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

[knowlewj01]

Homework Statement

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:

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

Homework 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:

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 can't see it, anyone have any ideas?

//Edit: I have put in the correct latex code so you can see my calculations ;)

 

[kuruman]

We cannot read your equations. Please try to use LateX.

 

[knowlewj01]

ah, ok sorry about that ill try edit it

 

[knowlewj01]

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

 

[rdl]

Your approach is correct, but there is a small error in your algebra. When you rearranged the equation, you forgot to include the "n" in the denominator on the right side. It should be:

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

Then, when you substitute into the original formula, you get:

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

Simplifying, this becomes:

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

which is the correct answer.