Given a sample of radius -226 having half-life of 4 days. Find, the probability, a nucleus disintegrates after 2 half lifes.

Options

A. 1
B. `1/2`
C. `1.5`
D. `3/4`

Answer

To solve the problem of finding the probability that a nucleus disintegrates after 2 half-lives, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Half-Life**:
The half-life of a radioactive substance is the time required for half of the radioactive nuclei in a sample to undergo disintegration. In this case, the half-life is given as 4 days.

2. **Determine the Number of Half-Lives**:
We are interested in the probability of disintegration after 2 half-lives. Since one half-life is 4 days, two half-lives will be:
\[ 2 \times 4 \text{ days} = 8 \text{ days} \]

3. **Initial Number of Nuclei**:
Let's assume we start with \( N_0 \) nuclei at \( t = 0 \).

4. **Calculate Remaining Nuclei After 2 Half-Lives**:
After one half-life, the number of remaining nuclei is:
\[ N_1 = \frac{N_0}{2} \]
After the second half-life, the number of remaining nuclei is:
\[ N_2 = \frac{N_1}{2} = \frac{N_0}{4} \]

5. **Calculate the Number of Disintegrated Nuclei**:
The number of nuclei that have disintegrated after 2 half-lives is:
\[ N_{\text{disintegrated}} = N_0 - N_2 = N_0 - \frac{N_0}{4} = \frac{3N_0}{4} \]

6. **Calculate the Probability of Disintegration**:
The probability \( P \) that a nucleus disintegrates after 2 half-lives is given by the ratio of the number of disintegrated nuclei to the initial number of nuclei:
\[ P = \frac{N_{\text{disintegrated}}}{N_0} = \frac{\frac{3N_0}{4}}{N_0} = \frac{3}{4} \]

### Final Answer:
The probability that a nucleus disintegrates after 2 half-lives is \( \frac{3}{4} \).