For a first order reaction, calculate the ratio between the time taken to complete three fourth of the reaction and the time taken to complete half of the reaction.

Explanation

To solve the problem of finding the ratio between the time taken to complete three-fourths of a first-order reaction and the time taken to complete half of the reaction, we can follow these steps:

### Step 1: Understand the first-order reaction kinetics
For a first-order reaction, the time taken to reach a certain fraction of completion can be calculated using the first-order rate equation:
\[ t = \frac{2.303}{k} \log \left( \frac{[A_0]}{[A]} \right) \]
where:
- \( t \) is the time taken,
- \( k \) is the rate constant,
- \( [A_0] \) is the initial concentration,
- \( [A] \) is the concentration at time \( t \).

### Step 2: Calculate time to complete half of the reaction
For half completion, \( [A] = \frac{[A_0]}{2} \):
\[ t_{1/2} = \frac{2.303}{k} \log \left( \frac{[A_0]}{\frac{[A_0]}{2}} \right) \]
This simplifies to:
\[ t_{1/2} = \frac{2.303}{k} \log(2) \]

### Step 3: Calculate time to complete three-fourths of the reaction
For three-fourths completion, \( [A] = \frac{[A_0]}{4} \):
\[ t_{3/4} = \frac{2.303}{k} \log \left( \frac{[A_0]}{\frac{[A_0]}{4}} \right) \]
This simplifies to:
\[ t_{3/4} = \frac{2.303}{k} \log(4) \]
Since \( \log(4) = 2 \log(2) \), we can write:
\[ t_{3/4} = \frac{2.303}{k} \cdot 2 \log(2) = \frac{4.606}{k} \log(2) \]

### Step 4: Calculate the ratio of the two times
Now, we can find the ratio of the time taken to complete three-fourths of the reaction to the time taken to complete half of the reaction:
\[ \text{Ratio} = \frac{t_{3/4}}{t_{1/2}} = \frac{\frac{4.606}{k} \log(2)}{\frac{2.303}{k} \log(2)} \]

### Step 5: Simplify the ratio
The \( k \) and \( \log(2) \) terms cancel out:
\[ \text{Ratio} = \frac{4.606}{2.303} = 2 \]

### Final Answer
The ratio of the time taken to complete three-fourths of the reaction to the time taken to complete half of the reaction is **2**.

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