Structure of Atom — Notes

Electron (Thomson, cathode rays; $e/m = 1.76 \times 10^{11}\ \text{C kg}^{-1}$; charge by Millikan), proton (canal rays) and neutron (Chadwick, 1932). Atomic number $Z$ = protons; mass number $A$ = protons + neutrons. Isotopes: same $Z$, different $A$. Isobars: same $A$, different $Z$.

Thomson: positive sphere with embedded electrons. Rutherford ($\alpha$-scattering on gold foil): most particles undeflected (empty space), few deflected (concentrated positive charge), very few rebound (small dense nucleus, about $10^{-15}$ m versus atom $10^{-10}$ m). Limitations: a circling electron should spiral inward; line spectra unexplained.

$c = \nu\lambda$ with $c = 3.0 \times 10^{8}\ \text{m s}^{-1}$; wavenumber $\bar{\nu} = 1/\lambda$. Planck: $E = h\nu = hc/\lambda$, $h = 6.626 \times 10^{-34}\ \text{J s}$.

Photoelectric effect: no emission below the threshold frequency $\nu_0$, whatever the intensity; above it, $h\nu = h\nu_0 + \text{KE}$. Intensity changes the number of electrons, frequency changes their kinetic energy. This supports the photon (particle) picture.

Rydberg: $\bar{\nu} = 1.097 \times 10^{7}\ \text{m}^{-1}\,(1/n_1^2 - 1/n_2^2)$. Series: Lyman ($n_1 = 1$, UV), Balmer (2, visible), Paschen (3, IR), Brackett (4), Pfund (5).

Bohr: stationary orbits, $m_\text{e} v r = nh/2\pi$; $E_n = -2.18 \times 10^{-18}\,Z^2/n^2$ J; $r_n = 52.9\,n^2/Z$ pm; emission or absorption of $h\nu = E_2 - E_1$. Works for H, He⁺, Li²⁺; fails for many-electron atoms, line splitting in fields, and ignores wave nature.

de Broglie: $\lambda = h/mv$; significant for electrons and other microscopic particles, negligible for a cricket ball ($\sim 10^{-34}$ m). Heisenberg: $\Delta x\,\Delta p \ge h/4\pi$; for an electron confined to about an atom the uncertainty in speed is of the order $10^{6}\ \text{m s}^{-1}$, so definite orbits cannot exist.

Orbital: region of high probability $\psi^2$ of finding the electron; each holds at most two electrons.

n = 1, 2, 3 … (shell); l = 0 to $n-1$ (s, p, d, f); mₗ = $-l$ to $+l$ ($2l + 1$ orbitals); mₛ = ±½. A shell holds $n^2$ orbitals and $2n^2$ electrons.

Shapes: s spherical; p dumbbell (3 orbitals); d four-lobed or lobes with ring (5 orbitals). Nodes: radial $= n - l - 1$; angular $= l$; total $= n - 1$.

Aufbau: fill in order of increasing $n + l$ (lower $n$ first on a tie): 1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s … Pauli: no two electrons share all four quantum numbers (max two per orbital, opposite spins). Hund: fill degenerate orbitals singly with parallel spins first.

Exceptions: Cr $[\text{Ar}]\,3d^5 4s^1$ and Cu $[\text{Ar}]\,3d^{10} 4s^1$ (half-filled and filled subshells are extra stable: symmetry and exchange energy). Cations lose the outermost (highest $n$) electrons first: Fe³⁺ is $[\text{Ar}]\,3d^5$.