Final answer:
To determine the age of the rock, we can use the concept of radioactive decay and the ratio of the amounts of Th-232 and Pb-208 present in the rock. By subtracting the amount of Pb-208 from the initial amount of Th-232, we can calculate the amount of Th-232 that has decayed over time. Using the half-life of Th-232, we can calculate the number of half-lives that have passed since the rock was formed. Finally, multiplying the number of half-lives by the half-life of Th-232 gives us the age of the rock.
Step-by-step explanation:
To determine the age of the rock, we can use the concept of radioactive decay and the ratio of the amounts of Th-232 and Pb-208 present in the rock. Since all the Pb-208 in the rock is produced from the decay of Th-232, we can assume that the initial amount of Pb-208 in the rock was zero. Therefore, the amount of Pb-208 present in the rock is equal to the amount of Th-232 that has decayed over time.
The half-life of Th-232 is about 14.05 billion years, meaning that half of the Th-232 will decay in that time. We can use the decay equation to determine the age of the rock:
- Amount of Th-232 at the beginning = 3.65 g
- Amount of Pb-208 at the end = 0.75 g
- Amount of Th-232 currently in the rock = 3.65 g - amount of Pb-208 = 3.65 g - 0.75 g = 2.9 g
Using the half-life of Th-232, we can calculate the number of half-lives that have passed since the rock was formed:
Number of half-lives = Log(0.5)(Initial amount / Current amount)
Number of half-lives = Log(0.5)(3.65 g / 2.9 g)
Number of half-lives ≈ 0.27
Since each half-life corresponds to a time period of 14.05 billion years, we can calculate the age of the rock:
Age of the rock ≈ Number of half-lives * Half-life of Th-232
Age of the rock ≈ 0.27 * 14.05 billion years
Age of the rock ≈ 3.79 billion years