Final answer:
The age of the mineral containing 61.0 mg of U-238 and 15.5 mg of Pb-206 is approximately 1.53 billion years, calculated using the half-life of U-238 and radioactive dating techniques.
Step-by-step explanation:
The age of the mineral can be determined by utilizing radioactive dating techniques based on the knowledge of the half-life of U-238, which decays into Pb-206. Given the half-life of U-238 is 4.5 × 10⁹ years, and the current amounts of U-238 and Pb-206 in the sample are 61.0 mg and 15.5 mg respectively, the age can be calculated as follows:
Let N0 be the original amount of U-238, and N be the present amount of U-238.
The formula for the number of half-lives (n) that have passed is: n = (log(N/N0))/log(1/2)
Since Pb-206 is the decay product of U-238, and assuming that all Pb-206 in the sample is from decay, N0 = N + Pb-206 amount.
Substituting the values: N0 = 61.0 mg + 15.5 mg
N0 = 76.5 mg
Using the formula: n = log(61.0 / 76.5) / log(1/2)
n ≈ 0.34
Therefore, the approximate age of the mineral is: Age = n × half-life of U-238
Age ≈ 0.34 × 4.5 × 10⁹ years
Age ≈ 1.53 × 10⁹ years
The age of the mineral is approximately 1.53 billion years.