Lesson 12 of 12 · 14 min
Chapter review
Watch a class
The whole chapter on YouTube
Bohr model and hydrogen spectrum, full chapter
Arvind Academy · Hinglish · Whole chapter · Open on YouTube
Full chapter one-sitting revision
Munil Sir · Hinglish · Whole chapter · Open on YouTube
Must-know facts
18 facts
- 1Thomson (1898): plum pudding model, positive charge spread through the atom with electrons embedded.
- 2Geiger–Marsden (about 1911): 5.5 MeV alpha-particles from ²¹⁴₈₃Bi on gold foil 2.1 × 10⁻⁷ m thick; ZnS screen and microscope.
- 3About 0.14% scattered by more than 1°; about 1 in 8000 by more than 90°.
- 4Nucleus 10⁻¹⁵ to 10⁻¹⁴ m; atom about 10⁻¹⁰ m, 10,000 to 100,000 times larger.
- 5Small impact parameter → large scattering; head-on (b smallest) → rebound, θ ≈ π.
- 6Closest approach d = 2Ze²/(4πε₀K); 7.7 MeV alpha on gold: d = 3.0 × 10⁻¹⁴ m = 30 fm; gold radius about 6 fm.
- 7Gold: Z = 79, about 50 times heavier than an alpha-particle.
- 8Hydrogen orbit: r = e²/(4πε₀mv²); E = −e²/(8πε₀r); K = −E, U = 2E.
- 9Ground state: r = 5.3 × 10⁻¹¹ m, v = 2.2 × 10⁶ m/s, E = −13.6 eV.
- 10Classical revolution frequency in hydrogen's ground orbit: about 6.6 × 10¹⁵ Hz.
- 11Bohr: stationary orbits, L = nh/2π, hν = Ei − Ef.
- 12rn ∝ n², vn ∝ 1/n, En = −13.6/n² eV = −2.18 × 10⁻¹⁸/n² J.
- 13E₂ = −3.40 eV, E₃ = −1.51 eV; excitation 10.2 eV (1→2) and 12.09 eV (1→3).
- 14Ionisation energy of hydrogen: 13.6 eV; n = ∞ has E = 0; continuum above E = 0.
- 15Absorption lines fall at the same wavelengths as the gas's emission lines.
- 16de Broglie (1923): 2πrn = nλ gives mvr = nh/2π; Davisson–Germer confirmed electron waves in 1927.
- 17Bohr model works only for hydrogenic atoms (H, He⁺, Li²⁺) and cannot give line intensities.
- 18Bohr received the Nobel Prize in Physics in 1922.
Common traps
Where marks are lost
Saying most alpha-particles bounced back from the gold foil.
Taking the distance of closest approach as the radius of the nucleus.
Thinking a larger impact parameter gives a larger deflection.
Writing the electron's total energy as positive, or U = −E.
Assuming the energy gaps grow as n grows.
Saying the radius goes as n and the speed as n.
Equating the emitted frequency with the electron's frequency of revolution in the Bohr model.
Applying Bohr's formula to neutral helium.
Formulas
9 to know
Coulomb force on the alpha-particle
F = (1/4πε₀)(2e)(Ze)/r²
Z = 79 for gold.
Distance of closest approach
d = 2Ze²/(4πε₀K)
Head-on; K is the alpha-particle's kinetic energy.
Orbit radius and speed
r = e²/(4πε₀mv²)
From mv²/r = e²/(4πε₀r²).
Energy of the orbiting electron
E = −e²/(8πε₀r)
K = −E, U = 2E.
Bohr's quantisation
L = mvr = nh/2π
n = 1, 2, 3 …
Photon from a transition
hν = Ei − Ef
Absorption: Ei + hν = Ef.
Radius of the nth orbit
rn = (n²/m)(h/2π)²(4πε₀/e²)
rn = n²a₀, a₀ = 5.3 × 10⁻¹¹ m.
Energy of the nth level
En = −me⁴/(8n²ε₀²h²) = −13.6/n² eV
= −2.18 × 10⁻¹⁸/n² J.
Standing-wave condition
2πrn = nλ
With λ = h/mv this gives mvr = nh/2π.
Key terms
15 terms
- Plum pudding model
- Thomson's atom: positive charge spread through the whole atom, electrons embedded in it.
- Nuclear model
- Rutherford's atom: all the positive charge and most of the mass in a tiny central nucleus.
- Scintillation
- A brief flash of light when an alpha-particle strikes a zinc sulphide screen.
- Impact parameter
- Perpendicular distance of an alpha-particle's initial velocity line from the centre of the nucleus.
- Scattering angle
- The angle θ between the alpha-particle's initial and final directions.
- Distance of closest approach
- Separation at which a head-on alpha-particle momentarily stops before turning back.
- Emission line spectrum
- Bright lines at specific wavelengths on a dark background, from an excited rarefied gas.
- Absorption spectrum
- Dark lines in a continuous spectrum at the wavelengths a gas absorbs.
- Stationary state
- An allowed orbit in which the electron does not radiate and the atom has a definite energy.
- Principal quantum number
- The integer n labelling Bohr's orbits and energy levels.
- Bohr radius
- Radius of hydrogen's innermost orbit, a₀ = 5.3 × 10⁻¹¹ m.
- Ground state
- The lowest-energy state, n = 1; −13.6 eV for hydrogen.
- Excited state
- Any state with n > 1, reached by collisions or by absorbing a photon.
- Ionisation energy
- Least energy to free the electron from the ground state: 13.6 eV for hydrogen.
- Hydrogenic atom
- A nucleus of charge +Ze with one electron, such as H, He⁺ or Li²⁺.