Final answer:
To find the decay time from 10,000 cpm to 8,500 cpm for p-32, we need the half-life and use the formula for exponential decay. As it's a fraction of a half-life, the precise time can be calculated; however, the provided options don't match such a calculation, suggesting an error in the question or answer choices.
Step-by-step explanation:
To determine how much time it takes for a sample of p-32 to decay from 10,000 counts per minute (cpm) to 8,500 cpm, we need to use its half-life and the nature of exponential decay. From the initial activity, we know that after one half-life, the activity will be halved. Using the information given:
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- initial activity (to) = 32,000 cpm
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- after one half-life = 16,000 cpm
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- after two half-lives = 8,000 cpm
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- after three half-lives = 4,000 cpm
Using this pattern, we can determine that it will take one half-life for the activity to decrease from 10,000 cpm to 5,000 cpm. However, as we need the time to reach 8,500 cpm (which is between 10,000 cpm and 5,000 cpm after one half-life), it will take less than one half-life. To calculate the exact time, we use the formula for radioactive decay:
N(t) = N0(1/2)^(t/T), where:
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- N(t) is the remaining counts per minute
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- N0 is the initial counts per minute
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- t is the time elapsed
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- T is the half-life period
Given that after one half-life (time T), the activity is at 16,000 cpm, and we need to find the time t when the activity is at 8,500 cpm, we can set up the equation:
8500 = 10,000(1/2)^(t/T)
Solving for t, we get:
t = T log(8500/10000) / log(1/2)
Assuming that one half-life is 10.25 minutes (given in one of the additional information pieces), then:
t = 10.25 log(8500/10000) / log(1/2) ≈ 3.39 minutes
Since it is just a small fraction of a half-life, we can look at the options provided and conclude that none of them are correct, indicating that the question might have some missing or incorrect information, or the provided answer options may be incorrect.