Question

Dissociation constant of propionic acid is $1.32 \times 10^{-5}$. Calculate the degree of dissociation of acid in $0.05\text{ M}$ solution.

• A. $2.6 \times 10^{-4}$
• B. $1.61 \times 10^{-2}$
• C. $1.90 \times 10^{-2}$
• D. $3.5 \times 10^{-5}$

Hint:

To easily calculate square roots of scientific notation manually, adjust the exponent so it is an even number before taking the root! For example, taking the root of $10^{-5}$ is messy, but converting it to $10 \times 10^{-6}$ gives a clean $10^{-3}$.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given the acid dissociation constant ($K_a$) and the molar concentration ($C$) of a weak acid (propionic acid). We need to find its degree of dissociation ($\alpha$).

Step 2: Key Formula or Approach:
According to Ostwald's dilution law for weak electrolytes, the degree of dissociation ($\alpha$) is related to the dissociation constant ($K_a$) and concentration ($C$) by the simplified equation:
$$K_a = C \alpha^2$$
Rearranging to solve for $\alpha$:
$$\alpha = \sqrt{\frac{K_a}{C}}$$
(Note: This approximation is valid because weak acids dissociate very little, so $1 - \alpha \approx 1$).

Step 3: Detailed Explanation:
Substitute the given values into the formula:
$K_a = 1.32 \times 10^{-5}$
$C = 0.05\text{ M}$
$$\alpha = \sqrt{\frac{1.32 \times 10^{-5}}{0.05}}$$
To make the math easier, convert the decimals to scientific notation carefully:
$$\alpha = \sqrt{\frac{13.2 \times 10^{-6}}{5 \times 10^{-2}}}$$
$$\alpha = \sqrt{2.64 \times 10^{-4}}$$
Take the square root of both parts:
$$\alpha = \sqrt{2.64} \times 10^{-2}$$
Since $\sqrt{2.25} = 1.5$ and $\sqrt{2.89} = 1.7$, the square root of $2.64$ must be approximately $1.62$.
$$\alpha \approx 1.62 \times 10^{-2}$$
Looking at the options, $1.61 \times 10^{-2}$ is the closest exact answer provided.

Step 4: Final Answer:
The calculated degree of dissociation is closest to $1.61 \times 10^{-2}$, which corresponds to option (B).