At least how many half-lives should elapse for a `1^(st)` order reaction `Ato` products so that the reaction is at least `95%` completed?
`(log2=0.3)`

Options

A. 4
B. 5
C. 6
D. 7

Explanation

To determine the minimum number of half-lives required for a first-order reaction to reach at least 95% completion, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Reaction Completion**:
- For a reaction to be 95% completed, 5% of the reactant A must remain. This means that 95% of A has been converted to products.

2. **Initial and Final Concentration**:
- Let the initial concentration of A be \( [A_0] \).
- After the reaction is 95% completed, the concentration of A will be \( [A] = 0.05 [A_0] \).

3. **Using the Half-Life Concept**:
- In a first-order reaction, the half-life (\( t_{1/2} \)) is constant and is given by:
\[ t_{1/2} = \frac{0.693}{k} \]
- Each half-life reduces the concentration of A by half.

4. **Calculating Concentration After Each Half-Life**:
- After 1 half-life:
\[ [A] = 0.5 [A_0] \]
- After 2 half-lives:
\[ [A] = 0.5 \times 0.5 [A_0] = 0.25 [A_0] \]
- After 3 half-lives:
\[ [A] = 0.5 \times 0.25 [A_0] = 0.125 [A_0] \]
- After 4 half-lives:
\[ [A] = 0.5 \times 0.125 [A_0] = 0.0625 [A_0] \]

5. **Determining When 95% Completion is Achieved**:
- After 1 half-life: 50% remains (50% completed).
- After 2 half-lives: 25% remains (75% completed).
- After 3 half-lives: 12.5% remains (87.5% completed).
- After 4 half-lives: 6.25% remains (93.75% completed).
- After 5 half-lives:
\[ [A] = 0.5 \times 0.0625 [A_0] = 0.03125 [A_0] \]
(which is 96.875% completed).

6. **Conclusion**:
- Therefore, at least **5 half-lives** are required for the reaction to be at least 95% completed.

### Final Answer:
The minimum number of half-lives required is **5**.

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