To determine the percentage of the original nuclei of a radioactive substance left after 5 half-lives, we first need to understand the concept of half-life and how it affects the quantity of a radioactive substance over time. ### Step-by-Step Explanation 1. **Understanding Half-Life**: - The half-life of a radioactive substance is the time required for half of the radioactive nuclei in a sample to decay. For example, if you start with 100 nuclei, after one half-life, you will have 50 nuclei remaining. 2. **Calculating Remaining Nuclei**: - After each half-life, the amount of the substance reduces to half of what it was at the beginning of that period. This can be expressed mathematically: - After 1 half-life: \( N = \frac{N_0}{2} \) - After 2 half-lives: \( N = \frac{N_0}{2^2} = \frac{N_0}{4} \) - After 3 half-lives: \( N = \frac{N_0}{2^3} = \frac{N_0}{8} \) - After 4 half-lives: \( N = \frac{N_0}{2^4} = \frac{N_0}{16} \) - After 5 half-lives: \( N = \frac{N_0}{2^5} = \frac{N_0}{32} \) 3. **Finding the Percentage Remaining**: - To find the percentage of the original nuclei remaining after 5 half-lives, we can use the formula: \[ \text{Percentage remaining} = \left( \frac{N}{N_0} \right) \times 100\% \] - Substituting \( N = \frac{N_0}{32} \): \[ \text{Percentage remaining} = \left( \frac{N_0/32}{N_0} \right) \times 100\% = \left( \frac{1}{32} \right) \times 100\% \approx 3.125\% \] 4. **Rounding**: - When rounding 3.125% to the nearest whole number, we get approximately 3%. Therefore, the correct answer is **C. 3%**. ### Why Other Options Are Incorrect - **Option A: 8%**: - This would imply that after 5 half-lives, more than 3% of the original nuclei remain. However, as shown in the calculations, only about 3.125% remains, which is significantly less than 8%. - **Option B: 5%**: - Similar to option A, this option suggests that a larger percentage of the original nuclei remains than what is actually calculated. The decay process is exponential, and after 5 half-lives, the remaining percentage is much lower than 5%. - **Option D: 1%**: - This option underestimates the remaining nuclei. While it is true that the amount decreases significantly, the calculation shows that approximately 3.125% remains, which is higher than 1%. ### Common Pitfalls - **Misunderstanding Half-Life**: Students often confuse the concept of half-life with a linear decay process. Remember, the decay is exponential, meaning the amount decreases rapidly at first and then slows down. - **Rounding Errors**: Be careful when rounding percentages. Always calculate the exact percentage before rounding to ensure accuracy. ### Revision Summary - The percentage of original nuclei remaining after 5 half-lives is approximately 3.125%. - The formula for remaining nuclei after \( n \) half-lives is \( N = \frac{N_0}{2^n} \). - After 5 half-lives, the remaining percentage is calculated as \( \left( \frac{1}{32} \right) \times 100\% \). - Common mistakes include misunderstanding the exponential nature of decay and incorrect rounding of percentages.