Open Question
A rock from Australia contains 0.438 g of Pb-206 for every 1.00 g of U-238. Assuming that the rock did not contain any Pb-206 at the time of its formation, how old is the rock?
21. Nuclear Chemistry
Radioactive Half-Life
Multiple Choice
How long will it take for 20% of the C-14 atoms in a sample of C-14 to decay, given that the half-life of C-14 is approximately 5730 years?
A
Approximately 5730 years
B
Approximately 1620 years
C
Approximately 11460 years
D
Approximately 2865 years
Verified step by step guidance1
Understand that the problem involves radioactive decay, specifically the decay of Carbon-14 (C-14) atoms. The half-life of C-14 is given as 5730 years, which is the time it takes for half of the C-14 atoms in a sample to decay.
Use the concept of half-life and the first-order decay equation to solve the problem. The decay of radioactive isotopes follows first-order kinetics, which can be described by the equation: \( N_t = N_0 e^{-kt} \), where \( N_t \) is the remaining quantity of the substance, \( N_0 \) is the initial quantity, \( k \) is the decay constant, and \( t \) is the time.
Calculate the decay constant \( k \) using the half-life formula: \( k = \frac{\ln(2)}{t_{1/2}} \). Substitute the given half-life of C-14 (5730 years) into the formula to find \( k \).
Determine the fraction of C-14 atoms remaining after 20% has decayed. If 20% has decayed, then 80% remains. Express this as a fraction: \( \frac{N_t}{N_0} = 0.80 \).
Substitute the values into the first-order decay equation: \( 0.80 = e^{-kt} \). Solve for \( t \) by taking the natural logarithm of both sides and rearranging the equation to find \( t = \frac{-\ln(0.80)}{k} \).
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