30.8 Quantum Numbers and Rules

Quantum numbers describe an electron's energy, angular momentum, orientation, and spin within an atom. They form the foundation for understanding how electrons are arranged and why atoms behave the way they do. This section covers the four quantum numbers, the rules governing electron configurations, and the quantum mechanical principles behind it all.

Quantum Numbers and Atomic States

Quantum numbers in atomic states

Every electron in an atom is described by a unique set of four quantum numbers. Think of them as an electron's "address" within the atom: the combination of all four tells you exactly which state that electron occupies.

The four quantum numbers are:

• Principal quantum number $n$
• Angular momentum quantum number $l$
• Magnetic quantum number $m_l$
• Spin quantum number $m_s$

The Pauli exclusion principle states that no two electrons in the same atom can share the same set of all four quantum numbers. This is what forces electrons into different orbitals and energy levels, giving atoms their structure.

Together, these quantum numbers let you predict electronic structure, atomic properties like ionization energy and atomic radius, and the spectral lines an atom produces (absorption and emission spectra).

Physical meaning of quantum numbers

Principal quantum number $n$

This tells you which main energy level (or shell) the electron is in. It takes positive integer values: 1, 2, 3, and so on. Higher $n$ means higher energy and a larger average distance from the nucleus.

• $n = 1$: K shell (lowest energy, closest to nucleus)
• $n = 2$: L shell
• $n = 3$: M shell
• $n = 4$: N shell

Angular momentum quantum number $l$

This describes the shape of the orbital and identifies the subshell. For a given $n$, $l$ can be any integer from 0 to $n - 1$.

• $l = 0$: s orbital (spherical shape)
• $l = 1$: p orbital (dumbbell shape)
• $l = 2$: d orbital (cloverleaf shape)
• $l = 3$: f orbital (more complex shape)

So if $n = 3$, the allowed values of $l$ are 0, 1, and 2, meaning the third shell contains s, p, and d subshells. The value of $l$ is also related to the magnitude of the electron's orbital angular momentum.

Magnetic quantum number $m_l$

This specifies the orientation of the orbital in space, which matters when an external magnetic field is present. For a given $l$, $m_l$ takes integer values from $-l$ to $+l$, including 0. That means:

• s subshell ($l = 0$): 1 orbital ($m_l = 0$)
• p subshell ($l = 1$): 3 orbitals ($m_l = -1, 0, +1$)
• d subshell ($l = 2$): 5 orbitals ($m_l = -2, -1, 0, +1, +2$)
• f subshell ($l = 3$): 7 orbitals ($m_l = -3, -2, -1, 0, +1, +2, +3$)

Each value of $m_l$ corresponds to a different spatial orientation (for example, the three p orbitals point along the x, y, and z axes).

Spin quantum number $m_s$

Electron spin is an intrinsic property with no classical equivalent. It can only take two values:

• $m_s = +\frac{1}{2}$ (spin up)
• $m_s = -\frac{1}{2}$ (spin down)

Because of this, each orbital holds a maximum of two electrons, and those two must have opposite spins. Electron spin is also responsible for the magnetic properties of materials: atoms with unpaired electrons are paramagnetic (attracted to magnetic fields), while atoms with all electrons paired are diamagnetic.

Electron Configurations and Quantum Principles

Electron configurations and quantum principles

Three key rules determine how electrons fill orbitals:

1. Pauli Exclusion Principle

No two electrons in the same atom can have identical sets of all four quantum numbers. In practice, this means each orbital holds at most two electrons, and they must have opposite spins. This principle is what gives the periodic table its structure.

2. Aufbau Principle

Electrons fill orbitals starting from the lowest available energy and working upward. The energy ordering is:

$1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f < 5d < 6p < 7s < 5f < 6d < 7p$

Notice that $4s$ fills before $3d$. This happens because the $4s$ orbital is slightly lower in energy for most atoms. There are a few well-known exceptions where half-filled or fully-filled d subshells are extra stable:

• Chromium: $1s^2\, 2s^2\, 2p^6\, 3s^2\, 3p^6\, 4s^1\, 3d^5$ (not $4s^2\, 3d^4$)
• Copper: $1s^2\, 2s^2\, 2p^6\, 3s^2\, 3p^6\, 4s^1\, 3d^{10}$ (not $4s^2\, 3d^9$)

3. Hund's Rule

When filling orbitals of the same energy (called degenerate orbitals), electrons spread out and occupy them singly before pairing up. All the singly occupied orbitals will have electrons with the same spin direction (parallel spins). This minimizes electron-electron repulsion and makes the configuration more stable.

For example, carbon's two 2p electrons each go into a separate p orbital with parallel spins rather than doubling up in one orbital.

Electron configuration notation

The shorthand looks like this: $1s^2\, 2s^2\, 2p^6\, 3s^2\, 3p^6$

• The number is the principal quantum number $n$
• The letter is the subshell (s, p, d, or f)
• The superscript is the number of electrons in that subshell

Some examples:

• Helium (2 electrons): $1s^2$
• Carbon (6 electrons): $1s^2\, 2s^2\, 2p^2$
• Neon (10 electrons): $1s^2\, 2s^2\, 2p^6$
• Sodium (11 electrons): $1s^2\, 2s^2\, 2p^6\, 3s^1$

Quantum Mechanical Principles

These are the deeper ideas from quantum mechanics that underpin everything above:

• Wave function: A mathematical function that describes the quantum state of an electron. You can't directly observe it, but it contains all the information about the electron's behavior.
• Schrödinger equation: The fundamental equation of quantum mechanics. Solving it for the hydrogen atom is what produces the quantum numbers $n$, $l$, and $m_l$ in the first place.
• Probability density: The square of the wave function at a given point. It tells you the likelihood of finding the electron in that region of space. Orbitals are typically drawn as surfaces enclosing regions where the probability density is high.
• Quantum superposition: A quantum system can exist in a combination of multiple states at once. The system "collapses" into a definite state only when a measurement is made.
• Uncertainty principle: You cannot simultaneously know both the exact position and exact momentum of an electron. The more precisely you measure one, the less precisely you can know the other. This is why we talk about probability distributions rather than fixed orbits.
• Degeneracy: When multiple quantum states share the same energy. For example, the three p orbitals within the same shell all have the same energy in the absence of an external field. Degeneracy is broken (the energies split apart) when a magnetic field is applied.