Atomic Orbitals and Quantum Numbers

Every electron in an atom can be fully described by four quantum numbers — a set of values that together specify which orbital an electron occupies and how it behaves within that orbital. This article covers what each quantum number describes, the range of values it can take, and how those rules combine to build the familiar picture of s, p, d, and f orbitals, shell by shell. That picture is the foundation for the next two subtopics in this chapter, which cover how orbitals overlap to form bonds and how they hybridize.

Key Takeaways

• Four quantum numbers fully describe any electron in an atom: n (principal), l (angular momentum), mₗ (magnetic), and mₛ (spin).

• The principal quantum number (n) gives the shell/energy level (positive integers 1, 2, 3...); a higher n means a larger shell, farther from the nucleus and higher in energy. It also sets the shell's maximum electron capacity.

• The angular momentum quantum number (l) defines the subshell and its shape, ranging from 0 to n − 1: l = 0 is s (spherical), l = 1 is p (dumbbell), l = 2 is d (cloverleaf), l = 3 is f (complex).

• The magnetic quantum number (mₗ) specifies orbital orientation within a subshell, ranging from −l to +l — for example, the p subshell's three values (−1, 0, +1) give the three p orbitals: pₓ, p_y, p_z.

• The spin quantum number (mₛ) describes electron spin, either +1/2 (spin-up) or −1/2 (spin-down) — this is why each orbital holds a maximum of two electrons.

• Building up shell by shell gives the standard capacities: n = 1 holds 2 electrons (1s), n = 2 holds 8 (2s + 2p), n = 3 holds 18 (3s + 3p + 3d), and n = 4 holds 32 (4s + 4p + 4d + 4f).

From Bohr's Model to the Quantum Mechanical Model

Earlier atomic models, like Bohr's, pictured electrons orbiting the nucleus along fixed paths, similar to planets orbiting the sun. The quantum mechanical model replaces that picture entirely. Instead of fixed orbits, electrons exist in orbitals — regions of space where there is a high probability of finding an electron.

These orbitals aren't arbitrary. They come directly from solving Schrödinger's equation, a fundamental equation of quantum mechanics. The math behind it isn't necessary for understanding what follows — what matters is that solving Schrödinger's equation produces the orbitals themselves, along with a set of rules that govern them. Those rules are summarized by four quantum numbers, each of which provides specific information about an electron's location and behavior.

The Four Quantum Numbers

Principal Quantum Number (n)

The principal quantum number, symbol n, can take on any positive integer value: 1, 2, 3, and so on. It identifies the main energy level, or shell, an electron occupies.

A larger value of n corresponds to a larger shell — the electron is, on average, farther from the nucleus. But n does more than describe size: it also sets the maximum number of electrons a shell can hold.

• n = 1 → shell holds up to 2 electrons

• n = 2 → shell holds up to 8 electrons

• n = 3 → shell holds up to 18 electrons

• n = 4 → shell holds up to 32 electrons

These totals aren't arbitrary either — the next sections show exactly where they come from.

Angular Momentum Quantum Number (l)

The angular momentum quantum number, symbol l, determines the shape of the subshell within a given shell. Its values range from 0 to n − 1.

Each value of l corresponds to a different type of subshell:

So while n tells you which shell an electron is in, l tells you which subshells exist within that shell.

Magnetic Quantum Number (mₗ)

The magnetic quantum number, symbol mₗ, specifies the orientation of an orbital in space and determines how many orbitals exist within a subshell.

For example, in a p subshell (l = 1), mₗ can equal −1, 0, or +1 — three possible values, meaning there are three p orbitals in the p subshell. These are labeled pₓ, p_y, and p_z, each aligned along a different spatial axis.

Spin Quantum Number (mₛ)

The spin quantum number, symbol mₛ, describes the spin direction of an electron. It has only two possible values: +1/2 or −1/2. Because of this, each orbital can hold a maximum of two electrons — one with spin up, one with spin down.

Building Up the Orbitals: Shells n = 1 Through n = 4

With the four quantum number rules in hand, the full orbital picture can be built up shell by shell.

Shell n = 1: Since n = 1, the angular momentum quantum number l can only equal 0, so there's just one possible subshell — 1s. The magnetic quantum number mₗ also has only one possible value here, meaning there's a single orbital in this subshell. Since each orbital holds a maximum of two electrons, the first shell's total capacity is 2 electrons.

Shell n = 2: Now l can be 0 or 1, giving two subshells — 2s and 2p. The 2s subshell works just like 1s: a single spherical orbital holding up to 2 electrons. The 2p subshell introduces three orbitals (mₗ = −1, 0, +1), together holding 6 electrons. Summing both subshells gives the second shell's total capacity of 8 electrons.

Shell n = 3: With three possible subshells — 3s, 3p, and 3d — the s subshell again contributes 1 orbital/2 electrons and the p subshell contributes 3 orbitals/6 electrons, just as before. Now, with l = 2, the d subshell is introduced: 5 orbitals holding up to 10 electrons. That brings the third shell's total capacity to 18 electrons.

Shell n = 4: A fourth subshell appears, the f subshell (l = 3), with 7 orbitals holding a total of 14 electrons. Added to the s, p, and d subshells already accounted for, this brings the fourth shell's maximum to 32 electrons.

Common MCAT Mistakes

• Confusing n and l. n tells you the shell (energy level); l tells you the subshell (shape) within that shell. A common slip is treating them as the same kind of "level" — they answer different questions.

• Forgetting the l range depends on n. l can only run from 0 to n − 1, not from 0 to n. For n = 2, l is limited to 0 and 1 (s and p) — there is no 2d subshell.

• Miscounting orbitals from mₗ. The number of mₗ values (and thus orbitals) in a subshell is 2l + 1, not l. For the d subshell (l = 2), that's 5 orbitals, not 2.

• Forgetting the electrons-per-orbital cap. No matter how many orbitals a subshell has, each individual orbital holds at most 2 electrons — one spin-up (mₛ = +1/2) and one spin-down (mₛ = −1/2). Total subshell capacity is orbitals × 2.

MCAT-Style Concept Check

Question: An electron has an angular momentum quantum number l = 2. How many orbitals exist within this subshell, and what is the subshell's maximum electron capacity?

• A) 2 orbitals, 4 electrons

• B) 3 orbitals, 6 electrons

• C) 5 orbitals, 10 electrons

• D) 7 orbitals, 14 electrons

Answer: C

Explanation: The number of orbitals in a subshell equals 2l + 1. For l = 2 (a d subshell), that's 2(2) + 1 = 5 orbitals. Since each orbital holds a maximum of 2 electrons, the subshell's total capacity is 5 × 2 = 10 electrons. (A describes an s-like doubling that doesn't match l = 2; B describes the p subshell, l = 1; D describes the f subshell, l = 3.)

FAQ

What are the four quantum numbers on the MCAT?

The principal quantum number (n), which gives the shell; the angular momentum quantum number (l), which gives the subshell shape; the magnetic quantum number (mₗ), which gives orbital orientation/count within a subshell; and the spin quantum number (mₛ), which gives electron spin direction.

What values can the angular momentum quantum number l take?

l ranges from 0 to n − 1. So for a given shell n, l can take on n total values: 0, 1, 2, ... up to n − 1. Each value corresponds to a subshell type (s, p, d, f).

Why can each orbital hold only two electrons?

Because the spin quantum number mₛ has only two possible values, +1/2 and −1/2. An orbital is fully occupied once it holds one electron of each spin.

How many electrons can the n = 3 shell hold?

18. The n = 3 shell has three subshells — 3s (2 electrons), 3p (6 electrons), and 3d (10 electrons) — which together sum to 18.