A source contains two phosphorous radio nuclides ${}_{15}{}^{32}\mathrm{P}\left({\mathrm{T}}_{1/2}=14.3 \mathrm{d}\right)$ and ${}_{15}{}^{33}\mathrm{P}\left({\mathrm{T}}_{1/2}=25.3 \mathrm{d}\right)$. Initially, $10\%$ of the decays come from ${}_{15}{}^{33}\mathrm{P}$. How long one must wait until $90\%$ do so?

(i) An atom has a nucleus. The nucleus is positively charged. The radius of the nucleus is smaller than the radius of an atom by a factor of 104. More than 99.9% mass of the atom is concentrated in the nucleus.

(ii) 1 atomic mass unit (1u) is 1/12th mass of one atom of 12C, $1 \mathrm{u}= 1.660563\times {10}^{-27}\mathrm{ }\mathrm{k}\mathrm{g}$.

(iii) The atomic number $Z$ is the number of protons in the atomic nucleus of an element. The mass number $A$ is the total number of protons and neutrons in the atomic nucleus; A=Z+N; Here $N$ denotes the number of neutrons in the nucleus.

(iv) A nuclear species or a nuclide is represented as ZAX, where $X$ is the chemical symbol of the species.

(v) Nuclides with the same atomic number, but different neutron number are called isotopes. Nuclides with the same mass number are isobars and those with the same neutron number are isotones.

2. Size of the nucleus:

(i) Radii of nuclei R=R0A1/3 where R0=1.2 fm and $A$ is mass number.

(ii) Nuclear density is independent of A. It is of the order of 1017kg/m3.

3. Nuclear force:

Neutrons and protons are bound in a nucleus by the short-range strong nuclear force. The nuclear force does not distinguish between neutron and proton.

4. Mass defect:

The nuclear mass M is always less than the total mass, of its constituents. The difference in mass of a nucleus and its constituents is called the mass defect. ΔM=Zmp+(A-Z)mn-M

5. Binding energy:

Binding energy is the energy released during the formation of nucleus, ΔEb=ΔMc2

6. Q-value:

The Q-value of a nuclear process is the amount of energy released or absorbed during the nuclear process.

Q= (sum of initial masses – sum of final masses) c2

7. Radioactivity:

(i) Law of radioactive decay: N(t)=N0e-λt where λ is the decay constant or disintegration constant.

(ii) The half-life T1/2=ln⁡2λ

(iii) The mean life $T=\frac{1}{\lambda }$

8. Nuclear fusion and nuclear fission:

(i) In fission, a heavy nucleus like 92235U breaks into two smaller fragments, e.g., 92235U+ 01n→ 51133Sb+ 4199Nb+4 01n

(ii) In fusion, lighter nuclei combine to form a larger nucleus. Fusion of hydrogen nuclei into helium nuclei is the source of energy of all stars including our sun.