Half-life of a radioactive substance A is 4 days. The probability that a nuclear will decay in two half-lives is

Options

A. (a) 1/4
B. (b) 3/4
C. (c) 1/2
D. (d) 1

Explanation

To solve the problem of finding the probability that a nucleus will decay in two half-lives, we can follow these steps:

### Step 1: Understand the concept of half-life
The half-life of a radioactive substance is the time required for half of the radioactive nuclei in a sample to decay. In this case, the half-life of substance A is given as 4 days.

### Step 2: Determine the number of nuclei remaining after one half-life
If we start with an initial number of nuclei \( N_0 \), after one half-life (4 days), the number of remaining nuclei will be:
\[ N_1 = \frac{N_0}{2} \]

### Step 3: Determine the number of nuclei remaining after two half-lives
After another half-life (total of 8 days), the number of remaining nuclei will be:
\[ N_2 = \frac{N_1}{2} = \frac{N_0/2}{2} = \frac{N_0}{4} \]

### Step 4: Calculate the number of decayed nuclei
The number of nuclei that have decayed after two half-lives can be calculated as:
\[ \text{Decayed nuclei} = N_0 - N_2 = N_0 - \frac{N_0}{4} = \frac{3N_0}{4} \]

### Step 5: Calculate the probability of decay
The probability that a nucleus will decay in two half-lives is given by the ratio of the number of decayed nuclei to the initial number of nuclei:
\[ P(\text{decay}) = \frac{\text{Decayed nuclei}}{N_0} = \frac{\frac{3N_0}{4}}{N_0} = \frac{3}{4} \]

### Conclusion
Thus, the probability that a nucleus will decay in two half-lives is:
\[ P(\text{decay}) = \frac{3}{4} \]