The big idea: In the Bohr model the electron in a hydrogen atom can only sit on a fixed set of energy levels — like rungs on a ladder, not a smooth ramp. Each level is numbered by a whole number n (the principal quantum number), n = 1, 2, 3, …
The energy of level n is quantized (it can only take these discrete values):
- energy of level n (eV)
- principal quantum number (1, 2, 3, …)
- the hydrogen constant (eV)
Two things to notice: (1) The energies are negative. The zero is set when the electron has just escaped the atom (n → ∞), so a bound electron has less energy than that — hence the minus sign.
(2) The levels bunch together as n rises: because of the 1/n², the gap from n = 1 to n = 2 is huge, but the gaps higher up shrink toward zero.
To find any level, just put its n into En = −13.6/n². The ground state (n = 1) is the lowest (most negative) — the most tightly bound. The numbers below are the ones you keep meeting in hydrogen problems.
| Level n | n² | En = −13.6/n² | Meaning |
|---|---|---|---|
| 1 | 1 | −13.6 eV | ground state (lowest, most bound) |
| 2 | 4 | −3.40 eV | first excited state |
| 3 | 9 | −1.51 eV | second excited state |
| 4 | 16 | −0.85 eV | third excited state |
| ∞ | ∞ | 0 eV | electron just free (ionised) |
Worked example — the n = 2 level
Use the Bohr formula to find the energy of the n = 2 level of hydrogen.
Solution
- Write the given formula first:
- Put in n = 2 (so n² = 4):
- Work it out — keep the unit and the sign:
Final answer
E₂ = −3.40 eV. (Negative because the electron is still bound to the atom.)
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How a photon is made: An electron can jump down from a higher level to a lower one. It loses energy, and that exact energy is carried off as a single photon of light. Because the levels are discrete, the photon energies are discrete too — that is why each gas emits a line spectrum (sharp coloured lines), not a continuous rainbow.
A jump up works in reverse: the atom absorbs a photon of exactly the right energy.
- Planck constant (J s)
- frequency of the photon (Hz)
- energy of the initial (higher) level (J or eV)
- energy of the final (lower) level (J or eV)
Worked example — the n = 3 → n = 2 photon
An electron in a hydrogen atom drops from n = 3 to n = 2. Find the energy of the emitted photon, in eV and in joules. (E₃ = −1.51 eV, E₂ = −3.40 eV; 1 eV = 1.60×10⁻¹⁹ J.)
Solution
- Photon energy = the drop in energy (given relation):
- Substitute the two levels (mind the signs):
- Add them — this is the photon energy in eV:
- Convert to joules (× 1.60×10⁻¹⁹):
Final answer
Photon energy = 1.89 eV = 3.0×10⁻¹⁹ J. (This is the red line of the hydrogen spectrum.)
Watch the signs: Use higher minus lower, Ei − Ef, so the photon energy comes out positive. Subtracting two negatives is where marks are lost: (−1.51) − (−3.40) = +1.89 eV, not −1.89 eV.
Pulling the electron right off: The ground state of hydrogen is n = 1 with E₁ = −13.6 eV. To ionise the atom you must lift the electron from this level all the way to E = 0 (n → ∞), where it is just free.
So the ionisation energy of hydrogen is the full 13.6 eV gap from the ground state up to zero.
- ground-state energy of hydrogen (eV)
- principal quantum number (here n = 1)
Worked example — ionising hydrogen
How much energy is needed to remove the electron from a hydrogen atom that starts in its ground state? Give the answer in eV.
Solution
- Energy needed = lift from the ground state (E₁) up to free (E = 0):
- Put in the ground-state energy E₁ = −13.6 eV:
- Work it out — keep the unit:
Final answer
Ionisation energy = 13.6 eV. (You add energy, so a positive amount.)
Emission (jump down)
- Electron falls to a lower level
- Atom gives out a photon
- Photon energy = Ei − Ef
- Makes the bright lines of a spectrum
Absorption / ionisation (jump up)
- Electron rises to a higher level (or escapes)
- Atom takes in a photon
- Needs exactly the right energy gap
- Full escape from n = 1 needs 13.6 eV
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Where it shows up: The Bohr model is HL only (E.1):
- Paper 1A — 'which transition gives the highest-energy photon?', 'what is the ionisation energy?', or read a level off En = −13.6/n². - Paper 2 — determine a photon's energy (eV → J), then often its frequency or wavelength, from a stated transition.
Three easy marks: (1) Photon energy = higher minus lower level, so it comes out positive. (2) Convert eV → J with × 1.60×10⁻¹⁹ before using hf or hc/λ. (3) The biggest energy drop (e.g. n = 2 → 1) gives the highest-frequency photon.
IB-style question — biggest jump, biggest photon
In hydrogen the levels are E₁ = −13.6 eV, E₂ = −3.40 eV, E₃ = −1.51 eV. An electron can fall by 3→2, 2→1, or 3→1. Determine which transition emits the highest-energy photon, and state that energy.
Solution
- Photon energy = the size of the drop (Ei − Ef). Compute each:
- And the other two drops:
- The largest drop is 3 → 1:
Final answer
The 3 → 1 transition emits the highest-energy photon, 12.1 eV (the largest energy gap).