Animated Solution for Chemistry - Chemical Thermodynamics: The Born-Haber cycle for KCl is evaluated with the following data : ${\Delta }_f{H}^{\circ }$ for KCl = $-436.7{\text{ kJ mol}}^{-1}$ ${\Delta }_{\text{sub}}{H}^{\circ }$ for K = $89.2{\text{ kJ mol}}^{-1}$, ${\Delta }_{\text{ionisation}}{H}^{\circ }$ for K = $419.0{\text{ kJ mol}}^{-1}$; ${\Delta }_{\text{electron gain}}{H}^{\circ }$ for Cl(g) = $-348.6{\text{ kJ mol}}^{-1}$, ${\Delta }_{\text{bond}}{H}^{\circ }$ for Cl${}_2$ = $243.0{\text{ kJ mol}}^{-1}$ The magnitude of lattice enthalpy of KCl in ${\text{kJ mol}}^{-1}$ is ......... (Nearest integer)

Imagine you are in a laboratory, and you want to create a crystal of potassium chloride ($\text{KCl}$) from solid potassium and chlorine gas. The total energy change in this direct process is called the enthalpy of formation, denoted as ${\Delta }_f{H}^{\circ }$. But what if we could break this down into a series of hypothetical, atomic-level steps? That's exactly what the Born-Haber cycle does! It's a beautiful application of Hess's Law, allowing us to visualize the energetic journey of these atoms from their standard states to a stable ionic lattice.

The Master Equation

According to Hess's Law, the total enthalpy change of a reaction is independent of the path taken. So, the direct formation enthalpy must equal the sum of all our hypothetical steps.

We first sublimate solid potassium into a gas, then ionize it to form ${\text{K}}^{+}$. Meanwhile, we break the chlorine molecule apart into individual atoms and add an electron to form ${\text{Cl}}^{-}$. Finally, these gaseous ions come together to form the solid lattice. The sum of all these energies equals the enthalpy of formation:

Now, let's carefully substitute the given values into our master equation. We have the formation enthalpy ($-436.7{\text{ kJ mol}}^{-1}$), sublimation energy ($89.2{\text{ kJ mol}}^{-1}$), ionization energy ($419.0{\text{ kJ mol}}^{-1}$), and electron gain enthalpy ($-348.6{\text{ kJ mol}}^{-1}$).

But wait, look closely at the bond dissociation energy! The value given ($243.0{\text{ kJ mol}}^{-1}$) is for a whole mole of chlorine gas (${\text{Cl}}_2$), but our balanced equation only requires half a mole to provide one mole of chlorine atoms. So, we must multiply the bond energy by $\frac{1}{2}$. This is a classic trap where many students make a silly mistake!

Let's simplify the numbers on the right side of our equation. Half of $243.0$ is exactly $121.5$. Now we have a clean, linear equation with just one unknown variable: the lattice enthalpy.

It's time for some basic arithmetic. Let's add up all the intermediate energy changes. Summing these up gives us $281.1{\text{ kJ mol}}^{-1}$. This positive value represents the net energy required to prepare the gaseous ions from their standard states.

We are almost there! To isolate the lattice enthalpy, we simply subtract $281.1$ from both sides.

The negative sign makes perfect physical sense—it tells us that a massive amount of energy is released when the oppositely charged ions crash together to form a stable crystal.

Finally, let's read the question one last time. It specifically asks for the magnitude of the lattice enthalpy, rounded to the nearest integer. The magnitude, or absolute value, is $717.8$. Rounding this to the nearest whole number gives us $718$. And there we have it, a perfect application of thermodynamics!