Question

The half-life of Radium-226 is 1590 years. If a sample contains 500 mg, how many mg will remain after 4000 years?

Answer 1

P = P₀*e^(kt).

Generally, for half-lives, I like using:

P = P₀*2^(kt).

Since the half life of Radium-226 is 1590 years, we see that P = (1/2)P₀ at t = 1590. This produces:

(1/2)P₀ = P₀*2^(1590k)
==> 1/2 = 2^(1590k)
==> 2^(-1) = 2^(1590k)
==> k = -1/1590.

So the require equation is:

P = P₀*2^(-t/1590).

Therefore, the amount remaining after 4000 years with a 500mg sample is:

P = 500 * 2^(-4000/1590) ≈ 87.4mg.