`_(87)^(221) `Ra is a radioactive substance having half life of `4` days .Find the probability that a nucleus undergoes decay after two half lives

Options

A. `1`
B. `(1)/(2)`
C. `(3)/(4)`
D. `(1)/(4)`

Explanation

To solve the problem, we need to determine the probability that a nucleus of radium-221 undergoes decay after two half-lives. Let's break down the solution step by step.

### 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 decay. For radium-221, the half-life is given as 4 days.

2. **Initial Number of Nuclei**:
- Let’s denote the initial number of nuclei as \( N_0 \).

3. **After One Half-Life**:
- After one half-life (4 days), the number of remaining nuclei will be:
\[ N_1 = \frac{N_0}{2} \]

4. **After Two Half-Lives**:
- After two half-lives (8 days), the number of remaining nuclei will be:
\[ N_2 = \frac{N_1}{2} = \frac{N_0}{4} \]

5. **Calculating the Number of Decayed Nuclei**:
- The number of nuclei that have decayed after two half-lives is:
\[ N_{\text{decayed}} = N_0 - N_2 = N_0 - \frac{N_0}{4} = \frac{3N_0}{4} \]

6. **Calculating the Probability of Decay**:
- The probability \( P \) that a nucleus has decayed after two half-lives is given by the ratio of the number of decayed nuclei to the initial number of nuclei:
\[ P = \frac{N_{\text{decayed}}}{N_0} = \frac{\frac{3N_0}{4}}{N_0} = \frac{3}{4} \]

7. **Final Answer**:
- Therefore, the probability that a nucleus undergoes decay after two half-lives is:
\[ P = \frac{3}{4} \]

### Summary:
The probability that a nucleus of radium-221 undergoes decay after two half-lives is \( \frac{3}{4} \).