Final answer:
To find the amount of Pb-206 in the rock sample after 4.6 × 10^8 years, we calculate that approximately 0.21 mg of the original 3.13 mg U-238 will have decayed to Pb-206, using the half-life of U-238 which is 4.51 × 10^9 years.
Step-by-step explanation:
To determine how much Pb-206 will be in the rock sample when it is 4.6 × 108 years old, we will use the concept of radioactive dating and the half-life of U-238 which is 4.51 × 109 years. The question is based on the decay of U-238 and its conversion to Pb-206 over time. Given that after one half-life, half of the original U-238 remains and the rest is converted into Pb-206, we can calculate the amount of decayed Uranium and thereby find the amount of Lead-206 formed.
First, we need to calculate the number of half-lives that have passed in 4.6 × 108 years:
Number of half-lives = (Age of rock) / (Half-life of U-238)
= (4.6 × 108 years) / (4.51 × 109 years)
≈ 0.102 or roughly 1/10th of a half-life.
Since it is less than one half-life, not even half of the original U-238 will have decayed. Therefore, to find the amount of U-238 that has decayed, we use the formula:
Decayed U-238 = Original amount of U-238 × (1 - (1/2)Number of half-lives)
= 3.13 mg × (1 - (1/2)0.102)
≈ 3.13 mg × (1 - 0.933)
≈ 3.13 mg × 0.067
≈ 0.21 mg
Therefore, after 4.6 × 108 years, approximately 0.21 mg of the original U-238 will have decayed to Pb-206.