A first order reaction completes 60% 20 minutes. The time required for the completion of 90% of the reaction is approx………..

Options

A. 30 minutes
B. 40 minutes
C. 50 minutes
D. 60 minutes

Answer

To solve the problem of finding the time required for the completion of 90% of a first-order reaction, given that it completes 60% in 20 minutes, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Reaction**:
We know that the reaction is first-order, and we are given that 60% of the reactant is converted to product in 20 minutes.

2. **Identify Variables**:
- Let the initial concentration of the reactant be \( A \).
- The amount reacted after 20 minutes (60% completion) is \( X = 0.6A \).
- The remaining concentration after 20 minutes is \( A - X = 0.4A \).

3. **Using the First-Order Rate Equation**:
The integrated rate law for a first-order reaction is given by:
\[ k = \frac{2.303}{t} \log\left(\frac{A}{A - X}\right) \]
Substituting the known values:
\[ k = \frac{2.303}{20} \log\left(\frac{A}{0.4A}\right) = \frac{2.303}{20} \log\left(\frac{1}{0.4}\right) \]

4. **Calculate the Logarithm**:
\[ \log\left(\frac{1}{0.4}\right) = \log(2.5) \quad (\text{since } \frac{1}{0.4} = 2.5) \]
Using a calculator, we find:
\[ \log(2.5) \approx 0.3979 \]

5. **Calculate the Rate Constant \( k \)**:
Now substituting back:
\[ k = \frac{2.303}{20} \times 0.3979 \approx 0.0289 \, \text{min}^{-1} \]

6. **Finding Time for 90% Completion**:
For 90% completion, \( X = 0.9A \) and the remaining concentration is \( A - X = 0.1A \).
Now, we use the rate equation again:
\[ t = \frac{2.303}{k} \log\left(\frac{A}{A - X}\right) = \frac{2.303}{k} \log\left(\frac{A}{0.1A}\right) = \frac{2.303}{k} \log(10) \]
Since \( \log(10) = 1 \):
\[ t = \frac{2.303}{k} \]

7. **Substituting the Value of \( k \)**:
\[ t = \frac{2.303}{0.0289} \approx 79.8 \, \text{minutes} \]

8. **Final Calculation**:
Since we need the approximate time for the completion of 90%, we round it to:
\[ t \approx 80 \, \text{minutes} \]

### Conclusion:
The time required for the completion of 90% of the reaction is approximately **80 minutes**.