Final answer:
To find the time it takes for 360.00 grams of chromium-48 to decay to 11.25 grams with a half-life of 21.6 hours, we calculate the number of half-lives using the logarithmic formula and find it to be approximately 5 half-lives. Multiplying by the half-life duration gives us a total time of 108 hours.
Step-by-step explanation:
The student is asking about the time it takes for a sample of chromium-48, which is a radioactive isotope, to decay from an initial mass of 360.00 grams to a final mass of 11.25 grams. Chromium-48 has a short half-life of 21.6 hours. To solve this, we use the concept of half-lives, which is the time it takes for half of the radioactive substance to decay.
First, we calculate how many half-lives it takes for 360.00 grams to decay to 11.25 grams using the formula:
n = log(final mass/initial mass) / log(0.5)
Substituting the given values:
n = log(11.25/360.00) / log(0.5)
n ≈ 5
Since each half-life is 21.6 hours, we then multiply the number of half-lives by the half-life duration:
Time = n * half-life duration
Time = 5 * 21.6 hours
Time = 108 hours
So, it will take approximately 108 hours for 360.00 grams of chromium-48 to decay to 11.25 grams.