can form different numbers (and types) of bonds:  one for hydrogen, four for carbon, three for nitrogen, two for oxygen, and so forth.  (The four bonds carbon can form will prove most convenient in coming chapters.)

There are numerous ways of portraying bonds.  In first year college chemistry courses the student is usually introduced to the Lewis dot structure.  Whereas before we represented bonds as bars between atoms, e.g., H-H, a better representation is to show the electrons themselves and where they are located in space:  H˸H.  Here it is clear that the two hydrogen atoms share their electrons in a region focused between the two nuclei.  You can see the stability this creates, and so why the bond is stable.

You’re already suspicious though, I’ll bet.  The Lewis dot structure still commits a cardinal sin when portraying electrons; for they are not hard little dots or spheres but spread out regions of uncertainty.  Remember how in the quantum world, we cannot say that objects cannot have exact locations, speeds, and directions; that if we do devise an experiment to measure one quantity exactly, we lose all information about the others.  Probability distributions rule the day.  Thus, an improvement on the dot structure representation would look something like this:

Figure XIV.

Here, the black spheres represent the hydrogen nuclei (usually just a proton), and the smeared out red region the electrons in the bonded state.  Note that we don’t even distinguish between one electron and the other, only that their combined region of highest density (probability distribution, that is) is directly between the nuclei.  This is about the best representation of how electrons behave in atoms and molecules.  To answer your next question, the dotted line surrounding the molecules encompasses ninety percent of the electrons’ distribution – in fact, this goes off to infinity.

*          *          *

What’s coming is, I think, rather grimmer than the material covered so far, but is so essential to describing atoms and molecules and bonds that I see no way of avoiding it.  Some advanced equations will be presented, but don’t worry, because there’s no need to understand them or know how to solve them.  I display them for you so that you will see there is a mathematical underpinning to all this talk of electron states and bonds.  Truth be told of course, there is a mathematical underpinning to all physics, so this should not come as a surprise.

But first, a switch in terminology.  Instead of states we will use the word orbital (both atomic and molecular) to describe an electron’s characteristics.  Orbital is, in my opinion, an unfortunate choice of word, as it implies electrons orbiting the nucleus, a fallacy I have struggled to help you overcome.  But it has been used so long now that we are apparently stuck with it; just remember that it is a synonym for state, not orbit, and you should be fine.  With this in mind, let’s plunge into the fray:

Figure XV.

This is the Schrödinger equation for the hydrogen atom.  You may recognize or know some of the symbols it contains, but the one I want you to concentrate on is y, which is the equation’s solution.  Interestingly, it is a wave function, i.e., a function that describes wave motion, such as that of a piano string vibrating back and forth.  The general form of this function, the solution to the equation, is:

Figure XVI.

Again, don’t worry about most of the details; deriving this solution requires some heavy duty math.  There are certain variables you should pay heed to, however.  These are n, ℓ, and m, known respectively as the principle quantum number, the angular momentum quantum number, and the magnetic moment quantum number.  The combined values of these numbers determines the exact form of y.  There are relationships among them as well:  while n can have any integer value from one on up (1, 2, 3, 4, 5, … ¥, ℓ can range only from 0 to n – 1, and m from – ℓ to + ℓ.  So for example, if n = 1 (the lowest energy states of the atom), then l and m can only be 0; while for n = 2 ℓ can have the values 0 and 1, while m be one of three values, -1, 0, and 1.

The orbitals I mentioned earlier are determined by their quantum numbers.  The number specify their shapes, the probability distributions of the electrons occupying them (a maximum of two, known as the Pauli Principle), and the energies of those electrons:

Figure VII. (repeated)

The elements in the periodic table are built up by adding additional protons (and neutrons) to the nucleus, and then filling the atomic orbitals from inwards to outwards with the required electrons to maintain the atoms electrical neutrality (atoms that are not neutral are called ions, but they are still atoms).  For the working chemist the electrons are described by orbital designations; say, for selenium, or element 34, the designation would be 1s²2s²2p⁶3s²3p⁶4s²3d¹⁰4p⁴, where 1, 2, 3, and 4 represent the shell or primary quantum number, s, p, and d the ℓ numbers in the shell, and the exponent the number of electrons occupying each orbital group in a shell (thus, there are 3 p orbitals, 5 d orbitals,  1 s orbital, and so on).  I won’t describe this scenario in any greater detail because:  a) it still befuddles me some after all these years and I doubt I could describe it clearly and simply; and  b) our main thrust will be bonding/molecular orbitals, and only a basic grasp of the atomic kind is needed to do so.

*          *          *

There is one more revelation about atoms that must be made before turning to molecules.  In truth, all the atoms that have been built up in the periodic table should be analyzed by constructing and solving a Schrödinger-type equation for each one, yet we have only done so for hydrogen and assumed the results applied to all atoms.  Amazingly, and fortunately, they almost do!  In fact, we could set up an equation for each elements; could set up, but never solve by any known mathematical technique known to Homo sapiens.  The reason?  In hydrogen we are dealing with only two interacting bodies, but for every other atom and molecule, we are working with more than two.  The so-called three-body equation is intractable to all math and physics; not just in the quantum world but everywhere, at all times, in the solar system and the universe as a whole – though whether it explains why there are only two sexes I have no idea:  nature seems to have no difficulty with it, but then nature is just doing what it does according to its laws; it (She?) isn’t trying to calculate anything in advance.

Returning to the equation of the hydrogen atom we are lucky indeed that its solution gives us a sound working model for the other atoms as well; only minor modification are needed to obtain excellent results for them.  However, and this is important to emphasize, any two or more solutions (functions) to the equation that have approximately equal energy and obey the mathematical requirement for “orthogonally” can be combined to make new functions/solutions.  For example, on carbon the single 2s and three 2p orbitals can be combined to create four new “sp³“ orbitals, which stick out at tetrahedral angles from the carbon nucleus:

Figure XVII.

I suspect I have presented you with a puzzle.  Why not use these orbitals to hold carbon’s four valence (outer) electrons, instead of the original one from solving the Schrödinger equation?  The answer is simply that the original orbitals are at lowest energy when considering lone carbon atoms; only when chemically bonded to other atoms – including other carbon atoms – is this configuration preferable, for reasons we shall see soon enough.

This is not the only possible combination.  Another “hybrid” (as they are know) orbital is the sp² orbital, in which one of the p orbitals – technically the p_z) is left out:

Figure XVIII.

Notice that the omission one a p orbital has changed the geometry of the atom from tetrahedral to a flat triangle; the new hybrid orbitals comprising the triangle is called, reasonably enough, sp².  Other hybrids or combinations produce a wide variety of configurations, especially when they involve d or higher orbitals.

*          *          *

I made the blithe statement that all of this has to do with bonding with other atoms to produce molecules, and it is time to explain why.

Atoms takes up considerable space in molecules, and therefor can not simply assume any arrangement.  A good example of this is the compound methane, or CH₄.  We should like to spread the hydrogen atoms and their electrons as far apart as necessary, in order to reduce interelectron repulsions.  As a first guess we could try a flat structure:

H

│

H ¾ C ¾ H

│

H

Figure XIX.

That this is inadequate should not be difficult to see.  We have here a two dimensional representation of methane; and although it is often convenient to draw it out this way, the lack of a third dimension means that the hydrogens are not spread as far apart as possible.

Go back to the tetrahedral arrangement of electrons around carbon, and you can see that the carbon’s electrons and the hydrogen’s electrons finally put as much space between each other as possible, to produce the configuration with lowest energy:

Figure XX.

Comparing this with Figure XVII. We see that the tetrahedral arrangements of hydrogens about the central carbon exactly matches its 3p³ bond arrangement.  This is why these are the preferred orbital pattern in the molecule.  The only thing left to do in our preliminary exploration is to describe just what a bond is.  I’ve used an equivalent phrase molecular orbital, but with out explanation what that means.

The fact that I’ve used bond and molecular orbital interchangeability should be suggestive however, implying that bonds are orbitals too, and this is quite right.  Just as atoms can combine their base orbitals into hybrid orbitals, so can orbitals on different atoms also merge into molecular orbitals, given the right conditions of fairly equal energy and orthogonality.  Remember the depiction of the hydrogen molecule earlier in the chapter:

Figure XIV. repeated

Of course, we really need to solve the Schrodinger equation for the system, but we can’t so we must find another way.  Again, that is the linear combination of orbitals technique.  The 1s orbitals on each hydrogen atom add together to produce the molecular orbital or bond, within which the two electrons have even lower energy than single atoms.  Bear in mind, though, that the number of hybrid or molecular orbitals must equal the number of base orbitals we began with.  Thus there are four sp³ orbitals about carbon (and nitrogen and oxygen, and many of atoms).  So combining the 1s orbitals on the two hydrogens must yield two new orbitals.  And indeed they do; in addition to the bonding orbital, there is also an “anti-bonding” orbital, an orbital which has the effect of repelling the hydrogens if there are electrons in them as much as the occupied bonding orbital which attracts the hydrogen:

Figure XXI.

Notice that the main electron density is on the outside of the atoms instead of between them in Figure XIV.  Some things ought to be starting to gel by now.  Normally, hydrogen exists as a diatomic molecule, H₂.  This is because the two electrons from the atoms can fill the bonding orbital, but the anti-bonding are empty.  On the other hand, the next element in the periodic table, helium or He, exists a monoatomic atoms.  He₂ , if it existed, would have two more electrons than H₂, and those two electrons would fill the anti-bonding orbital, pushing the heliums apart and breaking the bond.  Hence, only He atoms are found in nature.

*          *          *

There is one more step until we can start letting the horses out of the barn.  We’ve spoken so far of base atomic orbitals (1s, 2p, 3d, etc.), hybrid atomic orbitals (sp³ and sp²), and molecular orbitals resulting from combining base atomic orbitals.  Look back on the picture of the methane molecule (Figure XX).  Can you see how the sp³ orbitals on carbon are combining with the 1s on the four surrounding hydrogens?  Remember, any proper linear combination of orbitals (strictly speaking, their underlying wave functions) yields new orbitals.  So hybrid orbitals can combine with atomic, other molecular, or hybrid orbitals.  The possibilities are mind-boggling, perhaps even infinite, especially as you add more and more atoms to the potential molecules.  Many millions of molecules are already known to science.  Obviously, I can only cover a very tiny fraction of them, but fortunately that will do to get the idea in your mind clearly.

Let’s look at methane again, and try an experiment in our minds.  What if we were to take a methane and pop off one of the hydrogens?  Why then, we would have something the looks like this:

H

│

                H ¾ C ·

│

H

Figure XXII.

Forgive the flat structure, which we know not to be true, but there is method in madness here.  The first thing the hits your eye is that the right hydrogen has been replaced by a dot, which represents an unbonded electron.  In removing the hydrogen with its lone electron, we leave carbon’s contribution to that bond hanging out there in space, so to speak.  More importantly, it is now available for bonding with another atom or molecule with a lone electron. For example, the molecule here is actually called a methyl radical, radical meaning containing unbonded electrons.  We can certainly combine it with another methyl radical:

  H   H

    │   │

           H ¾ C ¾  C ¾ H

       │   │

    H   H

Figure XXIII.

Using the flat structure for methane makes it much easier to draw this new molecule, called ethane.  The bonds about each carbon are still tetrahedral, however.

Incidentally, bonds of this nature, with head-on overlap of the constituent orbitals are known as sigma (σ) bonds.  The way this chapter has progressed, you can be excused for thinking it is the only kind of bond.  There are several more, and right now I’d like to concentrate on a specific one:  the pi (π) bind.  In keeping with methane and ethane, I’ll show you the simplest case, that of ethylene:

Figure XXIII.

Before discussing the “double” bond between the carbon atoms in ethylene, I want to draw your attention to the flat, trigonal geometry about them.  We have already seen this, in Figure XVIII, the sp² hybrid orbitals which creates this geometry.  Four of these orbitals combine with the 1s orbitals on the hydrogens to create those bonds, while the two carbon pointing sp² overlap to create a sigma bond, just as with H₂.  But this leaves something unaccounted for.  Both carbon’s p_z orbitals are now free to make other combinations, and each contains a single electron it can use for bonding.  In fact, the most stable bond it can form is by a sideways overlap of the p_z orbitals.  Such resulting bonds are called pi bonds, and they look like this:

Figure XXIV.

Imagine the two orange balloons on the left as the carbons’ p_z orbitals; then the yellow region on the right is the region of maximum overlap or combination.  This orbital (although there are two lobes, it is one orbital, above and beneath the sigma, represented by the straight line) is the bonding; the ps also combine to make an anti-bonding orbital, which on ethylene is empty, stabilizing the pi structure.

*          *          *

This chapter could cover more, much, much more; and we will pursue more material along these lines.  Especially, our exploration will heavily revolve around hydrogen and carbon, as well as oxygen and nitrogen; for these are the main atoms which make up the molecular nature of life.  The basics have been laid down here.

Chapter Four: Quantum Cats

The last chapter, on relativity, was a warm up to the problem of thinking about weird things.  Remember, things only get weird when they fall outside, usually well outside, our common range of experiences.  We all have this biologically built-in understanding of the world, of physics more precisely, which we’ve acquired from our genetic ancestry and life’s experiences.  We rarely give it much thought, unless magicians play havoc with it to fool us, or physics teachers demand we absorb all the symbolic definitions and formalities and math we need if were going to – say, if we’re going to figure out how to send men and instruments to worlds millions of miles away.  Fortunately, few of us are called on for such flights of fancy.

Special relativity is weird because it involves travelling at speeds no one has ever come close to.  General relativity is weird because it tries to understand the universe as a whole, or at least things much more massive than ourselves; ambitions few of us have ever gotten the itch for.

It’s time now for a different direction.  From now on, we will be focussing on the realm of the vastly small and, often, extremely short-lived.  To give you an idea of just how small, how many molecules of water are in a glassful?  If the volume is about a cup, or, more usefully, let’s use the metric system and call it 100 milliliters or 0.1 liter, then that number is about 33,000,000,000,000,000,000,000,000 or 3.3×10²⁴ molecules.  To capture a feel for just how large a number than is (i.e., how small water molecules are), all of the waters of this planet Earth would take somewhere between 10¹⁹ and 10²⁰ glasses to hold it.  That number is at least ten thousand times smaller than the number of water molecules in an ordinary drinking glass.

My opening salvo is, objects that small definitely fall deep into our notions of weird.  You really can’t imagine it all, you can only do calculations with the numbers.  I’m tempted to try to give you a feeling for it; say, by going down to the beach and building a sand castle, and all that.  But sand grains themselves are so large that they fall into the range of common-sense physics, and are barely one iota along our quest for the ultra-small.  If sand grains behaved as atoms and molecules do, this would be a very, very different world indeed.

*          *          *

If you were to go down to the beach anyway, and try to follow the whims and whereabouts and whichways of a sand grain buffeted about by the sea breezes, you would probably feel like a cat chasing the spot of a laser pointer about the house (they go nuts chasing it).  Nevertheless, come, let me take you – a child version of you works best because of its innocence – down to the sea shore to chase a grain of sand.  You will no doubt quickly report to me that it is impossible, you will never be able to do it.  In response, I say,  “I know what you say sounds right, but it can’t be.  Think about it.  At every moment in time the sand grain has:  a) an exact mass; b) an exact speed; c) an exact direction it’s travelling; d) an exact momentum; e) an exact kinetic energy; f) an exact temperature; g) an exact albedo (how much of the sun’s light it reflects); and – I’m sure I could come up with other measurements, but you get the idea.  You just need to apply yourself more diligently to follow these quantities.”

A true child would not comprehend this quandary, but just find something else to play with.  But the adult you gets the point:  while in practice we simply cannot move quickly enough to follow a sand grain’s various dances, in theory it could be done.  Could be.  Are you certain?  What if I were to start shrinking that grain, ever smaller and smaller, until it was so small it fell beyond the range of the most powerful light microscopes to see it, down to something of the size of a water molecule?  What would happen then?

We suddenly feel as though we’re in the middle of Paul McCartney’s song “Lucy in the Sky With Diamonds” or perhaps “Penny Lane”; a surreal, dreamlike state where all the ordinary rules of reality have been suspended, and we have to figure out the new ones.

And in truth, that’s not just metaphor.  How reality looks and acts like at this scale is truly, and astonishingly, very, very little like sand grains and castles and rail cars.  And it is, at least at first, just a little unsettling.  For it means, for example, we have to throw away things like our previous, common-sense idea of exact measurements.  Oh, you can still make them, but at a price.  That price is complete loss of all information about something else equally measurable.

As a concrete example of this (always turns abstract concepts into concrete visions, when possible, remember), try to measure the position where our sand grain scaled down to water molecule can be found at any given moment.  Yes, you can still set up a measuring system that gives you a 100% accurate value for that position.  But only for that moment when you make the measurement.  And where has the grain gotten too after that?  Strangely, oddly, bizarrely, utterly incomprehensibly we have no idea!  In order to measure its position exactly, we have completely scrambled all information about its speed and direction from that position.  It, literally, and I mean literally in the literal sense here as in quite absolutely literally, could now be anywhere in the known universe!

You see, we must hand over our magnifying glasses and microscopes, and radar guns, and all other mechanisms for measuring the quantities of things, over to the police of this Brave (perhaps with a few shots of gin) New World, and accept new tools for spying on the world about us.  These tools only allow us to make probabilistic approximations on things.  By accepting limits on how well we can measure, we can measure other things too, albeit within their limits as well.

If I sound heavy-handed in repeating again that we are not dealing with metaphor or analogy, but with nature as it actually is, please be patient.  We are so accustomed to our “normal” rules of reality that different ones do sound like artistic inventions or flights of the imagination.  I merely want to emphasize over and over that what we are faced with is as real as the computer I’m typing these words on.  But it is only in the land of the ultra-miniature do our new rules come into play, that is, in a way that we cannot fail to notice even if we try.

*          *          *

I just used the word probabilistic, I suspect not for the first time.  In talking about probability we all, I think, have a good grasp of what this concept means.  We don’t know – sports analogies always work best – who is going to win the World Series next year but we can analyze the different teams and the caliber of their players and compute, even if roughly, probabilities – likelihoods, odds, where to place our money, and so on.  In a similar fashion we cannot predict the height of a human being chosen at random, but we can use probability to come up with a good ballpark (pardon the pun) figure.  In fact, the particular method of probability used here is simply called the Gaussian Distribution function, illlustrated below:

Figure V.

Don’t concern yourself with the symbols, like μ (mu) and σ (sigma), although we will return to them later.  Note simply that the distribution is highest in the middle of the curve, and drops off on either side, coming close to but never reaching zero.

A measurement of a large number of human heights, plotted on a graph, would look much like this.  There would be a high central region where most of the heights found themselves, and then declining tails of progressively shorter and taller – and fewer – people.  A person chosen at random has a 68.2% chance of falling within one standard deviation, or 1σ, of the highest, most central height, called μ or the norm or average; a 95.4% of coming within 2σ of the norm; and so forth.

And that’s it.  All we can say.  We simply cannot speak with certainty the height of an individual chosen randomly.  There is a big difference, however, between this problem, which is part of ordinary reality, and the “same” problem in the quantum world.  In the world we’re used to we can still measure the selected individual and reveal himself to be an individual (e.g., David Strumfels) with a height of six feet two inches.  But in the quantum world, no.  First, there are no individuals; and if we try to make measurements which with to create individuals, then other important information about them is irretrievably lost.  In the quantum world, probability rules with an iron fist.

We can place this in more prosaic contexts however.  According to the laws of the quantum world, we cannot say exactly where a water molecule is in a glass; but we can calculate a probability distribution, similar to the Gaussian but not eactly the same.  There  is, in fact, a famous calculation chemistry and physics students perform that is quite similar:  a lone electron in a (one-dimensional) box.  You cannot say precisely where the electron is, you can only construct an equation showing the probabilities of being in the box somewhere at any given moment.

Given that I’ve raised the subject of electrons …

*          *          *

Atoms.  We’ve all heard about them.  Unless you truly slept through all your schooling you heard that:  a) everything is made from them (not true); b) they’re so tiny nobody can even see them (also not true); and c) they’re composed of a tiny, dense nucleus made from protons and neutrons, with even tinier electrons spinning around it (partially true, but critically untrue for where this discussion is headed).

It is time to continue on our journey.  We were at the level of water molecules, and as as we work our way down past that, past individual atoms even, down finally to electrons, things change little.  Mainly, those probability curves just keep getting bigger.  They get so big, in fact, that by the time we’ve reached the level of electrons the curves are complete masters of their behaviors.

Our interest in electrons is simple enough; they are what make chemistry possible, and control most of what it can do.  Not everything – remember the discussion on organic chemistry and geometry before – but without electrons there is no such thing as chemistry at all.

Some further evidence of just how strange the quantum world is worth the effort here.  In the quantum world probability is truly just that; by that I mean that if, say, we knew all the forces and effects working on a coin toss we could predict, with absolute certainty, whether a given toss would be heads or tails.  And it is always one or the other, nothing … well, nothing in-between, curious as that may sound.  In quantum probabilities, no such luck however.  All we can do is make observations, in which we will see this or that or something else, an infinity of possibilities with each one having its given probability.

This might sound purely academic, but study the picture below:

Figure VI.

The gist of this, Erwin Schrödinger’s famous quantum cat experiment, is a sealable container (usually a box), within which resides:  a) a live cat; b) a vial of deadly poison; c) a hammer and a lever which can drop the hammer, breaking the vial and killing the cat; and d) a single radioactive atom with a 50% probability of decaying within an hour and which, if it does decay will trip the lever, again killing the cat.  Unlike the picture however, the box is not open but completely sealed so that no information on the cat’s state (alive or dead) can get out.

You see, the quantum world is even stranger than I’ve described.  I’ve spoken of making observations and getting results according to probabilities, as though the results were already there waiting for us to unveil them.  It’s worse than that.  The results don’t even exist until we make the observation!  Quantum physicists speak of things like “collapsing the state function” among other arcane words and phrases, but what they mean is making an observation.  The act of observation reduces the infinitely many probabilities into a single, actual result, and that result doesn’t exist until the observation is made.

Sounds like magic, I know, or some kind of tomfoolery.  But it is the simple truth.  A truth that has been demonstrated with absolute certainty by thousands of experiments, making measurements to many naughts beyond the decimal point.

*          *          *

Let’s return to our concrete example, the quantum cat.  Seal the box, wait an hour – remember, I said that there is a 50% probability of the atom within decaying within that time span – then open it again.  What will you see?

This part of the answer is simple.  Either the atom has decayed, the lever tripped, the vial containing the point released, and the cat dead; or the atom didn’t decay and the cat is very much alive.

The hard part of the answer is, what’s the state of the cat just before opening the box?  Your first reply I’ll bet is the same as before, though you’re probably suspicious by this point.  You smell a rat following the cat and you think I’m trying to pull a fast one on you.

If you feel that way, that’s actually good; it means you’re following the discussion.  (See, I told you you could do it!)  Remember I just said that, at the quantum level, the result of an observation doesn’t exist until it is made.  The cat isn’t dead or alive prior to opening the box – unless of course, you decide that the cat is an observer who saw the vial being smashed or not, which, to be honest, is a valid argument here.  So:  what does exist?

Recall that I said that the result of the coin toss experiment is never somethin “in-between?”  I meant that it’s in a head state or tail state even before we lift our covering hand and make the observation.  Again, no such luck with the quantum cat.  The convenient way of treating it, that is what scientists do, is to say that it is in a “superposition” of both states.  It’s both alive and dead; until you open the box that is, and collapse the state fuction into one of the other component states.

I advise not over-thinking about this.  Scientists and philosophers and other pundits to this day argue about what quantum physics implies about reality and our relation to it.  Even the cat experiment is taken seriously by some, while a joke by others.  My personal take about the experiment, by the way, is that it is not physically possible; you can’t construct a container which allows no information to leak out about the cat being dead or alive, meaning that it is always possible to make a measurement about the cat’s state without opening the box; whether or not the measurement is actually made or not (what exactly is a measurement or who or what makes it, anyway?) is irrelevant.  But their are problems with this stance too.  So please – don’t lose any sleep over it.

The pragmatic truth about quantum physics is that scientists use it because it works.  And there are very real versions of the cat experiment that can and have been done, experiments which have shown that the superposition of states with the result not existing until observation is made is correct.  As for the philosophical implications, most scientists are too practical-minded to worry about them.  Too much.

*          *          *

It is now time to return to electrons, and how they behave in atoms and molecules.  Remember, this is the fundamental requirement just to have chemistry, so we need a reasonable feel for this before we can continue.  Fortunately, a reasonable feel is all we need.  To get that, all is needed is to take what we learned about cats in the quantum world and apply them to this new arena.

Like cats in a box, electrons in atoms and molecules exist in quantum states; these states describe such (indirectly) observable quantities as their energies, magnetism, where they’re likely to be found, and so on. Also, like the cat’s fate, the states are quite distinct from each other, and the electron can only move from one to the other, never being found in-between.

If you have been wondering why the world of the infinitesimally small is called quantum, the answer lies here.  In the quantum world things generally exist in discrete states and not somewhere along a continuum.  To give another example, imagine getting in your car, pulling out along along a highway, and accelerating to 50 mph.  It seems to you that the acceleration is continuous, with no breaks anywhere; there are no discrete sepeed states, and you don’t make jumps from one state to another.  But if quantum rules applied strongly at “ordinary” levels of reality, you might find that you could only do the speeds 0, 10, 20, 30, 40, 50 mph exactly, jumping from 0 – 10 – 20 instantly.  You could not do 36.568 mph, for that “state” doesn’t exist.  Of course, it’s possible that the actual states don’t comprise a continuum anyway, they are just too close together to measure.  In fact, this is the actual case, and quantum physics applies at the medium, large, and enormous scales of reality too.  It’s just that as the masses involve increases, the states grow ever and ever close together.  Even at the level of sand grains they are far too close together to be observed.

So:  electrons in atoms and molecules do not orbit around nuclei, like planets around the sun, but exist in appropriate (usually lowest energy) quantum states.  All the images of atoms being like miniature solar systems are innacurate in this respect.  So why do we use them?  Simply, the actual picture of an atom is too difficult to illustrate at our reality level, and the solar system works best here.  It is accurate in a number of ways that justify its use.  Just bear in mind that, when it comes to electron behavior, and it’s relationship to chemistry, it really doesn’t work at all that way.

Before I go on, let me remind you about thermodynamics and entropy again.  This is a very complex subject, but all we need to know is that electrons, like everything else, “perfer” to be in their lowest energy states.  In a multi-atom molecular, electrons are attracted to more atomic nuclei (made of protons, and particles of opposite charge attract one another), and this alone would be large molecules the most stable states of matter.  Careful.  The larger an atom/molecule the more electrons there are too, and these tend to repel each other, thereby raising the energy.  Exactly what types and number atoms and molecules which from atoms is largely a balancing act between these two forces.

The simplest, and most abundant atom in the universe is hydrogen, consisting of a one proton nucleus and one electron distributed about it.  This is not the most stable atom; it’s abundance is purely the result of how the universe has evolved.  Next to hydrogen, helium, with a two-proton nuclear and two electrons about it, is most common; it is more stable that hydrogen, but is still far from maximum stability and again it’s abundance is explain by cosmic evolution.

As for molecules, hydrogen usually exists in diatomic molcules, in this case H₂.  This is because the two electrons are more strongly drawn toward the two nuclei than they repel each other.  Helium exists as monoatomic He atoms, however, because the four electrons exert a more profound effect than the two, albeit di-proton nuclei, can counter.

As we climb the periodic table of elements (just a diagrammic listing of all elements, to be read left to right and top to bottom) you see hydrogen and helium in the top row:

Figure VII.

The elements Li through Uuo don’t follow an easily made out trend when it comes to stability and molecule making, but there are general trends that can be worked out and understood, usually with a lot of hairy math and computer time.

As this section is in fact about molecules and not lone atoms, what we want is a feeling for how the constituent atoms form them, with the particular shapes they have.  A good place to start is to return to good old water, a simple molecule that can illustrate many of the principles.

Water is made from two hydrogen and one oxygen atoms:  the famous H₂O.  But why that particular formula, and why do water molecules take on a bent shape:

Figure VIII.

No need to go into the details here.  Basically, the available basic shapes of the oxygen atom make up a tetrahedron (remember, I said you need to recall some basic geometry):

Figure  IX.

      The ogygen nucleus sits at the center of the tetrahedron, and the electrons are “at” (actually around) the vertices (the connecting lines are the chemical bonds).  If you imagine sticking the hydrogen atoms (whose electrons make up spherical states) at two of the vertices, the water water molecule shape should pop out at you.  Each electron on the hydrogen atoms pair up with a “lone” electron on ogygens atoms pair up; this is another quantum oddity in which a state can hold only up to two electrons; any more and they spill over into other states, usually larger, more energetic ones.  This limitation, by the way, is why atoms and molecules can build up.

A more complex example of a molecule is glucose (blood sugar):

Figure X.

Here, the black spheres are carbon atoms, the red oxygen, and the white hydrogen.  Ignore the HC=O part at the upper right end (the = is called a double bond) and for the moment, concetrate your attention on the carbon atoms.  Notice how the bonds sticking out of them make for tetrahedrons too?  Carbon can bond with as many as four other atoms so it is very clear here.  But the H-O-C parts are also, like water, bent, indicating the same tetrahedral shape as in water.  But oxygen can bond with only a maximum of two other atoms, so the other points of the tetrahedra are empty.

I am not implying that all bonds have tetrahedral shapes.  Far from it.  It is merely one of a myriad shapes which is fairly common, especially among atoms in the top two rows of the periodic table.  The HC=O group, for example, is flat and triangular.  Another shape is represented below

Figure XI.

This molecule (which has a negative three charge) is comrpised of two shapes.  The 6 CºN portions are linear (the º is called a triple bond), while the central FeC₆ part is an octahedron, the carbon at the center:

Figure XII.

As said, there are numerous other shapes as well.  And shapes can be bonded to shapes, creating large, complex molecules, such as proteins and DNA and many other molecules found in our bodies.  Still, other molecules possess even simpler shapes, such as O=O (O₂), H-H (H₂), NºN (N₂), O=C=O (CO₂), and so on.  The possibilities are, for a practical purposes, infinite, and many millions of molecules are now known about.

Many of these molecules are called “organic” and it is time to know why.