Animated Solution for Chemistry - Solutions: If a compound $AB$ dissociates to the extent of $75\%$ in an aqueous solution, the molality of the solution which shows a $2.5\text{ K}$ rise in the boiling point of the solution is ......... molal. (Rounded off to the nearest integer) [$K_b=0.52{\text{ K kg mol}}^{-1}$]

Imagine you are standing in a laboratory, watching a beaker of water come to a rolling boil. You know that pure water boils at exactly $100^{\circ }\text{C}$ (or $373.15\text{ K}$) at standard atmospheric pressure. But what happens when you dissolve a salt into it? The boiling point rises! This fascinating phenomenon is known as Boiling Point Elevation, and it is a classic colligative property.

In this problem, we are dealing with an aqueous solution of a compound $AB$. When we heat this solution, we observe that its boiling point is elevated by $\Delta T_b=2.5\text{ K}$. Our mission is to find the molality of this solution.

The compound $AB$ is described as a binary electrolyte. This means that when it dissolves in water, it splits into exactly two ions:

So, the number of ions produced per molecule, $n$, is $2$. However, the problem throws a slight curveball: the compound doesn't dissociate completely. It only dissociates to the extent of $75\%$. This means our degree of dissociation, $\alpha$, is $0.75$.

Because the solute particles break apart, the total number of particles in the solution is greater than the number of formula units we initially dissolved. To account for this, we must calculate the van't Hoff factor ($i$), which represents the effective number of particles in the solution.

The formula linking the van't Hoff factor to the degree of dissociation is:

Let's substitute our known values into this elegant equation:

This tells us that for every mole of $AB$ we dissolve, we effectively get $1.75$ moles of particles floating around in our beaker.

Now, we bring in the heavy artillery—the formula for boiling point elevation:

We know the elevation $\Delta T_b=2.5\text{ K}$, the van't Hoff factor $i=1.75$, and the ebullioscopic constant for water $K_b=0.52{\text{ K kg mol}}^{-1}$. We need to isolate the molality, $m$:

Substituting the values, we get:

Let's compute the denominator first. Multiplying $1.75$ by $0.52$ gives us $0.91$. Now, we divide:

The question specifically asks us to round off our answer to the nearest integer. Looking at $2.747$, the closest integer is $3$.

Therefore, the molality of the solution is $3\text{ molal}$.

This problem beautifully illustrates how the microscopic behavior of molecules—their tendency to break apart—directly influences macroscopic, observable properties like the temperature at which a liquid boils. It's a perfect blend of stoichiometry and thermodynamics!