For a first order reaction the rate constant is `6.909 "min"^(-1)`. The time taken for 75 % conversion in minutes is

Options

A. 3/2 log 2
B. 2/3 log 3
C. 2/3 log 2
D. 3/2 log 3/4

Explanation

To find the time taken for 75% conversion in a first-order reaction with a given rate constant, we can use the first-order reaction rate equation. Here’s a step-by-step solution:

### Step 1: Understand the Formula
For a first-order reaction, the time taken (t) for a certain percentage of conversion can be calculated using the formula:
\[ t = \frac{2.303}{k} \log \left( \frac{a}{a - x} \right) \]
where:
- \( k \) is the rate constant,
- \( a \) is the initial concentration,
- \( x \) is the amount reacted.

### Step 2: Define the Variables
In this case:
- The rate constant \( k = 6.909 \, \text{min}^{-1} \).
- For 75% conversion, \( x = 0.75a \) (where \( a \) is the initial concentration).
- Therefore, \( a - x = a - 0.75a = 0.25a \).

### Step 3: Substitute the Values into the Formula
Now substitute \( a \) and \( a - x \) into the formula:
\[ t = \frac{2.303}{6.909} \log \left( \frac{a}{0.25a} \right) \]
This simplifies to:
\[ t = \frac{2.303}{6.909} \log \left( \frac{1}{0.25} \right) \]

### Step 4: Simplify the Logarithm
The logarithm can be simplified:
\[ \log \left( \frac{1}{0.25} \right) = \log(4) \]
Thus, we have:
\[ t = \frac{2.303}{6.909} \log(4) \]

### Step 5: Further Simplify
We can express \( \log(4) \) as:
\[ \log(4) = \log(2^2) = 2 \log(2) \]
So, the equation becomes:
\[ t = \frac{2.303}{6.909} \cdot 2 \log(2) \]
This simplifies to:
\[ t = \frac{2 \cdot 2.303}{6.909} \log(2) \]

### Step 6: Calculate the Final Value
Now, calculate the numerical value:
\[ t = \frac{4.606}{6.909} \log(2) \]
This gives us the time taken for 75% conversion.

### Final Answer
The final answer for the time taken for 75% conversion is:
\[ t \approx 0.666 \log(2) \text{ minutes} \]