6.4.P7

A variety of interesting behaviors can be described with different shapes of potential energy. Four qualitatively different PE shapes are shown in the figure below. In each figure, the horizontal axis is a position and the vertical axis is a potential energy. You can think of them as frictionless tracks with a small ball rolling on them. The energy they describe is like a gravitational potential energy of the ball. In this problem, we will neglect the repulsion between atoms when you try to push them too close together. Instead, we will assume they can just be pushed through (or past) each other. Thus, both ends of the graph -- large positive and large negative x -- represent the atoms being far apart.

• The graph labeled F₁ is a hill and represents an unstable equilibrium. A ball can sit at the top of this hill, but a small deviation from the exact center will lead to forces that will cause the ball to roll down the hill and convert the potential energy to kinetic.
• The graph labeled F₂ is a hill with a dip and represents a metastable state. If a ball is sitting in the center of the dip at the top, a very small deviation will just oscillate back and forth a tiny bit. But if you provide the ball with just a bit more of kinetic energy, it will have enough energy to go up over the lip of the inner dip and roll down the long hill, gaining lots of kinetic energy in the process.
• The graph labeled F₃ is a well. When the ball is at the bottom of the well, it is a bound state. You have to give the ball kinetic energy equal to the depth of well to get it out. If you do that, when it gets out, it would have 0 kinetic energy left, using it all up in climbing out of the well.
• The graph labeled F₄ is a double well. Suppose the ball is at the bottom of the left well. If we temporarily lend the ball enough kinetic energy to get it over the hump in the middle (an activation energy)-- it could roll freely into the deeper well on the right.  Then, taking the energy back that we had just lent the ball to get over the hump, we could trap the ball in the right well.  The ball is now at the same total energy as it had before, but is in the right well.  In that right well the ball has a lot of kinetic energy as it passes the lower dip than it had when it was in the left well. It could give that energy up (through a collision or emitting a photon) and come to rest in the bottom of the second well. This is how you can go (with a little bit of help from your friends) from one bound state to a more deeply bound one and get some additional energy out as a result.

Some of these situations are analogous to the situation of chemical bonding. Consider the following examples and discuss which if any of the potential energy shapes provide a useful analogy. Explain why you think so. (The energy going into or coming out of a chemical reaction may be in the form of kinetic energy -- seen as heat -- or in terms of electromagnetic energy -- a photon.)

A. Two moles hydrogen atoms may interact and form a mole of H₂ molecules. When they do so, they release 103 Cal (kcal).

B. Decomposing one mole of water molecules (H₂O) into its component atoms requires the input of 220 Cal.

C. The decomposition of one mole of water molecules into H₂ and O₂ molecules requires the input of 113 Cal.

D. The phosphorus in the head of a match burns in air, giving off energy when you strike the match.

Joe Redish 1/25/12