We know that radioactive reactions are first order reaction
For a first order reaction in CHEMICAL KINETICS
Rate Constant = 0.693/(Half Life)
we know that,
Half Life = 500 years
K = 0.693/500
K = 0.001386
For first order,
ln[A] = -Kt + ln[A*]
[A] = concentration of A after time, t
[A*] = initial concentration of A at time, t=0
Kt = 2.303 log {[A*]/[A]}
At half life [A] = [A*]/2
Kt = 2.303 log (2)
(0.001386)t = 2.303 × 0.3010
t = (2.303)(0.3010)/(0.001386)
t = 500.146 years
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