For a first order reaction, the ratio of the time taken for `7//8^(th)` of the reaction to complete to that of half of the reaction to complete is

Options

A. `3 : 1`
B. `1 : 3`
C. `2 : 3`
D. `3 : 2`

Explanation

To solve the problem, we need to determine the ratio of the time taken for \( \frac{7}{8} \) of the reaction to complete to that of half of the reaction to complete for a first-order reaction.

### Step-by-Step Solution:

1. **Understanding First-Order Reactions**:
For a first-order reaction, the time taken to complete a certain fraction of the reaction 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 reacted, and \( k \) is the rate constant.

2. **Calculating Time for \( \frac{7}{8} \) Completion**:
- If \( \frac{7}{8} \) of the reaction is complete, then \( x = A - \frac{7A}{8} = \frac{A}{8} \).
- Plugging into the formula:
\[ T_{\frac{7}{8}} = \frac{2.303}{k} \log \left( \frac{A}{\frac{A}{8}} \right) = \frac{2.303}{k} \log(8) \]

3. **Calculating Time for Half Completion**:
- For half of the reaction, \( x = A - \frac{A}{2} = \frac{A}{2} \).
- Plugging into the formula:
\[ T_{\frac{1}{2}} = \frac{2.303}{k} \log \left( \frac{A}{\frac{A}{2}} \right) = \frac{2.303}{k} \log(2) \]

4. **Finding the Ratio**:
Now, we need to find the ratio of the time taken for \( \frac{7}{8} \) of the reaction to complete to that for half of the reaction:
\[ \text{Ratio} = \frac{T_{\frac{7}{8}}}{T_{\frac{1}{2}}} = \frac{\frac{2.303}{k} \log(8)}{\frac{2.303}{k} \log(2)} \]
The \( \frac{2.303}{k} \) cancels out:
\[ \text{Ratio} = \frac{\log(8)}{\log(2)} \]

5. **Simplifying the Logarithm**:
Using the property of logarithms, \( \log(8) = \log(2^3) = 3 \log(2) \):
\[ \text{Ratio} = \frac{3 \log(2)}{\log(2)} = 3 \]

### Final Answer:
The ratio of the time taken for \( \frac{7}{8} \) of the reaction to complete to that for half of the reaction to complete is **3**.

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