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Chapter 38: Problem 57

Analysis of a Moon rock shows that \(82 \%\) of its initial \(\mathrm{K}-40\) has decayed to Ar-40, a process with a half-life of \(1.2 \times 10^{9}\) years. How old is the rock?

Short Answer

Expert verified
The age of the rock is \(2.4 \times 10^{9}\) years.

Step by step solution

01

Determine the number of half-lives passed

First, look at the percentage of K-40 remaining, which is 18%. To find out how many half-lives have occurred, start at 100% and keep halving until reaching a number less than or equal to 18%. Each halving represents one half-life. So, no half-life occurred when 50% of K-40 had decayed (50% remains). The first half-life occurred when another 25% had decayed (25% remains). The second half-life occurred when another 12.5% had decayed (approx. 12.5% remains). As 12.5% is less than 18%, two half-lives have occurred.
02

Calculate time passed

Now, knowing that two half-lives have passed, and given that the half-life of K-40 is \(1.2 \times 10^{9}\) years, multiply the number of half-lives by the half-life period to find the age of the rock. Age = Number of half-lives * half-life period = 2 * \(1.2 \times 10^{9}\) years.

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