For certain first order reaction, 75% of the reaction complete in 30 min. How much time it require to complete 99.9% of the reaction ?

Options

A. 150 min
B. 100 min
C. 90 min
D. 300 min

Answer

To solve the problem step by step, we will use the first-order reaction kinetics formula and the information provided in the question.

### Step 1: Understand the first-order reaction formula
For a first-order reaction, the time required to complete a certain percentage of the reaction can be calculated using the formula:
\[ t = \frac{2.303}{k} \log \left( \frac{[A_0]}{[A]} \right) \]
where:
- \( t \) is the time taken,
- \( k \) is the rate constant,
- \( [A_0] \) is the initial concentration,
- \( [A] \) is the concentration at time \( t \).

### Step 2: Set up the equation for 75% completion
Given that 75% of the reaction is complete in 30 minutes, we can set:
- \( [A_0] = 100\% \)
- \( [A] = 100\% - 75\% = 25\% \)

Substituting these values into the formula:
\[ 30 = \frac{2.303}{k} \log \left( \frac{100}{25} \right) \]

### Step 3: Simplify the equation
Calculating the logarithm:
\[ \log \left( \frac{100}{25} \right) = \log(4) \]
Thus, we can rewrite the equation as:
\[ 30 = \frac{2.303}{k} \log(4) \]

### Step 4: Set up the equation for 99.9% completion
Now, we need to find the time required to complete 99.9% of the reaction. For 99.9% completion:
- \( [A] = 100\% - 99.9\% = 0.1\% \)

Substituting these values into the formula:
\[ T_{99.9\%} = \frac{2.303}{k} \log \left( \frac{100}{0.1} \right) \]

### Step 5: Simplify the equation for 99.9% completion
Calculating the logarithm:
\[ \log \left( \frac{100}{0.1} \right) = \log(1000) = 3 \]
Thus, we can rewrite the equation as:
\[ T_{99.9\%} = \frac{2.303}{k} \cdot 3 \]

### Step 6: Divide the two equations
Now we will divide the equation for 75% completion by the equation for 99.9% completion:
\[ \frac{30}{T_{99.9\%}} = \frac{\frac{2.303}{k} \log(4)}{\frac{2.303}{k} \cdot 3} \]

The \( \frac{2.303}{k} \) terms cancel out:
\[ \frac{30}{T_{99.9\%}} = \frac{\log(4)}{3} \]

### Step 7: Solve for \( T_{99.9\%} \)
Now, rearranging the equation gives:
\[ T_{99.9\%} = \frac{30 \cdot 3}{\log(4)} \]

### Step 8: Calculate the value
Using the value of \( \log(4) \approx 0.602 \):
\[ T_{99.9\%} = \frac{90}{0.602} \approx 149.67 \text{ minutes} \]

Rounding this, we find:
\[ T_{99.9\%} \approx 150 \text{ minutes} \]

### Final Answer
The time required to complete 99.9% of the reaction is approximately **150 minutes**.

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