The initial number of atoms of a radioactive element with half life 100 days, is $9.6 × 10^{26}$. The number of atoms remaining undecayed after 500 days, will be :

Question

Accepted Answer

To determine the number of atoms remaining undecayed after 500 days, we need to use the concept of half-life. The half-life of a radioactive element is the time required for half of the radioactive atoms to decay.

Given:
- Initial number of atoms, $ N_0 = 9.6 \times 10^{26} $
- Half-life, $ t_{1/2} = 100 $ days
- Time elapsed, $ t = 500 $ days

The formula to calculate the remaining number of atoms after a certain period is:

$$ N = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}} $$

Substituting the given values into the formula:

$$ N = 9.6 \times 10^{26} \times \left(\frac{1}{2}\right)^{\frac{500}{100}} $$

$$ N = 9.6 \times 10^{26} \times \left(\frac{1}{2}\right)^5 $$

Calculating $\left(\frac{1}{2}\right)^5$:

$$ \left(\frac{1}{2}\right)^5 = \frac{1}{32} $$

Now, substitute back into the equation:

$$ N = 9.6 \times 10^{26} \times \frac{1}{32} $$

$$ N = \frac{9.6 \times 10^{26}}{32} $$

$$ N = 0.3 \times 10^{26} $$

$$ N = 3.0 \times 10^{25} $$

Thus, the number of atoms remaining undecayed after 500 days is $ 3.0 \times 10^{25} $, which corresponds to Option 4.

### Explanation of Incorrect Options:

1. **Option 1: $ 19.2 \times 10^{25} $**
   - This option suggests that more atoms remain undecayed than initially present, which is not possible. The calculation shows that only $ 3.0 \times 10^{25} $ atoms remain, making this option incorrect.

2. **Option 2: $ 9.6 \times 10^{25} $**
   - This option implies that a significant portion of the atoms remain undecayed, which is incorrect given the elapsed time of 500 days (5 half-lives). The correct calculation shows a much smaller number of remaining atoms.

3. **Option 3: $ 3.84 \times 10^{25} $**
   - Although closer to the correct answer, this option still overestimates the number of remaining atoms. The precise calculation yields $ 3.0 \times 10^{25} $, confirming that this option is incorrect.

In conclusion, the correct answer is Option 4: $ 3.0 \times 10^{25} $, as it accurately reflects the number of atoms remaining after 500 days based on the half-life decay formula.