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Math Help - Spherical geometry and the "phosphorus pentachloride paradox"

  1. #1
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    Spherical geometry and the "phosphorus pentachloride paradox"

    Ever since I took chemistry, this problem has boggled my mind:

    Why is it that molecules of every "electronic cross-section", as it were, have perfect symmetry up to and including 4, but then suddenly breaks down at 5?

    What I mean is this: In the simplest case of a molecule with one orbiting atom and one central atom (we're viewing it that way for sake of visual convenience), the angle of separation between one orbiting atom and any neighbouring orbiting atom is the precisely the same angle (i.e. = 360 degrees.....but in this case, of course, "any neighbouring orbiting atom" is simply that same atom). The same is true for a molecule with two orbiting atoms, where the angle of separation is 180 degrees. True for 3 with 120 degrees. True for 4 with 109 degrees (what an odd number to represent symmetry!).

    But at 5, it suddenly breaks down. This is precisely the case with the molecule PCl5. Not any given chloride ion has the same angle of separation from any other given chloride ion. Between some pairs, they are 120 degrees, and between other pairs they are 90 degrees. I call this a "paradox" because every ion is supposed to be pushing away from every neighbouring ion with the exact same force. So, in theory, there should be perfect symmetry. But there isn't. I have even suggested that, perhaps, there is perfect symmetry for PCl5 and other such molecules in a "fourth spatial dimension", with an angle of separation of 108 degrees in that dimension (since 108 is the weighted mean of the angles we perceive in our limited 3-dimensional view of PCl5).

    There must be some reason that is not purely empirical as to why symmetry breaks down at 5. I have never studied spherical geometry, but I assume that the answer lies there. At first I thought it might have to do with prime numbers, but then again, there is still symmetry at 2 and 3.

    Can anyone shed light on this matter?
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    Re: Spherical geometry and the "phosphorus pentachloride paradox"

    You might find this link interesting.

    Min-Energy Configurations of Electrons On A Sphere
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