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A sample taken from a layer of mica in a canyon has 2.10 grams of potassium-40. A test reveals it to be 2.6 billion years old. How much potassium-40 was in the sample originally if the half-life of potassium-40 is 1.3 billion years?

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Answer:

[tex]\boxed{ \text{8.40 g}}[/tex] 

Explanation:

The half-life of K-40 (1.3 billion years) is the time it takes for half of it to decay.  

After one half-life, half (50 %) of the original amount will remain.  

After a second half-life, half of that amount (25 %) will remain, and so on.  

We can construct a table as follows:  

 No. of                                 Fraction

half-lives          t/yr             Remaining  

      0               0                          1

      1                1.3  billion              ½

      2              2.6                          ¼

      3              3.9                          ⅛

We see that after 2 half-lives, ¼ of the original mass remains.

Conversely, if two half-lives have passed, the original mass must have been four times the mass we have now.

Original  mass = 4 × 2.10 g = [tex]\boxed{ \text{8.40 g}}[/tex]

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