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Chapter 21: Problem 68

The radioactive potassium- 40 isotope decays to argon- 40 with a half-life of \(1.2 \times 10^{9}\) yr. (a) Write a balanced equation for the reaction. (b) A sample of moon rock is found to contain 18 percent potassium40 and 82 percent argon by mass. Calculate the age of the rock in Years.

Short Answer

Expert verified
The age of the moon rock is approximately \(4.18 \times 10^{9}\) years.

Step by step solution

01

Write the balanced radioactive decay reaction

A balanced radioactive decay reaction conserves mass numbers and atomic numbers. Potassium-40 to Argon-40 involves beta decay, where an electron (or beta particle) is emitted from the nucleus. The balanced equation is: \(^{40}_{19}K \rightarrow ^{40}_{18}Ar + ^{0}_{-1}\beta + \gamma\). Here \(\gamma\) represents the gamma radiation released during decay.
02

Understand the concept of Half-life

Half-life is the time required for half of the atoms in a radioactive sample to decay. In this exercise, the half-life of potassium-40 is given to be \(1.2 \times 10^{9}\) years.
03

Use the decay relation

We know that the remaining fraction of the isotope after time t is given by \(f = \frac{1}{2}^{(t/T)}\), where T is the half-life of the isotope. From the question, we know that 82% of the original potassium-40 has decayed, meaning that 18% remains. Hence, \(f = 0.18\).
04

Solve for the time (t) of decay

Substitute \(f = 0.18\) and \(T = 1.2 \times 10^{9}\) years into the decay relation, then solve for \(t\). On solving this equation, \(t = -1.2 \times 10^{9} \ln(0.18) / \ln(0.5)\) years. The value of \(t\) is approximately \(4.18 \times 10^{9}\) years.

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