- #1

jisbon

- 476

- 30

- Homework Statement
- The repulsive potential energy of 2 ions ##A^{2+}## and ##B^{2-}## is given as ##3.6*10^{-2} r^{-5}## eV.

i) Derive the attractive force between the two ions, and subsequently the attractive potential energy.

ii) Calculate the bonding energy of the 2 ions

- Relevant Equations
- -

I seem to be having problems with part two in general, but here's my solution.

Attractive force:

##F_{A}=\dfrac {k\left( 2\right) \left( -2\right) }{r^{2}}=\dfrac {-4k}{r^{2}} ##

##E_{A}=-\int ^{r}_{\infty }F_{A}\cdot dr ##

##\begin{aligned}E_{A}=\int ^{r}_{\infty }\dfrac {4k}{r^{2}}dr\\ =\left[ \dfrac {4k}{r}\right] ^{r}_{\infty }=\dfrac {4k}{r}\end{aligned} ##

(ii) This is where I started having trouble. I understand that to find the bonding energy, I need to find the condition where the attractive force + repulsive force = 0 and find out the radius from there. However, I can't seem to find the radius.

This is what I attempted,

Bonding energy when F =0,

##\begin{aligned}E_{r}=\dfrac {3.6\times 10^{-2}}{r^{5}}\\ F_{r}=\int E_{r}dr\end{aligned} ##

##\begin{aligned}F_{r}=-\dfrac {0\cdot 18}{r^{6}}\\ F_{0}=0=-\dfrac {0.18}{r^{6}}+\dfrac {-4k}{r^{2}}\end{aligned} ##Can anyone show me where I did wrong? Thanks

Attractive force:

##F_{A}=\dfrac {k\left( 2\right) \left( -2\right) }{r^{2}}=\dfrac {-4k}{r^{2}} ##

##E_{A}=-\int ^{r}_{\infty }F_{A}\cdot dr ##

##\begin{aligned}E_{A}=\int ^{r}_{\infty }\dfrac {4k}{r^{2}}dr\\ =\left[ \dfrac {4k}{r}\right] ^{r}_{\infty }=\dfrac {4k}{r}\end{aligned} ##

(ii) This is where I started having trouble. I understand that to find the bonding energy, I need to find the condition where the attractive force + repulsive force = 0 and find out the radius from there. However, I can't seem to find the radius.

This is what I attempted,

Bonding energy when F =0,

##\begin{aligned}E_{r}=\dfrac {3.6\times 10^{-2}}{r^{5}}\\ F_{r}=\int E_{r}dr\end{aligned} ##

##\begin{aligned}F_{r}=-\dfrac {0\cdot 18}{r^{6}}\\ F_{0}=0=-\dfrac {0.18}{r^{6}}+\dfrac {-4k}{r^{2}}\end{aligned} ##Can anyone show me where I did wrong? Thanks