Chemistry · Ch 2 — Structure of Atom

Summary

Cathode rays are streams of electrons ( $e^-$ ), with charge-to-mass ratio $\frac{e}{m} = 1.76 \times 10^{11} \text{ C kg}^{-1}$ ; anode rays (positive ions) depend on the gas in the tube.
Rutherford’s $\alpha$ -particle scattering experiment showed that the atom has a tiny, dense, positively charged nucleus (radius $\approx 10^{-15}$ m) surrounded by mostly empty space (radius $\approx 10^{-10}$ m).
Atomic number $Z$ = number of protons; mass number $A$ = protons + neutrons. Isotopes have same $Z$ , different $A$ ; isobars have same $A$ , different $Z$ .
Electromagnetic radiation is described by wavelength $\lambda$ , frequency $\nu$ , and wave number $\bar{\nu} = 1/\lambda$ . The speed of light $c = \nu \lambda = 3 \times 10^8 \text{ m s}^{-1}$ .
Planck’s quantum theory: Energy is emitted/absorbed in discrete quanta: $E = h\nu$ , where $h = 6.626 \times 10^{-34} \text{ J s}$ .
Bohr’s model for hydrogen-like atoms: electrons revolve in fixed circular orbits with quantised angular momentum $mvr = \frac{nh}{2\pi}$ . Energy of level $n$ : $E_n = -13.6 \frac{Z^2}{n^2} \text{ eV}$ . Transition energy: $\Delta E = h\nu = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)$ , where $R_H = 1.097 \times 10^7 \text{ m}^{-1}$ .
Limitations of Bohr’s model: fails for multi-electron atoms, cannot explain fine structure, Zeeman effect, or the uncertainty principle.
Dual nature of matter (de Broglie): every moving particle has a wavelength $\lambda = \frac{h}{mv}$ . Heisenberg’s uncertainty principle: $\Delta x \cdot \Delta p \ge \frac{h}{4\pi}$ — position and momentum cannot both be known precisely.
Quantum mechanical model: atom described by a wavefunction $\psi$ $ψ$ , whose square $|\psi|^2$ $∣ ψ ∣^{2}$ gives the probability density of finding an electron. Orbitals are defined by three quantum numbers:
- Principal quantum number $n$ (shell, $n = 1,2,3,\dots$ )
- Azimuthal quantum number $l$ (subshell, $l = 0,1,\dots,n-1$ ; $s,p,d,f$ )
- Magnetic quantum number $m_l$ (orbital orientation, $-l$ to $+l$ )
- Spin quantum number $m_s$ ( $+\frac12$ or $-\frac12$ )
Shapes of orbitals: $s$ is spherical; $p$ are dumbbell-shaped ( $p_x, p_y, p_z$ ); $d$ are cloverleaf (four lobes) with one $d_{z^2}$ having a torus. …

Start your 14-day free trial to unlock the full solution →

← Previous · 2.6.6aCauses of Stability of Completely Filled and Half-filled Subshells Next · exercises →Exercises