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

1. May 17, 2010

knowlewj01

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:

$U(R) = \frac{A}{R^n} - \frac{\alpha e^2}{4 \pi \epsilon R}$

show that at equilibrium:

$U(R) = \frac{\alpha e^2}{4 \pi \epsilon R}(1 - \frac{1}{n})$

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

$U(R) = \frac{A}{R^n} - \frac{\alpha e^2}{4 \pi \epsilon R}$

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

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

now rearrange to get:

$\frac{n A}{R^{n+1}} = \frac{\alpha e^2}{4 \pi \epsilon R^2}$

Multiply through by R and divide through by n:

$\frac{A}{R^n} = \frac{\alpha e^2}{4 n \pi \epsilon R}$

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

$U(R) = \frac{\alpha e^2}{4 \pi \epsilon R}(\frac{1}{n} - 1)$

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. May 17, 2010

kuruman

3. May 17, 2010

knowlewj01

ah, ok sorry about that ill try edit it

4. May 18, 2010

knowlewj01

Nevermind, I just found out that this question was a misprint making it impossible. Thanks