In order to dcrease radioactive nuclei to one million of its initial number, number of half - lives required is

Options

A. 20
B. 40
C. 30
D. 10

Answer

To solve the problem of how many half-lives are required to decrease the number of radioactive nuclei to one millionth of its initial number, we can follow these steps:

### Step-by-Step Solution:

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. After each half-life, the quantity of the substance is halved.

2. **Establish the Relationship**:
The relationship between the initial number of nuclei (N₀), the remaining number of nuclei (N_t), and the number of half-lives (n) is given by the formula:
\[ N_t = \frac{N_0}{2^n} \]
Rearranging this gives:
\[ \frac{N_0}{N_t} = 2^n \]

3. **Set Up the Equation**:
We want to find n when \( N_t = \frac{N_0}{10^6} \). This means:
\[ \frac{N_0}{N_t} = 10^6 \]
Therefore, we can set up the equation:
\[ 2^n = 10^6 \]

4. **Convert to Logarithmic Form**:
To solve for n, we can take the logarithm of both sides:
\[ n \log(2) = \log(10^6) \]
Since \( \log(10^6) = 6 \), we have:
\[ n \log(2) = 6 \]

5. **Calculate n**:
Now, we can solve for n:
\[ n = \frac{6}{\log(2)} \]
Using the approximate value \( \log(2) \approx 0.301 \):
\[ n \approx \frac{6}{0.301} \approx 19.93 \]
Rounding this gives us \( n \approx 20 \).

6. **Conclusion**:
Therefore, the number of half-lives required to decrease the radioactive nuclei to one millionth of its initial number is approximately 20.

### Final Answer:
The number of half-lives required is **20**.

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