Final answer:
In 50 minutes, which encompasses three full half-lives of Francium-222, 1.5 grams would remain from an initial 12 grams. For a precise calculation at 50 minutes, an exponential decay formula is required.
Step-by-step explanation:
The student's question involves calculating the remaining quantity of a radioactive isotope after a certain period of time, given its half-life. Specifically, it is about Francium-222, which has a half-life of 15 minutes and determining how much remains after 50 minutes.
To solve this, first calculate the number of half-lives that have passed in the given time. In 50 minutes, there are ⅓ half-lives (50 / 15 = ⅓). Starting with 12 grams, after each half-life, the amount of the isotope is halved:
- After the first 15 minutes, 6 grams remain.
- After the second 15 minutes (30 minutes total), 3 grams remain.
- After the third 15 minutes (45 minutes total), 1.5 grams remain.
However, since 50 minutes does not perfectly fit into an exact number of half-lives, we must calculate the remaining amount after two-thirds of the fourth half-life. This calculation would require a more complex formula involving exponential decay, which we're not covering in this straightforward example. Assuming you want the amount just after the three full half-lives (45 minutes), 1.5 grams of Francium-222 would remain.
If you need the exact amount at 50 minutes, you would need to perform a calculation based on the exponential decay formula: remaining amount = initial amount × (1/2)^(time passed / half-life).