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CLASS 12 • PHYSICS
Nuclei
1. Nuclear Structure
Radius of Nucleus
$R = R_0 A^{1/3}$
$R_0 = 1.2 \times 10^{-15}$ m (Fermi)
Nuclear Density
$\rho \approx 2.3 \times 10^{17} \text{ kg/m}^3$
Independent of Mass Number (A)
Volume
$V \propto A$
2. Binding Energy (BE)
Mass Defect ($\Delta m$)
$\Delta m = [Z m_p + (A-Z) m_n] - M$
Energy Equivalent
$E = (\Delta m)c^2$
If mass in u: $E = \Delta m \times 931.5 \text{ MeV}$
BE Per Nucleon
$BE/A = \frac{\text{Total BE}}{A}$
Higher BE/A = More Stable
3. Radioactivity Law
Decay Equation
$N = N_0 e^{-\lambda t}$
Exponential Decay
Activity (R)
$R = -\frac{dN}{dt} = \lambda N$
Unit: Becquerel (Bq)
Half Life ($T_{1/2}$)
$T_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{\lambda}$
Mean Life ($\tau$)
$\tau = \frac{1}{\lambda} = 1.44 T_{1/2}$
4. Decay Types
Alpha Decay ($\alpha$)
$^A_Z X \to ^{A-4}_{Z-2} Y + ^4_2 He$
A decreases by 4, Z by 2
Beta Decay ($\beta^-$)
$^A_Z X \to ^A_{Z+1} Y + e^- + \bar{\nu}$
Neutron converts to Proton
Beta Decay ($\beta^+$)
$^A_Z X \to ^A_{Z-1} Y + e^+ + \nu$
Proton converts to Neutron
Gamma Decay ($\gamma$)
No change in A or Z (Energy release only)
5. Fission & Fusion
Nuclear Fission
Heavy Nucleus $\to$ 2 Lighter Nuclei
Example: Uranium-235 (Power Plants)
Nuclear Fusion
2 Light Nuclei $\to$ Heavy Nucleus
Example: Sun (4H $\to$ He)
Q-Value
$Q = (m_{reactants} - m_{products})c^2$
6. Common Exam Traps
- ⚠️ Density is Constant: Nuclear density ($\rho$) does NOT depend on size (A). It is same for Hydrogen and Uranium ($1:1$).
- ⚠️ Half-Life Math: Remaining Amount $N = N_0 (\frac{1}{2})^n$ where $n = t/T_{1/2}$. Don't use linear subtraction!
- ⚠️ Mass Units: In nuclear physics, use 'u' (amu). $1u = 931.5 \text{ MeV}/c^2$. Don't use kg unless asked.
7. Golden Theory Rules
- Nuclear Force: Short range, Charge independent, Spin dependent, Non-central. It is the strongest force in nature.
- Stability Belt: For light nuclei, $N \approx Z$. For heavy nuclei, $N > Z$ (neutrons needed to overcome proton repulsion).
- Iron Peak: Fe-56 has max BE/nucleon ($\approx 8.8$ MeV). It is the most stable element. Elements left of it Fuse, elements right of it Fission.
