Step 1: Understanding the Concept:
The Van't Hoff factor ($i$) corrects colligative property equations for solutes that dissociate or associate. The degree of dissociation ($\alpha$) relates directly to $i$ for weak electrolytes.
Step 2: Key Formula or Approach:
1. Calculate $i$: $i = \frac{\Delta T_{f} (\text{observed})}{\Delta T_{f} (\text{calculated})}$
2. Relate $i$ to $\alpha$: For acetic acid dissociating into 2 ions ($CH_{3}COOH \rightleftharpoons CH_{3}COO^{-} + H^{+}$), $n=2$. The formula is $i = 1 + \alpha(n - 1)$, which simplifies to $i = 1 + \alpha$.
Step 3: Detailed Explanation:
1. Calculate Van't Hoff factor ($i$):
Given observed depression = 0.0205 K.
Given calculated (theoretical) depression = 0.0197 K.
\[ i = \frac{0.0205}{0.0197} \approx 1.0406 \approx 1.041. \]
2. Calculate degree of dissociation ($\alpha$):
Using the relation $i = 1 + \alpha$:
\[ 1.041 = 1 + \alpha \]
\[ \alpha = 1.041 - 1 = 0.041. \]
Step 4: Final Answer:
The Van't Hoff factor is 1.041 and the degree of dissociation is 0.041.