The half-life of phosphorus-32 is 14.3 days, which means that after each 14.3-day period, the amount of phosphorus-32 remaining will be reduced by half. We can use the following equation to calculate the amount of phosphorus-32 remaining after a certain number of half-lives:
N = N0 * (1/2)^(t/t1/2)
where:
N = the amount of radioactive material remaining
N0 = the initial amount of radioactive material
t = the time elapsed
t1/2 = the half-life of the radioactive material
In this case, we know that:
N0 = 1 kg (the initial amount)
t1/2 = 14.3 days (the half-life)
t = 100.1 days (time elapsed)
Plugging these values into the equation, we get:
N = 1 kg * (1/2)^(100.1/14.3)
Simplifying this expression yields:
N = 1 kg * 0.0684
Thus, the amount of phosphorus-32 remaining after 100.1 days is:
N = 0.0684 kg
Therefore, at the end of the 100.1-day period, approximately 68.4 grams of phosphorus-32 remains.