Prerequisites
In our discussion of Atomic and Molecular forces we learned that the typical potential energy between two atoms includes two parts: an attractive part that falls off rapidly with increasing distance (proportional to 1/r^{6}), and a repulsive part that is very short ranged and dominates when the atoms try to get too close. The commonly used LennardJones model of this repulsion goes like 1/r^{12}, so the total PE is modeled by the equation
where A and B are constants whose values depend on the specific types of atoms. 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. The shape of the PE curve looks like the graph shown in the figure at the right.


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. At small r, the graph slopes very steeply down and to the right, indicating that there is a very strong repulsive force at close range.
Since the strong positive repulsion dominates at short distances, while the longer range negative attraction dominates at longer distances, in between the graph has to turn around. As the atoms come towards each other from far away, the potential energy slopes downward going negative, and then becomes repulsive and goes positive. Somewhere in between, it has to reach a minimum potential energy, before going back up again. The slope of the graph at this point is zero, so two atoms located at this distance experience no force. The attractive and repulsive forces balance exactly. This is a stable equilibrium: if you move the atom away from this point in either direction, it will experience a force pushing it back towards the equilibrium point. Therefore, atoms that have this potential energy interaction can form stable molecules! The value of r, where the potential energy is at a minimum, tells us the bond length for that particular bond.
Here's a simulation of two atoms coming together and forming a bond. (In this simulation, the atoms don't stay together! Why not? What would need to happen for them to stay together?)
An example of a pair of atoms in a bound state is shown in the figure at the right by the heavy black line. This represents a particular case of a state of the total mechanical (kinetic plus potential) of the atoms' relative motion.
Note that the total energy of the atoms is NEGATIVE. "Negative relative to what?", you might ask. Relative to our "zero" potential energy, which we take to be when the atoms are far apart and at rest. So there is LESS total mechanical energy when atoms are bonded together than when they are separated. This has two important consequences:
 If atoms start out bonded together, you have to ADD energy just to get them back to "zero" potential energy, i.e. "breaking" the bond requires an input of energy.
 In reverse: If a bond is formed (between atoms which were previously separate), the result is less potential energy than they started with, but by the principle of conservation of energy, we know this energy had to go somewhere else (it doesn't just disappear). Thus, when bonds are formed, energy is released. (Where does the energy go when a bond is formed? We'll get into some answers to this question later on, but in the meantime, think about this question and try to come up with some possible answers yourself.)


Ben Dreyfus 10/30/11 and Joe Redish 11/17/11
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