If the half-life of an element is 430 years & you had 2000 of this element, how long will it take to be stable?
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Radioactive decay proceeds by first order kinetics. There are two equations which we can use to solve this problem.
t1/2 = 0.693/k
where t1/2 is the half life and k is the rate constant. Once we know the value of k, we can use the formula below to solve for the answer.
ln([A]t/[A]o) = - kt
where [A] is the concentration at time t or time zero, k is the rate constant previously determined, and t is the time. Although we talk about A in terms of concentration, we can actually put a variety of values in there (mass, percent, etc) as long as the two "concentrations" are in the same units.
First to find k
430 years = 0.693/k
k = 0.00161 yr^-1
Now, we can use this to find the time. Since we can't find the natural log of zero, we will assume that [A]t = 1 atom.
ln (1/2000) = -0.00161 yr^-1 * t
t = 4721 years
So it will take 4721 years for sample to decay to the point where only 1 atom of it remains.
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