Upon irradiating californium with neutrons, a scientist discovered a new nuclide having mass number of 250 and a half-life of 30 min. After 90 min. of irradiation, the observed radioactivity due to nuclied was 100 dis/min. How many atoms of the nucliede were prepared intially?

Options

A. `2.4xx10^(4)`
B. `3.46xx10^(4)`
C. 1900
D. 800

Answer

To solve the problem, we will use the principles of radioactive decay and the relationship between the initial quantity of a radioactive substance, its half-life, and the remaining quantity after a certain time.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Half-life (t_half) = 30 minutes
- Time of irradiation (t) = 90 minutes
- Observed radioactivity after 90 minutes (R) = 100 dis/min

2. **Calculate the Decay Constant (λ):**
The decay constant (λ) can be calculated using the formula:
\[ \lambda = \frac{0.693}{t_{1/2}} \]
Substituting the given half-life:
\[ \lambda = \frac{0.693}{30} \approx 0.0231 \text{ min}^{-1} \]

3. **Determine the Number of Half-Lives:**
The number of half-lives (n) that have passed in 90 minutes can be calculated as:
\[ n = \frac{t}{t_{1/2}} = \frac{90}{30} = 3 \]

4. **Use the Radioactive Decay Formula:**
The relationship between the initial activity (A₀), remaining activity (A), and the number of half-lives is given by:
\[ A = A_0 \left(\frac{1}{2}\right)^n \]
Rearranging this to find A₀:
\[ A_0 = A \left(2^n\right) \]
Substituting the values:
\[ A_0 = 100 \left(2^3\right) = 100 \times 8 = 800 \text{ dis/min} \]

5. **Relate Activity to Number of Atoms:**
The activity (A) is also related to the number of atoms (N) and the decay constant (λ) by:
\[ A = \lambda N \]
Rearranging this to find N:
\[ N = \frac{A}{\lambda} \]
Substituting A₀ and λ:
\[ N_0 = \frac{800}{0.0231} \approx 34625.5 \]

6. **Final Answer:**
Since the number of atoms must be a whole number, we round it to:
\[ N_0 \approx 34626 \text{ atoms} \]

### Conclusion:
The initial number of atoms of the nuclide prepared is approximately **34626 atoms**.