The half-life of radioactive Polonium `(Po)` is `138.6` days. For ten lakh Polonium atoms, the number of disintegrations in `24` hours is.

Options

A. 2000
B. 3000
C. 4000
D. 5000

Explanation

To solve the problem, we need to determine the number of disintegrations of Polonium atoms in 24 hours given its half-life. Here's a step-by-step solution:

### Step 1: Understand the half-life
The half-life of Polonium (Po) is given as 138.6 days. This means that after 138.6 days, half of the original amount of Polonium will have decayed.

### Step 2: Convert the half-life into hours
Since we need to find the number of disintegrations in 24 hours, we first convert the half-life from days to hours:
\[ \text{Half-life in hours} = 138.6 \text{ days} \times 24 \text{ hours/day} = 3326.4 \text{ hours} \]

### Step 3: Calculate the decay constant (λ)
The decay constant (λ) can be calculated using the formula:
\[ \lambda = \frac{\ln(2)}{T_{1/2}} \]
Where \( T_{1/2} \) is the half-life in hours.
\[ \lambda = \frac{\ln(2)}{3326.4} \approx \frac{0.693}{3326.4} \approx 0.0002087 \text{ hours}^{-1} \]

### Step 4: Calculate the number of disintegrations in 24 hours
The number of disintegrations (N) in a given time can be calculated using the formula:
\[ N = N_0 (1 - e^{-\lambda t}) \]
Where:
- \( N_0 \) = initial number of atoms = 10,00,000 (10 lakh)
- \( t \) = time in hours = 24 hours

Substituting the values:
\[ N = 10,00,000 \times (1 - e^{-0.0002087 \times 24}) \]
Calculating \( e^{-0.0002087 \times 24} \):
\[ e^{-0.0002087 \times 24} \approx e^{-0.0050048} \approx 0.995 \]
Now substituting back:
\[ N = 10,00,000 \times (1 - 0.995) = 10,00,000 \times 0.005 \approx 5000 \]

### Final Result
The number of disintegrations in 24 hours is approximately **5000**.

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