Class content > Energy: The Quantity of Motion
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/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.
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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 pushing them together is stronger.
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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).
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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. In other words, we could draw a horizontal zero line at any PE position in the graphs above! But there is one very reasonable choice which we will generally make: Let's set the potential energy to be zero when the two charges are so far away from each other that they don't noticeably interact - "infinitely" far apart.
Why do we only introduce two electrostatic interactions? Because the electrostatic potential energy, just like any potential energy from multiple interactions can be simply added up (and it's a scalar so its not as difficult to add up as vector forces).
Electric potential energy between neutral objects: Van der Waals forces
The explanation above shows why ions, which are charged, attract or repel, 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 a rubbed balloon stuck to the wall, even though the wall was neutral. (See the PheT simulation.) 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.
Though it is much less dramatic than a balloon sticking to a wall, the same process also happens if two neutral objects are brought close to each other. Lets look at this at the atomic scale, and consider a pair of 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.
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Image from CLUE (Chemistry,
Life, the Universe, and Everything)
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Like Coulomb's law, we expect the Van der Waals 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/r2. If we have two dipoles, the potential falls like 1/r3. 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/r6. That's why Van der Waals forces are only noticeable on atomic scales or when two objects actually touch.
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.
While the van der Waals attraction can be at least qualitatively explained by physics you have already learned - charges and induced dipoles, the interatomic repulsion relies on physics you have not learned about yet in this intro physics class. Lets simply state that interatomic repulsion has to be even stronger than the attraction at very short distances (preventing all matter from collapsing), but weaker than the van der Waals attraction at larger distances. But what could possibly yield a larger PE at short distances (and a smaller PE at long distances) than at PE that changes as 1/r6 ? We had not emphasized this before: while the sixth power makes the potential very small very quickly with increasing distance r, it also makes the potential very strong at very small distances! So to generate a potential that is even stronger at small distances and even weaker at large distance all we have to do is go to a higher "power"! Let's just say we choose as our interatomic repulsion potential a function that goes as 1/r12
(NOTE: While the physics of the repulsion potential is beyond what we learned in physics so far, you may have learned the physics behind the repulsion potential in your chemistry class! The Pauli exclusion principle states basically that you can't have two electrons in the same state (with the same spin orientation) at the same time. This principle is the basis for putting electrons into 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.)
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A model for the atom-atom potential was constructed by John Lennard-Jones (1894-1954). It is therefore known as the Lennard-Jones 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 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. What you get is shown in the figure at the right.
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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 (= 26). 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
Wolfgang Losert 11/27/12
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