A polystyrene having formula `Br_(3)C_(6)H_(2)(C_(8)H_(8))_(n)` was prepared by heating styrene with tribromobenzoyl peroxide in absence of air. If it was found to contain `10.46%` Bromine by mass, what will be the value of n.

Options

A. 19
B. 20
C. 15
D. 10

Answer

To find the value of \( n \) in the formula \( \text{Br}_3\text{C}_6\text{H}_2(\text{C}_8\text{H}_8)_n \) given that the compound contains \( 10.46\% \) Bromine by mass, we can follow these steps:

### Step 1: Determine the mass of Bromine in the compound
The formula indicates that there are 3 Bromine atoms in the compound. The atomic mass of Bromine (Br) is approximately \( 80 \, \text{g/mol} \). Therefore, the total mass of Bromine in the compound is:
\[ \text{Mass of Bromine} = 3 \times 80 = 240 \, \text{g} \]

### Step 2: Write the expression for the total mass of the compound
The total mass of the compound can be expressed as:
\[ \text{Mass of compound} = 314 + 104n \]
where \( 314 \, \text{g/mol} \) is the mass of the non-repeating units (Br, C, and H) in the formula and \( 104n \) accounts for the repeating units of \( \text{C}_8\text{H}_8 \).

### Step 3: Set up the equation using the percentage of Bromine
According to the problem, the percentage of Bromine in the compound is given by:
\[ \frac{\text{Mass of Bromine}}{\text{Mass of compound}} \times 100 = 10.46 \]
Substituting the values we have:
\[ \frac{240}{314 + 104n} \times 100 = 10.46 \]

### Step 4: Solve for \( n \)
Rearranging the equation gives:
\[ 240 = 10.46 \times \left(314 + 104n\right) / 100 \]
Multiplying both sides by \( 100 \):
\[ 24000 = 10.46 \times (314 + 104n) \]
Now, divide both sides by \( 10.46 \):
\[ \frac{24000}{10.46} = 314 + 104n \]
Calculating the left side:
\[ 2291.55 \approx 314 + 104n \]
Now, subtract \( 314 \) from both sides:
\[ 2291.55 - 314 = 104n \]
\[ 1977.55 = 104n \]
Now, divide by \( 104 \):
\[ n \approx \frac{1977.55}{104} \approx 19.05 \]

### Step 5: Round to the nearest whole number
Since \( n \) must be a whole number, we round \( 19.05 \) to \( 19 \).

### Conclusion
The value of \( n \) is approximately \( 19 \).