The dissociation constant of two weak acids are `k_(a_(1)) & k_(a_(2))` respectively. Their relative strength is -

Options

A. `(K_(a_(2))/(K_(a_(1)))`
B. `(K_(a_(1))/(K_(a_(2))))^(1/2)`
C. `(K_(a_(1))/(K_(a_(2)))`
D. `(K_(a_(1)) xx K_(a_(2))/K_(a_(1)))^(1/2)`

Answer

To determine the relative strength of two weak acids based on their dissociation constants \( K_{a1} \) and \( K_{a2} \), we can follow these steps:

### Step 1: Understand the concept of dissociation constant
The dissociation constant \( K_a \) for a weak acid \( HA \) is given by the expression:
\[ K_a = \frac{[H^+][A^-]}{[HA]} \]
where \([H^+]\) and \([A^-]\) are the concentrations of the ions produced, and \([HA]\) is the concentration of the undissociated acid.

### Step 2: Define degree of ionization
The degree of ionization \( \alpha \) is defined as the fraction of the acid that has dissociated into ions. For a weak acid with an initial concentration \( C \):
- The concentration of \( H^+ \) and \( A^- \) at equilibrium will be \( C\alpha \).
- The concentration of undissociated acid \( [HA] \) will be \( C(1 - \alpha) \).

### Step 3: Substitute into the \( K_a \) expression
Substituting these values into the expression for \( K_a \):
\[ K_a = \frac{(C\alpha)(C\alpha)}{C(1 - \alpha)} = \frac{C^2\alpha^2}{C(1 - \alpha)} = \frac{C\alpha^2}{1 - \alpha} \]

### Step 4: Simplify for weak acids
For weak acids, \( \alpha \) is very small (i.e., \( \alpha \ll 1 \)). Thus, we can approximate \( 1 - \alpha \approx 1 \):
\[ K_a \approx C\alpha^2 \]

### Step 5: Solve for \( \alpha \)
From the above equation, we can express \( \alpha \):
\[ \alpha \approx \sqrt{\frac{K_a}{C}} \]

### Step 6: Compare the degrees of ionization of two acids
Let \( \alpha_1 \) and \( \alpha_2 \) be the degrees of ionization for acids with dissociation constants \( K_{a1} \) and \( K_{a2} \) respectively:
\[ \alpha_1 \approx \sqrt{\frac{K_{a1}}{C_1}} \quad \text{and} \quad \alpha_2 \approx \sqrt{\frac{K_{a2}}{C_2}} \]

### Step 7: Determine the relative strength
To find the relative strength of the two acids, we can take the ratio:
\[ \frac{\alpha_1}{\alpha_2} \approx \frac{\sqrt{K_{a1}/C_1}}{\sqrt{K_{a2}/C_2}} = \sqrt{\frac{K_{a1}}{K_{a2}}} \cdot \sqrt{\frac{C_2}{C_1}} \]

### Conclusion
The relative strength of the two acids is directly related to the square root of their dissociation constants. Therefore, if \( K_{a1} > K_{a2} \), then \( \alpha_1 > \alpha_2 \), indicating that the first acid is stronger than the second.