Class content > Energy: The Quantity of Motion > Atomic and Molecular forces
Prerequisites
Electric potential energy between charges
We know from Coulomb's law that two charged objects attract or repel each other with a force proportional to 1/r^{2}, 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.
We can also describe this in terms of potential energy. Let's say two charges attract. Their electric potential energy falls as 1/r as shown at the right. As the charges get closer to each other, their potential energy gets more negative. 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. The functional form is the same  1/r  but the sign of the product of the charge is now positive rather than negative, so the situation is reversed. As the charges 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).


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".)
Electric potential energy between neutral objects: Van der Waals forces
That explanation works for ions, which are charged, but doesn't explain how neutral atoms attract each other! Atoms 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 balloon 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.)
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.
How can we model this quantitatively? If we have two bare charges, we know the electric potential goes like 1/r. If we have one bare charge and a dipole (neutral but with + and  charges not in the same place) the potential falls like 1/r^{2}. If we have two dipoles, the potential falls like 1/r^{3}. The fact that our dipoles are not fixed but fluctuating, sometimes looking like dipoles, sometimes not, makes the result fall off even faster  like 1/r^{6}.
Interatomic repulsions
But there must be more to the story. 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 that prevents 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 atomatom 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?
A model for the atomatom potential was constructed by, John LennardJones (18941954). It is therefore known as the LennardJones potential and it is a reasonable approximation for the regions that atoms typically explore in biological systems. (Atoms don't get close enough to each other to see the details of the short ranged repulsion at biological temperatures.)
The LennardJones potential includes two parts: an attraction proportional to 1/r^{6}, and a repulsion proportional to 1/r^{12}. We can write this as PE = A/r^{12}  B/r^{6} , 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. What you get is shown in the figure at the right.


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. This is a very short ranged interaction! If you double the distance between two atoms, the potential energy associated with their attraction is divided by 64 (= 2^{6}). At small r, the graph climbs very steeply down as you approach the origin, indicating that there is a very strong repulsive force at close range.
Ben Dreyfus 10/30/2011 and Joe Redish 11/15/11
Comments (3)
Catherine Crouch said
at 6:44 pm on Nov 15, 2011
Here are a few bigpicture comments:
Fourth paragraph of first section: "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."
Will "much more so" be clear for the students, namely that you're talking about the distance dependence being a steeper function of r? I would be more specific. It may be sufficient to drop the phrase "Except much more so!". However, you may want to think carefully about a way to say this more clearly/precisely.
Next paragraph: "The fact that our dipoles are not fixed but fluctuating, sometimes looking like dipoles, sometimes not," Unless they are familiar with the idea that the dipoles are fluctuating in time from a previous reading, this is the first time that time dependence has been explicitly stated.
(Also, "fluctuating" may be an unfamiliar term for this group unless you've been using it?)
Catherine Crouch said
at 6:50 pm on Nov 15, 2011
What do the chemists think of the discussion of the Pauli exclusion principle (2nd paragraph of "Interatomic repulsions") making no mention of spin? Remember that two electrons can be in the same state with opposite spins ... The simplification "you can't have two electrons in the same place at the same time" seems to me to vastly oversimplify because my experience is that these students always know about the spin aspect of the Pauli principle even from high school chemistry. (They often don't *understand* it but they remember it.) Furthermore if you are going to discuss covalent bonding, then of course you have to deal with the idea that atoms can indeed share electrons ...
Catherine Crouch said
at 1:38 pm on Nov 21, 2011
I think this discussion does a good job of motivating qualitatively what the mathematical form of the potential needs to be, and then also interpreting that form. My question is: what are you going to do with the mathematical form of the LennardJones potential? Is it valuable for them to know the equation, as opposed to just seeing the graph? What will they do with it subsequently  will they calculate something with it? Compare it to another form?
A final observation: In optimizing the content of this course for the life sciences, we quickly get into many more complex situations than in typical introductory physics. As you've probably already considered, I think it's always important to ask what the goals are of introducing any particular mathematical treatment of a topic, and equally importantly, what you will expect the students to do with these ideas and mathematical formulas. If you have clear in your heads what you want the students to be able to do after instruction, then I think it is easier to decide whether it's worth developing the mathematics, or teach the idea only conceptually.
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