Radioactivity problem?

Question

A 5.4 microgram sample of 226RaCl2 has a radioactivity of 1.5x10^5 Bq. Calculate the half-life of 226Ra. And how do you come up with the answer?

Answer 1

First let's find how many nuclei there are in the sample.

5.4 ug RaCl2 x (1 g / 1 x 10^6 ug) x (1 mole RaCl2 / 226 g RaCl2) x (6.02 x 10^23 nuclei / 1 mole RaCl2) = 1.4 x 10^16 nuclei.

The rate of disintegration is 1.5 x 10^5 Bq.

1.5 x 10^5 Bq x ((1 nucleus / s) / 1 Bq)) = 1.5 x 10^5 nuclei / s

rate = k x Nt where k is the decay constant and Nt is the number of nuclei.

1.5 x 10^5 = k x (1.4 x 10^16)

1.1 x 10^-11 s^-1 = k

Recall that t 1/2 = 0.693 / k

t 1/2 = 0.693 / (1.1 x 10^-11 s^-1) = 6.5 x 10^10 s = 2000 years!