STRUCTURE OF ATOM (QUICK REVISION)

1. Subatomic Particles & Atomic Species

Particle	Charge (C)	Mass (kg)	Mass (u)	Key Exam Fact
Electron ($e^-$)	$-1.602 \times 10^{-19}$	$9.1 \times 10^{-31}$	$0.00054$	Thomson ($e/m$ is constant for all gases)
Proton ($p$)	$+1.602 \times 10^{-19}$	$1.672 \times 10^{-27}$	$1.00727$	Goldstein ($e/m$ depends on gas; max for $\text{H}_2$)
Neutron ($n$)	$0$	$1.675 \times 10^{-27}$	$1.00867$	Chadwick ($^9_4\text{Be} + ^4_2\text{He} \rightarrow ^{12}_6\text{C} + ^1_0\text{n}$)

Definitions Isotopes: Same $Z$, diff $A$ ($^1_1\text{H}, ^2_1\text{H}$) • Isobars: Same $A$, diff $Z$ ($^{14}_6\text{C}, ^{14}_7\text{N}$) • Isotones: Same neutrons ($A-Z$) • Isoelectronic: Same $e^-$ ($\text{N}^{3-}, \text{O}^{2-}, \text{F}^-, \text{Na}^+, \text{Mg}^{2+} \implies 10e^-$).

2. Electromagnetic Radiation & Photoelectric Effect

Wave & Photon Energy

$$ c = \nu \lambda \implies \bar{\nu} = \frac{1}{\lambda} $$ $$ E = h\nu = \frac{hc}{\lambda} \quad (h = 6.626 \times 10^{-34} \text{ J s}) $$ $$ hc \approx 12400 \text{ eV \AA} \approx 1240 \text{ eV nm} $$

Photoelectric Equation

$$ E_{\text{photon}} = W_0 + KE_{\max} $$ $$ h\nu = h\nu_0 + \frac{1}{2}m_e v_{\max}^2 = e V_s $$ ($W_0 = h\nu_0$: Work Function, $V_s$: Stopping Potential)

3. Bohr's Model & Hydrogen Spectrum

Quantization & Radius

$$ mvr = \frac{nh}{2\pi} $$ $$ r_n = 0.529 \times \frac{n^2}{Z} \text{ \AA} \quad (r_n \propto \frac{n^2}{Z}) $$

Velocity & Energy

$$ v_n = 2.18 \times 10^6 \times \frac{Z}{n} \text{ m/s} $$ $$ E_n = -13.6 \times \frac{Z^2}{n^2} \text{ eV} = -2.18 \times 10^{-18} \frac{Z^2}{n^2} \text{ J} $$

Energy Relations & Period

$$ KE = -E_n, \quad PE = 2E_n = -27.2 \frac{Z^2}{n^2} \text{ eV} $$ $$ Time Period \quad T_n \propto \frac{n^3}{Z^2}, \quad f_n \propto \frac{Z^2}{n^3} $$

Rydberg Formula & Spectral Series $$ \bar{\nu} = \frac{1}{\lambda} = R_H Z^2 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \quad (R_H = 109677 \text{ cm}^{-1}, \quad 1/R_H \approx 912 \text{ \AA}) $$ Lyman: $n_1=1$ (UV) • Balmer: $n_1=2$ (Visible) • Paschen: $n_1=3$ (IR) • Brackett: $n_1=4$ (IR) • Pfund: $n_1=5$ (IR)
Total Spectral Lines ($n_2 \rightarrow n_1$): $N = \frac{(n_2 - n_1)(n_2 - n_1 + 1)}{2}$. (Drop to ground state: $N = \frac{n(n-1)}{2}$).

4. Dual Nature & Uncertainty Principle

de Broglie Equation

$$ \lambda = \frac{h}{mv} = \frac{h}{p} = \frac{h}{\sqrt{2m(KE)}} = \frac{h}{\sqrt{2mqV}} $$ $$ \text{Electron Shortcut: } \lambda_e = \frac{12.27}{\sqrt{V}} \text{ \AA} \quad (2\pi r = n\lambda) $$

Heisenberg Uncertainty Principle

$$ \Delta x \cdot \Delta p \ge \frac{h}{4\pi} \implies \Delta x \cdot (m\Delta v) \ge \frac{h}{4\pi} $$ $$ \text{Energy-Time Form: } \Delta E \cdot \Delta t \ge \frac{h}{4\pi} $$

5. Quantum Mechanical Model & Quantum Numbers

Schrödinger Equation: $\hat{H}\psi = E\psi$. ($\mathbf{\psi^2}$ = probability density of finding electron in orbital).

Node Formulas (100% Exam Favorite) $$ \text{Radial Nodes (Spherical)} = n - l - 1 \quad \mid \quad \text{Angular Nodes (Nodal Planes)} = l \quad \mid \quad \text{Total Nodes} = n - 1 $$

Quantum Number	Symbol	Allowed Values	Physical Significance & Formulas
Principal	$n$	$1, 2, 3...$	Shell, Size & Energy. Orbitals in shell $= n^2$, Max $e^- = 2n^2$.
Azimuthal	$l$	$0 \dots (n-1)$	Subshell ($s=0, p=1, d=2, f=3$) & Shape. Orbital Angular Momentum: $L = \sqrt{l(l+1)} \frac{h}{2\pi}$.
Magnetic	$m_l$	$-l \dots +l$	Orbital orientation. Orbitals per subshell $= 2l+1$. Max $e^- = 2(2l+1)$.
Spin	$m_s$	$+\frac{1}{2}, -\frac{1}{2}$	Electron spin direction. Spin Magnetic Moment: $\mu = \sqrt{n(n+2)} \text{ BM}$ ($n$=unpaired $e^-$).

6. Electronic Configuration Rules & Must-Know Exceptions

1. Aufbau Principle & (n+l) Rule

Lower $(n+l)$ fills first. If $(n+l)$ is same, lower $n$ fills first.
Filling Order: $1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f...$

2. Pauli & Hund's Rules

Pauli: No 2 $e^-$ have all 4 QN same (Max 2 $e^-$/orbital with $\uparrow\downarrow$).
Hund's: Pairing in degenerate orbitals ($p,d,f$) occurs only after each is singly filled ($\uparrow$).

High-Yield Exam Exceptions & Ion Formation

Chromium ($\text{Cr}, Z=24$): Expected $[\text{Ar}] 4s^2 3d^4 \implies$ Actual: $[\text{Ar}] 4s^1 3d^5$ (Stable half-filled $d^5$).
Copper ($\text{Cu}, Z=29$): Expected $[\text{Ar}] 4s^2 3d^9 \implies$ Actual: $[\text{Ar}] 4s^1 3d^{10}$ (Stable fully-filled $d^{10}$).
Ion Formation Rule: Remove $e^-$ from outermost principal shell $n$ first! (e.g. $\text{Fe}^{3+} [Z=26] \implies [\text{Ar}] 3d^5$).
Ion Stability: $\text{Mn}^{2+} (3d^5)$ is more stable than $\text{Fe}^{2+} (3d^6)$ due to half-filled $d$-subshell exchange energy.