Quantum Numbers and Electronic Configuration

For any given $n$, $\ell$ can take integer values ranging from $0$ to $n-1$. These values correspond to specific orbital types, which are commonly assigned letters:

• $\ell = 0$: s orbital (spherical)
• $\ell = 1$: p orbital (dumbbell-shaped)
• $\ell = 2$: d orbital
• $\ell = 3$: f orbital

Consequently, the number of sublevels in a principal level is equal to $n$. For example, the 3rd energy level ($n=3$) contains three sublevels: $3s$, $3p$, and $3d$.

3. Magnetic Orbital Quantum Number ($m_\ell$)

The magnetic orbital quantum number, $m_\ell$, describes the orientation of an orbital in 3D space. Within a subshell, there are multiple orbitals that are degenerate, meaning they share the same size and energy level.

The values for $m_\ell$ depend on $\ell$ and range from $-\ell$ to $+\ell$, including zero. This determines how many individual orbitals exist in each subshell:

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

4. Magnetic Spin Quantum Number ($m_s$)

The final number, $m_s$, describes the electron spin—the direction the electron rotates on its axis. Unlike the other three numbers, $m_s$ does not describe an orbital, but rather the individual electron within that orbital. There are only two allowed values:

• $+\frac{1}{2}$ (often represented as an upward arrow $\uparrow$, indicating clockwise spin)
• $-\frac{1}{2}$ (often represented as a downward arrow $\downarrow$, indicating counter-clockwise spin)

Exam Tip: Validating Quantum Numbers To check if a set of quantum numbers $(n, \ell, m_\ell, m_s)$ is valid, always verify these three rules:

1. $\ell$ must be less than $n$ ($\ell < n$).
2. $m_\ell$ must be between $-\ell$ and $+\ell$.
3. $m_s$ must be exactly $+\frac{1}{2}$ or $-\frac{1}{2}$. Example: $(2, 2, 0, +\frac{1}{2})$ is invalid because $\ell$ cannot equal $n$.

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Rules for Electronic Configuration

Electronic configuration is the distribution of electrons among the various atomic orbitals. To write these configurations correctly, we must follow three fundamental principles.

The Aufbau Principle

The word "Aufbau" is German for "building up." This principle states that electrons occupy the available subshells of the lowest energy first. Once a subshell is filled, electrons move to the next highest energy level. The standard filling order is: $1s \rightarrow 2s \rightarrow 2p \rightarrow 3s \rightarrow 3p \rightarrow 4s \rightarrow 3d \rightarrow 4p \dots$ Note that the $4s$ orbital is filled before the $3d$ orbital because it is lower in energy.

The Pauli Exclusion Principle

This principle states that no two electrons in an atom can have the same set of four quantum numbers. In practical terms, this means an individual orbital can hold a maximum of two electrons, and they must have opposite spins.

Hund’s Rule

When filling degenerate orbitals (orbitals of the same energy, like the three $p$ orbitals), electrons will occupy empty orbitals singly with parallel spins before they start pairing up. This minimizes electron-electron repulsion and results in the lowest energy configuration.

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Advanced Concepts: Exceptions and Ions

Configuration Exceptions

While the Aufbau principle is a reliable guide, some elements deviate from the expected pattern to achieve greater stability. Orbitals are more stable when they are empty, half-filled, or completely filled.

For example, Chromium (Cr) and Copper (Cu) are classic exceptions:

• Chromium (Cr): Instead of the expected $[Ar] 4s^2 3d^4$, it adopts $[Ar] 4s^1 3d^5$ to achieve a stable, half-filled $d$ subshell.
• Copper (Cu): Instead of $[Ar] 4s^2 3d^9$, it adopts $[Ar] 4s^1 3d^{10}$ to achieve a stable, completely filled $d$ subshell.

Writing Ion Configurations

When an atom becomes an ion, electrons are either added (anions) or removed (cations).

1. Representative Elements: Electrons are added or removed from the highest energy level to reach the nearest noble gas configuration.
2. Transition Metals: When transition metals form cations, electrons are always removed from the highest energy level first (the valence electrons), even if those electrons were added after the $d$ electrons.

Example: Iron (Fe) to $Fe^{2+}$ and $Fe^{3+}$

• Fe atom: $1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^6$
• $Fe^{2+}$ cation: Lose 2 electrons from the $4s$ orbital $\rightarrow 1s^2 2s^2 2p^6 3s^2 3p^6 3d^6$.
• $Fe^{3+}$ cation: Lose 1 more electron from the $3d$ orbital $\rightarrow 1s^2 2s^2 2p^6 3s^2 3p^6 3d^5$.

Magnetism

The presence of unpaired electrons determines a substance's magnetic properties:

• Paramagnetic: Atoms with at least one unpaired electron; these are attracted by a magnet.
• Diamagnetic: Atoms with no unpaired electrons (all electrons are paired); these are slightly repelled by a magnet.