Half life of radioactive sample undergoing alpha decayis 1.4 x 10^17s. If number of nuclei in sample is 2.0 x 10^21 activity of sample is nearly

Options

A. `10^4 Bq`
B. `10^5Bq`
C. `10^6Bq`
D. `10^3`Bq`

Answer

To find the activity of a radioactive sample undergoing alpha decay, we can use the relationship between half-life, the number of nuclei, and the decay constant. Here’s a step-by-step solution:

### Step 1: Understand the relationship between half-life and decay constant
The decay constant (λ) is related to the half-life (t_half) by the formula:
\[ \lambda = \frac{\ln(2)}{t_{half}} \]
where \(\ln(2) \approx 0.693\).

### Step 2: Calculate the decay constant
Given that the half-life \(t_{half} = 1.4 \times 10^{17} \, \text{s}\), we can substitute this value into the formula:
\[ \lambda = \frac{0.693}{1.4 \times 10^{17}} \]

### Step 3: Simplify the decay constant calculation
Calculating the decay constant:
\[ \lambda \approx \frac{0.693}{1.4} \times 10^{-17} \]
\[ \lambda \approx 0.495 \times 10^{-17} \, \text{s}^{-1} \]
\[ \lambda \approx 4.95 \times 10^{-18} \, \text{s}^{-1} \]

### Step 4: Use the decay constant to find activity
The activity (R) of a radioactive sample is given by:
\[ R = \lambda N \]
where \(N\) is the number of nuclei in the sample. Given \(N = 2.0 \times 10^{21}\), we can substitute the values:
\[ R = (4.95 \times 10^{-18}) \times (2.0 \times 10^{21}) \]

### Step 5: Calculate the activity
Now we can calculate the activity:
\[ R = 4.95 \times 2.0 \times 10^{3} \, \text{Bq} \]
\[ R = 9.9 \times 10^{3} \, \text{Bq} \]

### Final Answer
The activity of the sample is approximately:
\[ R \approx 9.9 \times 10^{3} \, \text{Bq} \text{ or } 9900 \, \text{Bq} \]