Atomic Structure

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Atomic Structure — Key Facts for CUET Bohr model: $E_n = -\frac{13.6}{n^2}$ eV for hydrogen; angular momentum quantised: $m_evr = \frac{nh}{2\pi}$ de Broglie wavelength: $\lambda = \frac{h}{mv}$; Heisenberg: $\Delta x \cdot \Delta p \geq \frac{h}{4\pi}$ Photoelectric effect: $KE_{\max} = h\nu - \phi$; threshold frequency $\nu_0 = \frac{\phi}{h}$ Hund’s rule: maximum multiplicity in degenerate orbitals; Pauli exclusion: no two electrons can have all four quantum numbers identical Quantum numbers: $n$ (1–7), $l$ (0 to $n-1$), $m_l$ ($-l$ to $+l$), $m_s$ (±½) ⚡ Exam tip: For electron configuration, fill orbitals by increasing energy (Aufbau principle): 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding of atomic models and quantum mechanics.

Atomic Structure — CUET Chemistry Study Guide

The journey from Dalton’s solid sphere to Bohr’s quantised orbits to Schrödinger’s wave equation represents one of the greatest intellectual achievements in science. Each model addressed limitations of its predecessor.

Thomson’s Plum Pudding Model (1897): Discovered electron (charge $e = 1.6 \times 10^{-19}$ C, mass $m_e = 9.1 \times 10^{-31}$ kg). Model: positive sphere with electrons embedded. Disproven by Rutherford’s gold foil experiment (1911).

Rutherford’s Nuclear Model: Most alpha particles passed through gold foil undeflected, but ~1 in 20,000 deflected at large angles. This proved the atom is mostly empty space with a dense, positively charged nucleus. Limitation: orbiting electrons should spiral into nucleus (accelerating charges radiate energy).

Bohr’s Model (1913): Introduced quantisation of electron energy and angular momentum. Postulates:

1. Electrons revolve in stable circular orbits (no energy radiation)
2. Angular momentum is quantised: $L = m_evr = \frac{nh}{2\pi}$ where $n = 1, 2, 3…$
3. Energy is emitted/absorbed when electrons transition between orbits: $\Delta E = h\nu = E_f - E_i$

For hydrogen: $E_n = -\frac{13.6}{n^2}$ eV. The Rydberg constant $R_H = \frac{me^4}{8\varepsilon_0^2 h^2} = 1.097 \times 10^7$ m⁻¹. Spectral lines: Lyman series (UV, $n_i \to n_f = 1$), Balmer series (visible, $n_i \to n_f = 2$), Paschen series (IR, $n_i \to n_f = 3$).

de Broglie Hypothesis (1924): All matter has wave nature. $\lambda = \frac{h}{mv}$. For macroscopic objects, $\lambda$ is negligibly small. For electrons at 100 eV: $\lambda = \frac{12.27}{\sqrt{V}} = 0.123$ nm — comparable to atomic spacing, observable in electron microscopes.

Heisenberg’s Uncertainty Principle: $\Delta x \cdot \Delta p \geq \frac{h}{4\pi}$. We cannot simultaneously know exact position and momentum. For electrons, this makes “orbits” meaningless — we speak of probability distributions (orbitals).

Quantum Numbers:

• Principal ($n$): determines energy shell; $n = 1, 2, 3…$ (K, L, M…)
• Azimuthal ($l$): subshell shape; $l = 0$ (s), $l = 1$ (p), $l = 2$ (d), $l = 3$ (f)
• Magnetic ($m_l$): orbital orientation in magnetic field; $m_l = -l, …, 0, …, +l$
• Spin ($m_s$): electron spin; $m_s = +\frac{1}{2}$ or $-\frac{1}{2}$

Example: Write electron configuration of Iron (Z = 26) and Fe³⁺. Fe: [Ar] 4s² 3d⁶. Note: 4s fills before 3d (lower energy). Fe³⁺: remove from outermost shell: [Ar] 3d⁵ (the 4s electrons are removed first).

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Atomic Structure — Complete CUET Chemistry Notes

Schrödinger Wave Equation (1926): The complete quantum mechanical description: $\hat{H}\psi = E\psi$, where $\hat{H}$ is the Hamiltonian operator, $\psi$ is the wave function, and $E$ is energy. $|\psi|^2$ gives probability density of finding the electron. Solutions give discrete energy levels — quantisation emerges naturally.

Hydrogen Atom Wave Functions: For hydrogen, the energy depends only on $n$. Orbitals are labelled by $n$, $l$, $m_l$. The radial probability distribution shows that for s-orbitals ($l=0$), there is a non-zero probability at the nucleus (radial node at $r = 0$ is not possible). For n=2, we get 2s and 2p orbitals; for n=3, we get 3s, 3p, and 3d.

Pauli Exclusion Principle: No two electrons in an atom can have identical values for all four quantum numbers. This limits each orbital (defined by $n, l, m_l$) to maximum 2 electrons with opposite spins. In the periodic table, this explains why each successive element has a different electron configuration.

Hund’s Rule of Maximum Multiplicity: In degenerate orbitals (same $n$ and $l$), electrons fill singly with parallel spins before pairing. Example: Carbon (1s² 2s² 2p²) has configuration with two unpaired electrons in 2p orbitals with parallel spins.

Aufbau Principle and Exceptions: Orbitals fill in order of increasing $n + l$ value. When $n + l$ is equal, lower $n$ fills first. Exceptions: Chromium (Cu) uses [Ar] 4s¹ 3d⁵ instead of 4s² 3d⁴ (3d⁵ is half-filled, more stable). Similarly, Copper [Ar] 4s¹ 3d¹⁰.

Zeeman Effect: When atoms are placed in a magnetic field, spectral lines split because $m_l$ degenerate orbitals have different energies in the field. The normal Zeeman effect (singlet lines) involves no spin; the anomalous Zeeman effect involves electron spin.

Stern-Gerlach Experiment: Demonstrated space quantisation of electron spin. A beam of silver atoms passing through a non-uniform magnetic field splits into two beams (spin up and spin down), confirming $m_s = \pm\frac{1}{2}$.

Quantum Mechanical Tunnelling: The phenomenon where a particle can tunnel through a potential barrier even if its energy is less than the barrier height. Relevant in nuclear fusion (Sun’s energy production), scanning tunnelling microscopes (STM), and semiconductor devices.

CUET Exam Patterns (2022–2024):

• Quantum numbers and electronic configuration are most frequently tested (1–2 marks)
• Bohr model energy calculations (spectral lines) appeared in 2023
• de Broglie wavelength problems are common in Section B
• Heisenberg uncertainty principle tested in 2022
• Common mistakes: confusing $n$, $l$, $m_l$ ranges; wrong order of orbital filling; forgetting that 4s fills before 3d

⚡ Key insight: Remember the mnemonic for orbital energy order: “Some Men Have More Money Than My Sister” (4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f…). Also, when removing electrons to form cations, remove from the outermost shell first — for transition metals, this means removing 4s electrons before 3d electrons since 4s has higher energy in the ion.

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