Animated Solution for Chemistry - Solutions: A solute A dimerises in water. The boiling point of a 2 molar solution of A is $100.52^{\circ }\text{C}$. The percentage association of A is ......... . (Round off to the nearest integer) [Use : $K_b$ for water $=0.52{\text{ K kg mol}}^{-1}$, boiling point of water $=100^{\circ }\text{C}$]

Imagine you are a chemist looking at a beaker of water. You dissolve a solute, let's call it $A$, into this water. But solute $A$ has a secret: it doesn't like to be alone. In the aqueous environment, two molecules of $A$ find each other and bind together to form a single, larger molecule, $A_2$. This process is called dimerisation.

Because colligative properties like boiling point elevation depend strictly on the number of particles in the solution, this dimerisation is going to play a massive trick on our calculations. Let's unravel the math behind it.

First, we need to determine exactly how much the boiling point of the water has increased. The problem states that the boiling point of the solution is $100.52^{\circ }\text{C}$. We know the boiling point of pure water is exactly $100^{\circ }\text{C}$.

The elevation in boiling point, denoted as $\Delta T_b$, is simply the difference between these two temperatures:

To account for the fact that our molecules are pairing up, we must use the van't Hoff factor, $i$. This factor represents the ratio of the actual number of particles in solution to the number of particles we initially dissolved.

For an association process where $n$ molecules join together, the formula for $i$ is:

Here, $\alpha$ is the degree of association (the fraction of molecules that actually pair up). Since our solute is dimerising, $n=2$. Substituting this into our formula gives:

Now, we bring in the fundamental equation for boiling point elevation:

Let's substitute all our known values into the master equation:

Notice how beautifully the math simplifies. We can divide both sides by $0.52$:

Expanding the right side:

Rearranging to solve for $\alpha$:

An $\alpha$ value of $1$ means that 100% of the molecules have undergone dimerisation. There are no single $A$ molecules left; they are all paired up as $A_2$.

Therefore, the percentage association of the solute is 100.