To determine how long it would take for a 1 kg radioactive substance to decay into 12.5 g, we can use the concept of half-lives. The half-life of a radioactive substance is the time it takes for half of the material to decay. In this case, the half-life is 100 years.
Starting with 1 kg (or 1000 g) of the substance, we need to find out how many half-lives it takes to reduce 1000 g to 12.5 g.
Here’s the calculation:
- After 1 half-life (100 years): 1000 g / 2 = 500 g
- After 2 half-lives (200 years): 500 g / 2 = 250 g
- After 3 half-lives (300 years): 250 g / 2 = 125 g
- After 4 half-lives (400 years): 125 g / 2 = 62.5 g
- After 5 half-lives (500 years): 62.5 g / 2 = 31.25 g
- After 6 half-lives (600 years): 31.25 g / 2 = 15.625 g
- After 7 half-lives (700 years): 15.625 g / 2 = 7.8125 g
As we can see, it takes 6 complete half-lives to get below 12.5 g, where we reach about 15.625 g. To reach exactly 12.5 g, we are halfway through the seventh half-life.
The total time to decay from 1 kg to 12.5 g is therefore:
- 6 half-lives = 600 years
- Half of the 7th half-life (50 years)
So, the time taken to decay from 1 kg to 12.5 g is:
600 + 50 = 650 years
This means it would take approximately 650 years for 1 kg of the substance to decay to 12.5 g.