Carbon-11 is a radioactive isotope with medical applications. The half-life of Carbon-11 is 20 minutes. - How many half-lives have occurred after 100 minutes? - After 100 minutes, what fraction of the original amount would remain? - If there were originally $34.20 \mathrm{~g}$ of Carbon-11, what mass would remain after 100 minutes? Submit Question
Solution
Step 1 ：Given that the half-life of Carbon-11 is 20 minutes, we can calculate the number of half-lives that have occurred after 100 minutes using the formula: Number of half-lives = Total time / Length of one half-life.
Step 2 ：Substituting the given values into the formula, we get: Number of half-lives = 100 / 20 = 5.0.
Step 3 ：So, the number of half-lives that have occurred after 100 minutes is \(\boxed{5}\).
Step 4 ：After each half-life, the amount of Carbon-11 remaining is halved. So, after 5 half-lives, the fraction of the original amount that would remain is \( \frac{1}{2^5} = \frac{1}{32} \).
Step 5 ：If there were originally $34.20 \mathrm{~g}$ of Carbon-11, the mass that would remain after 100 minutes can be calculated by multiplying the original mass by the fraction remaining: $34.20 \mathrm{~g} \times \frac{1}{32} = 1.06875 \mathrm{~g}$.
