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Homework Help: Find equilibrium bond potential, given energy as a function of atomic separation

  1. May 17, 2010 #1
    1. The problem statement, all variables and given/known data

    Given that the total cohesive energy, U, in an ionic crystal as a function of nearest neighbor distance, R, between two ions +e and -e is given by:

    [itex]U(R) = \frac{A}{R^n} - \frac{\alpha e^2}{4 \pi \epsilon R}[/itex]

    show that at equilibrium:

    [itex] U(R) = \frac{\alpha e^2}{4 \pi \epsilon R}(1 - \frac{1}{n})[/itex]

    2. Relevant equations

    differentiate U with respect to R and set to zero to find the equilibrium bond length and substitute it into the origonal formula. I think this is the right way to do it but i keep getting the wrong answer, here is my best attempt:

    3. The attempt at a solution

    [itex]U(R) = \frac{A}{R^n} - \frac{\alpha e^2}{4 \pi \epsilon R}[/itex]

    differentiate w.r.t. R and equate to 0:

    [itex]\frac{dU}{dR} = 0 = -\frac{n A}{R^{n+1}} + \frac{\alpha e^2}{4 \pi \epsilon R^2}[/itex]

    now rearrange to get:

    [itex]\frac{n A}{R^{n+1}} = \frac{\alpha e^2}{4 \pi \epsilon R^2}[/itex]

    Multiply through by R and divide through by n:

    [itex]\frac{A}{R^n} = \frac{\alpha e^2}{4 n \pi \epsilon R}[/itex]

    Notice that the term [itex]\frac{A}{R^n}[/itex] appears in the original formula, so substitute to get:

    [itex] U(R) = \frac{\alpha e^2}{4 \pi \epsilon R}(\frac{1}{n} - 1)[/itex]

    the 1/n and 1 are the wrong way round, i have a feeling its a problem with my substitution but i cant see it, anyone have any ideas?

    //Edit: I have put in the correct latex code so you can see my calculations ;)
    Last edited: May 17, 2010
  2. jcsd
  3. May 17, 2010 #2


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    We cannot read your equations. Please try to use LateX.
  4. May 17, 2010 #3
    ah, ok sorry about that ill try edit it
  5. May 18, 2010 #4
    Nevermind, I just found out that this question was a misprint making it impossible. Thanks
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