What is the half life, in minutes, of a sample that shows a decay from 87.12 grams of radioactive material to 21.87 grams in a time period of 19.4 minutes? (answer to two decimal places)

Question

What is the half life, in minutes, of a sample that shows a decay from 87.12 grams of radioactive material to 21.87 grams in a time period of 19.4 minutes? (answer to two decimal places)

BlackTom AI · Accepted Answer

To analyze the decay, we use the exponential decay model N(t) = N0 * (1/2)^(t / T1/2), where N0 is the initial amount, N(t) is the amount after time t, and T1/2 is the half-life.

Step 1: Determine the ratio after 19.4 minutes.
Calculate N(t)/N0 = 21.87 / 87.12 ≈ 0.2510. This means the sample has about 25.10% of its initial mass after 19.4 minutes.

Step 2: Relate the ratio to the half-life.
We have (1/2)^(t / T1/2) = 0.2510. Taking natural logs on both sides gives (t / T1/2) * ln(1/2) = ln(0.2510).

Step 3: Solve for t / T1/2.
(t / T1/2) = ln(0.2510) / ln(0.5). Since ln(0.2510) ≈ -1.383 and ln(0.5) ≈ -0.6931, the quotient is ≈ 1.994.

Step 4: Solve for the half-life T1/2.
T1/2 = t / 1.994 = 19.4 / 1.994 ≈ 9.73 minutes.

Step 5: Round appropriately.
Rounding to two decimal places yields T1/2 ≈ 9.73 minutes.

Summary: The half-life of the sample is about 9.73 minutes based on the observed decay from 87.12 g to 21.87 g in 19.4 minutes.

B27FF 11. Forces of Nature

What is the half life, in minutes, of a sample that shows a decay from 87.12 grams of radioactive material to 21.87 grams in a time period of 19.4 minutes? (answer to two decimal places)

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