Animated Solution for Chemistry - Electrochemistry: Comprehension Passage

Imagine you are standing in a chemistry lab with two beakers. In the first beaker, you have an aqueous solution of a weak monobasic acid at $298\text{ K}$. Let's call its initial concentration $C$. The problem tells us that its molar conductivity, ${\Lambda }_m$, is $y\times 10^2{\text{ S cm}}^2{\text{ mol}}^{-1}$.

Now, you take this solution and dilute it $20$ times with water. This means the new concentration in the second beaker drops to $C/20$. Interestingly, the molar conductivity shoots up to $3y\times 10^2{\text{ S cm}}^2{\text{ mol}}^{-1}$. We are also given the limiting molar conductivity, ${\Lambda }_m^{\circ }$, which is a constant $4\times 10^2{\text{ S cm}}^2{\text{ mol}}^{-1}$.

Our mission is to find the initial degree of dissociation, $\alpha$, and the unknown multiplier, $y$.

To bridge the gap between molar conductivity and the degree of dissociation, we use a fundamental relationship:

For our initial solution, the degree of dissociation is:

For the diluted solution, the degree of dissociation becomes:

Notice how dilution increases the degree of dissociation! Now, here is the secret weapon: Ostwald's Dilution Law. The acid dissociation constant, $K_a$, is a thermodynamic property that depends only on temperature. Since the temperature is fixed at $298\text{ K}$, $K_a$ must be identical for both beakers.

The formula for $K_a$ is:

Let's plug our values into the $K_a$ formula for both states and equate them.

For the initial state:

For the diluted state:

Equating $K_{a1}$ and $K_{a2}$:

This equation looks intimidating, but it simplifies beautifully. We can cancel out $C{y}^2$ from both sides:

Now, we just cross-multiply to solve for $y$:

With $y$ in our hands, finding the initial degree of dissociation, $\alpha$, is a walk in the park. We know that $\alpha =y/4$:

Rounding off to two decimal places, we get $\alpha \approx 0.22$.

And there we have it! By trusting the constancy of $K_a$ and carefully tracking our variables through dilution, we've cracked the code.