A freshly prepared sample of a radioisotope of half - life `1386 s ` has activity `10^(3) ` disintegrations per second Given that `ln 2 = 0.693` the fraction of the initial number of nuclei (expressed in nearest integer percentage ) that will decay in the first `80 s ` after preparation of the sample is

Explanation

To solve the problem, we need to find the fraction of the initial number of nuclei that will decay in the first 80 seconds after the preparation of the sample. We will use the concepts of radioactive decay and the relationship between activity, decay constant, and half-life.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Half-life (\(T_{1/2}\)) = 1386 seconds
- Activity (\(A\)) = \(10^3\) disintegrations per second
- \( \ln 2 \) = 0.693
- Time (\(t\)) = 80 seconds

2. **Calculate the Decay Constant (\(\lambda\)):**
The decay constant (\(\lambda\)) can be calculated using the formula:
\[ \lambda = \frac{\ln 2}{T_{1/2}} \]
Substituting the values:
\[ \lambda = \frac{0.693}{1386} \approx 5.00 \times 10^{-4} \, \text{s}^{-1} \]

3. **Determine the Initial Number of Nuclei (\(N_0\)):**
The activity \(A\) is related to the decay constant and the initial number of nuclei by:
\[ A = \lambda N_0 \]
Rearranging gives:
\[ N_0 = \frac{A}{\lambda} = \frac{10^3}{5.00 \times 10^{-4}} = 2.00 \times 10^6 \]

4. **Calculate the Number of Nuclei Remaining after 80 seconds (\(N_t\)):**
The number of nuclei remaining after time \(t\) can be calculated using the formula:
\[ N_t = N_0 e^{-\lambda t} \]
Substituting the values:
\[ N_t = 2.00 \times 10^6 \cdot e^{-5.00 \times 10^{-4} \cdot 80} \]
First, calculate \(e^{-\lambda t}\):
\[ e^{-5.00 \times 10^{-4} \cdot 80} \approx e^{-0.04} \approx 0.9608 \]
Now calculate \(N_t\):
\[ N_t \approx 2.00 \times 10^6 \cdot 0.9608 \approx 1.9216 \times 10^6 \]

5. **Calculate the Number of Nuclei Decayed (\(N_0 - N_t\)):**
The number of nuclei that have decayed in the first 80 seconds is:
\[ N_0 - N_t = 2.00 \times 10^6 - 1.9216 \times 10^6 \approx 0.0784 \times 10^6 \]

6. **Calculate the Fraction of Initial Nuclei Decayed:**
The fraction of the initial number of nuclei that have decayed is given by:
\[ \text{Fraction decayed} = \frac{N_0 - N_t}{N_0} = \frac{0.0784 \times 10^6}{2.00 \times 10^6} \approx 0.0392 \]
To express this as a percentage:
\[ \text{Percentage decayed} \approx 0.0392 \times 100 \approx 3.92\% \]
Rounding to the nearest integer gives us approximately **4%**.

### Final Answer:
The fraction of the initial number of nuclei that will decay in the first 80 seconds is approximately **4%**.