4. One model for the potential energy of a two-atom molecule, where the atoms are separated by a distance r, is U(E) = Us [e)* - (÷)] a. What units should U, and ro have? b. Sketch a graph of this potential energy function vs. r, assuming r > 0. c. Show that the distance between the atoms when the molecule is in stable equilibrium is given by Teq = 24/6 ro

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a, b, and c

4. One model for the potential energy of a two-atom molecule, where the atoms are separated by a
distance r, is
12
UG) = Us [* - ))
a. What units should Uo and ro have?
b. Sketch a graph of this potential energy function vs. r, assuming r > 0.
c. Show that the distance between the atoms when the molecule is in stable equilibrium is given
by
req = 21/6 ro
d. If the distance between the atoms increases from equilibrium to r = 31/6 ro. Show that the
force from one atom on the other associated with this potential energy is given by
= -
2U0
37/6r0
e. Assume the atoms are oscillating so their maximum separation is the same as described in the
previous part, rmax = 31/6 ro. This will be one of the turning points on your graph for the potential
energy function. Go to the sketch you made in part (b.) and indicate this turning point as well as the
other turning point on your graph.
f. Show that the other turning point will occur at r =
1/6
O ro. To help you with this, I would
recommend that you introduce a variable called B
You will then be solving a quadratic
equation for ß. For the special case of the rmax that we are using, you will not end up needing to
use the quadratic formula in solving for ß as your equation can be factored.
g. Show that the maximum kinetic energy of the system will be given by
U.
KEmax
36
h. If the molecules begin at rest withr =r, what would be the eventual fate of the molecule? As
a hint, you should go back to your graph to help you answer this question.
Transcribed Image Text:4. One model for the potential energy of a two-atom molecule, where the atoms are separated by a distance r, is 12 UG) = Us [* - )) a. What units should Uo and ro have? b. Sketch a graph of this potential energy function vs. r, assuming r > 0. c. Show that the distance between the atoms when the molecule is in stable equilibrium is given by req = 21/6 ro d. If the distance between the atoms increases from equilibrium to r = 31/6 ro. Show that the force from one atom on the other associated with this potential energy is given by = - 2U0 37/6r0 e. Assume the atoms are oscillating so their maximum separation is the same as described in the previous part, rmax = 31/6 ro. This will be one of the turning points on your graph for the potential energy function. Go to the sketch you made in part (b.) and indicate this turning point as well as the other turning point on your graph. f. Show that the other turning point will occur at r = 1/6 O ro. To help you with this, I would recommend that you introduce a variable called B You will then be solving a quadratic equation for ß. For the special case of the rmax that we are using, you will not end up needing to use the quadratic formula in solving for ß as your equation can be factored. g. Show that the maximum kinetic energy of the system will be given by U. KEmax 36 h. If the molecules begin at rest withr =r, what would be the eventual fate of the molecule? As a hint, you should go back to your graph to help you answer this question.
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