Law Of Radioactivity Decay - Practice Questions & MCQ

Radioactivity-

The phenomenon by virtue of which substance, spontaneously, disintegrate by emitting certain radiations is called radioactivity.

Activity (A)-

Activity is measured in terms of disintegration per second.

$
A=-\frac{d N}{d t}
$

Units of radioactivity:-
Its SI unit is ' Bq (Becquerel)'.
Curie (Ci):- Radioactivity of a substance is said to be one curie if its atoms disintegrate at the rate of $3.7 \times 10^{10}$ disintegrations per second. I.e

$
1 \mathrm{Ci}=3.7 \times 10^{10} \mathrm{~Bq}=37 \mathrm{GBq}
$

Rutherford (Rd):- Radioactivity of a substance is said to be 1 Rutherford if its atoms disintegrate at the rate of $10^6$ disintegrations per second.

The relation between Curie and Rutherford- $1 \mathrm{C}=3.7 \times 10^4 \mathrm{Rd}$

Laws of radioactivity-

Radioactivity is due to the disintegration of a nucleus. The disintegration is accompanied by the emission of energy in terms of α, β and γ-rays either single or all at a time. The rate of disintegration is not affected by external conditions like temperature and pressure etc.

According to Laws of radioactivity the rate of the disintegration of the radioactive substance, at any instant, is directly proportional to the number of atoms present at that instant.

$
\text { i.e }-\frac{d N}{d t}=\lambda N
$

where $\lambda=$ disintegration constant or radioactive decay constant
- Number of nuclei after the disintegration (N)

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

where $N_0$ is the number of radioactive nuclei in the sample at $t=0$.
Similarly Activity of a radioactive sample at time $t$

$
A=A_0 e^{-\lambda t}
$

where $A_0$ is the Activity of a radioactive sample at time $t=0$
- Half-life $\left(T_{1 / 2}\right)$ -

The half-life of a radioactive substance is defined as the time during which the number of atoms of the substance is reduced to half their original value.

$
\mathrm{T}_{1 / 2}=\frac{0.693}{\lambda}
$

Thus, the half-life of a radioactive substance is inversely proportional to its radioactive decay constant.

• Number of nuclei in terms of half-life-

$
N=\frac{N_0}{2^{t / T_{1 / 2}}}
$

Note- It is a very useful formula to determine the number of nuclei after the disintegration in terms of half-life
- Mean or Average life ( $T_{\text {mean }}$ )

Definition: The arithmetic mean of the lives of all the atoms is known as the mean life or average life of the radioactive substance.
$T_{\text {mean }}=$ sum of lives of all atoms / total number of atoms
Let $|\mathrm{dN}|$ is the number of nuclei decaying between $t, t+d t$; the modulus sign is required to ensure that it is positive.

$
\begin{aligned}
& \mathrm{dN}=-A \mathrm{~N}_0 \mathrm{e}^{-\lambda \mathrm{t}} \mathrm{dt} \\
& \text { and }|\mathrm{dN}|=A \mathrm{~N}_0 \mathrm{e}^{-\lambda \mathrm{t}} \mathrm{dt} \\
& T_{\text {mean }}=\frac{\int_0^{\infty} t|d N|}{\int_0^{\infty}|d N|}=\frac{\frac{1}{\lambda^2}}{\frac{1}{\lambda}}=\frac{1}{\lambda} \\
& \Rightarrow T_{\text {mean }}=\frac{1}{\lambda}
\end{aligned}
$

The average life of a radioactive substance is equal to the reciprocal of its radioactive decay constant.

The average life of a radioactive substance is also defined as the time in which the number of nuclei reduces to $\left(\frac{1}{e}\right)$ part of the initial number of nuclei.

The relation between $T_{1 / 2}$ and $T_{\text {mean }}$ :-

$
\Rightarrow \mathrm{T}_{1 / 2}=(0.693) T_{\text {mean }}
$

OR
Half-life $=(0.693)$ Mean life