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Energy Levels and Hydrogen Spectrum

The hydrogen atom has discrete energy levels given by En = -13.6/n^2 eV, and spectral lines are produced when electrons make transitions between these levels. Different final levels form different spectral series.

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Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

    The hydrogen atom has discrete energy levels given by En = -13.6/n^2 eV, and spectral lines are produced when electrons make transitions between these levels. Different final levels form different spectral series.

  2. 2

    Then show how to use it

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

  3. 3

    Add one concrete 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.

  4. 4

    Avoid this incomplete 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.

Definition

The hydrogen atom has discrete energy levels given by En = -13.6/n^2 eV, and spectral lines are produced when electrons make transitions between these levels. Different final levels form different spectral 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.

Rule to remember

Energy level formula: En = -13.6/n^2 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 x 10^-19 J when finding wavelength in SI units.

Memory hook

Series names are decided by where the electron lands.

Examples and method

Worked example

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

Method to apply

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

Diagram support

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.

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.

Avoid common mistakes

Common confusion

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

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.

Exam tip

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

Quick check

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

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.

Answer writing and exam use

1-mark answer

The hydrogen atom has discrete energy levels given by En = -13.6/n^2 eV, and spectral lines are produced when electrons make transitions between these levels. Different final levels form different spectral series.

2-mark answer

The hydrogen atom has discrete energy levels given by En = -13.6/n^2 eV, and spectral lines are produced when electrons make transitions between these levels. Different final levels form different spectral series. Energy level formula: En = -13.6/n^2 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 x 10^-19 J when finding wavelength in SI units. 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-mark answer

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. Energy level formula: En = -13.6/n^2 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 x 10^-19 J when finding wavelength in SI units. Find the photon energy for transition n = 4 to n = 2 in hydrogen. E4 = -13.6/16 = -0.85 eV. E2 = -13.6/4 = -3.40 eV. Emitted photon energy = E4 - E2 = (-0.85) - (-3.40) = 2.55 eV. Since the final level is n = 2, the line belongs to the Balmer series. Exam questions commonly ask students to calculate energy difference, identify the spectral series, compare wavelengths or frequencies, or interpret an energy-level diagram. 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.
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