The half life period of ${}_{36}^{85}Kr$ is 10 years. How long will it take for 99% of the ${}_{36}^{85}Kr$ to disintegrate?

Question

Accepted Answer

To determine how long it will take for 99% of ${}_{36}^{85}Kr$ to disintegrate, we need to use the concept of half-life and the formula for radioactive decay.

### Explanation for Option 2:

1. **Understanding Half-Life:**
   - The half-life of a radioactive substance is the time required for half of the radioactive atoms present to decay.
   - For ${}_{36}^{85}Kr$, the half-life is given as 10 years.

2. **Decay Formula:**
   - The formula for radioactive decay is:
     $$
     N = N_0 \left(\frac{1}{2}\right)^{\frac{t}{t_{1/2}}}
     $$
     Where:
     - $ N $ is the remaining quantity of the substance.
     - $ N_0 $ is the initial quantity of the substance.
     - $ t $ is the time elapsed.
     - $ t_{1/2} $ is the half-life of the substance.

3. **Calculating Time for 99% Disintegration:**
   - If 99% of the substance has disintegrated, 1% remains. Therefore, $ N = 0.01N_0 $.
   - Plugging into the decay formula:
     $$
     0.01N_0 = N_0 \left(\frac{1}{2}\right)^{\frac{t}{10}}
     $$
   - Simplifying, we get:
     $$
     0.01 = \left(\frac{1}{2}\right)^{\frac{t}{10}}
     $$
   - Taking the logarithm of both sides:
     $$
     \log(0.01) = \frac{t}{10} \log\left(\frac{1}{2}\right)
     $$
   - Solving for $ t $:
     $$
     t = 10 \times \frac{\log(0.01)}{\log(0.5)}
     $$
   - Calculating the values:
     $$
     \log(0.01) = -2 \quad \text{and} \quad \log(0.5) \approx -0.3010
     $$
     $$
     t = 10 \times \frac{-2}{-0.3010} \approx 66.4 \text{ years}
     $$

Thus, it will take approximately 66.4 years for 99% of ${}_{36}^{85}Kr$ to disintegrate, which corresponds to Option 2.

### Explanation for Why Other Options are Incorrect:

1. **Option 1 (33.2 years):**
   - This option suggests a much shorter time than calculated. Given the half-life of 10 years, 33.2 years would not be sufficient for 99% disintegration, as shown by the decay formula.

2. **Option 3 (99.6 years):**
   - This option suggests a longer time than necessary. The calculation shows that 66.4 years is sufficient for 99% disintegration, making 99.6 years an overestimate.

3. **Option 4 (132.8 years):**
   - This option is even longer than Option 3 and is clearly an overestimate. The calculation confirms that 66.4 years is adequate for 99% disintegration, so 132.8 years is unnecessarily long.

In conclusion, Option 2 is the correct answer as it accurately reflects the time required for 99% of ${}_{36}^{85}Kr$ to disintegrate based on the half-life and decay formula.