How old is a rock that contains 83.2% of the amount of uranium-238 it contained when it was formed?

Name: Chemistry: A Molecular Approach
Author: Tro
ISBN: 9780137832217

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insert step 1> Determine the decay constant (\( \lambda \)) for uranium-238 using its half-life. The half-life of uranium-238 is approximately 4.5 billion years. Use the formula \( \lambda = \frac{0.693}{\text{half-life}} \).

insert step 2> Use the decay formula \( N = N_0 e^{-\lambda t} \), where \( N \) is the remaining quantity of uranium-238, \( N_0 \) is the initial quantity, \( \lambda \) is the decay constant, and \( t \) is the time elapsed.

insert step 3> Since the rock contains 83.2% of the original uranium-238, set \( N = 0.832 N_0 \). Substitute this into the decay formula to get \( 0.832 N_0 = N_0 e^{-\lambda t} \).

insert step 4> Simplify the equation to \( 0.832 = e^{-\lambda t} \).

insert step 5> Solve for \( t \) by taking the natural logarithm of both sides: \( \ln(0.832) = -\lambda t \). Rearrange to find \( t = \frac{\ln(0.832)}{-\lambda} \).

주요 개념

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

Radioactive Decay

Radioactive decay is the process by which unstable atomic nuclei lose 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 atoms in a sample to decay. Understanding this concept is crucial for determining the age of a rock based on the remaining quantity of a radioactive isotope.

Half-Life

Half-life is a specific time period in which half of the original amount of a radioactive substance decays into a different element or isotope. For uranium-238, the half-life is approximately 4.5 billion years. By knowing the percentage of uranium-238 remaining in a rock, one can calculate how many half-lives have passed and thus estimate the rock's age.

Radiometric Dating

Radiometric dating is a technique used to date materials by comparing the abundance of a radioactive isotope to its decay products. In the case of uranium-238, scientists measure the ratio of uranium-238 to lead-206, its stable decay product, to determine the age of the rock. This method provides a reliable means of estimating geological time scales and the age of ancient materials.