A rock sample contains 25 percent potassium- 40 and 75 percent daughter product argon- 40 . If potassium- 40 has a half-life of 1.3 billion years, how old is the rock?
A) 2.6 billion years.
B) 3.9 billion years.
C) 5.2 billion years.
D) 1.3 billion years.
E) 0.65 billion years.
The Correct Answer and Explanation is :
The correct answer is A) 2.6 billion years.
Here’s the explanation:
The process of radiometric dating uses the known half-life of a radioactive isotope to determine the age of a sample. In this case, we are dealing with potassium-40 (K-40), which decays into argon-40 (Ar-40). The given information states that the rock contains 25% potassium-40 and 75% argon-40.
- Understanding the decay process:
Potassium-40 decays over time into argon-40. When the rock formed, it would have contained only potassium-40. As time passed, potassium-40 began to decay into argon-40, but argon is a gas that is trapped in the rock and doesn’t escape once it forms. - Ratio of Potassium-40 to Argon-40:
The ratio of potassium-40 to argon-40 is given as 25% K-40 and 75% Ar-40. This means that for every 1 mole of potassium-40, there are 3 moles of argon-40 (since the total amounts of K-40 and Ar-40 combined must be 100%). - Using the half-life:
The half-life of potassium-40 is 1.3 billion years. This means that for every 1.3 billion years that pass, half of the potassium-40 in the sample decays to argon-40. - Calculating the age:
Since the ratio of potassium-40 to argon-40 is 1:3, this implies that the potassium-40 has decayed through two half-lives. After one half-life, half of the potassium-40 would decay, leaving 50% of the original potassium-40. After the second half-life, 50% of the remaining potassium-40 would decay, leaving 25% of the original potassium-40. Thus, the rock is two half-lives old. Therefore, the rock’s age is 2 × 1.3 billion years = 2.6 billion years.
This calculation shows that the correct answer is 2.6 billion years.