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

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.

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

Related Practice

Textbook Question

A mammoth skeleton has a carbon-14 decay rate of 0.48 disintegration per minute per gram of carbon (0.48 dis/min • g C). When did the mammoth live? (Assume that living organisms have a carbon-14 decay rate of 15.3 dis/min • g C and that carbon- 14 has a half-life of 5715 yr.)

Textbook Question

Write the nuclear reaction for the neutron-induced fission of U-235 to form Xe-144 and Sr-90. How many neutrons are produced in the reaction?

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Textbook Question

Write the nuclear reaction for the neutron-induced fission of U-235 to produce Te-137 and Zr-97. How many neutrons are produced in the reaction?

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