Atomic Structure

Cheatsheet Content

1. Basic Atomic Model & Terminology Atom: Smallest particle of an element retaining its chemical identity. Nucleus: Central part, contains protons ($p^+$) and neutrons ($n^0$). Electrons ($e^-$): Orbit the nucleus. Atomic Number (Z): Number of protons. Defines element. For a neutral atom, $Z = \text{number of electrons}$. Mass Number (A): $A = \text{number of protons} + \text{number of neutrons}$. Nuclide Representation: $_Z^A X$. Isotopes: Same Z, different A (different number of neutrons). E.g., $_1^1 H, _1^2 H, _1^3 H$. Isobars: Same A, different Z (different elements). E.g., $_ {18}^{40} Ar, _ {19}^{40} K, _ {20}^{40} Ca$. Isotones: Same number of neutrons ($A-Z$ is same). E.g., $_6^{14} C, _8^{16} O$. Isoelectronic Species: Same number of electrons. E.g., $N^{3-}, O^{2-}, F^-, Ne, Na^+, Mg^{2+}$. 2. Early Atomic Models & Discoveries Cathode Ray Experiment (J.J. Thomson): Discovered electrons. $e/m$ ratio for electron determined. Oil Drop Experiment (R.A. Millikan): Determined charge of an electron ($1.6 \times 10^{-19} C$). Anode Ray Experiment (Goldstein): Discovery of protons. Rutherford's $\alpha$-scattering Experiment: Majority of atom is empty space. Positive charge and most mass concentrated in a small, dense nucleus. Electrons revolve around nucleus. Impact parameter ($b$): Perpendicular distance of the velocity vector of $\alpha$-particle from the center of the nucleus. Small $b \rightarrow$ large deflection. Large $b \rightarrow$ small deflection. $b=0 \rightarrow 180^\circ$ deflection (head-on collision). Distance of closest approach ($r_0$): For head-on collision, kinetic energy of $\alpha$-particle converted to potential energy. $$ \frac{1}{2}mv^2 = \frac{1}{4\pi\epsilon_0} \frac{(2e)(Ze)}{r_0} $$ Where $m$ is mass of $\alpha$-particle, $v$ its velocity, $2e$ its charge, $Ze$ the nuclear charge. Discovery of Neutron (Chadwick): Bombardment of Beryllium with $\alpha$-particles. 3. Electromagnetic Radiation (EMR) Wave nature: Wavelength ($\lambda$): distance between two consecutive crests/troughs. Frequency ($\nu$): number of waves passing a point per second. Speed of light ($c$): $c = \lambda \nu = 3 \times 10^8 \text{ m/s}$. Wave number ($\bar{\nu}$): $\bar{\nu} = \frac{1}{\lambda}$. Particle nature (Planck's Quantum Theory): Energy is quantized. $E = h\nu = \frac{hc}{\lambda}$. $h = 6.626 \times 10^{-34} \text{ J s}$ (Planck's constant). Photoelectric Effect: Ejection of electrons from a metal surface when light of sufficient frequency strikes it. $$ h\nu = h\nu_0 + KE_{max} $$ $$ KE_{max} = \frac{1}{2}mv^2_{max} $$ Where $\nu_0$ is threshold frequency, $h\nu_0$ is work function ($\Phi_0$). 4. Bohr's Model for H-like Atoms Applicable to single-electron species ($H, He^+, Li^{2+}$, etc.). Postulates: Electrons revolve in fixed circular orbits (stationary states). Angular momentum is quantized: $mvr = n\frac{h}{2\pi}$. Energy is absorbed/emitted only when electron jumps between orbits. Formulas for $n^{th}$ orbit (for H-like species with atomic number Z): Radius ($r_n$): $$ r_n = 0.529 \frac{n^2}{Z} \mathring{A} $$ ($r_1$ for H is $0.529 \mathring{A}$, Bohr radius). Velocity ($v_n$): $$ v_n = 2.18 \times 10^6 \frac{Z}{n} \text{ m/s} $$ Energy ($E_n$): $$ E_n = -13.6 \frac{Z^2}{n^2} \text{ eV/atom} $$ (Lowest energy for $n=1$). Ionization Energy (IE): Energy to remove electron from ground state ($n=1$ to $n=\infty$). $$ IE = +13.6 Z^2 \text{ eV/atom} $$ Transition Energy: $$ \Delta E = E_f - E_i = 13.6 Z^2 \left(\frac{1}{n_i^2} - \frac{1}{n_f^2}\right) \text{ eV} $$ Wavelength of emitted/absorbed photon: $$ \frac{1}{\lambda} = R_H Z^2 \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) $$ Where $R_H = 109677 \text{ cm}^{-1}$ (Rydberg constant). $n_1 5. Hydrogen Emission Spectrum Lyman Series: $n_f = 1$ ($n_i = 2, 3, ...$). UV region. Balmer Series: $n_f = 2$ ($n_i = 3, 4, ...$). Visible region. Paschen Series: $n_f = 3$ ($n_i = 4, 5, ...$). IR region. Brackett Series: $n_f = 4$ ($n_i = 5, 6, ...$). IR region. Pfund Series: $n_f = 5$ ($n_i = 6, 7, ...$). IR region. Number of spectral lines from $n_2$ to $n_1$: $\frac{(n_2-n_1)(n_2-n_1+1)}{2}$. 6. Dual Nature of Matter De Broglie Wavelength: $$ \lambda = \frac{h}{p} = \frac{h}{mv} $$ For an electron accelerated through potential $V$: $$ \lambda = \frac{12.27}{\sqrt{V}} \mathring{A} $$ Heisenberg's Uncertainty Principle: Cannot simultaneously determine both position and momentum of a particle with absolute certainty. $$ \Delta x \cdot \Delta p \ge \frac{h}{4\pi} \quad \text{or} \quad \Delta x \cdot m\Delta v \ge \frac{h}{4\pi} $$ $$ \Delta E \cdot \Delta t \ge \frac{h}{4\pi} $$ 7. Quantum Mechanical Model & Quantum Numbers Schrödinger Equation: Describes wave behavior of electrons. Solution gives wave functions ($\Psi$) and energies. Atomic Orbital: Region of space where probability of finding an electron is maximum ($|\Psi|^2$). Quantum Numbers: Principal ($n$): $1, 2, 3, ...$. Shell, energy, size. Azimuthal/Angular Momentum ($l$): $0, 1, ..., (n-1)$. Subshell, shape. $l=0 \rightarrow s$ (spherical), $l=1 \rightarrow p$ (dumbbell), $l=2 \rightarrow d$ (double dumbbell), $l=3 \rightarrow f$. Magnetic ($m_l$): $-l, ..., 0, ..., +l$. Orientation in space. Number of orbitals in a subshell $= (2l+1)$. Number of orbitals in a shell $= n^2$. Spin ($m_s$): $+1/2, -1/2$. Electron spin direction. Nodes: Regions where probability of finding electron is zero. Total nodes $= n-1$. Radial nodes $= n-l-1$. Angular nodes $= l$. 8. Electronic Configuration Rules Aufbau Principle: Fill orbitals in increasing order of $(n+l)$ value. If $(n+l)$ is same, fill orbital with lower $n$ first. $$ 1s Pauli Exclusion Principle: No two electrons can have all four identical quantum numbers. An orbital can hold max 2 electrons with opposite spins. Hund's Rule of Maximum Multiplicity: Pair electrons only after all degenerate (same energy) orbitals are singly occupied with parallel spins. Exceptions: Cr ($[Ar] 3d^5 4s^1$), Cu ($[Ar] 3d^{10} 4s^1$) due to stability of half-filled/fully-filled orbitals. 9. Stability of Orbitals Completely filled and half-filled subshells are more stable due to: Symmetry: More symmetrical distribution of electrons. Exchange Energy: Electrons with parallel spins in degenerate orbitals can exchange positions, leading to stabilization. Greater number of exchanges, greater stability.