DDT on exposure to water decomposes. How much time will it take for its 90% decomposition? (Half-life = 20 years)

Options

A. 50 years
B. 67 years
C. 500 years
D. 700 years

Explanation

To solve the problem of how much time it will take for DDT to decompose by 90%, given that its half-life is 20 years, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Reaction Order**:
Since the problem states that DDT decomposes over a long period, we can assume this is a first-order reaction.

2. **Use the Half-Life Formula**:
For a first-order reaction, the relationship between the half-life (t_half) and the rate constant (k) is given by:
\[ k = \frac{0.693}{t_{half}} \]
Given that \( t_{half} = 20 \) years, we can calculate k:
\[ k = \frac{0.693}{20} \approx 0.03465 \, \text{years}^{-1} \]

3. **Set Up the Decomposition Equation**:
The first-order rate law can be expressed as:
\[ k = \frac{2.303}{T} \log \left( \frac{A_0}{A} \right) \]
Where:
- \( A_0 \) = initial amount (100% or 100)
- \( A \) = remaining amount after time T
- We need to find T when 90% has decomposed, meaning 10% remains. Thus, \( A = 10 \).

4. **Substitute Values into the Equation**:
Now substituting the values into the equation:
\[ 0.03465 = \frac{2.303}{T} \log \left( \frac{100}{10} \right) \]
Since \( \frac{100}{10} = 10 \), we have:
\[ \log(10) = 1 \]
Therefore, the equation simplifies to:
\[ 0.03465 = \frac{2.303}{T} \cdot 1 \]

5. **Solve for T**:
Rearranging the equation to solve for T gives:
\[ T = \frac{2.303}{0.03465} \]
Calculating this yields:
\[ T \approx 66.5 \, \text{years} \]

6. **Conclusion**:
Thus, the time required for 90% decomposition of DDT is approximately 67 years.

### Final Answer:
**67 years**