Atomic Spectra
When an atom emits or absorbs light it does so only at sharply defined wavelengths — producing a line spectrum rather than a continuous one. Niels Bohr's 1913 model explained the hydrogen line spectrum as transitions between quantised orbits. The PMDC MDCAT 2026 syllabus places Atomic Spectra in the modern-physics block; expect 1-2 high-yield MCQs from this chapter.
Atomic Spectra and Line Spectrum
A spectrum is the range of EM wavelengths emitted or absorbed by a substance. Hot solids emit a continuous spectrum; rarefied gases excited by an electric discharge emit a line emission spectrum — a set of bright lines on a dark background. A cool gas placed between a continuous source and a spectrometer absorbs the same wavelengths, producing a line absorption spectrum (dark lines on bright background, e.g. Fraunhofer lines in sunlight).
Bohr's model of the hydrogen atom
Bohr combined Rutherford's nuclear atom with Planck's quantum hypothesis. His three key postulates are:
- Stationary states: Electrons revolve only in certain allowed circular orbits without radiating energy.
- Quantisation of angular momentum: mvr = nh/(2π), where n = 1, 2, 3, ... is the principal quantum number.
- Quantum jumps: Energy is emitted or absorbed only when an electron jumps between orbits, with photon energy hf = Ei − Ef.
The radius of the n-th orbit is rn = n2 a0, where a0 = 0.529 Å is the Bohr radius (the n = 1 orbit). The total energy of the electron in the n-th level is
En = −13.6 / n2 eV
Negative sign → the electron is bound. As n → ∞, E → 0 (free electron). The ground state (n = 1) lies at −13.6 eV.
Hydrogen line series and the Rydberg equation
Each transition n2 → n1 emits a photon whose wavenumber is given by the Rydberg formula:
1/λ = R (1/n12 − 1/n22)
with R = 1.097 × 107 m−1 (Rydberg constant) and n2 > n1.
| Series | Transitions n → | Region of EM spectrum | Notable lines |
|---|---|---|---|
| Lyman | any → 1 | Ultraviolet | Highest-energy series |
| Balmer | any → 2 | Visible | Hα 656 nm (red), Hβ 486 nm (cyan), Hγ 434 nm (blue) |
| Paschen | any → 3 | Near infrared | — |
| Brackett | any → 4 | Infrared | — |
| Pfund | any → 5 | Far infrared | Lowest-energy series |
| n | Energy (eV) | Orbit radius | Status |
|---|---|---|---|
| 1 | −13.6 | a0 = 0.529 Å | Ground state |
| 2 | −3.40 | 4 a0 | 1st excited |
| 3 | −1.51 | 9 a0 | 2nd excited |
| 4 | −0.85 | 16 a0 | 3rd excited |
| ∞ | 0 | ∞ | Ionised (free electron) |
Ionisation energy of hydrogen
The energy needed to remove the electron from the ground state (n = 1) to n = ∞ is
Eion = E∞ − E1 = 0 − (−13.6) = 13.6 eV.
This is the standard ionisation energy of atomic hydrogen and a fundamental constant used throughout modern physics MCQs.
Emission vs absorption spectra
- Emission: excited atom drops from a higher to a lower level, releasing a photon. Bright line on dark background.
- Absorption: a cool gas absorbs photons of specific wavelengths from a continuous source, lifting electrons to higher levels. Dark line on bright background.
- Both spectra share the same wavelengths for a given element — a chemical fingerprint.
X-ray production
X-rays are produced when high-speed electrons are decelerated on a metal target inside an evacuated tube. The output has two components:
- Continuous (Bremsstrahlung) spectrum: caused by sudden deceleration of electrons. Has a sharp minimum wavelength λmin = hc/(eV), where V is the accelerating potential.
- Characteristic spectrum: sharp peaks (Kα, Kβ, ...) appearing when an inner-shell vacancy is filled by an outer-shell electron. Wavelengths are characteristic of the target material.
Worked MCQs
Five MCQs that capture the high-yield testing patterns for this chapter. Read the explanation even when you get the answer right — it's where the deeper concept lives.
Q1. The ionisation energy of a hydrogen atom in its ground state is:
E1 = −13.6 eV; ionisation energy = E∞ − E1 = 13.6 eV. The other values correspond to higher levels: E2 = −3.4 eV, E3 = −1.51 eV.
Q2. Spectral lines of the Balmer series of hydrogen lie in which region of the spectrum?
All Balmer transitions terminate at n = 2 and lie in the visible (Hα 656 nm red, Hβ 486 nm). The Lyman series is UV; Paschen, Brackett and Pfund all lie in the IR region.
Q3. According to Bohr's model, the angular momentum of an electron in the n-th orbit is:
Bohr's quantisation condition: mvr = nh/(2π) = nℏ. This is the cornerstone postulate that produces the discrete orbits and energies.
Q4. The minimum wavelength of X-rays produced in a Coolidge tube depends on:
λmin = hc/(eV). Higher accelerating voltage gives more energetic electrons and shorter minimum wavelength. The target material affects the characteristic peaks, not λmin.
Q5. The radius of the n-th Bohr orbit in a hydrogen atom is proportional to:
rn = n2 a0 with a0 = 0.529 Å. Energy varies as 1/n2; orbital radius varies as n2. So n = 2 is four times bigger than n = 1, etc.
Quick Recap
- Bohr quantises angular momentum: mvr = nh/(2π).
- rn = n2 a0, a0 = 0.529 Å.
- En = −13.6/n2 eV; ionisation energy of H = 13.6 eV.
- Photon emitted in transition: hf = Ei − Ef.
- 1/λ = R(1/n12 − 1/n22), R = 1.097 × 107 m−1.
- Lyman (UV) → 1, Balmer (visible) → 2, Paschen/Brackett/Pfund (IR) → 3/4/5.
- X-rays: continuous (Bremsstrahlung, λmin = hc/eV) + characteristic (target-specific).