Answer the following question:
(i)State the law of radioactive decay. Write the SI unit of 'activity'.
(ii) There are $$4\sqrt{2} \times 10^6$$ radioactive nuclei in a given radioactive sample. If half-life of the sample is $$20\,s,$$ how many nuclei will decay in $$10\,s ?$$
(i)The radioactive decay law states that the probability per unit time that a nucleus will decay is a constant, independent of time. This constant is called the decay constant and is denoted by λ, “lambda”. This constant probability may vary greatly between different types of nuclei, leading to the many different observed decay rates. The radioactive decay of certain number of atoms (mass) is exponential in time.
The SI unit of activity is the becquerel (Bq)
(ii) Given $$t_{1/2} = 20\,s$$
Also, $$t_{1/2} = \dfrac{lm \,2}{\lambda} \Rightarrow \lambda = \dfrac{ln \,2}{t_{1/2}} \Rightarrow \lambda = \dfrac{ln \,2}{20}$$
Also, according to equation of radioactivity
$$N = N_{0}e^{-\lambda t}$$
$$N = 4\sqrt{2} \times 10^6 \times e^{-=\dfrac{ln \,2}{20} \times 10}$$
$$ = 4\sqrt{2} \times 10^6 \times \dfrac{1}{\sqrt{2}} = 4 \times 10^6 \,Nuclei$$