Question

If AB is an ionic solid and the ionic radii of ${A^ + }$ and ${B^ - }$ are respectively ${R_c}$ and ${R_a}$ then the lattice energy of AB is proportional to which of the following?
A. $\dfrac{{{R_c}}}{{{R_a}}}$
B. ${R_c} + {R_a}$
C. $\dfrac{{{R_a}}}{{{R_c}}}$
D. $\dfrac{1}{{{R_c} + {R_a}}}\;$

Answer 1

Hint:Use Coulomb’s law to calculate the lattice energy of AB. Lattice energy is also inversely proportional to the internuclear distance of the given ionic solid. Since the given solid is ionic, the internuclear distance is equal to the sum of the radii of the individual ions.

Complete answer:
It is given that AB is an ionic solid. Since the ions ${A^ + }$ and ${B^ - }$ share an ionic bond between them, we can calculate the internuclear distance as the sum of the radii of both the individual ions.
${r_0} = {R_c} + {R_a}$
Now, we can use the modified version of Coulomb’s Law to calculate the lattice energy of AB.
$U = - \dfrac{{k{Q_1}{Q_2}}}{{{r_0}}}$
Where ${Q_1}$ and ${Q_2}$ are the ionic charges, ${r_0}$ is the internuclear distance, $k$ is a constant.
The charge on ${A^ + }$ is ${Q_1} = + 1$ and the charge on ${B^ - }$ is ${Q_2} = - 1$
Now, we substitute these values in our Coulomb’s equation:
$U = - \dfrac{{k \times ( + 1) \times ( - 1)}}{{{R_c} + {R_a}}}$
By solving, we get $U = \dfrac{k}{{{R_c} + {R_a}}}$

Therefore, the lattice energy is proportional to $\dfrac{1}{{{R_c} + {R_a}}}$ i.e. option D.

Additional information:
Lattice energy is defined as the amount of energy required to convert one mole of an ionic solid into gaseous ions. It is also the amount of bond strength between the ionic compounds. Lattice energy depends on two main factors: charge of the individual ions, size of the ions.


Note:
The lattice energy is inversely proportional to the internuclear distance of the given ionic solid. The value of lattice energy is always a positive value. The internuclear distance for an ionic solid is the sum of the radii of the individual ions.