Which of the following represent the expression for `(3)/(4)`th life of first order reaction

Options

A. `(k)/(2.303)log4//3`
B. `(2.303)/(k)log3//4`
C. `(2.303)/(k)log4`
D. `(2.303)/(k)log3`

Answer

To determine the expression for the \( \frac{3}{4} \)th life of a first-order reaction, we can follow these steps:

### Step 1: Understand the concept of half-life in first-order reactions
For a first-order reaction, the half-life (\( t_{1/2} \)) is given by the formula:
\[ t_{1/2} = \frac{0.693}{k} \]
where \( k \) is the rate constant.

### Step 2: Define the \( \frac{3}{4} \)th life
The \( \frac{3}{4} \)th life refers to the time taken for the concentration of the reactant to decrease to \( \frac{1}{4} \) of its initial concentration. This means that if the initial concentration is \( A_0 \), after \( t \) time, the concentration \( A_t \) will be:
\[ A_t = A_0 - x = \frac{A_0}{4} \]
where \( x \) is the amount of reactant that has reacted.

### Step 3: Calculate the amount reacted
From the above equation, we can find \( x \):
\[ x = A_0 - A_t = A_0 - \frac{A_0}{4} = \frac{3A_0}{4} \]

### Step 4: Use the first-order reaction formula
The integrated rate law for a first-order reaction is given by:
\[ t = \frac{2.303}{k} \log \left( \frac{A_0}{A_t} \right) \]
Substituting \( A_t = \frac{A_0}{4} \):
\[ t = \frac{2.303}{k} \log \left( \frac{A_0}{\frac{A_0}{4}} \right) \]

### Step 5: Simplify the logarithmic expression
This simplifies to:
\[ t = \frac{2.303}{k} \log \left( 4 \right) \]

### Step 6: Final expression for the \( \frac{3}{4} \)th life
Thus, the expression for the \( \frac{3}{4} \)th life of a first-order reaction is:
\[ t_{\frac{3}{4}} = \frac{2.303}{k} \log 4 \]

### Conclusion
The correct option that represents the expression for the \( \frac{3}{4} \)th life of a first-order reaction is:
\[ \frac{2.303}{k} \log 4 \]