The rate constant of the production of 2B (g) by the reaction , `A(g)overset(Delta)rarr2B(g)` is `2.48xx10^(-4)s^(-1)` A 1 : 1 molar ratio of A to B in the reaction mixture is attained after

Options

A. 26.25 minute
B. 27.25 minute
C. 28.25 minute
D. 0 minute

Answer

To solve the problem, we need to determine the time required for the molar ratio of A to B to become 1:1 in the reaction \( A(g) \rightarrow 2B(g) \) given the rate constant \( k = 2.48 \times 10^{-4} \, s^{-1} \).

### Step-by-Step Solution:

1. **Understanding the Reaction**:
The reaction is \( A(g) \rightarrow 2B(g) \). For every mole of A that reacts, 2 moles of B are produced.

2. **Setting Up Initial Conditions**:
Let the initial concentration of A be \( [A_0] \) and the initial concentration of B be 0. At time \( t \), let \( x \) be the amount of A that has reacted. Therefore, the concentrations at time \( t \) will be:
- Concentration of A: \( [A] = [A_0] - x \)
- Concentration of B: \( [B] = 2x \)

3. **Establishing the Molar Ratio**:
We want the molar ratio of A to B to be 1:1:
\[ [A] = [B] \]
Substituting the expressions we have:
\[ [A_0] - x = 2x \]
Rearranging gives:
\[ [A_0] = 3x \quad \Rightarrow \quad x = \frac{[A_0]}{3} \]

4. **Using the First-Order Kinetics Equation**:
The reaction is first-order with respect to A. The integrated rate law for a first-order reaction is given by:
\[ k t = 2.303 \log \left( \frac{[A_0]}{[A]} \right) \]
Substituting \( [A] = [A_0] - x = [A_0] - \frac{[A_0]}{3} = \frac{2[A_0]}{3} \):
\[ k t = 2.303 \log \left( \frac{[A_0]}{\frac{2[A_0]}{3}} \right) = 2.303 \log \left( \frac{3}{2} \right) \]

5. **Calculating the Logarithm**:
Calculate \( \log \left( \frac{3}{2} \right) \):
\[ \log \left( \frac{3}{2} \right) \approx 0.176 \]

6. **Substituting Values into the Rate Equation**:
Now substituting \( k = 2.48 \times 10^{-4} \, s^{-1} \):
\[ t = \frac{2.303 \times 0.176}{2.48 \times 10^{-4}} \]

7. **Calculating Time \( t \)**:
\[ t = \frac{0.405568}{2.48 \times 10^{-4}} \approx 1635.23 \, s \]

8. **Converting Time to Minutes**:
To convert seconds to minutes:
\[ t \approx \frac{1635.23}{60} \approx 27.25 \, \text{minutes} \]

### Final Answer:
The time required for the molar ratio of A to B to become 1:1 is approximately **27.25 minutes**.