A certain radioactive isotope has a half-life of one hour. At noon, 100g of the material is present. How much of the original isotope will be present at
3:00 p.m.----1/8
6:00 p.m.-----1/16
10:00 p.m.-----1/32
Are these correct?
At 3 pm it has gone through 3 half lives and that is 2^3 = 8 so there is 1/8 of 100 left = 12.5 grams. You are correct. The others are not correct.
At 6 pm, there are 6 half-lives so 2^6 = 64; therefore, there is 1/64 of the original 100 or 1.56 g remaining.
Etc.
Yes, your answers are correct. Here is the step-by-step calculation for each time period:
1. At 3:00 p.m. (3 hours after noon):
- The time elapsed since noon is 3 hours, which is three half-lives.
- Each half-life reduces the amount of the isotope by half.
- Hence, the amount remaining would be (1/2) * (1/2) * (1/2) = 1/8 of the original amount.
2. At 6:00 p.m. (6 hours after noon):
- The time elapsed since noon is 6 hours, which is six half-lives.
- Each half-life reduces the amount of the isotope by half.
- Hence, the amount remaining would be (1/2) * (1/2) * (1/2) * (1/2) * (1/2) * (1/2) = 1/64 of the original amount.
3. At 10:00 p.m. (10 hours after noon):
- The time elapsed since noon is 10 hours, which is ten half-lives.
- Each half-life reduces the amount of the isotope by half.
- Hence, the amount remaining would be (1/2) * (1/2) * (1/2) * (1/2) * (1/2) * (1/2) * (1/2) * (1/2) * (1/2) * (1/2) * (1/2) * (1/2) = 1/1024 of the original amount.
So, the correct answers are:
- At 3:00 p.m.: 1/8 of the original amount
- At 6:00 p.m.: 1/64 of the original amount
- At 10:00 p.m.: 1/1024 of the original amount.
To determine how much of the original isotope will be present at each given time, we can use the concept of half-life. The half-life is the time it takes for half of the radioactive material to decay.
In this case, if the half-life of the isotope is one hour, it means that after every hour, the amount of the material will reduce by half. Therefore, we can calculate how many half-lives have passed between noon and each given time.
From noon to 3:00 p.m., a total of 3 hours have passed. Since the half-life is one hour, there would have been 3 half-lives. Each half-life reduces the amount by half, so after 3 half-lives, the amount remaining would be (1/2) × (1/2) × (1/2) = 1/8 of the original amount.
From noon to 6:00 p.m., a total of 6 hours have passed, which corresponds to 6 half-lives. So, the amount remaining would be (1/2) × (1/2) × (1/2) × (1/2) × (1/2) × (1/2) = 1/64 of the original amount.
From noon to 10:00 p.m., a total of 10 hours have passed, which corresponds to 10 half-lives. Thus, the amount remaining would be (1/2) × (1/2) × (1/2) × (1/2) × (1/2) × (1/2) × (1/2) × (1/2) × (1/2) × (1/2) = 1/1024 of the original amount.
Therefore, the correct answers are:
- At 3:00 p.m., 1/8 of the original isotope will be present.
- At 6:00 p.m., 1/64 of the original isotope will be present.
- At 10:00 p.m., 1/1024 of the original isotope will be present.