The rate of first-order reaction is `1.5 xx 10^(-2) M "min"^(-1)` at `0.5 M` concentration of reactant. The half-life of reaction is

Options

A. `0.383min`
B. `23.1min`
C. `8.73 min`
D. `7.53 min`

Explanation

To find the half-life of the first-order reaction, we can follow these steps:

### Step 1: Identify the rate equation for a first-order reaction.
The rate of a first-order reaction is given by the equation:
\[ \text{Rate} = k \cdot [A] \]
where \( k \) is the rate constant and \([A]\) is the concentration of the reactant.

### Step 2: Substitute the given values into the rate equation.
We know the rate is \( 1.5 \times 10^{-2} \, \text{M min}^{-1} \) and the concentration \([A]\) is \( 0.5 \, \text{M} \). Therefore, we can write:
\[ 1.5 \times 10^{-2} = k \cdot 0.5 \]

### Step 3: Solve for the rate constant \( k \).
Rearranging the equation to solve for \( k \):
\[ k = \frac{1.5 \times 10^{-2}}{0.5} \]
Calculating this gives:
\[ k = 3.0 \times 10^{-2} \, \text{min}^{-1} \]

### Step 4: Use the half-life formula for first-order reactions.
The half-life (\( t_{1/2} \)) for a first-order reaction is given by the formula:
\[ t_{1/2} = \frac{0.693}{k} \]

### Step 5: Substitute the value of \( k \) into the half-life formula.
Now substituting \( k = 3.0 \times 10^{-2} \):
\[ t_{1/2} = \frac{0.693}{3.0 \times 10^{-2}} \]

### Step 6: Calculate the half-life.
Calculating this gives:
\[ t_{1/2} = 0.693 \times \frac{1}{3.0 \times 10^{-2}} = 0.693 \times 33.33 \approx 23.1 \, \text{min} \]

### Final Answer:
The half-life of the reaction is approximately \( 23.1 \, \text{minutes} \).

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