PhysicsCBSE Class 12 · Atoms

Energy Levels and Hydrogen Spectrum Class 12: Transitions and Lines

Updated 15 October 2026 · Fact-checked

A hydrogen atom has discrete energy levels Eₙ = −13.6/n² eV. When its electron drops from a higher level to a lower one, it emits a photon of energy hν = Eᵢ − E_f, giving a bright line at one definite wavelength. Absorbing a photon of exactly such an energy lifts it up. These discrete jumps explain hydrogen's line spectrum.

Understand Energy Levels and Hydrogen Spectrum

Pass light from a glowing tube of hydrogen through a prism and you do not see a rainbow; you see a few sharp coloured lines on a dark background. Each line is the fingerprint of a jump between two energy levels. The 2026-27 syllabus treats hydrogen line spectra qualitatively, using Bohr's energy levels.

The energy level diagram. From Bohr's model, $E_{n} = - \frac{13.6}{n ^{2}}$ eV:

n = 1: −13.6 eV (ground state, the most stable)
n = 2: −3.4 eV
n = 3: −1.51 eV
n = 4: −0.85 eV
n = 5: −0.54 eV
n → ∞: 0 eV (the electron is free).

The levels get closer together as n increases. States above the ground state are excited states.

Excitation and ionisation. The excitation energy is the energy needed to lift the electron from the ground state to a higher level: 10.2 eV to n = 2, 12.09 eV to n = 3. The ionisation energy is the energy needed to free it from the ground state: 13.6 eV. The corresponding potentials (10.2 V, 13.6 V) are the voltages through which an electron must be accelerated to supply these energies in a collision.

Emission spectrum. Atoms excited by heating or an electric discharge stay excited only for about $1 0^{- 8}$ s and then drop to lower levels, emitting photons with $h ν = E_{i} - E_{f}$ , i.e. $λ = \frac{h c}{E _{i} - E _{f}}$ (λ in nm ≈ 1240/ΔE in eV). Only certain energy differences are possible, so only certain wavelengths appear: a line spectrum. Large jumps give short wavelengths; small jumps give long wavelengths. An electron in level n can return by several routes, so up to $\frac{n ( n - 1 )}{2}$ different lines can come from atoms excited to level n.

Groups of lines (qualitative). Jumps that end on n = 1 give ultraviolet lines (the Lyman series); jumps ending on n = 2 give the visible lines (the Balmer series), such as the red line from 3 → 2 (about 656 nm); jumps ending on n = 3 give infrared lines (the Paschen series). Only the names and regions are needed; the series formulae are not in the 2026-27 syllabus.

Absorption spectrum. When white light passes through cool hydrogen gas (atoms mostly in the ground state), the atoms absorb only photons whose energies exactly match jumps upward from n = 1. Those wavelengths are missing, so dark lines appear on a continuous background. A photon with the wrong energy (for example 11 eV) is not absorbed by a ground-state atom; a photon above 13.6 eV can ionise it, the extra energy becoming the kinetic energy of the freed electron.

Photons versus electrons as exciters. A photon must have exactly the right energy to be absorbed (or enough to ionise). An electron colliding with the atom can hand over just part of its kinetic energy, so an 11 eV electron can excite the atom to n = 2 and keep 0.8 eV.

Energy Levels and Hydrogen Spectrum: formulas and key facts

Energy levels of hydrogen: $E_{n} = - \frac{13.6}{n ^{2}} eV$; −13.6, −3.4, −1.51, −0.85, … eV.
Photon emitted or absorbed: $h ν = E_{i} - E_{f}, λ = \frac{h c}{E _{i} - E _{f}}$; λ(nm) ≈ 1240/ΔE(eV).
Excitation and ionisation energies (from n = 1): $E_{1 \to 2} = 10.2 eV, E_{1 \to 3} = 12.09 eV, E_{i o n} = 13.6 eV$; Ionisation from level n: 13.6/n² eV.
Number of possible lines from level n: $N = \frac{n ( n - 1 )}{2}$; For many atoms, all routes down.
Regions of the lines (qualitative): $\to n = 1 : UV; \to n = 2 : visible; \to n = 3 : IR$; Lyman, Balmer, Paschen groups by name only.

How to solve Energy Levels and Hydrogen Spectrum questions

Use this method for energy-level and transition questions.

1Write the energies of the levels involved from Eₙ = −13.6/n² eV.
2Find ΔE = Eᵢ − E_f (positive for emission from a higher to a lower level).
3Convert to wavelength: λ (nm) ≈ 1240/ΔE (eV); or to frequency ν = ΔE/h.
4Longest wavelength = smallest jump; shortest wavelength = largest jump.
5For excitation by electrons, find the highest level whose excitation energy is below the electron's energy, then count the possible downward jumps.
6For photons, check whether the energy exactly matches a jump (absorbed) or exceeds 13.6 eV from n = 1 (ionisation).
7Name the region: ending on n = 1 is UV, on n = 2 visible, on n = 3 infrared.

Energy Levels and Hydrogen Spectrum in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Colours of a hydrogen lamp

Suppose you look at a hydrogen discharge tube through a simple spectroscope and see a bright red line and a fainter blue-green line.

(a) Show that the jump 3 → 2 gives the red line. (b) Find the wavelength for the jump 4 → 2. (c) Why does the tube give separate lines and not a rainbow?

Try it first, then show the solutionSolution

3 → 2: ΔE = −1.51 − (−3.4) = 1.89 eV; λ ≈ 1240/1.89 ≈ 656 nm (red).
4 → 2: ΔE = −0.85 − (−3.4) = 2.55 eV; λ ≈ 1240/2.55 ≈ 486 nm (blue-green).
Electrons can jump only between fixed levels, so only certain photon energies, and hence certain wavelengths, are produced.

Answer: (a) 1.89 eV gives about 656 nm; (b) about 486 nm; (c) the energy levels are discrete.

Memory hook: Each line is one jump; the colour tells you which jump.

Absorption by hydrogen clouds

Imagine light from a distant hot star passing through a cool cloud of hydrogen gas on its way to a telescope above the Earth's atmosphere.

(a) Which absorption line does the jump 1 → 2 produce, and in which region of the spectrum is it? (b) Why would the cool cloud absorb mainly the lines that start from n = 1? (c) Why must such a telescope be above the atmosphere?

Try it first, then show the solutionSolution

ΔE = 10.2 eV, λ ≈ 1240/10.2 ≈ 122 nm: ultraviolet.
In a cool gas almost all atoms are in the ground state, so absorption starts from n = 1.
Ultraviolet of this wavelength is absorbed by the atmosphere (the ozone layer absorbs most solar UV), so it cannot be observed from the ground.

Answer: (a) about 122 nm, ultraviolet; (b) the atoms are mostly in the ground state; (c) the atmosphere absorbs this ultraviolet.

Memory hook: Cool gas steals exactly the colours its atoms can jump with.

Tricks to remember Energy Levels and Hydrogen Spectrum

Short memory aids. Say each one aloud once, then use it on a question the same day.

13.6, 3.4, 1.51, 0.85
Learn the first four levels; divide 13.6 by 1, 4, 9, 16.
Use it when: Quick energy differences.
Big jump, short wave
λ ∝ 1/ΔE: the largest jump gives the shortest wavelength.
Use it when: Longest and shortest line questions.
Lines = n(n − 1)/2
Handshake count of all pairs of levels from 1 to n.
Use it when: Number of spectral lines.
UV-Visible-IR = 1-2-3
Ending on n = 1 gives UV, on n = 2 visible, on n = 3 infrared.
Use it when: Naming the region of a line.

Energy Levels and Hydrogen Spectrum: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Wavelength of an emitted photon · 2 marks · Very short answerBoard level

Find the energy and wavelength of the photon emitted when the electron in a hydrogen atom jumps from n = 3 to n = 2. In which region of the spectrum is it?

Try it first, then show the solutionSolution

ΔE = E₃ − E₂ = −1.51 − (−3.4) = 1.89 eV. (1)
λ ≈ 1240/1.89 ≈ 656 nm: visible (red). (1)

Answer: 1.89 eV, about 656 nm, red visible light.

Example 2: Energy level diagram and line spectrum · 3 marks · Short answerBoard level

Draw the energy level diagram of hydrogen for n = 1 to 4 and ∞, marking energies. Use it to explain (a) why hydrogen gives a line spectrum and (b) the difference between emission and absorption spectra.

Try it first, then show the solutionSolution

Diagram: horizontal lines at −13.6 (n = 1), −3.4 (n = 2), −1.51 (n = 3), −0.85 (n = 4) and 0 eV (n = ∞), getting closer together upwards; downward arrows for transitions. (1)
(a) The electron can have only these energies, so emitted photons have only the energies of the gaps, hν = Eᵢ − E_f, giving sharp lines. (1)
(b) Emission: excited atoms drop down and give bright lines on a dark background. Absorption: atoms in low levels absorb photons of matching energy from white light, giving dark lines on a bright background. (1)

Answer: Discrete levels allow only definite photon energies; emission gives bright lines, absorption gives dark lines at matching wavelengths.

Example 3: Lines from the level n = 4 · 3 marks · Short answerHarder board level

Hydrogen atoms are excited to n = 4. (a) How many different spectral lines can they emit? (b) Find the longest and shortest of these wavelengths.

Try it first, then show the solutionSolution

(a) N = 4 × 3/2 = 6 lines. (1)
(b) Longest: smallest jump 4 → 3: ΔE = 1.51 − 0.85 = 0.66 eV; λ ≈ 1240/0.661 ≈ 1875 nm (infrared). (1)
Shortest: largest jump 4 → 1: ΔE = 13.6 − 0.85 = 12.75 eV; λ ≈ 97.3 nm (ultraviolet). (1)

Answer: (a) 6; (b) about 1875 nm (4 → 3) and 97.3 nm (4 → 1).

Example 4: Excitation by an electron beam · 3 marks · Short answerHarder board level

A beam of electrons of energy 12.5 eV passes through hydrogen gas in its ground state. (a) Up to which level can the atoms be excited? (b) Which lines will be emitted, and of what wavelengths?

Try it first, then show the solutionSolution

(a) Excitation energies: to n = 3 is 12.09 eV (possible), to n = 4 is 12.75 eV (not possible). So up to n = 3. (1)
(b) Possible jumps: 3 → 2 (1.89 eV, ≈ 656 nm), 3 → 1 (12.09 eV, ≈ 102.6 nm), 2 → 1 (10.2 eV, ≈ 121.6 nm). (2)

Answer: (a) n = 3; (b) three lines: about 656 nm, 102.6 nm and 121.6 nm.

Example 5: Which photons are absorbed? · 5 marks · Long answerHOTS / competency

Hydrogen atoms in the ground state are exposed in turn to (i) photons of 10.2 eV, (ii) photons of 11 eV, (iii) photons of 15 eV and (iv) electrons of 11 eV.
(a) In each case, say what happens to the atoms, with reasons. (4)
(b) Why does case (iv) differ from case (ii)? (1)

Try it first, then show the solutionSolution

(i) 10.2 eV exactly equals E₂ − E₁: the photon is absorbed and the atom goes to n = 2. (1)
(ii) 11 eV matches no gap from n = 1 (10.2 or 12.09 eV) and is below 13.6 eV: the photon is not absorbed; it passes through. (1)
(iii) 15 eV exceeds the ionisation energy: the atom is ionised and the electron leaves with 15 − 13.6 = 1.4 eV of kinetic energy. (1)
(iv) An 11 eV electron can give 10.2 eV to excite the atom to n = 2 and move on with 0.8 eV. (1)
(b) A photon is absorbed whole (all or nothing), but a colliding electron can transfer only part of its kinetic energy. (1)

Answer: (i) absorbed, to n = 2; (ii) not absorbed; (iii) ionised, electron with 1.4 eV; (iv) excited to n = 2, electron keeps 0.8 eV. Photons give all their energy; electrons can give part.

Example 6: Spectrum of a cool gas · 4 marks · Case-basedHarder board level

Read the passage and answer.

White light passes through a tube of cool hydrogen gas and then through a prism.
(a) What does the spectrum look like? (1)
(b) Find the wavelength of the absorption line due to the jump 1 → 3. (1)
(c) Will the visible lines (that end on n = 2 in emission) appear strongly as dark lines here? Why? (1)
(d) What is the longest wavelength that can ionise these atoms? (1)

Try it first, then show the solutionSolution

(a) A continuous spectrum crossed by dark lines at particular wavelengths. (1)
(b) ΔE = 12.09 eV: λ ≈ 1240/12.09 ≈ 102.6 nm (ultraviolet). (1)
(c) No, hardly: those lines need atoms starting in n = 2, but in a cool gas nearly all atoms are in n = 1. (1)
(d) λ_max ≈ 1240/13.6 ≈ 91 nm. (1)

Answer: (a) a continuous spectrum with dark lines; (b) about 102.6 nm; (c) no, atoms are in the ground state; (d) about 91 nm.

Where students lose marks in Energy Levels and Hydrogen Spectrum

The slips examiners see most often, and what to write instead.

Mistake: Using the Rydberg series formula in a qualitative question.
Older notes include it.
Do this: For 2026-27, work from energy levels: λ = hc/(Eᵢ − E_f).
Usually costs: Time; the energy method is expected.
Mistake: Taking ΔE as the difference of n values.
Confusing quantum numbers with energies.
Do this: Compute ΔE from −13.6/n² for each level.
Usually costs: Most of the numerical.
Mistake: Saying any photon with more than 10.2 eV is absorbed by a ground-state atom.
Thinking atoms take what they need.
Do this: Below 13.6 eV, only photons matching an exact gap are absorbed.
Usually costs: 1 mark.
Mistake: Mixing up longest and shortest wavelength jumps.
Forgetting λ ∝ 1/ΔE.
Do this: Smallest energy gap gives the longest wavelength.
Usually costs: 1 mark.

Exam tips

Energy levels and hydrogen spectra were set in 2023 (55/1/1 Q11 and Q30(a)), 2024 (55/1/1 Q9 and 55/5/1 Q20), 2025 (55/7/1 Q10) and 2026 (55/1/1 Q16).
Draw a neat energy level diagram with arrows for the jumps asked; it organises the whole answer.
Use λ(nm) ≈ 1240/ΔE(eV) for speed; state the region (UV, visible, IR).
For electron-beam excitation, compare the beam energy with 10.2, 12.09 and 12.75 eV.
Series formulae are not required for 2026-27; name the series only when asked.

Practice questions on Energy Levels and Hydrogen Spectrum

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
The minimum energy needed to excite a ground-state hydrogen atom is
(a) 3.4 eV (b) 10.2 eV (c) 12.09 eV (d) 13.6 eV
Hint
n = 1 to n = 2.
Answer
(b) 10.2 eV.
Q2 · 2 marks · Very short answerBoard level
Why does hydrogen give a line spectrum rather than a continuous one?
Hint
Discrete levels.
Answer
The electron can have only certain discrete energies. Photons are emitted only in jumps between these levels, with energies equal to the gaps, so only certain wavelengths appear as lines.
Q3 · 1 mark · Very short answerHarder board level
Find the wavelength of the photon emitted in the jump n = 2 → n = 1.
Hint
ΔE = 10.2 eV.
Answer
λ ≈ 1240/10.2 ≈ 121.6 nm (ultraviolet).
Q4 · 1 mark · Very short answerHarder board level
Find the longest wavelength of light that can ionise a hydrogen atom whose electron is in n = 2.
Hint
Ionisation energy from n = 2 is 3.4 eV.
Answer
λ ≈ 1240/3.4 ≈ 365 nm.
Q5 · 1 mark · Assertion-reasonHarder board level
Assertion (A): Hydrogen gas in its ground state absorbs light only at certain wavelengths.
Reason (R): A photon is absorbed only if its energy equals the difference between the ground state and a higher level (or exceeds the ionisation energy).
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Hint
All-or-nothing absorption.
Answer
(a). (Above 13.6 eV all energies can be absorbed, giving a continuous absorption beyond the ionisation limit.)
Q6 · 2 marks · Very short answerHOTS / competency
Hydrogen atoms are excited to n = 5. How many different lines can they emit, and which jump gives the longest wavelength?
Hint
n(n − 1)/2; smallest gap.
Answer
5 × 4/2 = 10 lines; the longest wavelength comes from the smallest gap, 5 → 4.
Q7 · 2 marks · Very short answerHOTS / competency
Among the visible lines of hydrogen (jumps ending on n = 2), which jump gives the longest wavelength and which the shortest? Explain.
Hint
Smallest and largest gaps to n = 2.
Answer
Longest: 3 → 2 (smallest gap, 1.89 eV, about 656 nm, red). Shortest: from very high levels (n → ∞) to n = 2 (gap approaching 3.4 eV, about 365 nm). Wavelength is inversely proportional to the energy gap.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Energies of the first three levels of hydrogen?Show answer
−13.6, −3.4 and −1.51 eV.
2.Ionisation energy of hydrogen?Show answer
13.6 eV.
3.Photon energy in a jump?Show answer
hν = Eᵢ − E_f.
4.Number of lines from level n?Show answer
n(n − 1)/2.
5.Region of lines ending on n = 1?Show answer
Ultraviolet.
6.Wavelength of the 3 → 2 line?Show answer
About 656 nm (red).

Energy Levels and Hydrogen Spectrum: frequently asked questions

What is the difference between emission and absorption spectra?

An emission spectrum comes from excited atoms dropping to lower levels: bright lines on a dark background. An absorption spectrum comes from atoms absorbing light of matching energies from a continuous source: dark lines on a bright background at the same wavelengths.

Why is the hydrogen spectrum in the syllabus only qualitative?

For 2026-27 CBSE lists hydrogen line spectra as qualitative only. Use energy levels and hν = Eᵢ − E_f; the series formulae (Lyman, Balmer, Paschen) are not to be assessed quantitatively.

Why do levels crowd together near zero energy?

Eₙ = −13.6/n² changes rapidly for small n and very slowly for large n, so the gaps between successive levels shrink as n increases.

Can a ground-state hydrogen atom absorb an 11 eV photon?

No. 11 eV does not match the gap to any level (10.2 or 12.09 eV) and is less than the 13.6 eV ionisation energy, so the photon passes through.

What is excitation energy?

The energy needed to raise an atom from its ground state to an excited state, for hydrogen 10.2 eV to n = 2 and 12.09 eV to n = 3.