Question

A radioactive nucleus can decay simultaneously by two different process. The half life for

Answer 1

The radioactive element decays simultaneously by alpha decay with a half life of 4 years, and beta decay with a half- life of 12 years. What percentage of the radio-active element after the expiry of 12 year.

In this situation, we need to calculate the decay constant of alpha decay, the decay constant if beta decay,and the combined decay constant for the two modes of decay. Decay constants merely add up. Effective decay constant for both decays is decay constant for alpha decay plus decay constant for beta decay.

Now decay constant is inverse of mean life. Mean life = Half-life/0.693.

So decay constant of alpha decay = 0.693/4 in year^-1

And decay constant of beta decay = 0.693/12 in years^-1

And decay constant of both decay modes= 0.693[(1/4) +(1/12)]= 0.693(4/12) in years^-1= 0.693/3 in year^-1

So mean life of combined modes of decay = 3 year/0.693

Half-life for combined decay=( 3/0.693)×0.693= 3 year

Number of half-lives expired in 12 years= 12/3 = 4 half-lives

Radioactive element remaining after 4 half-lives =(½)⁴= 1/16= 100/16%= 6. 25%.