The rate of a first order reaction is `1.5xx10^(-2) and L^(-1) "min"^(-1)` at 0.8 M concentration of the reactant. The half - life of the reaction is

Options

A. 0.383 min
B. 36.97 min
C. 8.73 min
D. 7.53 min

Answer

To find the half-life of a first-order reaction given the rate and concentration of the reactant, we can follow these steps:

### Step 1: Write 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: Rearrange the equation to solve for \( k \).
From the equation, we can express \( k \) as:
\[ k = \frac{\text{Rate}}{[A]} \]

### Step 3: Substitute the given values into the equation.
We know:
- Rate = \( 1.5 \times 10^{-2} \, \text{L}^{-1} \, \text{min}^{-1} \)
- Concentration, \([A] = 0.8 \, \text{M}\)

Substituting these values into the equation for \( k \):
\[ k = \frac{1.5 \times 10^{-2}}{0.8} \]

### Step 4: Calculate \( k \).
Calculating \( k \):
\[ k = \frac{1.5 \times 10^{-2}}{0.8} = 1.875 \times 10^{-2} \, \text{min}^{-1} \]

### Step 5: Use the half-life formula for a first-order reaction.
The half-life (\( t_{1/2} \)) for a first-order reaction is given by the formula:
\[ t_{1/2} = \frac{0.693}{k} \]

### Step 6: Substitute \( k \) into the half-life formula.
Now substituting the value of \( k \):
\[ t_{1/2} = \frac{0.693}{1.875 \times 10^{-2}} \]

### Step 7: Calculate \( t_{1/2} \).
Calculating \( t_{1/2} \):
\[ t_{1/2} = \frac{0.693}{0.01875} \approx 36.96 \, \text{minutes} \]

### Step 8: Round the answer to the appropriate number of significant figures.
Thus, the half-life of the reaction is approximately:
\[ t_{1/2} \approx 36.97 \, \text{minutes} \]

### Final Answer:
The half-life of the reaction is **36.97 minutes**.

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