Question

Q.4
(a). Derive the cohesive energy for ionic crystals. Also explain the significance of Madelung’s constant using suitable diagram

Answer 1

Cohesive energy of ionic solid

$F=F_{att}+F_{rep}$

$F=\frac{A}{r^m}-\frac{B}{r^n}$

$F_{att}=\frac{Ae^2}{4\pi\epsilon r^2}$

$F_{rep}=\frac{B}{r^n}$

Work done by system of atom equal to negative of potential energy

$W=-P.E,P.E=-W$

$PE=-\frac{Ae^2}{4\pi\epsilon r}+\frac{B}{r^n}$

Equilibrium$r=r_0$

PE=-W=U(r)=min

Differenciate U(r) at $r=r_0$ W.r.t. to r is zero

$\frac{dU(r)}{dr}|_{r=r_0}=0$

$\frac{d}{dr}(-\frac{Ae^2}{4\pi\epsilon r^2})+\frac{d}{dr}(\frac{B}{r_0^n})=0$

$\frac{Ae^2}{4\pi\epsilon}(-\frac{1}{r_0^2})-nBr_0^{n-1}=0$

$\frac{Ae^2}{4\pi\epsilon r_0}\alpha\frac{1}{n}=\frac{B}{r_0^n}$

Now potential energy

$-\frac{Ae^2}{4\pi\epsilon r_0}+\frac{Ae^2}{4\pi \epsilon r_0}\times\frac{1}{n}=P E$

$P.E=-\frac{Ae^2}{4\pi \epsilon r_0}(1-\frac{1}{n})$

Unit are ev and Jules

Cohesive energy ionic solid per molecule per ion

$PE=-\frac{Ae^2}{4\pi \epsilon r_0}(1-\frac{1}{n})\times\frac{1}{2}$