Which of the following represents the expression for `3//4`th life of first order reaction?

Options

A. `k/(2.303)log4//3`
B. `(2.303)/(k)(log 3//4)`
C. `2.303/klog4`
D. `2.303/klog3`

Explanation

To find the expression for the `3/4`th life of a first-order reaction, we can follow these steps:

### Step 1: Understand the first-order reaction kinetics
The rate of a first-order reaction can be described by the equation:
\[ T = \frac{2.303}{k} \log \frac{A}{A - x} \]
where:
- \( T \) is the time taken for the concentration to change from \( A \) to \( A - x \),
- \( k \) is the rate constant,
- \( A \) is the initial concentration,
- \( x \) is the amount reacted.

### Step 2: Determine the value of \( x \) for `3/4`th life
For `3/4`th life of the reaction, we need to find the time taken for the concentration to decrease from \( A \) to \( A - x \) where \( x = \frac{3}{4}A \). This means that \( 3/4 \) of the initial concentration has reacted.

### Step 3: Calculate \( A - x \)
Substituting \( x \) into the equation:
\[ A - x = A - \frac{3}{4}A = \frac{1}{4}A \]

### Step 4: Substitute into the first-order rate equation
Now, we can substitute \( A \) and \( A - x \) into the first-order reaction equation:
\[ T = \frac{2.303}{k} \log \frac{A}{A - x} = \frac{2.303}{k} \log \frac{A}{\frac{1}{4}A} \]

### Step 5: Simplify the logarithm
This simplifies to:
\[ T = \frac{2.303}{k} \log \frac{A}{\frac{1}{4}A} = \frac{2.303}{k} \log 4 \]

### Conclusion
Thus, the expression for the `3/4`th life of a first-order reaction is:
\[ T = \frac{2.303}{k} \log 4 \]

### Final Answer
The correct option representing the `3/4`th life of a first-order reaction is:
\[ \frac{2.303}{k} \log 4 \]

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