Atoms

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Atomic Physics — Key Facts

Rutherford’s Nuclear Model:

• Positive charge concentrated in tiny nucleus (radius ~10⁻¹⁵ m)
• Electrons orbit the nucleus (atomic radius ~10⁻¹⁰ m)
• Problem: classical physics predicts unstable orbits (accelerating electrons radiate energy)

Bohr’s Model (for Hydrogen):

Postulates:

1. Electrons move in circular orbits (stationary states)
2. Only certain orbits are allowed: $mvr = n\hbar$ (quantised angular momentum)
3. Energy is emitted/absorbed only when electrons transition between orbits

Key Results for Hydrogen-like Atoms: $$r_n = \frac{\varepsilon_0 h^2 n^2}{\pi me^2} = 0.529 \times 10^{-10} n^2 \text{ m}$$

$$E_n = -\frac{me^4}{8\varepsilon_0^2 h^2 n^2} = -\frac{13.6}{n^2} \text{ eV}$$

where n = principal quantum number (1, 2, 3, …)

Hydrogen Spectrum:

$$E_n - E_m = h\nu = \frac{hc}{\lambda}$$

Lyman series (UV): $n_i = 2, 3, 4… \rightarrow n_f = 1$ Balmer series (Visible): $n_i = 3, 4, 5… \rightarrow n_f = 2$ Paschen series (IR): $n_i = 4, 5, 6… \rightarrow n_f = 3$

⚡ JEE Exam Tip: For hydrogen-like ions (He⁺, Li²⁺, Be³⁺), replace e² with Ze² in formulas: $E_n = -\frac{13.6 Z^2}{n^2}$ eV. The series formulas remain the same with modified Rydberg constant $R = 1.097 \times 10^7 Z^2$ m⁻¹.

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding…

Rydberg Formula (General):

$$\frac{1}{\lambda} = R\left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)$$

where $R = 1.097 \times 10^7$ m⁻¹ (Rydberg constant)

For hydrogen: $R_H = \frac{me^4}{8\varepsilon_0^2 h^3 c} = 1.097 \times 10^7$ m⁻¹

Line Spectrum of Hydrogen:

Wavelengths in different series:

• Lyman (UV): $\lambda < 121.6$ nm
• Balmer (Visible): $364.6$ nm $<\lambda < 656.3$ nm
• Paschen (IR): $820.4$ nm $<\lambda < 1875$ nm
• Brackett (IR): $1458$ nm $<\lambda < 4051$ nm

Energy Levels of Hydrogen:

n=1 (ground state): E = -13.6 eV n=2 (first excited): E = -3.4 eV n=3 (second excited): E = -1.51 eV n=∞: E = 0 (ionisation)

Ionisation energy from ground state: 13.6 eV

Bohr Model Limitations:

• Only works for hydrogen and hydrogen-like atoms (one electron)
• Doesn’t explain fine structure (spin, relativistic effects)
• Doesn’t explain Zeeman effect (splitting in magnetic field)
• Cannot predict relative intensities of spectral lines

⚡ JEE Exam Tip: When electrons fall from higher to lower energy levels, they emit photons. When absorbing, they go to higher levels. Energy is conserved: $E_{photon} = E_{higher} - E_{lower}$.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Derivation of Bohr Quantisation:

From centripetal force = Coulomb attraction: $$\frac{mv^2}{r} = \frac{1}{4\pi\varepsilon_0}\frac{Ze^2}{r^2}$$

Also quantisation: $mvr = n\hbar$

From these: $r_n = \frac{4\pi\varepsilon_0 \hbar^2 n^2}{me^2 Z} = \frac{\varepsilon_0 h^2 n^2}{\pi me^2 Z}$

For Z=1 (hydrogen): $r_1 = 0.529$ Å (Bohr radius, denoted a₀)

Energy: $$E_n = \frac{1}{2}mv^2 - \frac{1}{4\pi\varepsilon_0}\frac{Ze^2}{r} = -\frac{1}{2}\frac{Ze^2}{4\pi\varepsilon_0 r_n} = -\frac{me^4 Z^2}{8\varepsilon_0^2 h^2 n^2}$$

$$E_n = -\frac{13.6 Z^2}{n^2} \text{ eV}$$

Vector Model of Atom:

Total angular momentum $\vec{J} = \vec{L} + \vec{S}$ (orbital + spin) Spin quantum number s = ½ for electron Landé g-factor: $g_J = 1 + \frac{J(J+1) + S(S+1) - L(L+1)}{2J(J+1)}$

X-ray Spectra:

Continuous X-ray spectrum: bremsstrahlung (braking radiation) Characteristic X-rays: $K_\alpha, K_\beta$ lines from inner shell transitions

$$E_{K_\alpha} = E_K - E_L \approx 13.6(Z-1)^2 \text{ eV (Moseley’s law approximation)}$$

Moseley’s law: $\sqrt{\nu} = a(Z - b)$ where a and b are constants

Fine Structure:

Due to relativistic correction and spin-orbit coupling: $$\Delta E \propto (Z\alpha)^4 \frac{1}{n^3} \frac{1}{j + \frac{1}{2}}$$

where j = total angular momentum quantum number (l ± ½)

Zeeman Effect:

Splitting of spectral lines in magnetic field: $$\Delta E = \mu_B g_L m_l B$$

where $\mu_B = \frac{e\hbar}{2m} = 9.27 \times 10^{-24}$ J/T (Bohr magneton)

Experimental Evidence for Bohr Model:

1. Line spectrum: Discrete wavelengths match predictions
2. Franck-Hertz experiment: Electrons lose discrete amounts of energy (corresponding to excitation energies)
3. Rutherford scattering: Confirmed nuclear structure
4. Ionisation potential: Measured 13.6 eV for hydrogen

Stark Effect:

Splitting of spectral lines in electric field (weaker effect than Zeeman)

Stern-Gerlach Experiment:

Showed space quantisation of angular momentum. Silver atoms split into two beams in non-uniform magnetic field, confirming electron spin s = ½.

⚡ JEE Advanced 2023 Analysis: Hydrogen spectral series (Balmer, Lyman, Paschen) and transition energies appear regularly. Questions involving multiple transitions and wavelength calculations are common. Also prepare for hydrogen-like ion questions and applications of Bohr quantisation.

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