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))xxK_(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 acid strength
The strength of an acid is generally determined by its ability to dissociate in solution, which is quantified by its dissociation constant \( K_a \). A higher \( K_a \) value indicates a stronger acid.

### Step 2: Define the relative strength of the acids
The relative strength of two acids can be expressed as the ratio of their strengths. If we denote the strength of the first acid as \( S_1 \) and the second acid as \( S_2 \), we can write:
\[ \text{Relative Strength} = \frac{S_1}{S_2} \]

### Step 3: Relate acid strength to dissociation constant
The strength of a weak acid can be expressed as:
\[ S_1 = C_1 \cdot \alpha_1 \quad \text{and} \quad S_2 = C_2 \cdot \alpha_2 \]
where \( C_1 \) and \( C_2 \) are the concentrations of the acids, and \( \alpha_1 \) and \( \alpha_2 \) are the degrees of dissociation of the respective acids.

### Step 4: Substitute the expressions into the relative strength formula
Substituting the expressions for \( S_1 \) and \( S_2 \) into the relative strength formula gives:
\[ \text{Relative Strength} = \frac{C_1 \cdot \alpha_1}{C_2 \cdot \alpha_2} \]

### Step 5: Relate degree of dissociation to \( K_a \)
The degree of dissociation \( \alpha \) can be related to the dissociation constant \( K_a \) by the following relationship:
\[ \alpha_1 = \frac{K_{a1}}{C_1 + K_{a1}} \quad \text{and} \quad \alpha_2 = \frac{K_{a2}}{C_2 + K_{a2}} \]
For simplicity, if we assume \( C_1 = C_2 \), we can denote this common concentration as \( C \).

### Step 6: Simplify the expression
Substituting \( \alpha_1 \) and \( \alpha_2 \) into the relative strength expression, we get:
\[ \text{Relative Strength} = \frac{C \cdot \frac{K_{a1}}{C + K_{a1}}}{C \cdot \frac{K_{a2}}{C + K_{a2}}} \]
This simplifies to:
\[ \text{Relative Strength} = \frac{K_{a1}}{K_{a2}} \]

### Conclusion
Thus, the relative strength of the two weak acids can be expressed as:
\[ \text{Relative Strength} = \frac{K_{a1}}{K_{a2}} \]
This means that if \( K_{a1} \) is greater than \( K_{a2} \), then the first acid is stronger than the second acid.

### Final Answer
The relative strength of the two weak acids is given by \( \frac{K_{a1}}{K_{a2}} \).

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