The rate of a first-order reaction is `1.5xx10^(-2)"mol"L^(-1)` at 0.5 M concentration of the reactant. The half-life of the reaction is

Options

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

Answer

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

### Step 1: Understand the relationship between rate and concentration
For a first-order reaction, the rate of the reaction can be expressed as:
\[ \text{Rate} = k \cdot [A] \]
where:
- \( k \) is the rate constant,
- \( [A] \) is the concentration of the reactant.

### Step 2: Rearrange the equation to find the rate constant \( k \)
Given the rate of the reaction is \( 1.5 \times 10^{-2} \, \text{mol L}^{-1} \) and the concentration \( [A] \) is \( 0.5 \, \text{M} \):
\[ 1.5 \times 10^{-2} = k \cdot 0.5 \]

### Step 3: Solve for \( k \)
To find \( k \), we can rearrange the equation:
\[ k = \frac{1.5 \times 10^{-2}}{0.5} \]
Calculating this gives:
\[ k = 3.0 \times 10^{-2} \, \text{L mol}^{-1} \text{s}^{-1} \]

### Step 4: Use the half-life formula for a first-order reaction
The half-life \( t_{1/2} \) of 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
Now perform the calculation:
\[ t_{1/2} = \frac{0.693}{0.03} \approx 23.1 \, \text{minutes} \]

### Conclusion
The half-life of the reaction is approximately **23.1 minutes**.

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