Difficulty: Introductory | Prerequisites: basic algebra, scientific notation, unit conversions.
This topic sits at the start of atomic theory in general chemistry. It bridges classical physics (wave behaviour of light) and quantum mechanics (quantised energy levels, electron behaviour). You need it to make sense of electron configurations, periodic trends, and bonding later in the course. If you have not yet reviewed electromagnetic radiation basics or the relationship between wavelength and frequency, start there.
TL;DR
Light behaves as both a wave and a particle. Its energy is quantised and calculated from wavelength or frequency using Planck's constant. The Bohr model uses quantised energy levels to explain hydrogen's emission spectrum, and the energy difference between levels determines the wavelength of light emitted or absorbed.
Wavelength (λ)
The distance between two consecutive peaks (or troughs) of a wave, measured in metres (often nanometres for visible light). Think of it as the length of one full wave cycle.
Frequency (ν)
The number of wave cycles that pass a fixed point per second, measured in hertz (Hz, which is s⁻¹). In simple terms, this is how fast the wave oscillates.
Speed of light (c)
The speed at which electromagnetic radiation travels in a vacuum: 3.00 × 10⁸ m/s. Wavelength and frequency are inversely related through this constant.
Photon
A discrete packet (quantum) of light energy. Think of it as the smallest possible "piece" of light.
Planck's constant (h)
A fundamental constant (6.626 × 10⁻³⁴ J·s) that relates the energy of a photon to its frequency. It is the bridge between the wave picture and the particle picture of light.
Quantised energy levels
The idea that electrons in atoms can only occupy specific, fixed energy values, not anything in between. Think of it as a staircase rather than a ramp: the electron can stand on step 1 or step 2, but not halfway between.
Bohr model
A model of the hydrogen atom in which the electron orbits the nucleus in fixed circular paths ("orbits"), each with a defined energy. It correctly predicts hydrogen's emission spectrum but fails for multi-electron atoms.
Emission spectrum
The set of specific wavelengths of light given off when excited electrons drop to lower energy levels. Each element has a unique emission spectrum, sometimes called its "fingerprint."
Energy level (n)
The principal quantum number labelling each allowed orbit in the Bohr model, where n = 1 is closest to the nucleus and lowest in energy. Higher n means farther from the nucleus and higher (less negative) energy.
Ionisation energy (for hydrogen)
The energy required to completely remove the electron from the atom (n = 1 to n = ∞). For a single hydrogen atom this is 2.18 × 10⁻¹⁸ J.
Wave description of light
The classical equation is c = λν, where c is the speed of light, λ is wavelength, and ν is frequency.
Because c is constant, wavelength and frequency are inversely proportional: when wavelength increases, frequency decreases (and vice versa).
Longer wavelength = lower frequency. Shorter wavelength = higher frequency.
Particle (photon) description of light
Energy of a single photon is given by E = hν, or equivalently E = hc/λ.
Because E and λ are inversely related, longer wavelengths correspond to lower energy photons and shorter wavelengths to higher energy photons.
The unit of energy is the joule (J), where 1 J = 1 kg·m²/s².
Worked example: energy of a 656 nm photon
Convert wavelength to metres: 656 nm = 656 × 10⁻⁹ m = 6.56 × 10⁻⁷ m.
Apply E = hc/λ: E = (6.626 × 10⁻³⁴ J·s)(3.00 × 10⁸ m/s) / (6.56 × 10⁻⁷ m) = 3.03 × 10⁻¹⁹ J.
This is the energy of one photon. For one mole of photons, multiply by Avogadro's number: (3.03 × 10⁻¹⁹ J)(6.022 × 10²³ mol⁻¹) = 1.82 × 10⁵ J/mol (about 182 kJ/mol).
656 nm falls in the red region of the visible spectrum, which is consistent with it being relatively low energy visible light.
Energy of each orbit
For a hydrogen-like atom (one electron), the energy of the nth level is: Eₙ = −2.18 × 10⁻¹⁸ J × (z²/n²), where z is the nuclear charge (z = 1 for hydrogen) and n is the principal quantum number.
All allowed energies are negative, reflecting that the electron is bound to the nucleus. n = 1 is the ground state (most negative, lowest energy). As n increases, energy becomes less negative (higher).
Energy level values for hydrogen (z = 1)
n | Energy (J) |
|---|---|
1 | −2.18 × 10⁻¹⁸ |
2 | −5.45 × 10⁻¹⁹ |
3 | −2.42 × 10⁻¹⁹ |
4 | −1.36 × 10⁻¹⁹ |
5 | −8.72 × 10⁻²⁰ |
6 | −6.06 × 10⁻²⁰ |
∞ | 0 |
Key observations about these levels
The levels are not evenly spaced. The gap between n = 1 and n = 2 is much larger than the gap between n = 5 and n = 6. The levels converge (get closer together) as n increases.
The electron never stops between levels. It can only occupy the defined energy states.
Energy differences and photon emission
When an electron transitions between levels, the energy difference is: ΔE = E_final − E_initial = −2.18 × 10⁻¹⁸ J × (z²/n_f² − z²/n_i²).
If the electron drops from a higher level to a lower level, ΔE is negative (energy is released as a photon). The absolute value of ΔE equals the photon's energy.
The wavelength of the emitted photon is found from |ΔE| = hc/λ, so λ = hc/|ΔE|.
Worked example: n = 4 to n = 2 transition in hydrogen
ΔE = −2.18 × 10⁻¹⁸ J × (1/4 − 1/16) = −2.18 × 10⁻¹⁸ J × (3/16) = −4.09 × 10⁻¹⁹ J.
|ΔE| = 4.09 × 10⁻¹⁹ J.
λ = hc/|ΔE| = (6.626 × 10⁻³⁴)(3.00 × 10⁸) / (4.09 × 10⁻¹⁹) = 4.86 × 10⁻⁷ m = 486 nm.
486 nm is blue-green (cyan) visible light. This is one of the lines in hydrogen's Balmer series (transitions ending at n = 2).
Ionisation of hydrogen
Ionisation means removing the electron entirely: n = 1 → n = ∞.
E_∞ = 0, so ΔE = 0 − (−2.18 × 10⁻¹⁸ J) = +2.18 × 10⁻¹⁸ J per atom.
Per mole: (2.18 × 10⁻¹⁸ J)(6.022 × 10²³ mol⁻¹) = 1.31 × 10⁶ J/mol = 1310 kJ/mol.
Formula | What it does | Key variables |
|---|---|---|
c = λν | Relates wavelength and frequency of light | c = 3.00 × 10⁸ m/s |
E = hν | Energy of one photon from its frequency | h = 6.626 × 10⁻³⁴ J·s |
E = hc/λ | Energy of one photon from its wavelength | Combines both constants |
Eₙ = −2.18 × 10⁻¹⁸ J (z²/n²) | Energy of the nth Bohr orbit | z = nuclear charge, n = level |
ΔE = Ef − Ei | Energy change in a transition | Positive = absorbed, negative = emitted |
Unit reminders: always convert nm to m before plugging into formulas (1 nm = 10⁻⁹ m). Energy per mole = single-photon energy × Avogadro's number (6.022 × 10²³ mol⁻¹).
Emission spectra are how astronomers determine the chemical composition of distant stars: each element produces a unique pattern of spectral lines. Closer to home, neon signs and sodium street lamps produce their characteristic colours because electrons in those atoms emit photons at specific wavelengths when they drop between energy levels.
Students often think higher wavelength means higher energy. It does not. Wavelength and photon energy are inversely related.
Students sometimes assume that energy levels in the Bohr model are evenly spaced (like rungs on a ladder with equal gaps). They are not: the spacing decreases as n increases.
Students confuse frequency (ν, nu) with speed. Frequency is cycles per second; the speed of light (c) is constant for all wavelengths in a vacuum.
Students often forget the sign convention for ΔE. A negative ΔE means the atom released energy (emission). A positive ΔE means the atom absorbed energy (absorption). The sign tells you the direction of the transition.
⚠️ You will almost certainly be asked to calculate photon energy from a wavelength (E = hc/λ) or the reverse. Practise the unit conversion from nm to m until it is automatic.
⚠️ Expect a question on calculating ΔE for a specific Bohr transition (e.g., n = 3 → n = 1) and then converting that to a wavelength.
⚠️ Know what happens qualitatively when wavelength increases: frequency decreases, energy decreases.
⚠️ Be ready to convert single-photon energy to per-mole energy using Avogadro's number, and to identify whether a wavelength falls in the visible, UV, or IR range.
⚠️ The ionisation energy of hydrogen (2.18 × 10⁻¹⁸ J per atom, or 1310 kJ/mol) is a commonly tested value.
True or false: A photon with a wavelength of 400 nm has more energy than a photon with a wavelength of 700 nm. (True.)
Fill in the blank: When wavelength increases, frequency ______ . (Decreases.)
True or false: The energy levels in the Bohr model are evenly spaced. (False; they converge as n increases.)
Fill in the blank: The energy of the n = ∞ level in the Bohr model is ______ . (Zero.)
True or false: A negative ΔE for an electron transition means the atom absorbed a photon. (False; negative ΔE means emission.)
Q: A photon has a wavelength of 434 nm. Calculate its energy in joules.
A: Convert 434 nm to metres: 4.34 × 10⁻⁷ m. Then E = hc/λ = (6.626 × 10⁻³⁴)(3.00 × 10⁸) / (4.34 × 10⁻⁷) = 4.58 × 10⁻¹⁹ J.
Q: An electron in a hydrogen atom drops from n = 5 to n = 2. Calculate the energy of the emitted photon and its wavelength.
A: ΔE = −2.18 × 10⁻¹⁸ (1/4 − 1/25) = −2.18 × 10⁻¹⁸ (0.21) = −4.58 × 10⁻¹⁹ J. The photon energy is 4.58 × 10⁻¹⁹ J. λ = hc/E = (6.626 × 10⁻³⁴)(3.00 × 10⁸) / (4.58 × 10⁻¹⁹) = 4.34 × 10⁻⁷ m = 434 nm (violet).
Q: Why does the Bohr model work for hydrogen but not for helium?
A: The Bohr model accounts only for the interaction between a single electron and the nucleus. In helium (and all multi-electron atoms), electron-electron repulsion adds complexity that the model cannot handle.
Q: How much energy is needed to ionise one mole of hydrogen atoms from the ground state?
A: Per atom: 2.18 × 10⁻¹⁸ J. Per mole: (2.18 × 10⁻¹⁸)(6.022 × 10²³) = 1.31 × 10⁶ J/mol = 1310 kJ/mol.
Q: If the frequency of a photon is 6.00 × 10¹⁴ Hz, what is its wavelength and in what region of the electromagnetic spectrum does it fall?
A: λ = c/ν = (3.00 × 10⁸) / (6.00 × 10¹⁴) = 5.00 × 10⁻⁷ m = 500 nm. This is green visible light.
This material connects directly to electron configurations and periodic trends: once you understand that electrons occupy quantised energy levels, the aufbau principle and orbital filling order follow naturally. It also connects to bonding (CHEM 1015 later chapters), because the energy an atom needs to lose or gain electrons (ionisation energy, electron affinity) depends on these same energy-level ideas. The wave-particle duality theme continues in the next section with the de Broglie wavelength and the quantum mechanical model.
Wave-particle duality, electromagnetic radiation, electromagnetic spectrum, visible light spectrum, photon energy, Planck's constant, speed of light, wavelength-frequency relationship, c = lambda nu, E = hv, E = hc/lambda, Bohr model, Bohr atom, hydrogen emission spectrum, Balmer series, energy levels, principal quantum number, quantised orbits, line spectrum, atomic emission, ionisation energy of hydrogen, ground state, excited state, continuous spectrum vs line spectrum, CHEM 1015, general chemistry atomic structure.