Result for the 8th energy level
How many sublevels would you expect in the 8th energy level? Eight sublevels, because the allowed azimuthal quantum numbers are \(l=0,1,2,3,4,5,6,7\) when \(n=8\).
Quantum-number rule for sublevels
The principal quantum number \(n\) labels the principal energy level (shell). Sublevels (subshells) inside a shell are determined by the azimuthal quantum number \(l\), which must satisfy \[ 0 \le l \le n-1. \]
The count of allowed \(l\)-values is therefore \(n\), so the number of sublevels in the \(n\)-th energy level equals \(n\). For \(n=8\), the allowed set has 8 values and produces 8 sublevels.
Names of the 8 sublevels
The spectroscopic letter sequence begins s, p, d, f and continues alphabetically after f (with j typically skipped in this convention). For \(n=8\), the sublevels are:
- 8s (\(l=0\))
- 8p (\(l=1\))
- 8d (\(l=2\))
- 8f (\(l=3\))
- 8g (\(l=4\))
- 8h (\(l=5\))
- 8i (\(l=6\))
- 8k (\(l=7\))
Orbital counts as a consistency check
Each sublevel with quantum number \(l\) contains \(2l+1\) orbitals, so the \(n=8\) shell contains \[ \sum_{l=0}^{7}(2l+1)=1+3+5+7+9+11+13+15=64 \] orbitals in total, consistent with the shell rule \(n^2=64\). The maximum electron capacity is \(2n^2=128\).
Reference table
| Sublevel | \(l\) | Orbitals (\(2l+1\)) | Max electrons \(2(2l+1)\) |
|---|---|---|---|
| 8s | \(0\) | \(1\) | \(2\) |
| 8p | \(1\) | \(3\) | \(6\) |
| 8d | \(2\) | \(5\) | \(10\) |
| 8f | \(3\) | \(7\) | \(14\) |
| 8g | \(4\) | \(9\) | \(18\) |
| 8h | \(5\) | \(11\) | \(22\) |
| 8i | \(6\) | \(13\) | \(26\) |
| 8k | \(7\) | \(15\) | \(30\) |
Common interpretation notes
“Sublevels” and “subshells” refer to the same \(l\)-classified groups within a principal energy level. The rule “number of sublevels equals \(n\)” applies to the quantum-mechanical model of atomic orbitals and is independent of whether those very high-\(n\) orbitals are occupied in ground-state electron configurations of known elements.