Calculate in how many months , `(3/4)^(th) `of the substance will dacay, If half-life of the radioactive substance is 2 months.

Options

A. 3 months
B. 4 months
C. 8 months
D. 12 months

Answer

To solve the problem of how many months it takes for \( \frac{3}{4} \) of a radioactive substance to decay, given that its half-life is 2 months, we can follow these steps:

### Step 1: Understand the concept of half-life
The half-life of a substance is the time required for half of the substance to decay. In this case, the half-life is given as 2 months.

### Step 2: Determine the remaining substance after decay
If \( \frac{3}{4} \) of the substance has decayed, then the amount of substance remaining is:
\[ \text{Remaining substance} = 1 - \frac{3}{4} = \frac{1}{4} \]

### Step 3: Relate the remaining substance to half-lives
The remaining amount of substance can be expressed in terms of the initial amount using the half-life formula:
\[ n = n_0 \left( \frac{1}{2} \right)^t \]
where:
- \( n \) is the remaining quantity of the substance,
- \( n_0 \) is the initial quantity,
- \( t \) is the number of half-lives that have passed.

### Step 4: Set up the equation
Since we have determined that \( n = \frac{1}{4} n_0 \), we can set up the equation:
\[ \frac{1}{4} n_0 = n_0 \left( \frac{1}{2} \right)^t \]

### Step 5: Simplify the equation
Dividing both sides by \( n_0 \) (assuming \( n_0 \neq 0 \)):
\[ \frac{1}{4} = \left( \frac{1}{2} \right)^t \]

### Step 6: Express \( \frac{1}{4} \) in terms of \( \frac{1}{2} \)
We know that:
\[ \frac{1}{4} = \left( \frac{1}{2} \right)^2 \]
Thus, we can equate:
\[ \left( \frac{1}{2} \right)^t = \left( \frac{1}{2} \right)^2 \]

### Step 7: Solve for \( t \)
From the equation above, we can see that:
\[ t = 2 \]
This means that 2 half-lives have passed.

### Step 8: Calculate the total time
Since each half-life is 2 months, the total time taken for \( \frac{3}{4} \) of the substance to decay is:
\[ \text{Total time} = t \times \text{half-life} = 2 \times 2 \text{ months} = 4 \text{ months} \]

### Final Answer
It takes **4 months** for \( \frac{3}{4} \) of the substance to decay.

---