If the solubility product of `AgBrO_(3)`, and `AgSO_(4)` are `5.5 xx 10^(-5)` and `2 xx 10^(-5)` respectively, the relationship between the solubility of these salts can be correctly represented as

Options

A. `S_(AgBrO_(3)) gt S_(Ag_(2)SO_(4))`
B. `S_(AgBrO_(3)) = (S_(Ag_(2))SO_(4))`
C. `S_(AgBrO_(3)) lt S_(Ag_(2)SO_(4))`
D. Unpredictable

Answer

To determine the relationship between the solubility of `AgBrO3` and `Ag2SO4`, we will follow these steps:

### Step 1: Write the dissociation equations
For `AgBrO3`:
\[ \text{AgBrO}_3 \rightarrow \text{Ag}^+ + \text{BrO}_3^- \]

For `Ag2SO4`:
\[ \text{Ag}_2\text{SO}_4 \rightarrow 2\text{Ag}^+ + \text{SO}_4^{2-} \]

### Step 2: Write the expressions for solubility products (Ksp)
For `AgBrO3`, if the solubility is \( S \):
\[ K_{sp} = [\text{Ag}^+][\text{BrO}_3^-] = S \cdot S = S^2 \]
Given \( K_{sp} = 5.5 \times 10^{-5} \):
\[ S^2 = 5.5 \times 10^{-5} \]
\[ S = \sqrt{5.5 \times 10^{-5}} \]

For `Ag2SO4`, if the solubility is \( S \):
\[ K_{sp} = [\text{Ag}^+]^2[\text{SO}_4^{2-}] = (2S)^2 \cdot S = 4S^3 \]
Given \( K_{sp} = 2 \times 10^{-5} \):
\[ 4S^3 = 2 \times 10^{-5} \]
\[ S^3 = \frac{2 \times 10^{-5}}{4} = 5 \times 10^{-6} \]
\[ S = \sqrt[3]{5 \times 10^{-6}} \]

### Step 3: Calculate the solubility of each salt
1. For `AgBrO3`:
\[ S = \sqrt{5.5 \times 10^{-5}} \approx 7.41 \times 10^{-3} \, \text{mol/L} \]

2. For `Ag2SO4`:
\[ S = \sqrt[3]{5 \times 10^{-6}} \approx 1.70 \times 10^{-2} \, \text{mol/L} \]

### Step 4: Compare the solubility values
- Solubility of `AgBrO3` is approximately \( 7.41 \times 10^{-3} \, \text{mol/L} \)
- Solubility of `Ag2SO4` is approximately \( 1.70 \times 10^{-2} \, \text{mol/L} \)

### Conclusion
Since \( 1.70 \times 10^{-2} > 7.41 \times 10^{-3} \), we can conclude:
\[ \text{Solubility of } Ag_2SO_4 > \text{Solubility of } AgBrO_3 \]

### Final Answer
The relationship between the solubility of these salts can be correctly represented as:
\[ \text{Solubility of } AgBrO_3 < \text{Solubility of } Ag_2SO_4 \]

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