Developments Leading to Bohr’s Model: Quanta, Photoelectric Effect & Spectra
Bohr’s model did not appear from nowhere — it was built on three experimental earthquakes: Planck’s quanta, Einstein’s photoelectric effect, and the fingerprint lines of atomic spectra. Walk the evidence, learn the equations, and run the threshold experiment yourself.
What Led to Bohr’s Model? — Complete Theory
By 1900, light was confidently a wave. Maxwell had described electromagnetic radiation as coupled oscillating electric and magnetic fields travelling through vacuum at c = 3 × 10⁸ m/s — no medium needed. Every wave is described by the same four dials, and every numerical in this section runs on them:
| Parameter | Symbol · unit | Meaning | Working relation |
|---|---|---|---|
| Wavelength | λ · m, nm, Å | Peak-to-peak distance | 1 Å = 10⁻¹⁰ m · 1 nm = 10⁻⁹ m |
| Frequency | ν · Hz (s⁻¹) | Cycles per second | ν = c/λ |
| Wavenumber | ν̄ · cm⁻¹ | Cycles per cm — spectroscopist’s favourite | ν̄ = 1/λ (λ in cm) |
| Amplitude | a · — | Height of the wave | intensity ∝ a² (NOT energy per photon) |
Across the spectrum, energy climbs as wavelength shrinks: radio → microwave → IR → visible → UV → X → γ. The visible window is a sliver from ~400 nm (violet) to ~750 nm (red) — remember the direction: violet light packs more energy than red.
Earthquake 1 — the black body problem. Heat any object and it glows; the emitted colours shift with temperature in a way classical wave theory could not reproduce — its equations predicted absurd infinite ultraviolet output (the “ultraviolet catastrophe”). In 1900 Max Planck fixed it with a heretical assumption: matter emits or absorbs energy only in discrete packets — quanta — never continuously:
E = hν
Total energyE = nhν, n = 1, 2, 3…
Photon energy shortcutE(eV) = 1240/λ(nm)
with h = 6.626 × 10⁻³⁴ J s — the quantum of action, and the constant that powers the rest of this chapter.
Earthquake 2 — the photoelectric effect. Shine light on a metal; electrons pop out. The classical wave picture made three confident predictions — all wrong:
| Question | Classical wave theory | Experiment says |
|---|---|---|
| Emission at low frequency? | Yes, if intensity is high enough | No — below threshold ν₀, nothing, ever |
| KE of electrons vs intensity? | Increases with intensity | Independent of intensity; depends on frequency |
| Number of electrons vs intensity? | — | Proportional to intensity |
| Time lag for faint light? | Predicted delay | Instantaneous (< 10⁻⁹ s) |
In 1905 Einstein explained everything by promoting Planck’s quanta to real particles of light — photons, each carrying hν. One photon interacts with one electron; either it carries enough energy to free the electron or the electron never sees it:
hν = hν₀ + ½mv²
Work functionφ = hν₀ = 1240/λ₀ (eV)
Max KEKEmax = hν − φ
Here φ (work function) is the metal’s personal escape fee — the minimum photon energy that frees the laziest electron, and ν₀ (threshold frequency) = φ/h its frequency version. Below ν₀: no emission at any intensity. Above it: every extra joule goes to the electron’s motion.
Earthquake 3 — atomic spectra. Pass an element’s emission through a prism: not a rainbow, but discrete coloured lines — an emission spectrum. Pass white light through cool vapour: the same wavelengths go missing — dark lines on a continuum, the absorption spectrum, at exactly the emission wavelengths. Every element carries a unique line fingerprint (this is how helium was discovered in the Sun before it was found on Earth). For hydrogen, the visible lines fit a stunningly simple formula — the Balmer series:
ν̄ = RH(1/2² − 1/n²)
Lines656, 486, 434, 410 nm (n = 3,4,5,6)
Why discrete?← Bohr’s job (§ 2.4)
Discrete lines meant discrete energy changes meant discrete energy levels inside the atom. Rutherford’s model had no mechanism for that. Bohr read Planck + Einstein + Balmer together and quantized the orbit itself — that is the story of § 2.4. And radiation’s dual nature, once proven, planted the question § 2.5 answers: if waves can be particles, can particles be waves?
Stopping potential. The kinetic energy of photoelectrons is measured by applying a reverse (retarding) potential between the metal and the collector. The smallest reverse potential that stops even the fastest electrons is the stopping potential V₀; at that point the maximum kinetic energy is used up against the field: eV₀ = KEmax = hν − φ. So V₀ rises linearly with frequency — a graph of V₀ against ν is a straight line of slope h/e, meeting the ν-axis at the threshold frequency ν₀ — and it does not change with intensity. Handy form: V₀ in volts equals KEmax in eV. For sodium lit by 500 nm light, KEmax = 0.12 eV, so V₀ = 0.12 V.
Visualising the Spectrum & Running the Photoelectric Experiment
Ek ladder, ek experiment — the EM ladder first, then a metal plate you can fire any wavelength at, at any intensity you like.
Photoelectric Lab
Intensity slider ka khel: jab emission ho rahi hai, dots badhte-ghatate hain lekin har electron ka KE same rehta hai. Threshold se neeche intensity 100% kar ke dekho — kuch nahi hota. Wahi NTA trap, live.
Solved Examples (Step-by-Step)
Given → formula → substitute → verdict. Jo chain yahan chalti hai, wahi lab me live chalti hai.
Energy of 600 nm light, two ways
Calculate the energy of one photon of yellow-orange light of wavelength 600 nm, in joules and in eV. (h = 6.626 × 10⁻³⁴ J s, c = 3 × 10⁸ m/s)
- Formula
E = hc/λ - Substitute
E = (6.626×10⁻³⁴ × 3×10⁸) ÷ (600×10⁻⁹) - Result (J)
E = 3.31 × 10⁻¹⁹ J - Shortcut check
E = 1240/600 = 2.07 eVand 2.07 × 1.602×10⁻¹⁹ ≈ 3.31×10⁻¹⁹ J ✓
3.31 × 10⁻¹⁹ J = 2.07 eV per photon
Sodium meets 500 nm light
Sodium has a work function of 2.36 eV. Light of wavelength 500 nm falls on it. Find (a) the threshold wavelength, (b) the maximum KE of ejected electrons, (c) what happens if intensity is doubled.
- Threshold
λ₀ = 1240/φ = 1240/2.36 = 525 nm - Compare500 nm < 525 nm → photon energy exceeds φ → emission occurs.
- KE
KEmax = 1240/500 − 2.36 = 2.48 − 2.36 = 0.12 eV= 1.92 × 10⁻²⁰ J - Double intensityTwice as many electrons, same 0.12 eV each — KE reads frequency, not intensity.
λ₀ = 525 nm · KE = 0.12 eV · intensity doubles count only
Frequency and wavenumber of 480 nm light
Light of wavelength 480 nm (blue) — find its frequency (ν) and wavenumber (ν̄) in cm⁻¹.
- Frequency
ν = c/λ = 3×10⁸ ÷ 480×10⁻⁹ = 6.25 × 10¹⁴ Hz - Wavenumber
ν̄ = 1/λ = 1/(480×10⁻⁷ cm) = 2.083 × 10⁴ cm⁻¹ - Energy
E = hν = 6.626×10⁻³⁴ × 6.25×10¹⁴ = 4.14 × 10⁻¹⁹ J= 2.58 eV ✓ (1240/480) - Unit careλ in cm for ν̄ (spectroscopy convention), in m for ν — mixing the two is the classic slip.
ν = 6.25 × 10¹⁴ Hz · ν̄ = 2.08 × 10⁴ cm⁻¹
Key Formulas & Takeaways
Eight lines that solve this topic
Constants: h = 6.626 × 10⁻³⁴ J s · c = 3 × 10⁸ m/s · 1 eV = 1.602 × 10⁻¹⁹ J · photon shortcut 1240 nm·eV · Work functions: Cs 1.90 · K 2.26 · Na 2.36 · Zn 4.31 eV
- Three earthquakes built Bohr’s model — Planck’s quanta (black body), Einstein’s photons (photoelectric), and Balmer’s discrete lines (spectra).
- Intensity counts, frequency energises — below threshold nothing works at any intensity; above it, KE = hν − φ regardless of brightness.
- 1240/λ is the working constant — wavelength in, electron-volts out; every photoelectric numerical begins here.
- Line spectra are energy-level evidence — discrete lines forced discrete levels, the exact gap Bohr’s quantized orbits filled in § 2.4.
FAQs
What is the photoelectric effect and what is Einstein’s equation for it?
The photoelectric effect is the ejection of electrons from a metal surface when light of sufficiently high frequency falls on it. Einstein’s equation is hν = hν₀ + ½mv², where hν is the photon energy, hν₀ is the work function (minimum energy to remove the electron), and ½mv² is the maximum kinetic energy of the ejected electron.
Why did classical wave theory fail to explain the photoelectric effect?
Wave theory predicted that electrons should be ejected at any frequency if the intensity were high enough, that kinetic energy should grow with intensity, and that faint light would need a delay. Experiments showed the opposite: no emission below a threshold frequency regardless of intensity, kinetic energy depending on frequency alone, and ejection without any time lag. Only the photon picture of light explains this.
What is Planck’s quantum hypothesis?
Planck proposed that atoms and molecules can emit or absorb energy only in discrete quantities, not continuously. The smallest packet of energy is a quantum, E = hν, where h is Planck's constant (6.626 × 10⁻³⁴ J s) and ν is the frequency; total energy is an integral multiple, E = nhν. This single idea rescued the black body radiation curve and founded quantum theory.
What is the difference between emission and absorption spectra?
An emission spectrum shows bright coloured lines on a dark background — radiation emitted by excited atoms as electrons fall to lower levels. An absorption spectrum shows dark lines on a continuous background — the wavelengths absorbed by ground-state atoms as light passes through. The dark lines of an element's absorption spectrum fall at exactly the same wavelengths as its bright emission lines, making spectra a fingerprint for identifying elements.
Practice Questions (With Solutions)
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QCC Notes — Class 11 Chemistry
Strictly NCERT-aligned notes for JEE Main & NEET, prepared by QCC Notes (Padho Likho JEE). Content follows the latest NCERT edition and current NTA exam pattern.