A rock from Australia contains 0.438 g of Pb-206 for every 1.00 g of U-238. Assuming that the rock did not contain any Pb-206 at the time of its formation, how old is the rock?

Verified step by step guidance

1. Understand that the problem involves radioactive decay, specifically the decay of Uranium-238 (U-238) to Lead-206 (Pb-206).
2. Use the decay equation: \( N = N_0 e^{-\lambda t} \), where \( N \) is the number of undecayed nuclei, \( N_0 \) is the initial number of nuclei, \( \lambda \) is the decay constant, and \( t \) is the time.
3. Recognize that the decay constant \( \lambda \) is related to the half-life \( t_{1/2} \) of U-238 by the equation \( \lambda = \frac{\ln(2)}{t_{1/2}} \). The half-life of U-238 is approximately 4.468 billion years.
4. Set up the ratio of Pb-206 to U-238 to find the number of half-lives that have passed. Use the relationship: \( \frac{N_0 - N}{N} = \frac{\text{mass of Pb-206}}{\text{mass of U-238}} \).
5. Solve for \( t \) using the decay equation and the known values, including the calculated decay constant and the ratio of Pb-206 to U-238.