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

### Step 1: Understanding the Reaction
DDT decomposes in a first-order reaction. In a first-order reaction, the rate of reaction is directly proportional to the concentration of the reactant.

### Step 2: Determine the Amount Decomposed
Since we want to find the time for 90% decomposition, we can denote:
- Initial concentration (A0) = 100%
- Decomposed amount (x) = 90%
- Remaining concentration = A0 - x = 100% - 90% = 10%

### Step 3: Use the First-Order Kinetics Formula
The first-order kinetics formula is given by:
\[ t = \frac{2.303}{k} \log \left( \frac{A_0}{A_0 - x} \right) \]
Substituting the values we have:
\[ t = \frac{2.303}{k} \log \left( \frac{100}{10} \right) \]
\[ t = \frac{2.303}{k} \log(10) \]
Since \(\log(10) = 1\):
\[ t = \frac{2.303}{k} \]

### Step 4: Calculate the Rate Constant (k)
The half-life (\(t_{1/2}\)) for a first-order reaction is given by:
\[ t_{1/2} = \frac{0.693}{k} \]
We can rearrange this to find \(k\):
\[ k = \frac{0.693}{t_{1/2}} \]
Given that \(t_{1/2} = 20\) years:
\[ k = \frac{0.693}{20} \]
\[ k = 0.03465 \, \text{years}^{-1} \]

### Step 5: Substitute k Back into the Time Equation
Now substitute \(k\) back into the equation for \(t\):
\[ t = \frac{2.303}{0.03465} \]
Calculating this gives:
\[ t \approx 66.5 \, \text{years} \]

### Final Answer
The time it will take for DDT to decompose 90% is approximately **67 years**.

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