A radioactive substance has `10^8` nuclei. Its half life is 30 s . The number of nuclei left after 15 s is nearly

Options

A. `2 xx 10^5`
B. `3 xx 10^6`
C. `7xx 10^7`
D. `5 xx 10^8`

Answer

To solve the problem of finding the number of radioactive nuclei left after 15 seconds, we can follow these steps:

### Step 1: Understand the Given Information
We are given:
- Initial number of nuclei, \( N_0 = 10^8 \)
- Half-life, \( t_{1/2} = 30 \) seconds
- Time elapsed, \( t = 15 \) seconds

### Step 2: Determine the Number of Half-lives
To find out how many half-lives have passed in 15 seconds, we can use the formula:
\[ n = \frac{t}{t_{1/2}} \]
Substituting the values:
\[ n = \frac{15 \text{ s}}{30 \text{ s}} = \frac{1}{2} \]

### Step 3: Use the Formula for Remaining Nuclei
The formula to calculate the number of remaining nuclei after time \( t \) is:
\[ N = N_0 \left( \frac{1}{2} \right)^n \]
Substituting \( N_0 \) and \( n \):
\[ N = 10^8 \left( \frac{1}{2} \right)^{\frac{1}{2}} \]

### Step 4: Simplify the Expression
Now, we simplify \( \left( \frac{1}{2} \right)^{\frac{1}{2}} \):
\[ \left( \frac{1}{2} \right)^{\frac{1}{2}} = \frac{1}{\sqrt{2}} \approx 0.707 \]
Thus, we can write:
\[ N \approx 10^8 \times 0.707 \approx 7.07 \times 10^7 \]

### Step 5: Final Answer
Rounding the result, we find that the number of nuclei left after 15 seconds is approximately:
\[ N \approx 7 \times 10^7 \]

### Conclusion
The number of radioactive nuclei left after 15 seconds is nearly \( 7 \times 10^7 \).

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