Atomic Energy Levels and Line Spectra

Overview

Atomic spectra provide strong evidence that electrons in atoms occupy discrete energy levels.

Electrons do not possess arbitrary energies inside atoms. Instead, only certain allowed values exist.

When electrons move between these levels:

• energy is absorbed or emitted
• photons are involved
• characteristic spectral lines are produced

This is a key part of Atomic Structure and Quantum Physics.

Definition

Atomic energy levels are the allowed electron energies in an atom. Line spectra are the discrete wavelengths emitted or absorbed when electrons transition between these levels.

Why It Matters

This topic explains:

• why atoms do not emit continuous spectra
• why each element has a characteristic spectral fingerprint
• how photon energy is tied to level differences
• why ionisation corresponds to a specific energy threshold

Key Representations

Big Picture

Key ideas:

• electrons occupy discrete energies
• energies are quantised, not continuous
• transitions between levels involve photons
• line spectra arise from these transitions

If energy levels were continuous, atoms would produce continuous spectra instead of sharp lines.

Atomic Energy Levels

Using hydrogen as the standard example:

• lowest level = ground state
• higher levels = excited states
• highest limit = ionisation level

Important features:

• bound states have negative energy
• zero energy corresponds to a free electron far from the nucleus
• levels become closer together as $n$ increases

Meaning of Negative Energy

Negative energy means the electron is bound to the nucleus.

Energy must be supplied to remove it completely.

Example:

• $E=-13.6\text{eV}$ means the electron needs $13.6\text{eV}$ to reach $E=0$

Hydrogen Energy Formula

For hydrogen:

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

where:

• $n=1,2,3,\dots$
• $n=1$ is ground state
• $n\to \infty$ gives the ionisation limit

First few levels:

$n$	Energy (eV)
1	-13.6
2	-3.40
3	-1.51
4	-0.85

Electron Transitions

Upward Transition = Excitation

An electron gains energy and moves to a higher level.

Downward Transition = De-Excitation

An electron loses energy and moves to a lower level.

Energy released usually appears as a photon.

Photon Energy from Transitions

If the energy gap is $\Delta E$:

$$hf=\Delta E$$

Also:

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

Therefore:

• larger gap gives higher frequency
• larger gap gives shorter wavelength
• smaller gap gives longer wavelength

Excitation Methods

1. Photon Absorption

An atom absorbs a photon only if the photon energy exactly matches an allowed energy gap.

$$hf=\Delta E$$

If not matched, that transition does not occur.

2. Collision Excitation

Fast particles such as electrons, ions, or atoms collide with atoms and transfer energy.

This can excite electrons to higher levels.

Emission Line Spectrum

Produced by hot low-density gas.

Appearance:

• bright discrete lines
• dark background

Cause:

Excited electrons fall to lower levels and emit photons of specific energies.

Each line corresponds to one wavelength.

Absorption Line Spectrum

Produced when white light passes through cool low-density gas.

Appearance:

• continuous spectrum with dark lines at certain wavelengths

Cause:

Atoms absorb photons matching allowed upward transitions.

Re-emission occurs in all directions, so less light continues forward at those wavelengths.

Spectral Fingerprints

Each element has its own unique set of energy levels.

Therefore each element has its own unique line spectrum.

Uses:

• identifying gases
• astronomy and stellar composition
• discharge tubes
• laboratory spectroscopy

Hydrogen Series

Hydrogen spectral lines are grouped by final level.

Lyman Series

• transitions ending at $n=1$
• ultraviolet region

Balmer Series

• transitions ending at $n=2$
• visible region

Paschen Series

• transitions ending at $n=3$
• infrared region

Higher series exist for higher final levels.

Ionisation Energy

Ionisation energy is the minimum energy needed to remove the electron completely.

Final state:

$$E=0$$

Hydrogen ground state:

$$13.6\text{eV}$$

From excited states, less energy is required because the electron is already higher.

Typical Calculations

1. Wavelength from Transition

Use:

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

Then solve for $\lambda$.

2. Frequency from Photon

Use:

$$f=\frac{\Delta E}{h}$$

3. Excitation Energy

Find the difference between two levels:

$$\Delta E=E_{\text{final}}-E_{\text{initial}}$$

4. Possible Transitions

If an electron starts at level $n$, possible downward transitions are to any lower level.

5. Number of Spectral Lines

If an electron can fall from level $n$ to any lower level:

$$\text{Number of lines}=\frac{n(n-1)}{2}$$

for all possible downward transitions among levels up to $n$.

Summary

Atomic line spectra arise because electrons occupy quantised energy levels.

Key ideas:

• only certain electron energies are allowed
• transitions emit or absorb photons
• photon energy equals level difference
• each element has a unique spectrum
• ionisation corresponds to $E=0$

Core formulas:

$$E_n=-\frac{13.6}{n^2}\text{ eV}$$ $$hf=\Delta E$$ $$E=hf=\frac{hc}{\lambda }$$

Links

• Atomic Structure
• Atomic Structure Common Exam Traps
• Quantum Physics
• Wave-Particle Duality
• Uncertainty Principle