For a radioactive material, half-life is `10` minutes. If initially there are `600` number of nuclei, the time taken (in minutes) for the disintegration of `450` nuclei is.

Options

A. 30
B. 10
C. 20
D. 15

Explanation

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the Given Information
We know:
- The half-life (T) of the radioactive material is 10 minutes.
- The initial number of nuclei (N0) is 600.
- The number of nuclei that disintegrate (N') is 450.

### Step 2: Calculate the Remaining Nuclei
To find the remaining nuclei (N) after the disintegration, we can use the formula:
\[ N = N_0 - N' \]
Substituting the values:
\[ N = 600 - 450 = 150 \]

### Step 3: Use the Radioactive Decay Formula
The formula for radioactive decay is given by:
\[ \frac{N}{N_0} = \left( \frac{1}{2} \right)^{\frac{t}{T}} \]
Where:
- N is the remaining number of nuclei (150),
- N0 is the initial number of nuclei (600),
- t is the time taken (which we need to find),
- T is the half-life (10 minutes).

### Step 4: Substitute the Values into the Formula
Substituting the known values into the formula:
\[ \frac{150}{600} = \left( \frac{1}{2} \right)^{\frac{t}{10}} \]

### Step 5: Simplify the Left Side
Simplifying the left side gives:
\[ \frac{150}{600} = \frac{1}{4} \]
So we can rewrite the equation as:
\[ \frac{1}{4} = \left( \frac{1}{2} \right)^{\frac{t}{10}} \]

### Step 6: Express \(\frac{1}{4}\) as a Power of \(\frac{1}{2}\)
We know that:
\[ \frac{1}{4} = \left( \frac{1}{2} \right)^{2} \]
Thus, we can equate the exponents:
\[ \left( \frac{1}{2} \right)^{2} = \left( \frac{1}{2} \right)^{\frac{t}{10}} \]

### Step 7: Set the Exponents Equal to Each Other
Since the bases are the same, we can set the exponents equal:
\[ 2 = \frac{t}{10} \]

### Step 8: Solve for t
Multiplying both sides by 10 gives:
\[ t = 2 \times 10 = 20 \text{ minutes} \]

### Final Answer
The time taken for the disintegration of 450 nuclei is **20 minutes**.

---