What can be inferred about the age of a rock that contains equal amounts of potassium-40 and argon-40?

• A. The rock is newly formed and contains no radioactive decay products.
• B. The rock is two half-lives old for potassium-40.
• C. The rock is one half-life old for potassium-40.
• D. The rock is one quarter-life old for potassium-40.

Answer: (C)

Verified step by step guidance

1. Understand that potassium-40 (\(^{40}\mathrm{K}\)) is a radioactive isotope that decays into argon-40 (\(^{40}\mathrm{Ar}\)) over time through radioactive decay.
2. Recall the concept of half-life: the time required for half of the original amount of a radioactive isotope to decay into its product.
3. If a rock contains equal amounts of \(^{40}\mathrm{K}\) and \(^{40}\mathrm{Ar}\), it means half of the original \(^{40}\mathrm{K}\) has decayed into \(^{40}\mathrm{Ar}\).
4. Set up the relationship: initial amount of \(^{40}\mathrm{K}\) = amount of remaining \(^{40}\mathrm{K}\) + amount of produced \(^{40}\mathrm{Ar}\). Since they are equal, the remaining \(^{40}\mathrm{K}\) is half the initial amount.
5. Conclude that the rock must be one half-life old because one half-life corresponds to the time when half of the original radioactive isotope remains and an equal amount of decay product has formed.