For nuclei with $A>100$,

1. Atomic Nucleus:

The average radius of the nucleus may be written as

R=R0A1/3, $R_0=1.1\times {10}^{-15} \mathrm{m}$

A is mass number.

2. Mass Defect and Binding Energy:

The binding energy of nucleus of mass M, is given by $B=\left(Z{m}_{\mathrm{p}}+N{m}_{\mathrm{n}}-M\right)c^2$

For a+X→Y+b

$Q-\mathrm{value}=\left(M_{\mathrm{a}}+M_{\mathrm{x}}-M_{\mathrm{\gamma }}-M_{\mathrm{b}}\right)c^2$

3. Nuclear Reactions:

(i) Alpha - decay process:

XZA→z-2A-4Y+He24

Q-value $=\left[M\left({}_Z{}^{A}X\right)-M\left({}_{Z-2}{}^{A-4}Y\right)-M\left({}_2{}^{4}He\right)\right]c^2$

(ii) Beta- minus decay:

XZA→YZ+1A+β-+v-

Q-value $=\left[M\left({}_Z{}^{A}X\right)-M\left({}_{Z+1}{}^{A}Y\right)\right]c^2$

(iii) Beta plus decay:

XZA→YZ-1A+β++v

Q-value $= \left[M\left({}_Z{}^{A}X\right)-M\left({}_{Z-1}{}^{A}Y\right)-2m_{\mathrm{e}}\right]c^2$

(iv) When the atomic electron is captured, X-rays are emitted.

XZA+e→YZ-1A+v

Q-value $=\left[M\left({}_Z{}^{A}X\right)-M\left({}_{Z-1}{}^{A}Y\right)\right]c^2$

4. Nuclear Fission:

Discovered by Hahn and Strassmann in $1938.$ By attack of a particle on a heavy nucleus A>230 and splitting it into two or more lighter nuclei, a certain mass disappears which is obtained in the form of energy.

$A+p\to \mathrm{excited}\mathrm{ }\mathrm{nucleus} \to B+C+Q$

5. Nuclear Fusion:

It is the phenomenon of fusing two or more lighter nuclei to form a single heavy nucleus.

$A+B\to C+Q \left(\text{Energy}\right)$

The product $\left(C\right)$ is more stable than the reactants $\left(A \text{and} B\right)$.

mass defect, $\mathrm{ }\mathrm{\Delta }m=M_{\mathrm{A}}+M_{\mathrm{B}}-M_{\mathrm{C}}$

Energy released is $E=\mathrm{\Delta }m\times 931 \mathrm{MeV}$

The total binding energy and binding energy per nucleon of $C$, both are more than of $A$ and B.

$\mathrm{\Delta }E=E_{\mathrm{C}}-\left(E_{\mathrm{A}}+E_{\mathrm{B}}\right)$

6. Radioactive Decay Law:

(i) In radioactive decay, the number of nuclei at instant t is given by N=N0e-λt, λ- decay constant.

(ii) Activity of sample: A=A0e-λt

(iii) Activity per unit mass is called specific activity.

(iv) Half-life: T1/2=0.693λ

(v) Average life: $T_{\mathrm{av}}=\frac{T_{1/2}}{0.693}$

(vi) A radioactive nucleus can decay by two different processes having half-lives t1 and t2 respectively. Effective half-life of the nucleus is given by 1t=1t1+1t2.

(vii) Probability of a nucleus for survival of time t=NN0=N0e-λtN0=e-λt

7. Parallel radioactive disintegration:

Let the initial number of nuclei of A is N0 then at any time number of nuclei of A,B & Care given by

$N_0=N_{\mathrm{A}}+N_{\mathrm{B}}+N_{\mathrm{C}}$

$\Rightarrow \frac{\mathrm{d}{N}_{\mathrm{A}}}{\mathrm{d}t}=\frac{-\mathrm{d}}{\mathrm{d}t}\left(N_{\mathrm{B}}+N_{\mathrm{C}}\right)$

A disintegrates into $B$ and $C$ by emitting $\alpha , \beta$ particle.

Now, $\frac{\mathrm{d}{N}_B}{\mathrm{d}t}=-{\lambda }_1{N}_{\mathrm{A}} \text{and} \frac{\mathrm{d}{N}_c}{\mathrm{d}t}=-{\lambda }_2{N}_{\mathrm{A}} \Rightarrow \frac{\mathrm{d}}{\mathrm{d}t}\left(N_{\mathrm{B}}+N_{\mathrm{C}}\right)=-\left({\lambda }_1+{\lambda }_2\right)N_{\mathrm{A}}$

$\Rightarrow \frac{\mathrm{d}{N}_A}{\mathrm{d}t}=-\left({\lambda }_1+{\lambda }_2\right)N_{\mathrm{A}} \Rightarrow {\lambda }_{\mathrm{eff}}={\lambda }_1+{\lambda }_2 \Rightarrow t_{\mathrm{all}}=\frac{t_1{t}_2}{t_1+t_2}$

8. Radioactive Disintegration with Successive Production:

A is being produced at a rate α.

A disintegrates into B.

$\frac{\mathrm{d}{N}_A}{\mathrm{d}t}=\alpha -\lambda N_{\mathrm{A}} \dots \left(1\right)$

when $N_{\mathrm{A}}$ is maximum $\frac{\mathrm{d}{N}_{\mathrm{A}}}{\mathrm{d}t}=0 \Rightarrow \alpha -\lambda N_{\mathrm{A}}=0$

$N_{\mathrm{Amax}}=\frac{\alpha }{\lambda }=\frac{\mathrm{ }\text{rate of production}}{\lambda }$

By equation $\left(1\right)\text{,} {\int }_0^t\frac{\mathrm{d}{N}_{\mathrm{A}}}{\alpha -\lambda N_{\mathrm{A}}}={\int }_0^t\mathrm{d}t\text{,}$ Number of nuclei is $N_{\mathrm{A}}=\frac{\alpha }{\lambda }\left(1-e^{-\lambda t}\right)$