6.4 Quantum Numbers and Atomic Orbitals
PRE-HEALTH PREP
The 4 Quantum Numbers Simplified
General Chemistry 6.4
Quantum numbers are the four-number system (n, l, ml, ms) used to describe the energy, shape, orientation, and spin of every electron in an atom. This lesson covers the shapes of s, p, and d orbitals, the allowed values and hierarchical relationships among the four quantum numbers, and the Pauli exclusion principle that prevents any two electrons from sharing the same quantum state. Worked examples demonstrate how to assign quantum numbers from orbital diagrams and how to identify valid versus invalid quantum number sets.
Atomic Orbitals: Shapes and Types
Atomic orbitals are the three-dimensional regions of space where electrons are most likely to be found within an atom. Each orbital is described by a mathematical wave function—a solution to the Schrödinger equation—and the shapes you see in textbooks are simply the three-dimensional graphs of those equations.
Fundamental Rule: Any orbital, regardless of type, can hold a maximum of two electrons. An orbital may be empty, half-filled with one electron, or fully occupied with two.
The four main types of orbitals you will encounter are s, p, d, and f. Beyond these, g and h orbitals exist in theory, but no stable element places electrons in them under normal conditions.
3 Orbital Shapes: s, p, and d
An s orbital is spherical. Picture a three-dimensional ball centered on the nucleus—electrons in an s orbital have equal probability of being found in any direction at a given distance from the nucleus.
p orbitals are dumbbell-shaped, with two lobes extending in opposite directions from the nucleus. Remember that these are three-dimensional objects occupying real space, not flat figures on a page.
d orbitals come in two characteristic shapes. Four of the five d orbitals (dxy, dxz, dyz, and dx²−y²) resemble a four-leaf clover, each oriented differently in space. The fifth, dz², has two lobes along the z-axis with a donut-shaped (toroidal) ring in the xy-plane.
Example: d Orbital Orientations
The dxy orbital lies in the xy-plane with its lobes between the x and y axes. The dx²−y² orbital also occupies the xy-plane, but its lobes point directly along the x and y axes. Despite sharing the same plane, these two orbitals are distinct because of their different orientations. Each four-leaf-clover shape represents a single orbital that can hold up to two electrons.
How Many Orbitals Are in Each Subshell?
The number of orbitals in each subshell follows a simple odd-number pattern:
| Subshell | Number of Orbitals | Maximum Electrons |
|---|---|---|
| s | 1 | 2 |
| p | 3 | 6 |
| d | 5 | 10 |
| f | 7 | 14 |
Pattern to Remember: The orbital counts follow the sequence of odd numbers: 1, 3, 5, 7 (and theoretically 9 for g, 11 for h). You will see exactly why this pattern emerges when we examine the magnetic quantum number below.
Higher-Shell Orbitals and Radial Nodes
The shapes described above are the simplest versions of each orbital type. The lowest shell containing an s orbital is the first shell (1s); the lowest shell with p orbitals is the second shell (2p); and the lowest shell with d orbitals is the third shell (3d). As you move to higher shells, each orbital type reappears in a larger, more complex form.
For example, the 2s orbital is a bigger sphere than the 1s, but it also contains a radial node—a spherical shell within the orbital where the probability of finding an electron drops to zero. Picture a softball nested inside a basketball: electrons can be found in the inner region or the outer region, but not in the hollow gap between them.
Example: Counting Radial Nodes
The 1s orbital has zero radial nodes. The 2s orbital has one radial node. The 3s orbital has two radial nodes. The same pattern applies to p and d orbitals in higher shells—3p orbitals are larger and more complex than 2p orbitals, with an additional radial node incorporated into their shapes.
Note: Radial nodes are not a major focus in most general chemistry courses, but they become important in physical chemistry and upper-division courses. For now, just understand that higher-shell orbitals are larger and contain more nodes.
Orbital Size, Energy, and the Bohr Model
As you move from 1s to 2s to 3s, the orbitals grow larger. Larger orbitals place electrons farther from the nucleus on average, and electrons farther from the nucleus possess higher energy. This means the lower the shell number, the smaller the orbitals and the lower the energy of the electrons they contain.
This idea connects back to the Bohr model. Bohr envisioned electrons traveling around the nucleus in flat, circular orbits—like planets around the sun. The modern quantum mechanical model replaces those two-dimensional orbits with three-dimensional orbitals described by wave functions. Electrons exhibit wave-like properties that Bohr did not account for, and these wave functions describe the probability of finding an electron at any point in space.
Key Distinction: The Bohr model uses orbits (2D circular paths). The quantum mechanical model uses orbitals (3D probability distributions). Although Bohr’s shell numbers (n = 1, 2, 3…) carry over directly, the underlying physics is fundamentally different.
What Are the 4 Quantum Numbers?
Every electron in an atom is described by a unique set of four quantum numbers: n, l, ml, and ms. You need to know each quantum number’s name, what it tells you, and the range of values it can take.
| Symbol | Name | What It Tells You | Allowed Values |
|---|---|---|---|
| n | Principal | Shell (energy level) | 1, 2, 3, … ∞ |
| l | Azimuthal (angular momentum) | Subshell (orbital type) | 0, 1, 2, … (n − 1) |
| ml | Magnetic | Specific orbital (orientation) | −l, … 0, … +l |
| ms | Spin | Electron spin direction | +½ or −½ |
n — The Principal Quantum Number
The principal quantum number (n) identifies which shell an electron occupies. These are the same shell numbers that correspond to Bohr’s orbits: n = 1 is the first shell, n = 2 is the second, and so on.
The allowed values for n are positive integers starting at 1 and extending to infinity: 1, 2, 3, 4, … In practice, ground-state atoms only populate shells up to about n = 7, but higher shells are confirmed to exist through spectroscopic measurements of electron transitions.
Degeneracy in Hydrogen vs. Multi-Electron Atoms: In the hydrogen atom (or any one-electron ion), all orbitals with the same principal quantum number are degenerate—they have the same energy. For example, 2s and 2p have the same energy in hydrogen. In any atom with more than one electron, this degeneracy disappears because of electron–electron repulsions, and subshells within the same shell differ in energy.
l — The Azimuthal Quantum Number
The azimuthal quantum number (l) identifies the subshell, which tells you the type of orbital the electron occupies. Each numerical value of l corresponds to a letter code:
- l = 0 → s subshell
- l = 1 → p subshell
- l = 2 → d subshell
- l = 3 → f subshell
The allowed values for l range from 0 to (n − 1) in integer steps. This relationship between n and l explains why higher shells contain more subshell types:
Example: Subshells Available in Each Shell
- Shell 1 (n = 1): l = 0 only → s subshell only
- Shell 2 (n = 2): l = 0, 1 → s and p subshells
- Shell 3 (n = 3): l = 0, 1, 2 → s, p, and d subshells
- Shell 4 (n = 4): l = 0, 1, 2, 3 → s, p, d, and f subshells
Each higher shell introduces one new subshell type. The fifth shell would technically include g orbitals, but no stable element places electrons there.
Angular Nodes and the Azimuthal Quantum Number: The value of l also indicates the number of angular nodes in an orbital. An s orbital (l = 0) has no angular nodes. A p orbital (l = 1) has 1 angular node. A d orbital (l = 2) has 2 angular nodes. An f orbital (l = 3) has 3 angular nodes. Equivalently, l tells you the orbital’s angular momentum and classifies its shape.
ml — The Magnetic Quantum Number
The magnetic quantum number (ml) identifies a specific orbital within a subshell. Because each orbital within a subshell has a distinct orientation in three-dimensional space, specifying ml is equivalent to specifying which orbital the electron occupies.
The allowed values for ml range from −l to +l in integer steps. Just as l is bounded by n, ml is bounded by l:
| Subshell (l) | ml Values | Number of Orbitals |
|---|---|---|
| s (l = 0) | 0 | 1 |
| p (l = 1) | −1, 0, +1 | 3 |
| d (l = 2) | −2, −1, 0, +1, +2 | 5 |
| f (l = 3) | −3, −2, −1, 0, +1, +2, +3 | 7 |
Notice how this produces exactly the 1, 3, 5, 7 odd-number pattern we saw earlier. The formula is straightforward: the number of orbitals in any subshell equals 2l + 1.
ms — The Spin Quantum Number
The spin quantum number (ms) describes an intrinsic property of the electron called spin. Unlike the other three quantum numbers, ms does not come from solving the Schrödinger equation—it arises from a separate, fundamental property of matter.
Exam Tip: If you are asked which quantum number does not come from the Schrödinger equation, the answer is ms (the spin quantum number). The other three—n, l, and ml—emerge directly from its solutions.
Electron spin is not literally a spinning motion. It is a fundamental, intrinsic property—much like charge—that causes electrons to interact with magnetic fields. Just as charge comes in two types (positive and negative), spin comes in exactly two values: +½ and −½.
We represent electrons as arrows in orbital diagrams: an up arrow (↑) for one spin value and a down arrow (↓) for the other. When two electrons share an orbital, they must have opposite spins.
What Is the Pauli Exclusion Principle?
The Pauli exclusion principle states that no two electrons in the same atom can share the same set of all four quantum numbers. Since a set of four quantum numbers uniquely identifies an electron’s “address” within the atom, this principle means that no two electrons can occupy the exact same state.
In practical terms, two electrons can share the same orbital (same n, l, and ml), but they must differ in their spin quantum number (ms). This is precisely why each orbital holds a maximum of two electrons—one with spin +½ and one with spin −½.
Common Mistake: Students sometimes think the Pauli exclusion principle prevents electrons from being in the same orbital. It does not. It prevents them from having all four quantum numbers identical. Two electrons in the same orbital simply must have opposite spins.
The Address Analogy
Think of quantum numbers as a coded address system for electrons. Imagine someone gives you their address not as “123 Oak Street, Tempe, Arizona” but as a series of numbers: state 48, city 7, street 104, house 2. Each number narrows down the location, and the full set pinpoints one unique spot.
Quantum numbers work the same way:
Example: The Electron Address System
- n (principal) = the state — tells you the shell
- l (azimuthal) = the city — tells you the subshell type
- ml (magnetic) = the street — tells you the specific orbital
- ms (spin) = the house number — tells you which of the two possible electrons
The Pauli exclusion principle simply says no two electrons can live at the same address—every electron has its own unique combination of these four numbers.
How to Find Quantum Numbers: Worked Examples
Reading Quantum Numbers from an Orbital Diagram
Orbital diagrams represent each orbital as a box and each electron as an arrow (up or down) inside the box. The boxes are arranged in order of increasing energy, and from the position of an electron in the diagram, you can assign values to its four quantum numbers.
For most electrons, you can determine n and l with certainty. The magnetic quantum number ml may have several possible values (since you typically cannot tell which specific orbital within a subshell the electron occupies), and ms is simply +½ or −½ depending on the arrow direction.
Example: Quantum Numbers for a 1s Electron
Consider an electron in the 1s orbital:
- n = 1 — the electron is in the first shell
- l = 0 — it occupies an s subshell (l = 0 means s)
- ml = 0 — the only possible value when l = 0
- ms = +½ or −½ — depends on the electron’s spin
If both electrons in the 1s orbital are shown, one must be +½ and the other −½. The Pauli exclusion principle demands opposite spins—if both had the same spin, all four quantum numbers would be identical, which nature does not allow.
Energy Ordering of Subshells
In orbital diagrams, energy increases upward. Shell 1 is lower in energy than shell 2, which is lower than shell 3, and so on. Within a given shell, the subshells also differ in energy: s < p < d < f (lowest to highest).
Important Exception: In hydrogen (and any one-electron system), all subshells within the same shell have equal energy. For example, 3s, 3p, and 3d are all degenerate in hydrogen. This splitting into different energies only occurs in multi-electron atoms due to electron–electron interactions. You will explore this topic in greater detail in a later lesson.
Example: Quantum Numbers for a 3p Electron
Consider an electron placed in the 3p subshell:
- n = 3 — the electron is in the third shell
- l = 1 — it occupies a p subshell (l = 1 means p)
- ml = −1, 0, or +1 — any of the three p orbitals is possible
- ms = +½ or −½ — depends on the electron’s spin
From the orbital diagram alone, you cannot pinpoint which of the three p orbitals holds this electron. You know ml must be −1, 0, or +1, but you cannot narrow it further.
How to Identify Valid Quantum Number Sets
A common exam question provides several sets of four quantum numbers and asks you to identify which set is valid. Here is a systematic strategy for eliminating invalid choices.
4-Step Elimination Strategy
- Check ms first. If the spin quantum number is anything other than +½ or −½, eliminate that option immediately.
- Check n. The principal quantum number must be a positive integer (1, 2, 3, …). Any negative value, zero, or non-integer eliminates the option.
- Compare l to n. The azimuthal quantum number must satisfy 0 ≤ l ≤ n − 1. If l equals n or exceeds n − 1, the set is invalid.
- Compare ml to l. The magnetic quantum number must satisfy −l ≤ ml ≤ +l. If ml falls outside this range, eliminate it.
Example: Validating a Quantum Number Set
Is the set (n = 4, l = 2, ml = −1, ms = +½) valid?
- ms = +½ — valid (must be ±½) ✓
- n = 4 — valid (positive integer) ✓
- l = 2 — valid (l can range from 0 to 3 when n = 4) ✓
- ml = −1 — valid (ml can range from −2 to +2 when l = 2) ✓
All four checks pass. This set describes an electron in a 4d orbital and is perfectly valid.
Example: Catching an Invalid Set
Is the set (n = 4, l = 4, ml = −1, ms = +½) valid?
- ms = +½ — valid ✓
- n = 4 — valid ✓
- l = 4 — invalid! When n = 4, l can only go up to 3 (n − 1). ✗
This set fails at step 3. No need to check ml—the set is already impossible.
Exam Strategy: When a multiple-choice question asks for the quantum numbers of an electron in a specific orbital (e.g., 3p), first eliminate any option where n or l doesn’t match. Then remember that ml has multiple valid values—for a 3p electron, ml can be −1, 0, or +1, and ms can be +½ or −½. That gives six valid combinations, but only one will appear among the answer choices.
7 Key Takeaways
- Atomic orbitals are three-dimensional wave functions (solutions to the Schrödinger equation) that describe where electrons are likely to be found. The main types are s (spherical), p (dumbbell), d (four-leaf clover or toroidal), and f (complex).
- Each orbital holds a maximum of two electrons, and the number of orbitals per subshell follows the odd-number pattern: 1 (s), 3 (p), 5 (d), 7 (f).
- The principal quantum number (n) specifies the shell, the azimuthal quantum number (l) specifies the subshell type, the magnetic quantum number (ml) identifies the specific orbital, and the spin quantum number (ms) distinguishes between the two electrons an orbital can hold.
- Each quantum number’s range is bounded by the one before it: n ≥ 1, then 0 ≤ l ≤ n − 1, then −l ≤ ml ≤ +l, and ms = ±½.
- The Pauli exclusion principle states that no two electrons in an atom can share all four quantum numbers—which is why each orbital holds at most two electrons with opposite spins.
- The spin quantum number (ms) is the only quantum number that does not come from the Schrödinger equation; it describes an intrinsic property of the electron.
- To validate a set of quantum numbers on an exam, check ms first, then n, then l relative to n, and finally ml relative to l.