Final answer:
The decay rate of a radioactive substance with a half-life of 8,000 years is found using the first-order kinetics formula, yielding a rate constant (k) of 0.000086625 per year, corresponding to option a: 0.0865%.
Step-by-step explanation:
The rate at which a radioactive substance decays is related to its half-life. The half-life is the time it takes for one half of the radioactive atoms to decay. If the half-life of a substance is 8,000 years, we can use the relationship between the decay rate and half-life for first-order kinetics, which is given by the formula:
rate constant (k) = 0.693 / half-life (T1/2)
By substituting the half-life of the substance in the formula, we find the decay rate:
k = 0.693 / 8,000 years = 0.000086625 per year
Therefore, the the closest answer to the provided decay rate is option a, which is 0.0865%. To clarify, the decay rate in this case reflects the fraction of the substance that decays per year.