One of the challenging ideas of using energy at the atomic and the molecular lev

Question

One of the challenging ideas of using energy at the atomic and the molecular level is the idea of bound states. What this means is that you might start with two objects that have essentially zero kinetic energy, they get close and interact strongly in an attractive way. They find some way to emit energy into another form and wind up being stuck together—bound. You have to put energy in—do work on them —in order to get them apart. Let's work through the language of potential energy to see how to talk about this.

(a) Let's first talk about a simple problem that you have now had some practice with—the motion of a skateboarder on a track. Suppose the track looks like a dip in the ground as shown in the figure. A potential shape like this is often referred to as a potential well, since it looks like a dug-out area for a well.

Now suppose that the skateboarder approaches the dip from the left traveling with a positive kinetic energy. The figure below shows the skateboarder's total mechanical energy as a solid black line at a PE of 10 units (units unspecified).

a. Describe the motion of the skateboarder and how her potential and kinetic energies change as she moves through the well.

Now suppose that she starts inside the well at a zero velocity —say at point x = 2.5 units with a total energy as shown by the heavy solid line.

b. Describe the motion of the skateboarder and how her potential and kinetic energies change as she moves through the well.

c. Her total energy is shown in the figure as 10 units. How can this be? Is it reasonable for the total mechanical energy to be negative?

(d) If she wants to climb out of the well and be at 0 kinetic energy at the point x = 3 units, how much energy would she need to gain?

(e) The skateboarder is actually just an analogy for the cases we are interested in, which are interacting atoms. This is really too simple a model: the atoms are impenetrable and will repel if pushed too close together. Instead of the simple well shown above, the atom-atom potential looks more like the one shown in the figure below. When the atoms are far apart there is little to no interaction. When they are closer, they are attracted and pulled together. If they get too close they are pushed apart. The potential energy of the interaction looks like the figure below.

If the atoms have the energy of 7.5 units as shown by the solid line in the figure, describe their motion and how their potential and kinetic energies change as they move in the well.

(f) If the atoms have an energy of 7.5 units as shown by the solid line in the figure, would you have to put energy in to separate the atoms or by separating them would you gain energy? How much? Explain why you think so.

Explanation / Answer

She approaches the dip moving at a constant velocity. Her KE remains constant (and positive) and her PE remains 0. When she reaches the dip, her PE begins to become more negative. So to have her total energy constant, her KE becomes larger and she speeds up. She speeds up as the depth of the well increases, reaching a maximum speed at the well's bottom. As she begins to climb the right side of the well, the PE gets less negative so the KE gets less positive and she slows down. When she reaches the end of the well her KE is the same as it was before she entered so she continues on at the same speed she had before she reached the well.

b)

She starts at 0 velocity inside the well so with a 0 KE, a negative PE, and a negative total mechanical energy. The well slopes down to the right so there is a force pushing her to the right. She begins to accelerate in that direction and, as her KE increases, her PE gets more negative, keeping the total mechanical energy constant. She reaches her maximum speed at the bottom of the well and slows as she rises up on the right, coming to rest at the point x = +2.5 units. Her energy is still negative so she can't leave the well.

c)

It is. The PE is really only relative. We can decide where we take it to be 0. If we choose it to be 0 when she is outside the well and at rest (a convenient place), then if her total energy is positive she can leave the well with a positive speed. If her total energy is negative it means that positive energy has to be added to bring her to a point at rest outside the well (to the value of 0 energy).

d)

10 units of energy.

e)

The atoms will be oscillating and bound together since their total energy is negative. We can start looking at the oscillation at the point when their separation is 2 units and their KE is 0. They are therefore at rest. Since the well slopes down to the left, there is a force pushing the atoms together. They begin to speed up towards each other, their KE increasing and their PE decreasing until they are approaching each other at their fastest speed at the bottom of the well. At that point their PE is a large negative value and their KE is the positive value needed to bring their total

energy up to -7.5 units. They continue approaching each other (since the force at the minimum is 0) but as they get closer the PE rises creating a repulsive force which slows them down. Their PE rises so their KE drops until they reach about x = 0.5 units. There, their PE = their total E (-7.5 units) so their KE equals 0. They start being pushed back apart and the cycle reverses. When they get back to 2, they stop and the cycle begins again.

f)

The have a total energy of -7.5 units so to get them apart you would have to put in +7.5 units of energy. This would put them at rest (KE = 0) a large distance apart (PE = 0).

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