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CLASS 12 • PHYSICS
Atoms
1. Rutherford's Alpha Scattering
Distance of Closest Approach ($r_0$)
$r_0 = \frac{2kZe^2}{K_\alpha}$
When KE converts fully to PE
Impact Parameter ($b$)
$b \propto \cot(\theta/2)$
If $b=0$, scattering angle $\theta = 180^\circ$
Scattering Probability
$N(\theta) \propto \frac{1}{\sin^4(\theta/2)}$
2. Bohr's Postulates
Angular Momentum (Quantized)
$L = mvr = \frac{nh}{2\pi}$
n is Principal Quantum Number
Frequency Condition
$h\nu = E_2 - E_1$
Energy emitted during transition
Radius of Orbit
$r_n = 0.529 \frac{n^2}{Z} \mathring{A}$
Velocity in Orbit
$v_n = \frac{c}{137} \frac{Z}{n}$
3. Energy of Electron
Total Energy ($E_n$)
$E_n = -13.6 \frac{Z^2}{n^2} \text{ eV}$
Kinetic Energy (K)
$K = -E_n$
Always Positive
Potential Energy (U)
$U = 2E_n$
Always Negative
Ionization Energy
$E_{ion} = +13.6 \frac{Z^2}{n^2} \text{ eV}$
4. Spectral Series
Rydberg Formula
$\frac{1}{\lambda} = R Z^2 (\frac{1}{n_1^2} - \frac{1}{n_2^2})$
$R \approx 1.09 \times 10^7 m^{-1}$
Lyman Series (UV Region)
$n_1 = 1$, $n_2 = 2,3,4...$
Balmer Series (Visible)
$n_1 = 2$, $n_2 = 3,4,5...$
Paschen Series (IR)
$n_1 = 3$, $n_2 = 4,5,6...$
5. Proportionality Shortcuts
| Quantity | Depends on 'n' | Depends on 'Z' |
|---|---|---|
| Radius ($r$) | $\propto n^2$ | $\propto 1/Z$ |
| Velocity ($v$) | $\propto 1/n$ | $\propto Z$ |
| Energy ($E$) | $\propto 1/n^2$ | $\propto Z^2$ |
| Time Period ($T$) | $\propto n^3$ | $\propto 1/Z^2$ |
6. Common Exam Traps
- ⚠️ Ground State: "Ground State" means $n=1$. "First Excited State" means $n=2$. Don't use $n=1$ for excited states!
- ⚠️ Total Energy Sign: Total Energy is always NEGATIVE. Kinetic Energy is always POSITIVE.
- ⚠️ Number of Lines: Number of spectral lines emitted when electron jumps from $n$ to ground state is $\frac{n(n-1)}{2}$.
7. Golden Theory Rules
- Stability: An electron in a stable orbit does not radiate energy, even though it is accelerating (Centripetal). This contradicts Maxwell's theory.
- Limitation: Bohr's model is valid ONLY for single-electron species (H, $He^+$, $Li^{++}$). It fails for multi-electron atoms.
- Negative Energy: The negative sign indicates that the electron is BOUND to the nucleus. To free it, positive energy must be supplied.
