The half-life for the process 238U→206Pb is 4.5 * 10^9 yr. A mineral sample contains 75.0 mg of 238U and 18.0 mg of 206Pb. What is the age of the mineral?

Verified step by step guidance

1. Step 1: Understand the decay process. The decay of 238U to 206Pb is a first-order process, which means the rate of decay is proportional to the amount of 238U present.
2. Step 2: Use the concept of half-life. The half-life of 238U is given as 4.5 \times 10^9 years. The half-life is the time required for half of the radioactive nuclei in a sample to decay.
3. Step 3: Calculate the initial amount of 238U. Since the decay process converts 238U to 206Pb, the initial amount of 238U is the sum of the current amount of 238U and the amount of 206Pb formed. Therefore, initial 238U = 75.0 mg + 18.0 mg.
4. Step 4: Determine the number of half-lives that have passed. Use the formula: \( N_t = N_0 \times (1/2)^{n} \), where \( N_t \) is the remaining amount of 238U, \( N_0 \) is the initial amount, and \( n \) is the number of half-lives. Solve for \( n \).
5. Step 5: Calculate the age of the mineral. Multiply the number of half-lives \( n \) by the half-life of 238U (4.5 \times 10^9 years) to find the age of the mineral.

Key Concepts

Half-life

Half-life is the time required for half of a sample of a radioactive substance to decay into its daughter products. In this context, the half-life of uranium-238 (238U) is 4.5 billion years, meaning that after this period, half of the original 238U will have transformed into lead-206 (206Pb). Understanding half-life is crucial for calculating the age of geological samples through radiometric dating.

Radioactive Decay

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation, resulting in the transformation into a different element or isotope. In the case of uranium-238, it decays into lead-206 over time. The rate of decay is constant and can be quantified using the half-life, which is essential for determining the age of a mineral sample based on the ratio of parent to daughter isotopes.

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. By measuring the ratio of 238U to 206Pb in the mineral sample and knowing the half-life, one can calculate the time elapsed since the mineral formed. This method provides a reliable means of determining the age of rocks and fossils.