Question

What is radioactivity? State the law of radioactivity decay. Show that radioactive decay is exponential in nature.
The half life of radium is 1600 years. How much time does 1 g of radium take to reduce to 0.125 g?

Answer 1

Radioactivity is phenomenon of spontaneous disintegration of nucleus of an atom with emission of one or more radiations. The most common forms of radiation emitted are alpha ($\alpha$), beta ($\beta$), and gamma ($\gamma$) radiations.

Law of radioactive decay: According to this law, the rate of decay of radioactive atoms at any instant is directly proportional to the number of atoms present at that instant.

$\frac{dN}{dt}\propto N$

$\frac{dN}{dt}=\lambda N$

where $N$ is number of integrated nucleus present in the sample at any time t and $\lambda$ is decay constant.

By integrating the above equation,

$\int \frac{dN}{N}=\int \lambda t$

$lnN=\lambda t+C$

$N=N_0{e}^{\lambda t}$

where $N_0$ is original amount.

Therefore, radioactive decay is exponential in nature.

Initial mass of Radium $=1g$

Final mass of Radium $=0.125g$

Half life $t_{1/2}=1600$ years

The quantity remaining 'n' lifes is $\frac{1}{2^n}$ of initial quantity. So,

$\frac{1}{2^n}=\frac{Finalmass}{Initialmass}=\frac{0.125}{1}=\frac{1}{8}=\frac{1}{2^3}$

$\therefore n=3$

Now, time takes $=n\times t_{1/2}=2\left(1600\right)=4800$ years