Atomic Structure Formula Sheet

Every must-know formula, constant, and high-yield fact from the Atomic Structure chapter, grouped by sub-topic for last-minute revision.

Key Constants

Constant	Symbol	Value
Planck’s constant	$h$	$6.626 \times 10^{-34}$ J·s
Speed of light	$c$	$3 \times 10^8$ m/s
Rydberg constant	$R_H$	$1.097 \times 10^7$ m⁻¹
First Bohr radius	$a_0$	$0.529$ Å
Coulomb constant	$k = \frac{1}{4\pi\epsilon_0}$	$9 \times 10^9$ N·m²/C²
Electron mass	$m_e$	$9.1 \times 10^{-31}$ kg
eV to Joule	—	$1\text{ eV} = 1.6 \times 10^{-19}$ J

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Electromagnetic Radiation

Quantity	Formula	Notes
Wave relation	$c = \nu\lambda$	$c = 3\times10^8$ m/s in vacuum
Wave number	$\bar{\nu} = \dfrac{1}{\lambda}$	unit m⁻¹
Photon energy	$E = h\nu = \dfrac{hc}{\lambda}$	higher $\nu$ → higher $E$
Photon momentum	$p = \dfrac{h}{\lambda} = \dfrac{E}{c}$	photons are massless but carry momentum

$$\boxed{E = h\nu = \frac{hc}{\lambda}}$$

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Photoelectric Effect

$$\boxed{h\nu = \phi + \tfrac{1}{2}mv_{max}^2 = h\nu_0 + KE_{max}}$$

Quantity	Formula	Notes
Work function	$\phi = h\nu_0$	minimum energy to eject electron
Maximum KE	$KE_{max} = h\nu - \phi$	depends only on frequency
Stopping potential	$eV_0 = KE_{max} = h\nu - \phi$	$V_0 = \dfrac{h(\nu - \nu_0)}{e}$
Threshold wavelength	$\lambda_0 = \dfrac{hc}{\phi}$	maximum $\lambda$ that still ejects
Maximum velocity	$v_{max} = \sqrt{\dfrac{2(h\nu - \phi)}{m}}$	—

$V_0$ vs $\nu$ graph: slope $= \dfrac{h}{e}$, x-intercept $= \nu_0$, y-intercept $= -\dfrac{\phi}{e}$.

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Bohr’s Model (hydrogen-like, nuclear charge $Ze$)

Postulates: quantized circular orbits (no radiation in a stationary state), quantized angular momentum, and energy emitted/absorbed only on transitions.

$$\boxed{mvr = \frac{nh}{2\pi} = n\hbar} \qquad \boxed{\Delta E = E_2 - E_1 = h\nu}$$

Quantity	Formula	Scaling
Force balance	$\dfrac{mv^2}{r} = \dfrac{kZe^2}{r^2}$	—
Radius of nth orbit	$r_n = \dfrac{0.529\,n^2}{Z}$ Å	$r_n \propto \dfrac{n^2}{Z}$
Velocity in nth orbit	$v_n = \dfrac{2.18\times10^6\,Z}{n}$ m/s	$v_n \propto \dfrac{Z}{n}$
Energy of nth orbit	$E_n = -\dfrac{13.6\,Z^2}{n^2}$ eV	$= -\dfrac{2.18\times10^{-18}Z^2}{n^2}$ J
Energy ratio	$\dfrac{E_{n_1}}{E_{n_2}} = \dfrac{n_2^2}{n_1^2}$	—
Orbital period	$T_n = \dfrac{2\pi r_n}{v_n}$	$\propto \dfrac{n^3}{Z^2}$
Frequency of revolution	$f_n = \dfrac{v_n}{2\pi r_n}$	$\propto \dfrac{Z^2}{n^3}$
Ionization energy	$IE = 0 - E_1 = 13.6\,Z^2$ eV	from ground state to $\infty$

$$\boxed{r_n = \frac{0.529\,n^2}{Z}\ \text{Å} \qquad E_n = -\frac{13.6\,Z^2}{n^2}\ \text{eV}}$$

Hydrogen energy levels: $E_1 = -13.6$, $E_2 = -3.4$, $E_3 = -1.51$, $E_4 = -0.85$ eV; $E_\infty = 0$.

Ionization energies: H (Z=1) → 13.6 eV; He⁺ (Z=2) → 54.4 eV; Li²⁺ (Z=3) → 122.4 eV.

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Hydrogen Spectrum

$$\boxed{\frac{1}{\lambda} = \bar{\nu} = R_H Z^2\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)}$$

with $n_2 > n_1$ (final state $n_1$). Equivalently $h\nu = -13.6 Z^2\left(\frac{1}{n_2^2} - \frac{1}{n_1^2}\right)$ eV.

Series	$n_1$	Region	$\lambda_{limit}$ (nm)
Lyman	1	UV	91.2
Balmer	2	Visible	364.6
Paschen	3	Infrared	820.4
Brackett	4	Far IR	—
Pfund	5	Far IR	—

Quantity	Formula	Notes
Series limit	$\dfrac{1}{\lambda_{limit}} = R_H Z^2 \dfrac{1}{n_1^2}$	$n_2 \to \infty$ (shortest $\lambda$)
Lines from level $n$	$N = \dfrac{n(n-1)}{2}$	electron de-exciting from $n$ to ground
Lines for $n_2 \to n_1$	$N = \dfrac{(n_2-n_1)(n_2-n_1+1)}{2}$	—

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Dual Nature of Matter

$$\boxed{\lambda = \frac{h}{mv} = \frac{h}{p}}$$

Quantity	Formula	Notes
de Broglie (KE form)	$\lambda = \dfrac{h}{\sqrt{2m\,KE}}$	same KE → $\lambda \propto \dfrac{1}{\sqrt{m}}$
Accelerated charge $q$ through $V$	$\lambda = \dfrac{h}{\sqrt{2mqV}}$	$KE = qV$
Electron through $V$ (volts)	$\lambda = \dfrac{12.27}{\sqrt{V}}$ Å	quick-calc shortcut
Thermal wavelength	$\lambda = \dfrac{h}{\sqrt{3mkT}}$	particles at temperature $T$
Bohr quantization	$2\pi r = n\lambda$	gives $mvr = \dfrac{nh}{2\pi}$

Heisenberg Uncertainty Principle

$$\boxed{\Delta x \cdot \Delta p \geq \frac{h}{4\pi}}$$

Equivalent forms: $\Delta x \cdot m\Delta v \geq \dfrac{h}{4\pi}$ and $\Delta E \cdot \Delta t \geq \dfrac{h}{4\pi}$.

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Quantum Mechanical Model

Item	Expression	Notes
Schrödinger equation	$\hat{H}\psi = E\psi$	$\dfrac{\partial^2\psi}{\partial x^2}+\dfrac{\partial^2\psi}{\partial y^2}+\dfrac{\partial^2\psi}{\partial z^2}+\dfrac{8\pi^2 m}{h^2}(E-V)\psi = 0$
Probability density	$\lvert\psi\rvert^2$	$\psi$ must be single-valued, continuous, finite
Normalization	$\int \lvert\psi\rvert^2\,dV = 1$	total probability = 1
Orbital angular momentum	$L = \sqrt{l(l+1)}\cdot\dfrac{h}{2\pi}$	depends on $l$
Spin angular momentum	$S = \sqrt{s(s+1)}\cdot\dfrac{h}{2\pi} = \sqrt{\tfrac{3}{4}}\cdot\dfrac{h}{2\pi}$	$s = \tfrac{1}{2}$

Four Quantum Numbers

QN	Symbol	Allowed values	Determines
Principal	$n$	$1, 2, 3, \dots$	shell, size, energy
Azimuthal	$l$	$0$ to $(n-1)$	subshell, shape
Magnetic	$m_l$	$-l$ to $+l$ ($2l+1$ values)	orbital orientation
Spin	$m_s$	$+\tfrac{1}{2}$ or $-\tfrac{1}{2}$	electron spin

Counts: orbitals in a subshell $= 2l+1$; orbitals in a shell $= n^2$; max electrons in subshell $= 2(2l+1)$; max electrons in shell $= 2n^2$.

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Shapes of Orbitals & Nodes

Node type	Formula	Description
Radial nodes	$n - l - 1$	spherical surfaces
Angular nodes	$l$	planes or cones
Total nodes	$n - 1$	sum of both

Quantity	Formula	Notes
Radial probability	$P(r) = 4\pi r^2 \lvert\psi\rvert^2$	1s peak at $r = a_0 = 0.529$ Å

Subshell	$l$	Shape	Angular nodes	$m_l$ values
s	0	Sphere	0	0
p	1	Dumbbell (starts n=2)	1 plane	$-1, 0, +1$
d	2	4-lobed / ring (starts n=3)	2 planes	$-2 \dots +2$
f	3	Complex (starts n=4)	3 planes	$-3 \dots +3$

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Electronic Configuration

Filling order (Aufbau / $(n+l)$ rule)

$$1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f < 5d < 6p < 7s < 5f < 6d < 7p$$

Lower $(n+l)$ fills first; for equal $(n+l)$, lower $n$ fills first.

Rule	Statement
Aufbau	Fill orbitals in order of increasing energy
Pauli exclusion	No two electrons share all four quantum numbers (max 2 per orbital, opposite spins)
Hund’s rule	Singly fill degenerate orbitals with parallel spins before pairing

Subshell	Max electrons
s	2
p	6
d	10
f	14

Must-know exceptions (extra stability of half / fully-filled subshells)

Element	Z	Expected	Actual
Cr	24	[Ar] 4s² 3d⁴	[Ar] 4s¹ 3d⁵
Cu	29	[Ar] 4s² 3d⁹	[Ar] 4s¹ 3d¹⁰
Mo	42	[Kr] 5s² 4d⁴	[Kr] 5s¹ 4d⁵
Ag	47	[Kr] 5s² 4d⁹	[Kr] 5s¹ 4d¹⁰
Au	79	[Xe] 6s² 4f¹⁴ 5d⁹	[Xe] 6s¹ 4f¹⁴ 5d¹⁰

Stable configurations: fully filled s², p⁶, d¹⁰, f¹⁴; half filled p³, d⁵, f⁷.

Ions and magnetism

• Cations: remove electrons from the outermost shell first — for transition metals, 4s before 3d (in ions 3d lies below 4s). e.g. Fe ([Ar] 3d⁶ 4s²) → Fe²⁺ ([Ar] 3d⁶).
• Anions: add electrons to the lowest available orbital. e.g. O ([He] 2s² 2p⁴) → O²⁻ ([Ne]).
• Magnetism: unpaired electrons → paramagnetic; all paired → diamagnetic.

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Limitations of Bohr’s Model

graph TD
    A[Bohr's Model fails] --> B[Only single-electron species: H, He+, Li2+]
    A --> C[No fine structure of lines]
    A --> D[No Zeeman effect: magnetic field]
    A --> E[No Stark effect: electric field]
    A --> F[Violates Heisenberg uncertainty]
    A --> G[No relative line intensities]
    A --> H[No explanation of chemical bonding]

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One-Glance Headline Formulas

$$E = \frac{hc}{\lambda} \quad\bullet\quad h\nu = \phi + KE_{max} \quad\bullet\quad r_n = \frac{0.529\,n^2}{Z}\,\text{Å} \quad\bullet\quad E_n = -\frac{13.6\,Z^2}{n^2}\,\text{eV}$$$$\frac{1}{\lambda} = R_H Z^2\!\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right) \quad\bullet\quad \lambda = \frac{h}{mv} \quad\bullet\quad \Delta x\,\Delta p \geq \frac{h}{4\pi} \quad\bullet\quad N = \frac{n(n-1)}{2}$$