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These exercises use the radioactive decay model. Radium-221 has a half-life of 30 s. How long will it take for 95% of a sample to decay?

User Mahamoutou
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Final answer:

It will take approximately 180 seconds for 95% of a radium-221 sample to decay, as this is roughly six half-lives of the substance with each half-life being 30 seconds.

Step-by-step explanation:

The student's question is about how long it will take for 95% of a radium-221 sample to decay given that its half-life is 30 seconds. To solve this, we can use the concept of radioactive decay and half-lives.

The half-life is the time it takes for half of a radioactive substance to decay. Since 95% decay means that 5% remains, we need to calculate how many half-lives it takes to get from 100% to 5%. This calculation can be done using the formula for exponential decay:

N(t) = N0 * (1/2)^(t/h)

Where:

  • N(t) is the remaining quantity of the substance after time t,
  • N0 is the initial quantity of the substance,
  • t is the time elapsed,
  • h is the half-life of the substance.

Since we want to know when 5% of the original sample is left (N(t) = 0.05 * N0), we can set up the equation 0.05 = (1/2)^(t/30) and solve for t. Taking the logarithm of both sides, we get: t/30 = log2(0.05), which gives us t as approximately six half-lives or 180 seconds.

Thus, it will take approximately 180 seconds for 95% of a radium-221 sample to decay.

User Benjwadams
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