Half life of a first order reaction is `69.3` minutes. Time required to complete `99.9%` of the reaction will be `:`

Options

A. 693 minutes
B. 999 minutes
C. `99.9` minutes
D. 691 minutes

Explanation

To solve the problem of finding the time required to complete 99.9% of a first-order reaction given its half-life, we can follow these steps:

### Step 1: Determine the rate constant (K)
The half-life (\(T_{1/2}\)) of a first-order reaction is given by the formula:
\[ T_{1/2} = \frac{0.693}{K} \]
Given that \(T_{1/2} = 69.3\) minutes, we can rearrange the formula to solve for \(K\):
\[ K = \frac{0.693}{T_{1/2}} = \frac{0.693}{69.3} \]

### Step 2: Calculate K
Now, we can calculate \(K\):
\[ K = \frac{0.693}{69.3} \approx 0.01 \text{ min}^{-1} \]

### Step 3: Use the first-order reaction formula to find time (T)
For a first-order reaction, the time required to reach a certain percentage of completion can be calculated using the formula:
\[ T = \frac{2.303}{K} \log \left(\frac{A}{A - x}\right) \]
Where:
- \(A\) = initial concentration
- \(x\) = amount reacted

In this case, if we assume the initial concentration \(A = 100\) and \(x = 99.9\), we have:
\[ T = \frac{2.303}{K} \log \left(\frac{100}{100 - 99.9}\right) = \frac{2.303}{K} \log \left(\frac{100}{0.1}\right) \]

### Step 4: Calculate the log term
Calculating the log term:
\[ \log \left(\frac{100}{0.1}\right) = \log(1000) = 3 \]

### Step 5: Substitute K into the time equation
Now substituting \(K\) into the equation:
\[ T = \frac{2.303}{0.01} \times 3 \]

### Step 6: Calculate T
Calculating \(T\):
\[ T = \frac{2.303 \times 3}{0.01} = \frac{6.909}{0.01} = 690.9 \text{ minutes} \]

### Step 7: Round the result
Rounding \(690.9\) minutes gives us approximately \(691\) minutes.

### Final Answer
Thus, the time required to complete \(99.9\%\) of the reaction is approximately \(693\) minutes.

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