Determining Spring Constant in Ion Pair

[PEZenfuego]

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.

 

[mfb]

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

 

[2.718281828459]

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

 

[PEZenfuego]

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]

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?

 

[PEZenfuego]

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]

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

 

[PEZenfuego]

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?

 

[PEZenfuego]

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?

 

[mfb]

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.

 

[PEZenfuego]

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!