Problem $1$

We have seen in class that ionic bonds formed between two atoms can be described by the following

pair potential

$E_N=E_A+E_R=-\frac{A}{r}+\frac{B}{r^n},$

where A and $B$ are positive constants, and $r$ is the interatomic spacing. In this problem, we will explore

the implications of this energy to KCl $($potassium chloride$).$

a$)$ The net force between two atoms is given by the sum of the attractive $F_A$ and repulsive $F_R$

components. Based on $E_N,$ what are the expressions for $F_A$ and $F_R?$ What is the net force when two

atoms are separated by the equilibrium spacing $r_0?$

b$)$ Based on $E_N,$ derive an expression for the equilibrium spacing $r_0.$

c$)$ The constant A can be theoretically estimated as

$A=M\frac{|Z_1||Z_2|e^2}{4{}_0}$

where $M$ is the Madelung constant, $Z_i$ are the charges of the resulting ions forming the bond, ${}_0$ is

the vacuum permittivity, and $e=1\mathrm{.}60{10}^{-19}C$ is the electron charge. For KCl $,$ the exponent $n$

in the potential is estimated as $n=9.$ If $4{}_0=1\mathrm{.}78{10}^{-38}\frac{C^2}{\text{e}\text{V}\text{n}\text{m}},M=1\mathrm{.}75,$ and

$B=2\mathrm{.}70{10}^{-5}eVn{m}^9,$ calculate $r_0.$

Hint: $Z_1$ and $Z_2$ are the number of electrons $K$ and Cl gain or lose when forming the ionic bond to

attain a stable electron configuration. If you are unsure, check the periodic table.

d$)$ Calculate the bonding energy $E_0($in electronvolts, eV $)$ between the two ions.

e$)$ Plot $($or simply sketch$)$ the energies $E_A,E_R$ and $E_N$ from $r=0$ up to $r=1nm.$ Show in your plot

where is the equilibrium spacing $r_0$ location.