The time taken for the completion of 3/4 of a first order reaction is

Options

A. `(2.303//k)log 3//4`
B. `(2.303 //k)log 4`
C. `(2.303//k) log 1//4`
D. `(2.303//0.75)log k`

Explanation

To find the time taken for the completion of \( \frac{3}{4} \) of a first-order reaction, we can use the first-order reaction kinetics formula:

### Step 1: Understand the formula
The time \( T \) for a first-order reaction is given by:
\[ T = \frac{2.303}{k} \log \left( \frac{a}{a - x} \right) \]
where:
- \( T \) is the time taken,
- \( k \) is the rate constant,
- \( a \) is the initial concentration,
- \( x \) is the amount reacted.

### Step 2: Substitute values for \( x \)
For the completion of \( \frac{3}{4} \) of the reaction, the amount reacted \( x \) is \( \frac{3}{4}a \). Thus, the remaining concentration \( a - x \) becomes:
\[ a - x = a - \frac{3}{4}a = \frac{1}{4}a \]

### Step 3: Substitute into the formula
Now, substituting \( x \) into the formula:
\[ T = \frac{2.303}{k} \log \left( \frac{a}{\frac{1}{4}a} \right) \]

### Step 4: Simplify the logarithm
This simplifies to:
\[ T = \frac{2.303}{k} \log \left( \frac{a}{\frac{1}{4}a} \right) = \frac{2.303}{k} \log \left( 4 \right) \]

### Step 5: Final expression
Thus, the time taken for the completion of \( \frac{3}{4} \) of the reaction is:
\[ T = \frac{2.303 \log(4)}{k} \]