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.

### Step 1: Understand the problem
We know that in a first-order reaction, the time taken for a certain percentage of the reaction to complete can be calculated using the formula:
\[ t = \frac{2.303}{k} \log \left( \frac{A}{A - x} \right) \]
where:
- \( A \) is the initial concentration,
- \( x \) is the amount of reactant that has reacted,
- \( k \) is the rate constant,
- \( t \) is the time taken.

### Step 2: Calculate the rate constant \( k \)
Given that 75% of the reaction is complete in 30 minutes, we can set:
- \( A = 100 \) (initial concentration),
- \( x = 75 \) (amount reacted).

Substituting these values into the formula:
\[ t = 30 \text{ min} = 30 \times 60 \text{ seconds} = 1800 \text{ seconds} \]

Now, we can rearrange the formula to solve for \( k \):
\[ k = \frac{2.303}{t} \log \left( \frac{A}{A - x} \right) \]
\[ k = \frac{2.303}{1800} \log \left( \frac{100}{100 - 75} \right) \]
\[ k = \frac{2.303}{1800} \log \left( \frac{100}{25} \right) \]
\[ k = \frac{2.303}{1800} \log(4) \]

Using the logarithm value \( \log(4) \approx 0.602 \):
\[ k = \frac{2.303}{1800} \times 0.602 \]
\[ k \approx 0.0000077 \text{ s}^{-1} \]

### Step 3: Calculate time for 99.9% completion
Now we want to find the time required to complete 99.9% of the reaction. Here:
- \( x = 99.9 \) (amount reacted).

Using the same formula:
\[ t_{99.9\%} = \frac{2.303}{k} \log \left( \frac{A}{A - x} \right) \]
\[ t_{99.9\%} = \frac{2.303}{0.0000077} \log \left( \frac{100}{100 - 99.9} \right) \]
\[ t_{99.9\%} = \frac{2.303}{0.0000077} \log \left( \frac{100}{0.1} \right) \]
\[ t_{99.9\%} = \frac{2.303}{0.0000077} \log(1000) \]

Using \( \log(1000) = 3 \):
\[ t_{99.9\%} = \frac{2.303}{0.0000077} \times 3 \]
\[ t_{99.9\%} \approx \frac{2.303 \times 3}{0.0000077} \]
\[ t_{99.9\%} \approx 897,000 \text{ seconds} \]

### Step 4: Convert seconds to minutes
To convert seconds to minutes:
\[ t_{99.9\%} \approx \frac{897,000}{60} \approx 14,950 \text{ minutes} \]

### Final Answer
The time required to complete 99.9% of the reaction is approximately **14,950 minutes**.

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