Animated Solution for Chemistry - Solutions: Molal depression constant for a solvent is $4.0{\text{ K kg mol}}^{-1}$. The depression in the freezing point of the solvent for $0.03{\text{ mol kg}}^{-1}$ solution of ${\text{K}}_2{\text{SO}}_4$ is (Assume complete dissociation of the electrolyte)

Imagine you are standing in a freezing winter landscape, and you see salt being thrown on the icy roads. Have you ever wondered why that works? It all comes down to a fascinating phenomenon in chemistry known as Depression in Freezing Point.

When you add a non-volatile solute to a pure solvent, the solute particles disrupt the solvent's ability to form a solid crystal lattice. As a result, the temperature must drop even lower for the solution to freeze. This is a colligative property, meaning it depends strictly on the number of solute particles, not their identity.

The mathematical relationship that governs this temperature drop is beautifully simple:

Let's break down our cast of characters: - $\Delta T_f$: The depression in freezing point (how much the freezing point drops). - $K_f$: The molal depression constant, a unique fingerprint for every solvent. In our problem, $K_f=4.0{\text{ K kg mol}}^{-1}$. - $m$: The molality of the solution, which is given as $0.03{\text{ mol kg}}^{-1}$. - $i$: The van't Hoff factor. This is where the real chemistry happens!

If we were dissolving sugar, $i$ would just be $1$ because sugar doesn't break apart in water. But we are dealing with Potassium Sulphate (${\text{K}}_2{\text{SO}}_4$), a strong electrolyte.

The problem explicitly tells us to assume complete dissociation. Let's visualize what happens when a single molecule of ${\text{K}}_2{\text{SO}}_4$ dives into the solvent:

One molecule shatters into two potassium ions and one sulphate ion. That is a total of $3$ independent particles swimming around in the solution! Therefore, our van't Hoff factor is:

Now, we have all the pieces of our puzzle. Let's substitute them into our master equation:

First, multiply the van't Hoff factor by the constant:

Next, multiply by the molality:

And there we have it! The freezing point of the solvent will be depressed by exactly $0.36\text{ K}$.

A Word of Caution: Always read the question carefully. If the electrolyte was weak and did not dissociate completely, we would have to use the degree of dissociation ($\alpha$) to find $i$ using the formula $i=1+\alpha (n-1)$. But for strong electrolytes with $100\%$ dissociation, $i$ is simply the total number of ions!