Electrons don’t follow orbits
The Bohr model breaks down for atoms with more than one electron. In the 1920s, scientists discovered that electrons, like light, behave both as particles and as waves, and Werner Heisenberg showed that you can’t know both an electron’s exact position and its exact velocity at the same time (the uncertainty principle). So an electron has no definite path. In 1926, Erwin Schrödinger treated the electron in an atom as a wave, and the quantum mechanical model we use today was born.
Orbitals and the electron cloud
Since we can’t say which path an electron follows, we can only say how likely it is to be found at each place around the nucleus. Drawing that likelihood as a haze of dots, dense where the electron is likely to be, gives an electron cloud电子云.
In the quantum mechanical model, each allowed state of an electron in an atom is an orbital原子轨道. An orbital’s shape is usually drawn as the surface that encloses about 90% of the chance of finding the electron. Despite the name, an orbital isn’t an orbit: it’s a region where the electron is likely to be, not a path.
Energy levels and sublevels
Electrons are grouped by energy into principal energy levels能层, the “shells” of lesson 1.4.4, numbered n = 1, 2, 3, 4, and so on. These are the same energy levels n as in the Bohr model of lesson 2.2.3; what’s new here is that each level splits into sublevels. (Chinese textbooks also label the levels K, L, M, N.) Electrons in the same level can still differ in energy, so each level is divided into sublevels能级 (also called subshells), labeled s, p, d, and f:
- Level n has n sublevels: level 1 has only 1s; level 2 has 2s and 2p; level 3 has 3s, 3p, and 3d; level 4 has 4s, 4p, 4d, and 4f.
- An s, p, d, or f sublevel has 1, 3, 5, or 7 orbitals.
- Each orbital holds at most 2 electrons (the Pauli exclusion principle, lesson 2.2.5), so s, p, d, and f sublevels hold at most 2, 6, 10, and 14 electrons.
| Level n | Sublevels | Orbitals | Maximum electrons |
|---|---|---|---|
| 1 | 1s | 1 | 2 |
| 2 | 2s, 2p | 4 | 8 |
| 3 | 3s, 3p, 3d | 9 | 18 |
| 4 | 4s, 4p, 4d, 4f | 16 | 32 |
Level n has n² orbitals and holds up to 2n² electrons. That’s where lesson 1.4.4’s limit of 8 for the second shell comes from, and the third level can actually hold 18.
Orbital shapes
- s orbitals are spheres, larger for larger n.
- p orbitals are dumbbells. Every p sublevel has three, pointing along the x, y, and z axes and labeled px, py, and pz; all three have the same energy.
- There are five d orbitals, most shaped like four-leaf clovers, and seven f orbitals with more complex shapes.
Within a level, sublevel energy rises in the order s < p < d < f; for the same type of sublevel, larger n means higher energy: 1s < 2s < 3s. But sublevels from different levels overlap. For example, 4s is lower in energy than 3d, which matters when you place electrons in lesson 2.2.5.
Pick an energy level to see its sublevels, then click a sublevel to see its orbitals:
Each box stands for one orbital.
Example 1: The third energy level
Which sublevels does the third energy level have? How many orbitals, and how many electrons at most? And why did lesson 1.4.4 put only 8 electrons in potassium’s third shell?
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Sublevels. Level n has n sublevels, so n = 3 has 3: 3s, 3p, and 3d.
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Orbitals. s, p, and d sublevels have 1, 3, and 5 orbitals:
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Maximum electrons. Each orbital holds at most 2 electrons (with opposite spins):
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Why only 8 for potassium? Once 3s and 3p are full, with 8 electrons, the next sublevel by energy is 4s, not 3d. So potassium’s 19th electron goes into 4s, and 3d fills later. “At most 18” is the third level’s capacity, not what potassium actually puts there.
Example 2: Which sublevels don’t exist?
Which of these sublevels don’t exist: 1p, 2d, 3f, 4f?
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The rule. Level n has only n sublevels, appearing in the order s, p, d, f:
n Sublevels 1 s 2 s, p 3 s, p, d 4 s, p, d, f -
Check each one. 1p: level 1 has only s, so no. 2d: level 2 has only s and p, so no. 3f: level 3 stops at d, so no. 4f: f first appears at level 4, so it exists, with 7 orbitals.
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Conclusion. 1p, 2d, and 3f don’t exist.
- 2s
An s sublevel has just 1 orbital, a sphere.
- 3p
A p sublevel has 3 orbitals: px, py, and pz.
- 4d
A d sublevel has 5 orbitals.
- 4f
An f sublevel has 7 orbitals.
- 1s
Level 1 has only the 1s sublevel, with 1 orbital.
- 5p
Every p sublevel has 3 orbitals, whatever the level.
- 3d
A d sublevel has 5 orbitals and holds up to 10 electrons.
- 6s
An s sublevel has 1 orbital; larger n just makes the sphere bigger.
- 5f
An f sublevel has 7 orbitals and holds up to 14 electrons.
Common mistakes
Mistake 1: “An orbital is the circular path an electron travels around the nucleus.” Electrons have no definite path. An orbital is a state of the electron, drawn as the region where it’s likely to be found.
Mistake 2: “Every energy level holds at most 8 electrons.” Level n holds up to 2n² electrons: 18 for level 3, 32 for level 4. The limit of 8 applies to an atom’s outermost level.
Mistake 3: “A sublevel in a lower energy level always has lower energy.” Sublevels overlap: 4s is lower in energy than 3d, so 4s fills first.
Quick check
What is the maximum number of electrons the third energy level can hold?
Level 3 has 9 orbitals with up to 2 electrons each, for 18 (2n²). 9 is the number of orbitals, and 8 is the limit for an outermost level.
Which of these sublevels does not exist?
Level 2 has only the 2s and 2p sublevels; there is no 2d.
Summary
- Electrons have no definite paths, only probabilities: an electron cloud shows where an electron is likely to be, and an orbital is one allowed state of an electron.
- Energy level n has n sublevels (s, p, d, f), with 1, 3, 5, and 7 orbitals; level n has n² orbitals and holds up to 2n² electrons.
- s orbitals are spheres and the three p orbitals are dumbbells. Sublevel energy rises s < p < d < f, and sublevels from different levels overlap (4s < 3d).
On to AP (for later)
AP calls sublevels “subshells” and uses them to write electron configurations (topic 1.5) and to label the peaks of a PES spectrum (topic 1.6). AP doesn’t require the four quantum numbers, but the shapes of s and p orbitals help explain bonding and hybrid orbitals (topic 2.7).