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?

Name: Chemistry: A Molecular Approach
Author: Tro
ISBN: 9780134112831

검증된 단계별 안내

Understand that the problem involves radioactive decay, specifically the decay of Uranium-238 (U-238) to Lead-206 (Pb-206).

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.

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.

Set up the ratio of Pb-206 to U-238 to find the number of half-lives that have passed. Use the relationship: \( \frac{N_0 - N}{N} = \frac{\text{mass of Pb-206}}{\text{mass of U-238}} \).

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.

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. This decay occurs at a predictable rate, characterized by the half-life, which is the time required for half of the radioactive substance to decay. Understanding this concept is crucial for determining the age of rocks through radiometric dating.

Half-Life

Half-life is a specific time period in which half of a given quantity of a radioactive isotope decays into its daughter products. For U-238, the half-life is approximately 4.5 billion years. This concept allows scientists to calculate the age of geological samples by measuring the ratio of parent isotopes to daughter isotopes present in the sample.

Radiometric Dating

Radiometric dating is a technique used to date materials by comparing the abundance of a radioactive isotope within the sample to the abundance of its stable decay products. In this case, the ratio of Pb-206 to U-238 provides a means to calculate the time elapsed since the rock formed, assuming no lead was present initially.