Energy levels and photon emission

Welcome to Energy Levels and Photon Emission!

In the previous chapters, we looked at how light acts like a particle (the photoelectric effect) and how electrons can hit atoms to give them energy (excitation and ionisation). Now, we are going to look at what happens when those "excited" atoms settle back down. This chapter explains why certain gases glow with specific colors and how atoms reveal their inner structure through light.

1. Discrete Energy Levels

Inside an atom, electrons can't just be anywhere. They exist in specific, fixed orbits called energy levels. Think of these levels like the rungs on a ladder: you can stand on the first rung or the second rung, but you cannot hover in the space between them. Because these levels have specific, fixed values, we say they are discrete.

Key points to remember:

1. The ground state is the lowest energy level an electron can occupy (the "bottom rung").
2. Excited states are higher energy levels.
3. Energy levels are usually given in electron volts (\(eV\)) or Joules (\(J\)).

Quick Tip: When looking at energy level diagrams, the values are often negative. This is because they represent how much energy the electron is "missing" compared to being completely free from the atom. The ground state is the most negative value.

2. The Process of Photon Emission

When an electron has been moved to a higher energy level (through excitation), it is unstable. To become stable again, it must "fall" back down to a lower energy level. This process is called de-excitation.

When the electron falls, it must lose the extra energy it was carrying. It does this by spitting out a single photon of electromagnetic radiation. The energy of this photon is exactly equal to the difference between the two energy levels.

The Fundamental Equation

The energy of the emitted photon is calculated using this formula:

\(hf = E_1 - E_2\)

Where:
- \(h\) is Planck’s constant (\(6.63 \times 10^{-34} J s\)).
- \(f\) is the frequency of the emitted photon in Hertz (\(Hz\)).
- \(E_1\) is the energy of the higher level.
- \(E_2\) is the energy of the lower level.

Since we know from the photon model that \(c = f\lambda\), we can also write this as:
\(\frac{hc}{\lambda} = E_1 - E_2\)

This shows that a bigger "drop" in energy levels produces a photon with a higher frequency (and a shorter wavelength).

Key Takeaway:

One electron transition = One photon emitted. The "gap" between levels determines the "color" of the light.

3. Evidence: Line Spectra

If you pass the light from a glowing gas (like neon or hydrogen) through a diffraction grating, you don't see a rainbow. Instead, you see a series of sharp, bright lines of specific colors. This is called a line emission spectrum.

Why is this important?
Line spectra are the ultimate proof that energy levels are discrete. If electrons could exist at any energy, we would see a continuous smear of colors. Because we only see specific lines, we know that electrons can only make specific jumps between specific energy levels.

Did you know? Each element has its own unique set of energy levels, meaning every element has a unique "barcode" of spectral lines. This is how astronomers can tell what distant stars are made of without ever visiting them!

4. Working with Units: \(J\) and \(eV\)

In your exam, energy levels might be quoted in Joules (\(J\)) or electron volts (\(eV\)). You must be able to convert between them because Planck's constant (\(h\)) is almost always used with Joules.

The Conversion Factor:
\(1 eV = 1.60 \times 10^{-19} J\)

To convert \(eV\) to \(J\): Multiply by \(1.60 \times 10^{-19}\).
To convert \(J\) to \(eV\): Divide by \(1.60 \times 10^{-19}\).

Common Mistake: Forgetting to convert \(eV\) to Joules before using the formula \(hf = E_1 - E_2\). Always check your units before you start calculating!

Step-by-Step: Calculating Photon Frequency

If an electron falls from an energy level of \(-1.5 eV\) to a level of \(-3.4 eV\), what is the frequency of the photon emitted?

Step 1: Find the energy difference in \(eV\).
\(\Delta E = (-1.5) - (-3.4) = 1.9 eV\)

Step 2: Convert the energy difference to Joules.
\(E = 1.9 \times 1.60 \times 10^{-19} = 3.04 \times 10^{-19} J\)

Step 3: Use the equation \(E = hf\) to find frequency.
\(f = \frac{E}{h} = \frac{3.04 \times 10^{-19}}{6.63 \times 10^{-34}}\)
\(f \approx 4.59 \times 10^{14} Hz\)

Summary Review

- Discrete Levels: Electrons live on fixed "shelves" of energy.
- De-excitation: Electrons falling to lower levels emit photons.
- Energy Equation: The photon energy \(hf\) equals the difference between levels (\(E_1 - E_2\)).
- Evidence: Line spectra prove that these energy levels are discrete, not continuous.
- Units: Always convert \(eV\) to Joules before calculating frequency or wavelength.