Determining Spring Constant in Ion Pair

Homework Statement

The force between an ion pair is given by $F=-k\alpha\frac{e^{2}}{r^{2}}[1-\left(\frac{r_{\circ}}{r}\right)^{m-1}]$
Find the value of $$r$$ where the equilibrium position is.

Determine the effective spring constant for small oscillations from the equilibrium.

Using $m=8~\text{and}~\alpha=1.7476$ estimate the frequency of vibration of a Na+ ion in NaCl

Homework Equations

Binomial expansion theorem

The Attempt at a Solution

The first question is easy as you set the force equal to 0 and it is no surprise that the answer is $r=r_{\circ}.$ When I try using the binomial expansion theorem, I always end up with a dependence on $r_{\circ}$. But in the next portion, I have to find the frequency of vibration for Na+ ion and am only given alpha and m. Thanks for any help.

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mfb
Mentor
Where do you get a binomial expansions? Please show your work, otherwise it is hard to find out what went wrong.

Maybe use that $\left(1-x\right)^n\approx 1-nx$ where x is much less than 1.

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Maybe use that $\left(1-x\right)^n\approx 1-nx$ where x is much less than 1.
I did this! So I converted $r$ to $r_{\circ}+\Delta r$ and get $F=-k\alpha \frac{e^{2}}{r_{\circ}+\Delta r}[1-\left(1-\frac{\Delta r}{r_{\circ}}\right)^{1-m}]$ and say that $\frac{\Delta r}{r_{\circ}}$ is much less than 1 (which is reasonable for small angles). I was hoping that this would get rid of the r and $r_{\circ}$ dependence this way, but even if I do, what do I do about F?

mfb
Mentor
A right, with ##r \approx r_0## you get that binomial expansion, okay.

but even if I do, what do I do about F?
How does F vary for very small Δr? In particular, what about its derivatives?

Finding the derivative of force will show the points at which it is minimized and maximized, but I all ready know that. What good is it?

mfb
Mentor
It will also show you how the force varies for small deviations from the equilibrium point. This gives the effective spring constant.

It will also show you how the force varies for small deviations from the equilibrium point. This gives the effective spring constant.

Well, I end up with the spring constant being $\frac{k\alpha e^{2} \left(m-1\right)}{r_{0}^{3}}$ and plugging this into mathematica shows the tangent, which is in good agreement for small deviations. Great...so now what?

So I took r0 to be the ion separation between na and cl which was 0.28 nm. Using this and all of the other information, I ended up with a frequency of 1.19*10^13 hertz. Is this reasonable?

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mfb
Mentor
Well, I end up with the spring constant being $\frac{k\alpha e^{2} \left(m-1\right)}{r_{0}^{3}}$
I agree with that result.

Looks like you have to use the external value for r0.
I don't know about the frequency, but it does not look completely wrong.

For the units to work out I HAVE to know r0. Maybe my value is crap, but at least my solution shows the understanding is there. It turned out to not be too hard...maybe a little convoluted. Thanks for the help!