21.9 Discrete Energy Levels and Atomic Spectra

Discrete Electron Energy Levels

Experiments show that electrons in an atom can only occupy specific, fixed energy states — they cannot have arbitrary energies. These are called discrete energy levels.

• Each energy level is labelled by a principal quantum number $n=1,2,3,\dots$
• The ground state ($n=1$) is the lowest energy state; the electron is most tightly bound.
• Higher levels ($n=2,3,\dots$) are called excited states.
• For hydrogen, the energy of level $n$ is:

$E_n=-\frac{13.6}{n^2}\text{ eV}$

The negative sign indicates the electron is bound to the nucleus.

Excitation and Ionisation

• Excitation: An atom absorbs energy and an electron jumps to a higher level. The minimum energy needed to move from the ground state to a specific higher level is supplied by the excitation potential (the accelerating p.d. required).
• Ionisation: The electron is completely removed from the atom. For hydrogen from the ground state, the ionisation energy is $13.6\text{ eV}$.

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The Photon Energy Equation: $hf=\Delta E$

When an electron transitions between energy levels, a photon is emitted or absorbed whose energy exactly equals the energy difference:

$hf=E_{\text{higher}}-E_{\text{lower}}=\Delta E$

where $h=6.63\times 10^{-34}\text{ J s}$ is Planck's constant and $f$ is the photon frequency.

Equivalently, using wavelength:

$\frac{hc}{\lambda }=\Delta E$

Worked Example

An electron in hydrogen falls from $n=3$ to $n=2$.

$\Delta E=E_3-E_2=(-1.51)-(-3.4)=1.89\text{ eV}=1.89\times 1.6\times 10^{-19}\text{ J}$

$f=\frac{\Delta E}{h}=\frac{3.02\times 10^{-19}}{6.63\times 10^{-34}}\approx 4.56\times 10^{14}\text{ Hz}$

This lies in the visible (red) region — part of the Balmer series.

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Emission and Absorption Line Spectra

Emission Spectrum

• A hot, low-pressure gas is excited (by heat or electrical discharge).
• Electrons jump to higher levels, then de-excite, emitting photons of specific frequencies.
• Observed as bright coloured lines on a dark background.
• Each element has a unique set of lines — an atomic fingerprint.

Absorption Spectrum

• White (continuous) light passes through a cool gas.
• Atoms absorb photons whose energies match allowed transitions, exciting electrons to higher levels.
• Observed as dark lines on a continuous coloured background.
• The dark lines appear at exactly the same frequencies as the emission lines of the same element.

Why Does Hydrogen Produce Many Lines?

Although each hydrogen atom has only one electron, a sample contains vast numbers of atoms. When excited, different electrons transition between different pairs of energy levels, producing photons of many distinct frequencies.

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Spectral Series of Hydrogen

Transitions ending at the same lower level form a series:

The Balmer series is the only one partially in the visible range, producing the characteristic red, blue-green, and violet lines of hydrogen.