A first order reaction is half completed in `45` minutes. How long does it need `99.9%` of the reaction to be completed

Options

A. 5 hours
B. 7.5 hours
C. 10 hours
D. 20 hours

Answer

To solve the problem of how long it takes for 99.9% of a first-order reaction to be completed, we can follow these steps:

### Step 1: Understand the Half-Life of a First-Order Reaction
For a first-order reaction, the half-life (t₁/₂) is given by the formula:
\[ t_{1/2} = \frac{0.693}{k} \]
where \( k \) is the rate constant.

### Step 2: Calculate the Rate Constant (k)
Given that the half-life is 45 minutes, we can rearrange the formula to find \( k \):
\[ k = \frac{0.693}{t_{1/2}} = \frac{0.693}{45 \text{ min}} \]

Calculating this gives:
\[ k \approx 0.0154 \text{ min}^{-1} \]

### Step 3: Use the First-Order Reaction Formula
The time taken for a first-order reaction to reach a certain percentage completion can be calculated using the formula:
\[ t = \frac{2.303}{k} \log \left( \frac{[A_0]}{[A]} \right) \]
where:
- \( [A_0] \) is the initial concentration,
- \( [A] \) is the concentration at time \( t \).

### Step 4: Determine the Concentration Values for 99.9% Completion
If 99.9% of the reaction is completed, then only 0.1% of the reactant remains. This means:
- \( [A_0] = 100 \) (initial concentration),
- \( [A] = 0.1 \) (remaining concentration).

### Step 5: Substitute Values into the Formula
Now we can substitute these values into the equation:
\[ t = \frac{2.303}{0.0154} \log \left( \frac{100}{0.1} \right) \]

Calculating the logarithm:
\[ \log \left( \frac{100}{0.1} \right) = \log(1000) = 3 \]

Now substituting this back into the equation:
\[ t = \frac{2.303}{0.0154} \times 3 \]

### Step 6: Calculate the Time
Calculating this gives:
\[ t \approx \frac{2.303 \times 3}{0.0154} \approx 450 \text{ minutes} \]

### Final Answer
Therefore, the time required for 99.9% of the reaction to be completed is **450 minutes** or **7.5 hours**.

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