The half-life of ²³⁸U is 4.5⨉10⁹ yr. A sample of rock of mass 1.6 g produces 29 dis/s. Assuming all the radioactivity is due to ²³⁸U, find the percent by mass of ²³⁸U in the rock.

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Determine the decay constant (\( \lambda \)) using the half-life formula: \( \lambda = \frac{\ln(2)}{t_{1/2}} \), where \( t_{1/2} = 4.5 \times 10^9 \) years.

Convert the decay constant from per year to per second by using the conversion factor: 1 year = 3.15 \times 10^7 seconds.

Use the decay rate formula \( A = \lambda N \) to find the number of \( ^{238}U \) atoms (\( N \)) in the sample, where \( A = 29 \) disintegrations per second.

Calculate the mass of \( ^{238}U \) using the number of atoms \( N \) and Avogadro's number (\( 6.022 \times 10^{23} \) atoms/mol), along with the molar mass of \( ^{238}U \) (238 g/mol).

Determine the percent by mass of \( ^{238}U \) in the rock by dividing the mass of \( ^{238}U \) by the total mass of the rock (1.6 g) and multiplying by 100.

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Key Concepts

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

Half-life

Half-life is the time required for half of the radioactive nuclei in a sample to decay. For uranium-238, this period is approximately 4.5 billion years. Understanding half-life is crucial for calculating the remaining quantity of a radioactive isotope in a sample over time, which is essential for determining the mass percentage of uranium-238 in the rock.

Radioactivity and Decay Rate

Radioactivity refers to the process by which unstable atomic nuclei lose energy by emitting radiation. The decay rate, measured in disintegrations per second (dis/s), indicates how many atoms decay in a given time frame. In this problem, the decay rate of 29 dis/s provides the necessary information to calculate the amount of uranium-238 present in the rock sample.

Mass Percent Calculation

Mass percent is a way to express the concentration of a component in a mixture, calculated as the mass of the component divided by the total mass of the mixture, multiplied by 100. To find the percent by mass of uranium-238 in the rock, one must first determine the mass of uranium-238 based on its decay rate and then use the total mass of the rock sample to compute the percentage.