Final answer:
After 9 years, or 9 half-lives, you would have 2 grams remaining from the original 1024 grams of a radioactive element with a half-life of 1 year.
Step-by-step explanation:
The question at hand involves the concept of radioactive decay and the calculation of the amount of substance remaining after a given number of half-lives. A half-life is the period required for half of a quantity of a radioactive element to decay into its daughter elements. The original element, in the process of decay, gets transformed into a different element with the emission of radioactive particles.
If you start with 1024 grams of a radioactive element with a half-life of 1 year, you would apply the concept of half-lives to determine how much remains after 9 years. After the first year, half of it decays, leaving 512 grams. By the end of the second year, half of that amount decays, resulting in 256 grams remaining. This halving process continues with each passing year. After 9 half-lives (9 years), the amount remaining can be calculated using the formula:
Remaining amount = Initial amount * (1/2)n
Where n is the number of half-lives.
Plugging in the values gives us:
Remaining amount = 1024g * (1/2)9
Remaining amount = 1024g * 1/512
Remaining amount = 2 grams
Hence, after 9 years, you would have 2 grams of the original radioactive element remaining.