The radioisotope potassium-40 decays to argon-40 by positron emission with a half-life of 1.3 ã 109 yr. a sample of moon rock was found to contain 78 argon-40 atoms for every 22 potassium-40 atoms. the age of the rock is:

4.66 x 10^8 yr

Explanation:

The age of the rock can be calculated using the equation:

ln (N/N₀) = - kt    where N is the quantiy of radioisotope decayed and N₀ is the initially quantity present of the radioisotope; k is the decay constant, and t is the time.

Now from the data , we have 78 argon-40  atoms for every 22 potassium-40 atoms, we can deduce that originally we had 22 + 78 = 100 atoms of potassium-40 so this is our N₀.

When we look at the equation, we see that k is unknown, but we can calculate it from the half-life which is given by the equation:

k =  0.693/ t half-life = 0.693/ 1.3 x 10⁹ yr = 5.33 x 10⁻¹⁰ yr⁻¹

Now we are in position to answer the question.

ln ( 78/100 ) =  - (5.33 x 10⁻¹⁰ yr⁻¹ ) t

- 0.249 = - 5.33 x 10⁻¹⁰ yr⁻¹  t

0.249/ 5.33 x 10⁻¹⁰ yr⁻¹  = t

4.66 x 10^8 yr

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