Atomic Energy Levels & Spectra

The Quantum Atom

Before the 20th century, physicists imagined the atom as a miniature solar system, with electrons orbiting the nucleus at any distance they liked. However, classical physics predicted that such orbiting electrons would constantly radiate energy and spiral into the nucleus in a fraction of a second.

In 1913, Niels Bohr revolutionised physics by proposing that electrons can only exist in discrete, quantised energy levels (or shells).

• An electron sitting in an allowed energy level does not emit radiation.
• An electron cannot exist between these levels.
• Energy is only absorbed or emitted when an electron makes an instantaneous "jump" (transition) between two allowed levels.

Learning Goals: By the end of this guide, you should be able to:

1. Understand the difference between the ground state and excited states.
2. Relate electron transitions to the emission and absorption of photons.
3. Calculate transition energies and equivalent photon wavelengths.
4. Explain how atomic spectra act as "fingerprints" for elements.

Energy Level Mathematics

Energy levels are typically assigned a principal quantum number, $n = 1, 2, 3 \dots$

• $n=1$ (Ground State): The lowest possible energy level. Electrons normally reside here.
• $n > 1$ (Excited States): Higher energy states. Electrons are unstable here and will promptly fall back down.
• $n = \infty$ (Ionisation Limit): The energy level where the electron is completely removed from the atom's influence.

By convention, the ionisation limit is defined as having $0\ \text{eV}$ of energy. Therefore, all bound energy levels within the atom have negative values. For example, the ground state of Hydrogen is $-13.6\ \text{eV}$.

Electron Transitions and Photons

To jump to a higher level (e.g., $n=1 \rightarrow n=3$), the electron must absorb a photon carrying the exact energy difference. To drop to a lower level (e.g., $n=3 \rightarrow n=2$), the electron must emit a photon carrying the exact energy difference.

The energy of the photon ($E_{\text{photon}}$) is given by:

Emission and Absorption Spectra

Because a specific element's atoms only possess a unique set of allowed energy levels, they can only ever absorb or emit photons of very specific, corresponding energies.

1. Emission Spectra

If you heat a gas or pass high voltage through it, electrons are excited to higher levels. When they inevitably fall back down to lower levels, they emit photons. When this light is passed through a prism, you don't see a continuous rainbow. Instead, you see a black background with a few bright, distinct coloured lines. Each line corresponds to a specific downward $\Delta E$ transition.

2. Absorption Spectra

If you shine continuous white light (a full rainbow) through a cold gas, the electrons absorb the specific photon energies needed to jump to higher levels. When this light is viewed through a prism, you see a full continuous rainbow, but with dark missing lines. These dark lines match exactly the bright lines of the emission spectrum for that element.

This is how astronomers know what stars are made of! By looking at the dark absorption lines in starlight, they can identify the exact "barcode" fingerprint of elements like Hydrogen or Helium in the star's atmosphere.

Worked Examples

Example 1: Calculating Photon Wavelength from a Transition

Question: In a hydrogen atom, an electron falls from the $n=3$ level ($-1.51\ \text{eV}$) to the $n=2$ level ($-3.40\ \text{eV}$). What is the wavelength of the emitted photon? ($hc = 1240\ \text{eV}\cdot\text{nm}$)

Step 1: Calculate the energy difference ($\Delta E$):

(The energy difference is positive, representing the $1.89\ \text{eV}$ photon created).

Step 2: Relate energy to wavelength:

This precisely matches the distinct red line visible in the Hydrogen Balmer series.

Example 2: Ionisation Energy

Question: An electron is in the $n=2$ excited state of Hydrogen ($-3.40\ \text{eV}$). What is the minimum energy photon required to ionise the atom from this state?

Answer: Ionisation means promoting the electron to $n=\infty$, where $E = 0\ \text{eV}$.

The photon must possess exactly $3.40\ \text{eV}$ or more of energy.

Common Mistakes

1. Forgetting energy levels are Negative — Students often write the ground state as $+13.6\ \text{eV}$. It is $-13.6\ \text{eV}$. When calculating $\Delta E$, doing $E_{\text{upper}} - E_{\text{lower}}$ translates to $(-1.51) - (-3.40)$, which gives a positive energy difference.
2. "Close enough" photons — If an electron requires exactly $10.2\ \text{eV}$ to jump from $n=1$ to $n=2$, and a $10.5\ \text{eV}$ photon hits it, the electron will not absorb it. The photon will pass straight through. Bound electrons cannot absorb "partial" photons or take the $10.2\ \text{eV}$ and "throw away" the $0.3\ \text{eV}$. It must be an exact match. (Note: If the photon energy exceeds the ionisation energy, it will be absorbed, and the excess energy becomes the kinetic energy of the freed electron, as seen in the photoelectric effect).
3. Calculating collisions with massive particles incorrectly — Unlike photons, an incoming electron carrying $10.5\ \text{eV}$ of kinetic energy in a collision can excite the atom by transferring exactly $10.2\ \text{eV}$ of its mechanical energy to the atomic electron, bouncing away with the remaining $0.3\ \text{eV}$.

Exam Tips (A-Level / AP / IB)

• Count the transitions! If an exam asks "How many different spectral lines can be emitted by a gas of atoms excited to the $n=4$ level?", don't just guess 3. Calculate all possible downward pathways: $4\rightarrow 3$, $4\rightarrow 2$, $4\rightarrow 1$, $3\rightarrow 2$, $3\rightarrow 1$, $2\rightarrow 1$. That makes a total of 6 possible lines. Shortcut formula: $\frac{n(n-1)}{2}$
• Always be ready to convert $\text{eV}$ back to Joules ($1\ \text{eV} = 1.6 \times 10^{-19}\ \text{J}$) if you need to use the standard SI definition of $h = 6.63 \times 10^{-34}\ \text{J}\cdot\text{s}$ to find frequency in Hz.
• The Lyman series (transitions to $n=1$) is all Ultraviolet. The Balmer series (transitions to $n=2$) is Visible. The Paschen series (transitions to $n=3$) is Infrared.

Frequently Asked Questions

Why do the energy gaps get smaller at higher $n$ levels?

Because the electric force holding the electron to the nucleus follows an inverse-square law ($F \propto 1/r^2$). As the electron orbits further out, the nucleous' grip weakens dramatically, so the energy difference between orbit $n=10$ and $n=11$ is minuscule compared to the massive gap between $n=1$ and $n=2$.

Do energy levels exist if no electron is in them?

Yes. Energy levels are allowed "parking spaces" for electrons, defined by the quantum mechanical wavefunctions of the atom's electric field. A parking space exists whether a car is currently parked there or not.

Related Topics

• Photoelectric Effect — Understand what happens when the photon energy exceeds the final ionisation limit.
• Wave-Particle Duality — Explore De Broglie's theory that standing electron waves are the actual physical cause of these discrete energy levels.