Counting Spectral Lines
If atoms populate every level from 1 through N and all downward transitions occur, the maximum number of distinct lines is N choose 2. A single atom follows one cascade, whereas a sample can realize all allowed paths.
Why this shows up in the exam
Counting emission lines after excitation · Separating sample spectra from one-atom cascades · Inferring the highest populated level
Learn the idea
A set of N populated levels can produce at most N(N-1)/2 distinct downward transitions. Draw the accessible levels as floors. Every pair of floors can supply one energy gap, but the preparation process decides which floors are actually occupied.
🧠 Memory hook: Count level pairs, but first ask which levels are populated.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- N_lines,max = N(N-1)/2 — all pairwise transitions among N levels
- N_to_lower = n_i - 1 — direct downward choices from one specified level
How to approach it
- 1Find the highest accessible level
- 2Decide whether all intermediate populations occur in the sample
- 3Count distinct level pairs and remove forbidden or absent paths if stated
Common slip-ups that cost marks
- •Using N(N+1)/2
- •Assuming one atom emits every possible line in one cascade
- •Counting inaccessible levels
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
In hydrogen, an electron transitions from n = 2 to n = 1. Using E_n = -13.6/n^2 eV, find the emitted photon energy.
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