Chapter 20, Problem 14STP

To determine

To Calculate: The age of a rock.

Explanation of Solution

Give info:

Half-life of potassium-40= $1.28billionyears$

Formula used:

  $$ Age of rock= number of half-lives multiplied by half life

Calculation:

Half-life is the amount of time it takes for half the lives in a sample of an isotope to decay.

Following figure showing the proportion of decaying nuclei left after each half-life.

  

It can be clearly seen from above figure that after each half-life the number of remaining undecayed nuclei are decreasing rapidly.

• After one half-life the number of remaining undecayed nuclei reduced to half (50 percent) that means 50 percent (100-50) of nuclie have been decayed.
• And after two half-lives the number of remaining undecayed nuclei reduced to one-fourth (25 percent) that means 75 percent (100-25) of nuclie have been decayed.
• And after about 2.9 half-lives the number of remaining undecayed nuclei reduced to one-eighth (12.5 percent) that means 87.5 percent (100-12.5) of nuclie have been decayed as pointed in above graph.

Hence, about 2.9 half-lives passed to decay 87.5 percent of potassium-40 atoms.

Age of rock= number of half-lives multiplied by half life

Age of rock (in years) $=(2.9\times 1.28)billionyears$

Age of rock (in years) $=3.71billionyears$

Conclusion:

Thus,the age of a rock in which 87.5 percent of potassium-40 atoms have been decayed to argon-40 is about 3.71 billion years.