If a rock contained 12.0 g of ⁴⁰k when the earth was formed 4.54 billion years ago

Question: If a rock contained 12.0 g of ⁴⁰k when the earth was formed 4.54 billion years ago and the half-life of ⁴⁰k is 1.25 billion years, what mass of ⁴⁰k remains, in grams, in the rock today?

Final answer:

The question is about the concept of radioactive decay and half-life. The half-life of ⁴⁰k is 1.25 billion years, and since about 3.63 of these half-lives have passed since the Earth was formed, we multiply 12.0g of ⁴⁰k by 1/2 three times. This results in approximately 1.5 grams of ⁴⁰k remaining in the rock today.

Explanation:

This question is about understanding the concept of half-life, which is a physics-based concept with connections to Chemistry. The half-life of an isotope, in this case ⁴⁰k, is the time it takes for half of the isotope's atoms to decay. In this problem, ⁴⁰k has a half-life of 1.25 billion years.

To start the solution, you'd first need to determine how many half-lives have passed since the Earth was formed. This is done by diving the total time (4.54 billion years) by the half-life (1.25 billion years), which equals approximately 3.63. Since half-lives are exponential, we'd apply an exponential reduction. So, after one half-life, one has 1/2 the remaining original material, after two half-lives, one has 1/4, after three, 1/8 and so on. Therefore, to determine the amount of ⁴⁰k left, we multiply 12.0g by 1/2 three times. That will yield an approximate answer of 1.5 grams of ⁴⁰k remaining present-day.