# 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.

Therefore

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.