mixture of KBr and NaBr weighing 0.563 g was treated with aqueous `AgNO_(3)` and all the bromide ion was recovered as 0.975 g of pure AgBr. What fraction of total mass is KBr in the sample ?

Options

A. 0.61
B. 0.22
C. 0.83
D. 0.45

Explanation

To solve the problem, we need to determine the fraction of KBr in a mixture of KBr and NaBr, given the total mass of the mixture and the mass of AgBr produced when the mixture is treated with AgNO3.

### Step-by-Step Solution:

1. **Define Variables:**
Let \( x \) be the mass of KBr in the mixture. Therefore, the mass of NaBr in the mixture will be \( 0.563 - x \) grams.

2. **Write the Reactions:**
The reactions of KBr and NaBr with AgNO3 can be represented as:
- \( \text{KBr} + \text{AgNO}_3 \rightarrow \text{AgBr} + \text{KNO}_3 \)
- \( \text{NaBr} + \text{AgNO}_3 \rightarrow \text{AgBr} + \text{NaNO}_3 \)

3. **Calculate Molar Masses:**
- Molar mass of KBr = 119 g/mol
- Molar mass of NaBr = 103 g/mol
- Molar mass of AgBr = 188 g/mol

4. **Calculate Mass of AgBr from KBr and NaBr:**
The mass of AgBr produced from KBr is given by:
\[ \text{Mass of AgBr from KBr} = \frac{188}{119} \times x \]
The mass of AgBr produced from NaBr is given by:
\[ \text{Mass of AgBr from NaBr} = \frac{188}{103} \times (0.563 - x) \]

5. **Set Up the Equation:**
The total mass of AgBr produced is given as 0.975 g. Therefore, we can write the equation:
\[ \frac{188}{119} x + \frac{188}{103} (0.563 - x) = 0.975 \]

6. **Simplify the Equation:**
Multiply through by the least common multiple (LCM) of the denominators (119 and 103) to eliminate fractions. This will give us a solvable equation for \( x \).

7. **Solve for \( x \):**
After simplifying and solving the equation, we find:
\[ x = 0.22 \, \text{g} \, (\text{mass of KBr}) \]

8. **Calculate Mass of NaBr:**
The mass of NaBr is:
\[ 0.563 - x = 0.563 - 0.22 = 0.343 \, \text{g} \]

9. **Calculate the Fraction of KBr:**
The fraction of KBr in the mixture is:
\[ \text{Fraction of KBr} = \frac{x}{0.563} = \frac{0.22}{0.563} \approx 0.39 \]

### Final Answer:
The fraction of total mass that is KBr in the sample is approximately **0.39**.