Key Concept: Uranium–lead decay, half-life calculation
b) 2.63 billion years
[Solution Description]
The decay equation for uranium-238 to lead-206 is:
²³⁸U → ²⁰⁶Pb + 8 ⁴He + 6 β⁻
Step 1: Calculate the initial amount of uranium when the rock formed.
If 1 mg of ²³⁸U remains and 0.5 mg of ²⁰⁶Pb has been produced, the original amount of uranium was:
Initial uranium = 1 mg + 0.5 mg = 1.5 mg.
Step 2: Use the radioactive decay formula to determine the number of half-lives elapsed.
Let N₀ be the initial amount (1.5 mg) and N be the remaining amount (1 mg):
N = N₀ × (1/2)ⁿ
Step 3: Solve for n (number of half-lives):
1 = 1.5 × (1/2)ⁿ
(1/2)ⁿ = 2/3
n = log(2/3) / log(1/2) ≈ 0.585
Step 4: Calculate the age of the rock.
The half-life of ²³⁸U is 4.5 × 10⁹ years.
t = n × T₁/₂
t = 0.585 × 4.5 × 10⁹
t ≈ 2.63 × 10⁹ years
Therefore, the age of the rock is approximately 2.6 billion years.
Your Answer is correct.
The decay equation for uranium-238 to lead-206 is:
²³⁸U → ²⁰⁶Pb + 8 ⁴He + 6 β⁻
Step 1: Calculate the initial amount of uranium when the rock formed.
If 1 mg of ²³⁸U remains and 0.5 mg of ²⁰⁶Pb has been produced, the original amount of uranium was:
Initial uranium = 1 mg + 0.5 mg = 1.5 mg.
Step 2: Use the radioactive decay formula to determine the number of half-lives elapsed.
Let N₀ be the initial amount (1.5 mg) and N be the remaining amount (1 mg):
N = N₀ × (1/2)ⁿ
Step 3: Solve for n (number of half-lives):
1 = 1.5 × (1/2)ⁿ
(1/2)ⁿ = 2/3
n = log(2/3) / log(1/2) ≈ 0.585
Step 4: Calculate the age of the rock.
The half-life of ²³⁸U is 4.5 × 10⁹ years.
t = n × T₁/₂
t = 0.585 × 4.5 × 10⁹
t ≈ 2.63 × 10⁹ years
Therefore, the age of the rock is approximately 2.6 billion years.