For two weak acids A and B, the ratio of their percent ionization is 4 : 9. The ratio of their Ka would be–

Options

A. `4:9`
B. `2:3`
C. `16:81`
D. `3:2`

Answer

To find the ratio of the dissociation constants (Ka) of two weak acids A and B given the ratio of their percent ionization, we can follow these steps:

### Step 1: Understand the relationship between percent ionization and Ka
The percent ionization of a weak acid is related to its dissociation constant (Ka) and its concentration (C). The degree of ionization (α) is defined as the fraction of the acid that ionizes. The relationship can be expressed as:
\[ K_a = \frac{C \cdot \alpha^2}{1 - \alpha} \]

### Step 2: Assume that the ionization is very small
For weak acids, we often assume that the degree of ionization (α) is very small compared to 1, which simplifies our equation to:
\[ K_a \approx C \cdot \alpha^2 \]
This means that:
\[ \alpha \approx \sqrt{\frac{K_a}{C}} \]

### Step 3: Set up the ratios for the two acids
Let the percent ionization of acid A be α₁ and that of acid B be α₂. Given that the ratio of their percent ionization is:
\[ \frac{\alpha_1}{\alpha_2} = \frac{4}{9} \]
We can express this as:
\[ \alpha_1 = 4k \quad \text{and} \quad \alpha_2 = 9k \]
for some constant k.

### Step 4: Use the relationship for the two acids
Using the relationship derived earlier:
\[ \frac{\alpha_1}{\alpha_2} = \sqrt{\frac{K_a1}{K_a2}} \]
Substituting the values of α₁ and α₂:
\[ \frac{4k}{9k} = \sqrt{\frac{K_a1}{K_a2}} \]
This simplifies to:
\[ \frac{4}{9} = \sqrt{\frac{K_a1}{K_a2}} \]

### Step 5: Square both sides to eliminate the square root
Squaring both sides gives:
\[ \left(\frac{4}{9}\right)^2 = \frac{K_a1}{K_a2} \]
This results in:
\[ \frac{16}{81} = \frac{K_a1}{K_a2} \]

### Step 6: Write the final ratio
Thus, the ratio of the dissociation constants (Ka) of acids A and B is:
\[ \frac{K_a1}{K_a2} = \frac{16}{81} \]

### Conclusion
The ratio of Ka for the two weak acids A and B is \( \frac{16}{81} \).

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