Animated Solution for Chemistry - Solutions: Freezing point of a $4\%$ aqueous solution of $X$ is equal to freezing point of $12\%$ aqueous solution of $Y$. If molecular weight of $X$ is $A$, then molecular weight of $Y$ is

Imagine you are standing in a laboratory with two beakers in front of you. One contains a $4\%$ aqueous solution of a mysterious substance $X$, and the other holds a $12\%$ aqueous solution of another substance $Y$. You dip a thermometer into both and discover something fascinating: they freeze at the exact same temperature!

This simple observation is the key to unlocking the molecular weight of $Y$. Let's dive into the beautiful world of colligative properties to see how.

When two solutions share the same freezing point, it means their depression in freezing point, denoted by $\Delta T_f$, is identical. The formula for this depression is:

Since both are aqueous solutions, their solvent is water. This means the molal depression constant, $K_f$, is exactly the same for both. If we equate their $\Delta T_f$ values, the $K_f$ beautifully cancels out, leaving us with a profound realization: their molalities must be equal.

Now, we need to carefully extract the molality from the given percentages. A $4\%$ aqueous solution by mass means that for every $100\text{ g}$ of the solution, there are $4\text{ g}$ of solute $X$.

Here is where many students make a classic silly mistake: they take the mass of the solvent as $100\text{ g}$. But remember, the solvent mass is the total mass minus the solute mass! So, the mass of water is $100-4=96\text{ g}$.

The molality of $X$ is:

Similarly, for the $12\%$ solution of $Y$, we have $12\text{ g}$ of solute in $88\text{ g}$ of water ($100-12=88$). Its molality is:

We know that $m_X=m_Y$, and we are given that the molecular weight of $X$ is $A$ ($M_X=A$). Let's set up our master equation:

The $1000$s cancel out immediately. Simplifying the fractions, we get:

Now, it's just a matter of cross-multiplying to isolate $M_Y$:

When we calculate $\frac{36}{11}$, we get approximately $3.27$. So, $M_Y\approx 3.27A$.

Looking at our options, the closest integer multiple is $3A$. And just like that, by understanding the physical reality behind the equations, we've cracked the code!