Chapter 38: Problem 51

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 moon rock is approximately \(3.5 \times 10^{9}\) years. Please note that this is an approximate value, the actual answer might vary depending upon the decimal places used in calculation.

Step by step solution

Calculate the remaining percentage of K-40

Given that 82% of the K-40 has decayed, it means that 18% of the original K-40 remains in the rock. This is obtained by subtracting the decayed percentage from 100, i.e., \(100 - 82 = 18\% \) or \(0.18\) in decimal.

Represent the remaining amount with the decay formula

Our given problem can be represented by the formula \(N=N_{0}2^{-t/T}\) as \( 0.18 = 1 \times 2^{-t/T}\) where \(N = 0.18\) is the remaining amount of K-40, \(T = 1.2 \times 10^{9}\) years is the half-life and the \(N_{0} = 1\) assuming the initial amount was 100%.

Solve for t

Rearranging the equation to solve for \(t\), we have \(-t/T= \log_{2}(0.18)\), so \(t = -T\log_{2}(0.18)\). Substituting the given value for \(T\) we find that \(t = -1.2 \times 10^{9}\log_{2}(0.18)\). Evaluating this expression will give the age of the rock in years.

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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

Step by step solution

Calculate the remaining percentage of K-40

Represent the remaining amount with the decay formula

Solve for t

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