Physics (9702) | Energy levels in atoms and line spectra

Energy Levels in Atoms and Line Spectra (9702 Quantum Physics)

Hello future physicist! This chapter is one of the coolest parts of modern physics because it explains *why* atoms don't just collapse and *how* we know what stars are made of! We are moving away from the simple classical model (like planetary orbits) and diving into the quantum world where energy comes in neat, fixed packets.

Don't worry if this feels different from classical mechanics; the key is understanding the rules of the quantum world: electrons live on specific "floors" and must jump, they can’t float!

1. Discrete Energy Levels in Isolated Atoms

In classical physics, an electron orbiting a nucleus should continuously lose energy and spiral into the nucleus. But thankfully, they don't! The stability of atoms is explained by the idea of **discrete energy levels**.

Analogy: The Energy Ladder

Imagine an electron is climbing a ladder.

• The ladder has fixed rungs (steps). An electron can stand on rung 1 or rung 2, but it cannot float in the space *between* them.
• In an atom, these rungs represent **discrete energy levels**.
• Discrete means the energy can only take specific, fixed values, not a continuous range. This is often called **quantization**.

For an isolated atom (like atomic hydrogen):

1. The lowest possible energy level (the bottom rung) is called the ground state. This is the most stable state, usually denoted \(E_0\) or \(E_1\).
2. Any level above the ground state is called an excited state. The atom is unstable in this state and will quickly return to the ground state.
3. If an electron gains enough energy to leave the atom completely, this is called **ionization**. The energy required to remove an electron from the ground state is the ionization energy.

The potential energy of the electron is defined as zero when it is completely free (at infinity). Because electrons are *bound* to the nucleus by attractive forces, their energy values in the discrete levels are always **negative**.

For example, an electron might have energy levels of -13.6 eV, -3.4 eV, -1.5 eV, etc. The -13.6 eV level is the ground state (most negative, therefore lowest energy).

2. Transitions and Photon Energy

Electrons don't stay in excited states for long. They move between these discrete levels by either absorbing or emitting photons.

The Role of the Photon

When an electron changes energy levels, the energy difference is exactly equal to the energy of a single light packet (a photon).

The energy of a photon, \(E\), is related to its frequency, \(f\), by Planck's equation (which you covered in the photoelectric effect):
\[E = hf\]

Where \(h\) is the Planck constant.

Calculating Energy Transitions

If an electron jumps from a higher energy level \(E_H\) to a lower energy level \(E_L\), the energy released must equal the energy of the photon emitted:

The energy difference \(\Delta E\) is:
\[\Delta E = E_H - E_L\]

Therefore, the frequency of the emitted photon is given by:

Formula for Energy Transition:
\[hf = E_H - E_L\]

Remember, since the energy levels are fixed, the energy differences (\(E_H - E_L\)) are also fixed. This means that only photons with specific, fixed frequencies (\(f\)) can be emitted or absorbed.

Units Check: The Electronvolt (eV)

Atomic energy changes are very small when measured in Joules (J). We typically use the electronvolt (eV):

Definition: The energy gained by an electron when it is accelerated through a potential difference of 1 Volt.
\[1 \text{ eV} = 1.60 \times 10^{-19} \text{ J}\]

Tip: If your energy levels are given in eV, you must convert the difference (\(E_H - E_L\)) into Joules before you can use \(hf\) to calculate frequency \(f\).

3. Line Spectra: Atomic Fingerprints

Because only discrete energy transitions are possible, atoms only interact with light of specific frequencies. This produces **line spectra**—distinct bright or dark lines—rather than a continuous rainbow.

3.1 Emission Line Spectra (Bright Lines)

An emission spectrum occurs when a gas is hot or subjected to an electric discharge (like in a neon sign).

Appearance: Bright colored lines on a dark background.

Formation Process (Excitation followed by De-excitation):

1. Excitation: Energy (heat or electricity) is supplied to the atom, causing electrons to jump from the ground state up to various excited states (\(E_L \to E_H\)).
2. Emission (De-excitation): The electrons are unstable in these high levels and quickly fall back down to lower energy levels, including the ground state (\(E_H \to E_L\)).
3. Every time an electron drops, it emits a photon with energy equal to the energy gap \((E_H - E_L)\).
4. Since only certain fixed energy gaps exist, only specific frequencies (\(f = \Delta E / h\)) are emitted, resulting in the bright, distinct lines.

Example: A sodium lamp produces two very characteristic yellow lines.

3.2 Absorption Line Spectra (Dark Lines)

An absorption spectrum occurs when light containing a continuous range of frequencies (a continuous spectrum) passes through a cooler, dilute gas.

Appearance: Dark lines appearing within a continuous spectrum (like a rainbow with black bars).

Formation Process (Absorption):

1. Continuous Spectrum: White light (which contains all frequencies/energies) is shone through a cold gas.
2. Selective Absorption: Electrons in the gas atoms can only jump *up* to a higher energy level if they absorb a photon with an energy that *exactly matches* one of the atom's specific energy gaps.
3. These specific photons are removed from the white light passing through.
4. The missing photons correspond to specific frequencies, creating the **dark lines** in the spectrum observed on the other side.

Crucial Connection:

The energy required for an electron to jump UP from \(E_L\) to \(E_H\) is exactly the same as the energy released when it jumps DOWN from \(E_H\) to \(E_L\).

Therefore, the **dark lines in an absorption spectrum occur at the exact same frequencies/wavelengths as the bright lines in the emission spectrum for that element.** This is the atomic fingerprint!