The time needed for the completion of 2/3 of a 1st order reaction, when rate constant is `4.771xx10^(-2)"min"^(-1)` is

Options

A. 23.03 min
B. 2.303 min
C. 6.93 min
D. 69.3min

Answer

To solve the problem of finding the time needed for the completion of 2/3 of a first-order reaction with a given rate constant, we can follow these steps:

### Step 1: Understand the first-order reaction formula
For a first-order reaction, the relationship between the rate constant (k), time (t), and concentrations is given by the equation:
\[ k = \frac{2.303}{t} \log \left(\frac{A_0}{A_t}\right) \]
Where:
- \( A_0 \) = initial concentration
- \( A_t \) = concentration at time t

### Step 2: Determine the concentrations
In this case, we need to find the time for the completion of 2/3 of the reaction. This means that 1/3 of the reactant remains. If we assume the initial concentration \( A_0 = A \), then:
\[ A_t = \frac{A}{3} \]

### Step 3: Substitute into the equation
Substituting \( A_0 \) and \( A_t \) into the equation gives:
\[ k = \frac{2.303}{t} \log \left(\frac{A}{A/3}\right) \]
This simplifies to:
\[ k = \frac{2.303}{t} \log(3) \]

### Step 4: Rearrange the equation to solve for time
Rearranging the equation to solve for time \( t \):
\[ t = \frac{2.303 \log(3)}{k} \]

### Step 5: Plug in the values
Given:
- \( k = 4.771 \times 10^{-2} \, \text{min}^{-1} \)
- \( \log(3) \approx 0.4771 \)

Now substituting these values into the equation:
\[ t = \frac{2.303 \times 0.4771}{4.771 \times 10^{-2}} \]

### Step 6: Calculate the time
Calculating the numerator:
\[ 2.303 \times 0.4771 \approx 1.097 \]
Now, substituting this into the equation:
\[ t = \frac{1.097}{4.771 \times 10^{-2}} \approx \frac{1.097}{0.04771} \approx 23.03 \, \text{minutes} \]

### Conclusion
The time needed for the completion of 2/3 of the reaction is approximately **23.03 minutes**.