To tackle this problem, we need to understand the decay process of radioactive substances and how they transform from one element to another. In this scenario, we have three radioactive elements: A, B, and C, which decay into one another, and we want to determine the amounts of each after 30 days, starting with one mole of A.
Understanding Half-Lives
The half-life of a radioactive substance is the time it takes for half of the substance to decay. Here are the half-lives for each element:
- A: 4.5 seconds
- B: 15 days
- C: 1 second
Calculating Decay of A
Since the half-life of A is very short (4.5 seconds), we can calculate how many half-lives fit into 30 days. First, convert 30 days into seconds:
30 days = 30 × 24 × 60 × 60 = 2,592,000 seconds.
Next, determine how many half-lives of A occur in this time:
Number of half-lives = Total time / Half-life of A = 2,592,000 seconds / 4.5 seconds ≈ 576,000.
After each half-life, the amount of A is halved. Therefore, after 576,000 half-lives, the amount of A will be:
Remaining A = Initial amount × (1/2)^(number of half-lives) = 1 mole × (1/2)^(576,000) ≈ 0 moles.
Decay of A to B
Since A decays completely into B, we need to consider the decay of B. However, since A has decayed to nearly zero, we can assume that B will also be negligible. But let's calculate how much B would have been produced before it starts decaying.
For every mole of A that decays, one mole of B is produced. However, B has a much longer half-life of 15 days, so we need to calculate how much B remains after it has been produced from A.
Calculating B's Decay
After A decays, the amount of B produced is initially 1 mole. Now, let's calculate how many half-lives of B fit into 30 days:
Number of half-lives of B = 30 days / 15 days = 2.
After 2 half-lives, the remaining amount of B is:
Remaining B = Initial amount of B × (1/2)^(number of half-lives) = 1 mole × (1/2)^2 = 1 mole × 1/4 = 0.25 moles.
Decay of B to C
Next, we need to consider the decay of B into C. However, since B is also radioactive and has a half-life of 15 days, we need to calculate how much C is produced from B's decay.
Since B decays into C, we can calculate how much C is produced from B's decay over the 30 days. The half-life of C is very short (1 second), so it will decay rapidly.
Calculating C's Decay
Initially, when B decays, it produces C. The amount of C produced from B's decay can be calculated, but since C decays almost instantly (1 second), we can assume that after 30 days, C will also be negligible.
Final Moles of A, B, C, and D
After 30 days, we can summarize the amounts:
- A: 0 moles (decayed completely)
- B: 0.25 moles (after decay)
- C: 0 moles (decayed almost instantly)
- D: 0 moles (not produced from B or C)
In conclusion, after 30 days, you will have approximately 0 moles of A, 0.25 moles of B, 0 moles of C, and 0 moles of D. The rapid decay of A and C, combined with the longer half-life of B, leads to this result. If you have any further questions about radioactive decay or half-lives, feel free to ask!
