Question

An isotope of a radioactive element has half-life equal to 9 thousand years. Imagine a sample that is so old that most of its radioactive atoms have decayed, leaving just 10 percent of the initial quantity of the isotope remaining. How old is the sample? (Give your answer correct to at least one decimal place.)

Answer 1

The age of the sample in which only 10 percent of the radioactive isotope remains is slightly less than 27,000 years, based on its half-life of 9,000 years.

Step-by-step explanation:

To calculate the age of a sample in which only 10 percent of a radioactive isotope remains, we use the concept of half-life. The half-life is the time required for half of the atoms in a sample to decay. Given the half-life of the isotope is 9,000 years, we can follow a step-by-step approach to find out how many half-lives have passed to leave only 10 percent of the original isotope.

Starting with 100 percent of the isotope, after one half-life (9,000 years), 50 percent would remain. After the second half-life (18,000 years total), 25 percent would remain. Continuing this process, we can determine the number of half-lives needed to reach 10 percent:

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• After 3 half-lives (27,000 years), 12.5 percent remains.
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• To get to 10 percent, the time will be slightly less than 3 half-lives.

We know after three half-lives, 12.5 percent of the isotope remains, which is slightly more than the 10 percent remaining in the sample. Therefore, the age of the sample is a little less than 27,000 years. To approximate, we would solve the equation 0.5^n = 0.10 (where n represents the number of half-lives), but since the exact age in years is not asked for, our estimate is the sample is slightly less than 27,000 years old.