If concentration of two weak acids are different and D.O.D `(alpha)` are very less then their relative strngth can be compared by :

Options

A. `([H^(+)]_(1))/([H^(+)]_(2))`
B. `(alpha1)/(alpha2)`
C. `(C_(1)alpha_(1))/(C_(2)alpha_(2))`
D. `(K_(a1)C_(1))/(K_(a2)C_(2))`

Answer

To solve the problem of comparing the relative strength of two weak acids with different concentrations and very low degrees of dissociation, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Weak Acids**:
Let the first weak acid be represented as HA with an initial concentration \( C_1 \) and the second weak acid as HP with an initial concentration \( C_2 \).

2. **Dissociation of Weak Acids**:
The dissociation of HA can be represented as:
\[ HA \rightleftharpoons H^+ + A^- \]
At equilibrium, if the degree of dissociation is \( \alpha_1 \), the concentrations will be:
- \([HA] = C_1(1 - \alpha_1)\)
- \([H^+] = C_1\alpha_1\)
- \([A^-] = C_1\alpha_1\)

3. **Equilibrium Constant for HA**:
The dissociation constant \( K_{a1} \) for the first acid can be expressed as:
\[ K_{a1} = \frac{[H^+][A^-]}{[HA]} = \frac{C_1\alpha_1 \cdot C_1\alpha_1}{C_1(1 - \alpha_1)} = \frac{C_1\alpha_1^2}{1 - \alpha_1} \]
Since \( \alpha_1 \) is very small, \( 1 - \alpha_1 \approx 1 \). Thus,
\[ K_{a1} \approx C_1\alpha_1^2 \]

4. **Solving for \( \alpha_1 \)**:
Rearranging gives:
\[ \alpha_1 \approx \sqrt{\frac{K_{a1}}{C_1}} \]

5. **Repeat for the Second Acid (HP)**:
Similarly, for the second weak acid HP:
\[ HP \rightleftharpoons H^+ + B^- \]
At equilibrium, if the degree of dissociation is \( \alpha_2 \), the concentrations will be:
- \([HP] = C_2(1 - \alpha_2)\)
- \([H^+] = C_2\alpha_2\)
- \([B^-] = C_2\alpha_2\)

6. **Equilibrium Constant for HP**:
The dissociation constant \( K_{a2} \) for the second acid can be expressed as:
\[ K_{a2} = \frac{C_2\alpha_2^2}{1 - \alpha_2} \approx C_2\alpha_2^2 \]
Therefore,
\[ \alpha_2 \approx \sqrt{\frac{K_{a2}}{C_2}} \]

7. **Comparing the Strengths**:
The relative strength of the acids can be compared by looking at the concentration of \( H^+ \) ions produced:
\[ [H^+]_{acid1} = C_1\alpha_1 \quad \text{and} \quad [H^+]_{acid2} = C_2\alpha_2 \]

8. **Final Comparison**:
Thus, the ratio of the \( H^+ \) concentrations can be expressed as:
\[ \frac{[H^+]_{acid1}}{[H^+]_{acid2}} = \frac{C_1\alpha_1}{C_2\alpha_2} \]

Substituting the expressions for \( \alpha_1 \) and \( \alpha_2 \):
\[ \frac{C_1\sqrt{\frac{K_{a1}}{C_1}}}{C_2\sqrt{\frac{K_{a2}}{C_2}}} = \frac{\sqrt{K_{a1}C_1}}{\sqrt{K_{a2}C_2}} \]

### Conclusion:
The relative strength of the two weak acids can be compared using the ratio of their \( H^+ \) ion concentrations, which is given by:
\[ \frac{C_1\alpha_1}{C_2\alpha_2} = \frac{\sqrt{K_{a1}C_1}}{\sqrt{K_{a2}C_2}} \]