• If atoms are electrically neutral (with an equal number of protons and electrons in each atom), what makes them stick together to form molecules and larger structures?
•  Or conversely, if something makes atoms attract each other, what makes them stay some distance apart, instead of moving all the way together?  Why don't molecules (and everything made of molecules) implode?
• How can we model this interaction quantitatively?
• And what does all of this have to do with energy?

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A quick review:

We know from Coulomb's law that two charged objects attract or repel each other with a force proportional to 1/r², i.e. inversely proportional to the distance squared.  So when the charges get closer together, the force of attraction or repulsion gets stronger.  A lot stronger:  if you cut the distance between the charges in half, the force will be multiplied by four.

(Note: The following paragraphs can be edited based on how much was and wasn't covered in the electrical potential energy section.)

We can also describe this in terms of potential energy.  Let's say two charges attract.  Then as they get closer together, their potential energy decreases.  (This makes sense, because as they move closer together, they would accelerate, and therefore gain more kinetic energy, which means they're losing potential energy.  It also makes sense because you would have to do work to split them apart.)  Not only that, but the slope of the potential energy gets steeper as the charges get closer together, indicating that the force is stronger.

Now let's say two charges repel.  Then it's reversed:  As they are pushed closer together, their potential energy increases.  (This time, go through the reasoning on your own in reverse, to convince yourself that this makes sense.)  Again, the slope of the potential energy gets steeper as the charges get closer together, since the force is still stronger when the charges are closer (even though this force is now a repulsion rather than an attraction).

How does the potential energy of two charges change quantitatively with the distance between them?  We know that the force is proportional to 1/r², and the force is the derivative (slope) of the potential energy with respect to position, so the force is a change in energy divided by a change in position.  Therefore, the potential energy must be proportional to 1/r.

As you know, we can set the "zero" point of potential energy anywhere we want; all that really matters is the change in potential energy as something goes from point A to point B.  So we'll set the potential energy to be zero when the two charges are "infinitely" far apart. ("Infinite" in physics doesn't have the same meaning as "infinite" in math; it just means "far enough away that they don't noticeably interact".)

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But wait!  That explanation works for ions (which are charged), but doesn't explain how atoms attract each other!  Atoms are neutral (they have an equal number of protons and electrons, so the net charge is zero), so they don't experience electric forces!  OR DO THEY?

Recall how the balloon stuck to the wall, even though the wall was neutral.  What was going on there?  Let's say the balloon had a net negative charge.  Then the negative charges (electrons) in the wall are repelled by the negative charges in the balloon, and they move (slightly) farther away.  Now the part of the wall closest to the ballon has a (slightly) positive net charge, and the negative charge in the balloon is attracted to this positive charge.

That's basically what happens with neutral atoms.  Atoms are neutral overall, but as you know from chemistry, they are made up of a positive nucleus on the inside, and electrons on the outside.  Imagine you're an atom, approaching another atom.  You might find that, at a given time, the electrons in that other atom are not distributed completely symmetrically about the nucleus.  As a result, when you get close enough, the part of the other atom that happens to be closest to you might look positive or negative.  Let's say it looks negative (like the balloon).  Then your electrons get repelled to the other side, and the side of you closest to the other atom becomes more positive, and you are attracted to the other atom.  Or let's say the other atom (on the side closest to you) looks positive.  Then your electrons get attracted to that side, and the side of you closest to the other atom becomes more negative, and once again, you are attracted to the other atom.  Either way, the result is an attraction!  This net attractive force is known as a Van der Waals force (or specifically a London dispersion force) which you may have heard about in your chemistry class.  Here's an animation of this phenomenon, along with some questions to consider.  (Many thanks to the CLUE (Chemistry, Life, the Universe, and Everything) project for these simulations.)

How can we model this interaction quantitatively?  Like Coulomb's law, we expect the dispersion force to get stronger as the atoms get closer together, and weaker as they get farther apart.  Except MUCH MORE SO!  This attraction between atoms is only significant if the atoms are really close; otherwise they just look neutral.

Atoms are complicated (especially larger atoms, with large numbers of electrons), so we're not going to come up with one equation (like Coulomb's law) that exactly describes the attraction between them.  But we can come up with a rough approximation that is good enough for us to see the important features.  And actually, John Lennard-Jones (1894-1954) already came up with that approximation, known as the Lennard-Jones potential.

Unlike the Coulomb potential, which is proportional to 1/r, the Lennard-Jones potential models the attraction between two neutral atoms as proportional to 1/r⁶.  That's not a typo - it's inversely proportional to the SIXTH power!  So if you double the distance between two atoms, the potential energy associated with their attraction is divided by 64 (= 2⁶).  So this is a very short-range interaction.

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But that can't be all there is!  If atoms just attracted each other, and this attraction continued to get stronger as they got closer together, then everything would eventually attract to everything, and all matter would collapse!  So there has to be something else preventing atoms from getting too close.

In your chemistry classes, you've probably learned about the Pauli exclusion principle, which states basically that you can't have two electrons in the same place at the same time.  This principle is the basis for all of chemistry, because it makes electrons end up in different orbitals, which gives different elements their chemical properties.  The Pauli exclusion principle provides the answer to our puzzle:  atoms can't get too close together, or they'll run into the problem of having two electrons in the same state.

This Pauli repulsion is the other half of the Lennard-Jones potential.  To model this repulsion quantitatively, let's think about what properties it needs to have:  it needs to be even stronger than the attraction at very short distances (preventing all matter from collapsing), but even weaker at larger distances (so that there's still some attraction).  But what could possibly be stronger at short distances (and weaker at long distances) than the sixth power?  Oh, I don't know, how about THE TWELFTH POWER?

So the Lennard-Jones potential includes two parts:  an attraction proportional to 1/r⁶, and a repulsion proportional to 1/r¹².  We can write this as PE = A/r¹² - B/r⁶ , where A and B are constants whose values depend on the specific types of atoms.  (The positive term represents repulsion, and the negative term represents attraction.) To see what this looks like, you can try graphing it on a graphing calculator or spreadsheet, and experiment with different values of A and B.  Here's an example:

Let's see what we can conclude from this graph.  At large r, the potential energy graph looks flat.  The slope is just about zero.  Thus, atoms that are far apart feel just about no force.  At small r, the graph slopes very steeply down and to the right, indicating that there is a very strong repulsive force at close range.

"Ok," you say, "but we knew all of that without the graph or the mathematical model!  That was how we came up with the model in the first place!  Does this model tell us anything we didn't already know?"  Here's one thing we can see from the graph that wasn't obvious:  The graph slopes downward, and then reaches a minimum potential energy, before going back up again.  What's happening at this minimum point?  The slope of the graph at this point is zero, meaning that an atom located at this point experiences no net force.  This is where the attractive and repulsive forces (discussed above) balance exactly.  This is a stable equilibrium:  if you move the atom away from this point in either direction, it will experience a force pushing it back to the equilibrium point.  Therefore, atoms can form stable molecules!  This r, where the potential energy is at a minimum, tells us the bond length for that particular bond (which you may have heard about in chemistry, especially if you've taken orgo).

Here's a simulation of two atoms coming together and forming a bond.  (In this simulation, the atoms don't stay together!  Why not?  What would need to happen for them to stay together?)

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Another important point you can observe in this graph is that the potential energy at the equilibrium point is NEGATIVE.  "Negative relative to what?", you might ask.  Relative to our "zero" potential energy, which (as discussed above) is when the atoms are far apart.  So there is LESS potential energy when atoms are bonded together than when they are separated.

This has two important consequences:

1) If atoms start out bonded together, you have to ADD energy just to get them back to "zero" potential energy, i.e. "breaking" the bond requires an input of energy.

2) In reverse: If a bond is formed (between atoms which were previously separate), the result is less potential energy than they started with, but by the principle of conservation of energy, we know this energy had to go somewhere else (it doesn't just disappear).  Thus, when bonds are formed, energy is released.

"But wait!" you say. "What does 'released' mean?  Where does the energy go?"  We'll get into some answers to this question later on, but in the meantime, think about this question and try to come up with some possible answers yourself.

Ben Dreyfus 10/30/2011