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Wednesday, January 11, 2012

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, , 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, … ¥, can range only from 0 to n – 1, and m from – to +.  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 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 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 pz) 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 ¾ C ¾ 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 ¾ C ·
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 pz 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 pz 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’ pz 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.