1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    kuruman

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook