Lesson 12 of 12 · 17 min
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Must-know facts
19 facts
- 01e/mₑ = 1.758820 × 10¹¹ C kg⁻¹; e = 1.602176 × 10⁻¹⁹ C; mₑ = 9.1094 × 10⁻³¹ kg.
- 02Masses in u: electron 0.00054, proton 1.00727, neutron 1.00867; the neutron is slightly heavier than the proton.
- 03Chadwick (1932) found the neutron by hitting beryllium with α-particles.
- 04In the gold-foil experiment about 1 in 20,000 α-particles bounced back; nucleus about 10⁻¹⁵ m, atom about 10⁻¹⁰ m.
- 05Neutrons = A − Z for atoms and ions alike; isotopes share Z, isobars share A.
- 06Hydrogen isotopes: protium 99.985%, deuterium 0.015%, tritium in traces.
- 07c = νλ with c = 3.0 × 10⁸ m s⁻¹; visible light spans 400-750 nm, or 7.5 × 10¹⁴ to 4.0 × 10¹⁴ Hz.
- 08E = hν, h = 6.626 × 10⁻³⁴ J s; one mole of photons at 5 × 10¹⁴ Hz carries about 199.5 kJ.
- 09hν = W₀ + ½mₑv²; kinetic energy depends on frequency, number of electrons on brightness.
- 10Balmer series (n₁ = 2) is the only visible hydrogen series; Lyman (n₁ = 1) is ultraviolet; Paschen, Brackett, Pfund are infrared.
- 11Rydberg constant for hydrogen: 109,677 cm⁻¹; in energy form 2.18 × 10⁻¹⁸ J.
- 12Bohr: rₙ = 52.9 n²/Z pm and Eₙ = −2.18 × 10⁻¹⁸ Z²/n² J; He⁺ n = 1 has E = −8.72 × 10⁻¹⁸ J and r = 26.45 pm.
- 13The n = 5 to n = 2 hydrogen line has ν ≈ 6.91 × 10¹⁴ Hz (about 434 nm).
- 14λ = h/mv; Δx·Δp ≥ h/4π.
- 15Shell n has n subshells, n² orbitals and at most 2n² electrons; a subshell has 2l + 1 orbitals.
- 16Radial nodes = n − l − 1, angular nodes = l, total nodes = n − 1.
- 17(n + l) rule: lower sum first; tie goes to lower n (so 4s before 3d, 4f before 5d).
- 18Cr 3d⁵ 4s¹ and Cu 3d¹⁰ 4s¹ are the exceptions to learn.
- 19Hund's rule: pairing begins with the 4th p, 6th d and 8th f electron.
Common traps
Where marks are lost
Formulas
13 to know
Mass number
A = Z + (number of neutrons)
Z = protons = electrons in a neutral atom.
Wave relation
c = νλ; ν̄ = 1/λ
c = 3.0 × 10⁸ m s⁻¹; ν̄ usually in cm⁻¹.
Planck's quantum
E = hν = hc/λ
h = 6.626 × 10⁻³⁴ J s; multiply by Nₐ = 6.022 × 10²³ mol⁻¹ for a mole of photons.
Photoelectric equation
hν = hν₀ + ½mₑv²
hν₀ = W₀, the work function; no emission if ν < ν₀.
Rydberg formula (hydrogen)
ν̄ = 109,677 (1/n₁² − 1/n₂²) cm⁻¹
n₂ > n₁; n₁ = 1, 2, 3, 4, 5 gives Lyman, Balmer, Paschen, Brackett, Pfund.
Bohr angular momentum
mₑvr = nh/2π
n = 1, 2, 3 …
Bohr radius
rₙ = 52.9 n²/Z pm
Z = 1 for hydrogen; one-electron species only.
Bohr energy
Eₙ = −2.18 × 10⁻¹⁸ (Z²/n²) J
E = 0 at n = ∞; ionisation energy of H from n = 1 is 2.18 × 10⁻¹⁸ J per atom.
Transition energy and frequency
ΔE = 2.18 × 10⁻¹⁸ (1/nᵢ² − 1/n_f²) J; ν = 3.29 × 10¹⁵ (1/nᵢ² − 1/n_f²) Hz
Negative ΔE is emission, positive is absorption.
de Broglie wavelength
λ = h/mv = h/p
Detectable only for very small masses.
Uncertainty principle
Δx · Δp ≥ h/4π; Δx · Δv ≥ h/4πm
Negligible for everyday masses.
Node count
radial = n − l − 1; angular = l; total = n − 1
l = 0, 1, 2, 3 for s, p, d, f.
Capacity
orbitals per subshell = 2l + 1; orbitals per shell = n²; electrons per shell = 2n²
Two electrons of opposite spin per orbital.
Key terms
15 terms
- Cathode rays
- The stream of electrons that flows from cathode to anode in a low-pressure discharge tube.
- Canal rays
- Positive gaseous ions formed in a discharge tube, whose mass depends on the gas used.
- Isotopes
- Atoms of one element with the same atomic number but different mass numbers.
- Isobars
- Atoms of different elements that share the same mass number.
- Quantum
- The smallest packet of energy that can be emitted or absorbed as radiation, equal to hν.
- Work function
- The least energy a photon must bring to free an electron from a given metal surface.
- Threshold frequency
- The lowest light frequency that can eject electrons from a given metal.
- Line spectrum
- A spectrum of separate bright (or dark) lines at wavelengths characteristic of an element.
- Stationary state
- In Bohr's model, an allowed orbit in which the electron keeps a fixed energy.
- Atomic orbital
- A one-electron wave function ψ in an atom, labelled by n, l and mₗ.
- Probability density
- |ψ|², the probability per unit volume of finding the electron at a point.
- Node
- A surface on which the probability density of an orbital is zero.
- Degenerate orbitals
- Orbitals that have exactly the same energy.
- Effective nuclear charge
- The net positive pull felt by an electron after shielding by the other electrons.
- Exchange energy
- The stabilisation gained when electrons of the same spin in degenerate orbitals can swap places.