Quick Revision Notes: Class 12 Physics

Chapter: Nuclei

Nucleons: Protons and neutrons present inside the nucleus are collectively known as nucleons.
Atomic Number ($Z$): Number of protons in a nucleus. (Equal to the number of electrons in a neutral atom).
Mass Number ($A$): Total number of nucleons (protons + neutrons) in a nucleus.
- $A = Z + N$ (where $N$ is the number of neutrons).
Representation of a Nucleus: denoted as ^A_Z X, where $X$ is the chemical symbol of the element.

The volume of a nucleus is directly proportional to its mass number ($A$).
- V \propto A
If the nucleus is assumed to be spherical with radius $R$, then (4/3)\pi R^3 \propto A, which gives the relation for nuclear radius:
- R = R_0 A^{1/3}
- Where $R_0$ is a constant approximately equal to 1.2 \times 10^{-15} m or 1.2 fm.

Nuclear density is independent of mass number $A$, meaning all nuclei have approximately the same density.
Formula:
- \rho = \frac{\text{Mass of nucleus}}{\text{Volume of nucleus}} = \frac{m \cdot A}{(4/3)\pi R_0^3 A} = \frac{3m}{4\pi R_0^3}
Order of Nuclear Density: \approx 2.3 \times 10^{17} kg/m^3 (Extremely high density).

Einstein's Mass-Energy Relation:
- E = \Delta m \cdot c^2
- (where $c$ is the speed of light in vacuum, \approx 3 \times 10^8 m/s).
Atomic Mass Unit (amu or u):
- $1\text{ u} = \frac{1}{12} \times \text{mass of one C-12 atom} \approx 1.66 \times 10^{-27} kg$.
- Energy equivalent of $1\text{ u} = 931.5 MeV$.
Mass Defect ($\Delta m$): The difference between the rest mass of a nucleus and the sum of the rest masses of its constituent nucleons.
- \Delta m = [Z m_p + (A - Z)m_n] - M
- Where:
  - $m_p$ = mass of a proton
  - $m_n$ = mass of a neutron
  - $M$ = actual mass of the nucleus

Binding Energy ($BE$): The energy required to break a nucleus into its constituent nucleons, or the energy released when nucleons combine to form a nucleus.
- BE = \Delta m \cdot c^2
- BE = [Z m_p + (A - Z)m_n - M] \times 931.5 MeV (if masses are in amu).
Binding Energy per Nucleon ($\bar{BE}$):
- \bar{BE} = \frac{BE}{A}
- It is a measure of the stability of the nucleus. Higher the $\bar{BE}$ per nucleon, more stable is the nucleus.
- Note: Maximum $\bar{BE}$ per nucleon is for Iron (^56_{26}Fe), which is about 8.8 MeV/nucleon.

The forces that hold the protons and neutrons together in a nucleus are called nuclear forces.

Key Properties:
1. They are the strongest forces in nature (strong attractive force).
2. They are charge-independent (act equally between proton-proton, neutron-neutron, and proton-neutron).
3. They are short-range forces (effective only up to a distance of a few femtometres).
4. They are non-central forces and show saturation property.

The phenomenon of spontaneous emission of radiations ($\alpha, \beta, \gamma$) from unstable nuclei is called radioactivity.

Law of Radioactive Decay: The rate of disintegration of radioactive substance at any instant is directly proportional to the number of undecayed radioactive atoms present at that instant.
- -\frac{dN}{dt} \propto N => -\frac{dN}{dt} = \lambda N
- Where $\lambda$ is the decay constant or disintegration constant.
Exponential Decay Law (Integration form):
- N(t) = N_0 e^{-\lambda t}
- Where $N_0$ = initial number of nuclei at $t = 0$, and $N(t)$ = number of nuclei at time $t$.

Half-Life ($T_{1/2}$): The time interval in which the number of radioactive nuclei reduces to half of its initial value.
- T_{1/2} = \frac{\ln(2)}{\lambda} = \frac{0.693}{\lambda}
Mean Life ($\tau$): The average life time of all radioactive nuclei.
- \tau = \frac{1}{\lambda} = \frac{T_{1/2}}{0.693} = 1.44 \times T_{1/2}
Relation between remaining nuclei and half-life:
- N = N_0 \left(\frac{1}{2}\right)^n
- Where $n = \frac{t}{T_{1/2}}$ (number of half-lives).

Nuclear Fission: The process in which a heavy nucleus splits into two or more intermediate-mass nuclei with the release of a large amount of energy.
- Example: Fission of Uranium-235 by a slow neutron: ^1_0n + ^{235}_{92}U \rightarrow ^{144}_{56}Ba + ^{89}_{36}Kr + 3(^1_0n) + \text{Energy}
Nuclear Fusion: The process in which two or more light nuclei combine to form a single heavy nucleus, accompanied by a tremendous release of energy.
- Example: Fusion of hydrogen isotopes in the sun: ^2_1H + ^2_1H \rightarrow ^3_2He + ^1_0n + 3.27 MeV