Stare at the Graph for Awhile
The graph above shows a portion of a potential energy curve for atomic bonds within a hypothetical material. The diagram somewhat resembles a Lennard-Jones potential diagram .
The inter-atomic separation is plotted on the horizontal axis, and the bond potential energy is plotted on the vertical axis. The equilibrium separation is also shown.
When this material is heated, does it expand or contract?
Note:
Assume that the material is not heated enough to make the quasi-linear portion of the curve (on the right) relevant. As a bonus, what if we didn't impose this limitation?
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1 solution
How to read the graph
The graph of bond potential shows the relative energetic cost (vertical axis) of two atomic nuclei being placed the corresponding distance (on the horizontal axis) away from each other. At short distances there are repulsive forces that tend to keep nuclei from overlapping, and at long distances there are attractive nuclear forces that tend to bring them together.
A physical analogy for bond length
Instead of thinking about the two atoms, we can think about the bond like a mass on a spring. At low energies, the mass stays put at one end and the bond length is constant—this corresponds to the equilibrium separation on the graph of bond potential. As the energy of the mass grows, so does the amplitude of its vibration. If this were a perfect analogy, the bond potential graph would grow quadratically on both sides of the equilibrium solution, e.g. U ( x ) ∼ ( x − x eq ) 2 .
Putting it together
In this case though, the bond potential is not symmetric. The repulsive forces at low distance grow more slowly than the attractive forces at long distances. In other words, the curve is steeper to the right of the equilibrium point than it is to the left. This means that as the energy of the material is raised, the vibrations of the atomic bonds will be able to penetrate short distance (corresponding to compression) more easily than long distance (corresponding to stretches).
Putting it all together, this means that as the energy of the material is raised, the bond lengths will spend more time at short distances, moving the average bond length down, and leading to an overall compression of the hypothetical material.
Bonus
Once the energy of the atomic bond raises above the quasi-linear portion, the situation reverses. At that energy level, the bond potential energy rises more steeply on the left than it does on the right, and we expect the bond length to penetrate longer distances, spending more time in the stretched state and thus expect to see the hypothetical material expand.