Polychlorinated biphenyls, PCBs, known to be dangerous environment pollutants , are a group of compounds with the general empirical formula `C_12H_mCl_(10-m)`, where m is an integer. What is the value of m, if percentage of carbon atom in the compounds is 40 ?

Explanation

To solve the problem, we need to determine the value of \( m \) in the empirical formula \( C_{12}H_mCl_{10-m} \) given that the percentage of carbon in the compound is 40%.

### Step 1: Set up the equation for percentage of carbon
The percentage of carbon in a compound can be calculated using the formula:

\[ \text{Percentage of Carbon} = \left( \frac{\text{Molar mass of Carbon}}{\text{Total Molar Mass of Compound}} \right) \times 100 \]

### Step 2: Identify the molar masses
- The molar mass of carbon (C) is \( 12 \, \text{g/mol} \).
- The molar mass of hydrogen (H) is \( 1 \, \text{g/mol} \).
- The molar mass of chlorine (Cl) is \( 35.5 \, \text{g/mol} \).

### Step 3: Calculate the total molar mass of the compound
The total molar mass of the compound \( C_{12}H_mCl_{10-m} \) can be expressed as:

\[ \text{Total Molar Mass} = (12 \times 12) + (m \times 1) + ((10 - m) \times 35.5) \]

This simplifies to:

\[ \text{Total Molar Mass} = 144 + m + (10 - m) \times 35.5 \]

Expanding this gives:

\[ \text{Total Molar Mass} = 144 + m + 355 - 35.5m = 499 - 34.5m \]

### Step 4: Set up the equation with the given percentage
We know that the percentage of carbon is 40%, so we can set up the equation:

\[ 40 = \left( \frac{12 \times 12}{499 - 34.5m} \right) \times 100 \]

### Step 5: Solve for \( m \)
Rearranging the equation gives:

\[ 0.4 = \frac{144}{499 - 34.5m} \]

Cross-multiplying leads to:

\[ 0.4(499 - 34.5m) = 144 \]

Expanding this gives:

\[ 199.6 - 13.8m = 144 \]

Subtracting 199.6 from both sides results in:

\[ -13.8m = 144 - 199.6 \]

\[ -13.8m = -55.6 \]

Dividing both sides by -13.8:

\[ m = \frac{55.6}{13.8} \approx 4.03 \]

### Step 6: Round \( m \) to the nearest integer
Since \( m \) must be an integer, we round \( 4.03 \) to \( 4 \).

### Step 7: Write the final empirical formula
Substituting \( m = 4 \) back into the empirical formula gives:

\[ C_{12}H_4Cl_{10-4} = C_{12}H_4Cl_6 \]

### Final Answer
The value of \( m \) is \( 4 \) and the empirical formula is \( C_{12}H_4Cl_6 \).

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