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Chemical bonding

This version was saved 12 years, 4 months ago View current version     Page history
Saved by Joe Redish
on November 17, 2011 at 11:11:05 am
 

 

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/r6), and a repulsive part that is very short ranged and dominates when the atoms try to get too close. The commonly used Lennard-Jones model of this repulsion goes like 1/r12, so the total PE is modeled by the equation

 

We can write this as PE = A/r12 - B/r6 , where A and B are constants whose values depend on the specific types of atoms.  (The positive term represents repulsion, and the negative term represents attraction.) 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.  Here's an example:

 

 

Let's see what we can conclude from this graph.  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.

 

"Ok," you say, "but we knew all of that without the graph or the mathematical model!  That was how we came up with the model in the first place!  Does this model tell us anything we didn't already know?"  Here's one thing we can see from the graph that wasn't obvious:  The graph slopes downward, and then reaches a minimum potential energy, before going back up again.  What's happening at this minimum point?  The slope of the graph at this point is zero, meaning that an atom located at this point experiences no net force.  This is where the attractive and repulsive forces (discussed above) 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 to the equilibrium point.  Therefore, atoms can form stable molecules!  This r, where the potential energy is at a minimum, tells us the bond length for that particular bond (which you may have heard about in chemistry, especially if you've taken orgo).

 

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?)

 

***

 

Another important point you can observe in this graph is that the potential energy at the equilibrium point is NEGATIVE.  "Negative relative to what?", you might ask.  Relative to our "zero" potential energy, which (as discussed above) is when the atoms are far apart.  So there is LESS potential energy when atoms are bonded together than when they are separated.

 

This has two important consequences:

1) 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.

2) 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.

 

"But wait!" you say. "What does 'released' mean?  Where does the energy go?"  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 10/30/2011

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