A radioactive sample of half life 23.1 days is disintegrating continuously. The percentage decay of its in 15th to 16th days will be
[ Take `e^(0.03)` = 1.03]

Options

A. `5%`
B. `1%`
C. `2.9%`
D. `3.5%`

Explanation

To solve the problem of determining the percentage decay of a radioactive sample from the 15th to the 16th day, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Half-Life**:
The half-life of the radioactive sample is given as \( t_{1/2} = 23.1 \) days. This means that after 23.1 days, half of the original sample will have decayed.

2. **Calculate the Decay Constant (\( \lambda \))**:
The decay constant \( \lambda \) can be calculated using the formula:
\[ \lambda = \frac{0.693}{t_{1/2}} \]
Substituting the given half-life:
\[ \lambda = \frac{0.693}{23.1} \approx 0.03 \, \text{days}^{-1} \]

3. **Determine the Time Intervals**:
We are interested in the decay from day 15 to day 16. Therefore, \( t_1 = 15 \) days and \( t_2 = 16 \) days.

4. **Use the Exponential Decay Formula**:
The number of undecayed nuclei at any time \( t \) can be expressed as:
\[ N(t) = N_0 e^{-\lambda t} \]
For \( t_1 = 15 \):
\[ N(15) = N_0 e^{-\lambda \cdot 15} \]
For \( t_2 = 16 \):
\[ N(16) = N_0 e^{-\lambda \cdot 16} \]

5. **Calculate the Decay from Day 15 to Day 16**:
The change in the number of undecayed nuclei from day 15 to day 16 is:
\[ \Delta N = N(15) - N(16) = N_0 e^{-\lambda \cdot 15} - N_0 e^{-\lambda \cdot 16} \]
Factoring out \( N_0 e^{-\lambda \cdot 15} \):
\[ \Delta N = N_0 e^{-\lambda \cdot 15} (1 - e^{-\lambda}) \]

6. **Calculate the Percentage Decay**:
The percentage decay from day 15 to day 16 is given by:
\[ \text{Percentage Decay} = \frac{\Delta N}{N(15)} \times 100 = \frac{N_0 e^{-\lambda \cdot 15} (1 - e^{-\lambda})}{N_0 e^{-\lambda \cdot 15}} \times 100 \]
Simplifying this gives:
\[ \text{Percentage Decay} = (1 - e^{-\lambda}) \times 100 \]

7. **Substitute the Value of \( \lambda \)**:
Using the approximation \( e^{-\lambda} \approx 1 - \lambda \) for small \( \lambda \):
\[ 1 - e^{-\lambda} \approx 0.03 \]
Therefore:
\[ \text{Percentage Decay} \approx 0.03 \times 100 = 3\% \]

8. **Final Calculation**:
Since we are given \( e^{0.03} \approx 1.03 \), we can refine our calculation:
\[ \text{Percentage Decay} = (1 - \frac{1}{1.03}) \times 100 \approx 2.9\% \]

### Conclusion:
The percentage decay of the radioactive sample from the 15th to the 16th day is approximately **2.9%**.