Atomic Structure

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Atomic Structure — Key Facts for MDCAT

Subatomic Particles:

Particle	Symbol	Charge	Mass (kg)	Relative Mass
Proton	p or p⁺	+1	1.673 × 10⁻²⁷	1
Neutron	n or n⁰	0	1.675 × 10⁻²⁷	1
Electron	e⁻	-1	9.109 × 10⁻³¹	1/1836

Atomic Number (Z): Number of protons in the nucleus. Defines the element. Mass Number (A): Total number of protons + neutrons. Isotopes: Same Z, different A (e.g., ₁H², ₁H³ — deuterium and tritium). Isobars: Different Z, same A (e.g., ₆C¹⁴, ₇N¹⁴). Neutron Number (N): N = A – Z.

Bohr’s Model (for hydrogen-like species): $$E_n = -\frac{13.6Z^2}{n^2} \text{ eV} = -\frac{R_H Z^2}{n^2}$$ Where $n$ = principal quantum number (1, 2, 3…), $R_H$ = Rydberg constant = 13.6 eV.

Energy levels: $n=1$ (ground state), $n=2,3,4…$ (excited states). For hydrogen: $E_n = -13.6/n^2$ eV.

Wavelength of emitted photon: $$\frac{1}{\lambda} = R_H \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)$$ Where $n_2 > n_1$ and $R_H = 1.097 \times 10^7$ m⁻¹.

⚡ Exam tip: For hydrogen spectrum questions — Lyman series (UV, $n_1=1$), Balmer series (visible, $n_1=2$), Paschen series (IR, $n_1=3$). The Balmer formula with $n_1=2$ gives the visible hydrogen lines. Calculate the wavelength and identify the colour: 656 nm (Hα, red), 486 nm (Hβ, blue-green), 434 nm (Hγ, violet).

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students who want genuine understanding.

Atomic Structure — Complete Study Guide

Quantum Numbers: Every electron in an atom is described by four quantum numbers:

1. Principal quantum number (n): Determines energy level and average distance from nucleus. $n = 1, 2, 3…$
2. Azimuthal/angular momentum quantum number (ℓ): Determines subshell shape. For a given $n$: $\ell = 0$ to $(n-1)$. Subshell notation: s (ℓ=0), p (ℓ=1), d (ℓ=2), f (ℓ=3).
3. Magnetic quantum number ($m_\ell$): Orientation of orbital in space. For a given $\ell$: $m_\ell = -\ell$ to $+\ell$ (total $2\ell + 1$ values).
4. Spin quantum number ($m_s$): Direction of electron spin. $m_s = +\frac{1}{2}$ or $-\frac{1}{2}$ (clockwise or anticlockwise).

Maximum electrons per shell: $2n^2$

Shell	Subshells	Max electrons
K (n=1)	1s	2
L (n=2)	2s, 2p	2 + 6 = 8
M (n=3)	3s, 3p, 3d	2 + 6 + 10 = 18
N (n=4)	4s, 4p, 4d, 4f	2 + 6 + 10 + 14 = 32

Pauli’s Exclusion Principle: No two electrons in an atom can have the same set of all four quantum numbers. Each orbital can hold maximum 2 electrons with opposite spins.

Hund’s Rule: Electrons fill degenerate orbitals (same energy) one at a time with parallel spins before pairing. Example: Nitrogen (1s² 2s² 2p³) — the three 2p electrons have parallel spins in three separate p orbitals.

Aufbau Principle: Electrons fill orbitals in order of increasing energy: 1s → 2s → 2p → 3s → 3p → 4s → 3d → 4p → 5s → 4d → 5p → 6s → 4f → 5d → 6p → 7s → 5f

⚡ Common mistakes: Forgetting that 4s fills before 3d. Remember: energy of orbital depends on $(n+\ell)$ value — lower sum fills first. For 4s: (4+0)=4; for 3d: (3+2)=5, so 4s fills first. When writing electron configurations, remember Cr (₋Cr) is [Ar] 3d⁵ 4s¹ NOT [Ar] 3d⁴ 4s²; Cu (₂₉Cu) is [Ar] 3d¹⁰ 4s¹. This is due to extra stability of half-filled and fully-filled d subshells.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Atomic Structure — Advanced Notes

de Broglie Equation: Matter has wave-particle duality: $$\lambda = \frac{h}{mv} = \frac{h}{p}$$ Where $h$ = Planck’s constant (6.626 × 10⁻³⁴ J·s). This matters practically only for very small masses — for an electron at 100 eV, $\lambda \approx 0.12$ nm (comparable to atomic dimensions).

Heisenberg’s Uncertainty Principle: $$\Delta x \cdot \Delta p \geq \frac{\hbar}{2}$$ Where $\hbar = \frac{h}{2\pi}$. We cannot simultaneously know the exact position and momentum of a particle. For electrons, this means we cannot precisely trace their paths — only describe probability distributions (orbitals).

Schrödinger Equation: $$H\psi = E\psi$$ Where $H$ is the Hamiltonian operator, $\psi$ is the wave function (contains all information about the electron), $E$ is energy. Solving gives wave functions (orbitals), not orbits.

The probability density is $|\psi|^2$ (probability per unit volume). The radial probability distribution shows where the electron is most likely to be found at a given distance from the nucleus.

Quantum Numbers from Schrödinger Solution: For a given $n$: $\ell$ can be 0 to $n-1$; $m_\ell$ can be $-\ell$ to $+\ell$; $m_s = \pm\frac{1}{2}$.

Orbital Shapes:

• s orbital: Spherical (1 per subshell)
• p orbital: Dumbbell-shaped (3 per subshell, p$_x$, p$_y$, p$_z$)
• d orbital: Cloverleaf (except d$_{x^2-y^2}$ which is cloverleaf with donut) — 5 per subshell
• f orbital: Complex shapes — 7 per subshell

Photoelectric Effect: Einstein’s equation: $E_{photon} = h\nu = \phi + KE_{max}$ Where $\phi$ = work function (minimum energy to eject electron), $KE_{max}$ = maximum kinetic energy of ejected electron. Threshold frequency $\nu_0 = \phi/h$. Below this, no photoelectron emission regardless of light intensity.

Rydberg Formula Derivation: From Bohr model: $E_n = -\frac{R_H hc}{n^2}$ For transition from $n_2$ to $n_1$: $\Delta E = Rhc\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)$ Since $E = \frac{hc}{\lambda}$: $\frac{1}{\lambda} = R_H\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)$

Screening (Penetration) Effect: Inner electrons shield outer electrons from the full nuclear charge. Effective nuclear charge $Z_{eff} = Z - \sigma$ where $\sigma$ is the screening constant. Order of screening: f > d > p > s (f electrons are least effective at shielding).

MDCAT Question Patterns: MDCAT Pakistan atomic structure questions frequently test: (1) quantum number assignments and allowed values, (2) electron configuration including exceptions (Cr, Cu, Mo, Ag), (3) Hund’s rule application for p³, d⁵ configurations, (4) de Broglie wavelength calculations, (5) photoelectric effect equation, (6) Heisenberg uncertainty principle — which combinations of measurements are possible. 2–3 questions per paper. Quantum numbers are high-yield for MCQs.

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