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Energy Levels and Hydrogen Spectrum

The hydrogen atom has discrete energy levels given by En = -13.6/n2 eV, and spectral lines are produced when electrons make transitions between these levels. Different final levels form different spectral series.

In one line: Series names are decided by where the electron lands.

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Understand it

  • The negative energy shows that the electron is bound to the nucleus.
  • The ground state has n = 1 and energy -13.6 eV; higher levels have less negative energy and are closer to ionisation.
  • When an electron falls from a higher level to a lower level, the atom emits a photon whose energy equals the difference between the two levels.
  • Transitions ending at n = 1 form the Lyman series, ending at n = 2 form the Balmer series, and ending at n = 3 form the Paschen series.

Example

The transition from n = 3 to n = 2 belongs to the Balmer series and lies in the visible region. The transition from n = 2 to n = 1 belongs to the Lyman series and has higher photon energy.

Formula and rule

Energy level formula: En = -13.6/n2 eV for hydrogen, where n = 1, 2, 3,... is the principal quantum number.

Photon energy for emission: ΔE = Ei - Ef = hν = hc/λ.

Here h is in J s, ν in Hz, c in m s−1, λ in m.

If energy is in eV, convert using 1 eV = 1.6 × 10−19 J when finding wavelength in SI units.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Find the photon energy for transition n = 4 to n = 2 in hydrogen.

  1. 1E4 = -13.6/16 = -0.85 eV.
  2. 2E2 = -13.6/4 = -3.40 eV.
  3. 3Emitted photon energy = E4 - E2 = (-0.85) - (-3.40) = 2.55 eV.
  4. 4Since the final level is n = 2, the line belongs to the Balmer series.

Method you can reuse

  1. 1Write En = -13.6/n2 eV.
  2. 2Calculate initial and final energies.
  3. 3For emission, subtract final energy from initial energy.
  4. 4Classify the series using the final n.
  5. 5Convert units only if frequency or wavelength is required.

Sketch it in your answer

Use an energy-level diagram with horizontal lines labelled n = 1, 2, 3, 4 and energies becoming closer together near zero. Downward arrows show emission. Label final levels for Lyman, Balmer, and Paschen series.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Why are the lines in the hydrogen spectrum discrete rather than continuous?

Think first, then show answer
The hydrogen spectrum has discrete lines because the electron can have only certain allowed energies. A photon is emitted or absorbed only for a transition between two allowed levels, so only specific energy differences and wavelengths occur.

Common wrong answer

A common wrong answer is classifying n = 4 to n = 2 as Paschen because it starts from 4. Series name depends on the final level, so it is Balmer.

Don't mix this up

Students often think a higher orbit means a more negative energy. Actually, as n increases, energy becomes less negative and approaches zero.

Exam tip

For spectral series, classify by the final level: Lyman ends at 1, Balmer ends at 2, Paschen ends at 3.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Exam questions commonly ask students to calculate energy difference, identify the spectral series, compare wavelengths or frequencies, or interpret an energy-level diagram.
  1. 1

    Enough for 1 mark

    The hydrogen atom has discrete energy levels given by En = -13.6/n2 eV, and spectral lines are produced when electrons make transitions between these levels. Different final levels form different spectral series.

  2. 2

    Add for 2 marks

    Example: The transition from n = 3 to n = 2 belongs to the Balmer series and lies in the visible region. The transition from n = 2 to n = 1 belongs to the Lyman series and has higher photon energy.

  3. 3

    Add for 3 marks

    The negative energy shows that the electron is bound to the nucleus.

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