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, we will follow these steps:

### Step 1: Identify the initial conditions
We are given:
- Initial number of nuclei, \( N_0 = 600 \)
- Number of nuclei disintegrated, \( N_d = 450 \)

### Step 2: Calculate the remaining nuclei
The remaining nuclei after disintegration can be calculated as:
\[ N = N_0 - N_d = 600 - 450 = 150 \]

### Step 3: Use the half-life formula
The relationship between the remaining nuclei and the initial nuclei can be expressed using the half-life formula:
\[ N = N_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \]
where:
- \( N \) = remaining nuclei
- \( N_0 \) = initial nuclei
- \( t \) = time elapsed
- \( T_{1/2} \) = half-life of the material

Given that the half-life \( T_{1/2} = 10 \) minutes, we can substitute the known values into the equation:
\[ 150 = 600 \left( \frac{1}{2} \right)^{\frac{t}{10}} \]

### Step 4: Simplify the equation
Dividing both sides by 600 gives:
\[ \frac{150}{600} = \left( \frac{1}{2} \right)^{\frac{t}{10}} \]
This simplifies to:
\[ \frac{1}{4} = \left( \frac{1}{2} \right)^{\frac{t}{10}} \]

### Step 5: 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 rewrite the equation as:
\[ \left( \frac{1}{2} \right)^{2} = \left( \frac{1}{2} \right)^{\frac{t}{10}} \]

### Step 6: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal:
\[ 2 = \frac{t}{10} \]

### Step 7: Solve for \( t \)
Multiplying both sides by 10 gives:
\[ t = 20 \text{ minutes} \]

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

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