Answer: A) The decay constant (λ) is a measure of the rate at which a radioactive isotope decays. It is defined as the probability per unit time that an atom of the isotope will decay. The decay constant can be calculated using the formula:
λ = -ln(1 - x) / t
Where x is the fraction of the original activity that has decayed (in this case, 0.40), and t is the time over which the decay occurred (in this case, one week).
B) The half-life (T1/2) is the amount of time it takes for half of the original activity of a radioactive isotope to decay. It can be calculated using the formula:
T1/2 = ln(2) / λ
C) The mean lifetime (τ) is the average amount of time an atom of a radioactive isotope will survive before it decays. It can be calculated using the formula:
τ = 1 / λ
It's important to note that the above formulas are based on the exponential decay model, which assumes that the decay process is random and that the decay constant is constant over time. If the isotope does not decay in this way, these formulas may not give accurate results.
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