Chemistry · Ch 2 — Quantum Mechanical Model of Atom

Summary

Atoms were once believed to be non-divisible, until the discovery of sub-atomic particles. J.J. Thomson proposed the atom as a positively charged sphere with electrons embedded in it, but this could not explain why atoms are stable.
Rutherford's alpha-scattering experiment introduced the nucleus - a tiny, positively charged core with negatively charged electrons revolving around it at high speed. Bohr modified this picture by introducing quantised, non-radiating stationary orbits.
Louis de Broglie proposed that all matter has dual (wave and particle) character, with de Broglie wavelength $\lambda=h/mv=h/m_ev$ - significant only for a microscopic particle such as an electron. Davisson and Germer proved the electron's wave nature experimentally through electron diffraction.
For a microscopic particle, position and momentum cannot both be measured simultaneously with full precision - Heisenberg's uncertainty principle, $\Delta x\cdot\Delta p\geq h/4\pi$ .
De Broglie's concept and Heisenberg's principle together led to the quantum mechanical model of the atom. Schrödinger's wave equation, $\hat{H}\psi=E\psi$ , is exactly solvable for one-electron systems (H, $\text{He}^+$ , etc.) but is far too complex to solve exactly for multi-electron systems. It is solvable only for certain energy values (eigenvalues), whose corresponding wavefunctions are called atomic orbitals. $\psi$ itself has no physical meaning, but $|\psi|^2$ gives the probability of finding the electron. This is the origin of the orbital: the three-dimensional region of space where the probability of finding the electron is maximum.
An electron in an orbital is described by four quantum numbers: principal ( $n$ , the shell/energy level), azimuthal ( $l$ , the subshell/shape), magnetic ( $m$ , the spatial orientation) and spin ( $s$ , the electron's intrinsic spin).
The general one-electron solution in spherical polar coordinates is $\Psi(r,\theta,\varphi)=R(r)\cdot f(\theta)\cdot g(\varphi)$ . Plotting $4\pi r^2\cdot R(r)^2$ against $r$ gives the radial distribution curve, with $(n-l-1)$ radial nodes; the angular distribution gives the orbital's boundary shape, with $l$ angular nodes. s orbitals are spherical; p orbitals are dumbbell-shaped; d orbitals are clover-leaf shaped. …

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