Quantum Numbers & Atomic Orbitals Reference Cheat Sheet

Quantum numbers describe the allowed energy states and locations of electrons in atoms. This reference helps students connect atomic orbitals, subshells, electron configurations, and periodic table patterns. It is useful for solving problems about allowed quantum numbers, orbital capacity, and electron arrangement.

These ideas are essential for understanding atomic structure in chemistry and modern physics.

The four quantum numbers are the principal quantum number $n$ , angular momentum quantum number $\ell$ , magnetic quantum number $m_{\ell}$ , and spin quantum number $m_s$ . The value of $n$ gives the main energy level, while $\ell$ identifies the subshell shape such as $s$ , $p$ , $d$ , or $f$ . Each orbital holds at most $2$ electrons with opposite spins, so a subshell capacity depends on the number of orbitals it contains.

Electron configurations follow the Aufbau principle, Pauli exclusion principle, and Hund's rule.

Key Facts

The principal quantum number must be a positive integer: $n = 1, 2, 3, \ldots$ .
For a given energy level $n$ , the angular momentum quantum number can be $\ell = 0, 1, 2, \ldots, n - 1$ .
Subshell labels match angular momentum values as $\ell = 0$ for $s$ , $\ell = 1$ for $p$ , $\ell = 2$ for $d$ , and $\ell = 3$ for $f$ .
For a given subshell $\ell$ , the magnetic quantum number can be $m_{\ell} = -\ell, \ldots, 0, \ldots, +\ell$ .
The number of orbitals in a subshell is $2\ell + 1$ .
The maximum number of electrons in a subshell is $2(2\ell + 1)$ because each orbital holds $2$ electrons.
The total maximum number of electrons in shell $n$ is $2n^2$ .
Electron spin has two allowed values, $m_s = +\frac{1}{2}$ or $m_s = -\frac{1}{2}$ .

Vocabulary

Principal quantum number: The quantum number $n$ identifies the main energy level or shell of an electron.
Angular momentum quantum number: The quantum number $\ell$ identifies the subshell type and orbital shape within a shell.
Magnetic quantum number: The quantum number $m_{\ell}$ identifies a specific orbital orientation within a subshell.
Spin quantum number: The quantum number $m_s$ identifies the spin state of an electron as $+\frac{1}{2}$ or $-\frac{1}{2}$ .
Orbital: An orbital is a region of space described by quantum numbers where an electron is likely to be found.
Electron configuration: An electron configuration lists how electrons are arranged among shells, subshells, and orbitals.

Common Mistakes to Avoid

Using $\ell = n$ as an allowed value, which is wrong because $\ell$ can only range from $0$ to $n - 1$ .
Treating a subshell as one orbital, which is wrong because a subshell contains $2\ell + 1$ orbitals.
Putting more than $2$ electrons in one orbital, which violates the Pauli exclusion principle because paired electrons must have opposite spins.
Pairing electrons too early in equal-energy orbitals, which violates Hund's rule because electrons occupy separate orbitals with parallel spins first.
Confusing $m_{\ell}$ with $m_s$ , which is wrong because $m_{\ell}$ describes orbital orientation while $m_s$ describes electron spin.

Practice Questions

1 For $n = 4$ , list all allowed values of $\ell$ .
2 How many orbitals are in a $d$ subshell, and how many electrons can it hold at maximum?
3 Find the maximum number of electrons that can fit in the shell with $n = 3$ using $2n^2$ .
4 Explain why two electrons in the same orbital cannot have the same complete set of four quantum numbers.

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Quantum Numbers & Atomic Orbitals Reference Cheat Sheet

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