$($a$)(15$ points$)$

The net bonding energy EN between two isolated positive and negative ions is a function of interionic distance r as follows:

$$

$$ EN $=-$ Ar $+$ Bn $(1)$ r

$$

where A$,$ B$,$ and n are constants for the particular ion pair. Equation $(1)$ is also valid for the bonding energy between adjacent ions in solid materials. The modulus of elasticity E is proportional to the slope of the interionic force$$separation curve at the equilibrium interionic separation; that is$,$

$$

E$$dFdr $$r$0$

$$

Derive an expression for the dependence of the modulus of elasticity on these A$,$ B$,$ and n parameters, using the following procedure:

$1.$ Establish a relationship for the force F as a function of r$,$ realizing that

$$

dEN

F $=$ dr

$$

$2.$ Now take the derivative dF$/$dr$.$

$3.$ Develop an expression for r$0,$ the equilibrium separation. Because r$0$ corresponds to the value of r at the minimum of the EN$-$versus$-$r curve, take the derivative dEN$/$dr$,$ set it equal to zero, and solve for r$,$ which corresponds to r$0.$

$4.$ Finally, substitute this expression for r$0$ into the relationship obtained by taking dF$/$dr$.$