If the half-life of an element is 430 years & you had 2000 of this element, how long will it take to be stable?
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.