1.1 Quantum Mechanical Model of the Atom

Wave-Particle Duality and Quantum Mechanics

The quantum mechanical model replaced the classical picture of electrons orbiting the nucleus like tiny planets. Instead, it treats electrons as wave-like entities described by mathematical functions, and it predicts only the probability of finding an electron in a given region of space. Everything in this topic builds on that shift in thinking.

Fundamental Concepts of Wave-Particle Duality

Wave-particle duality means that matter and energy don't fit neatly into the "wave" or "particle" box. They behave as both, depending on the experiment.

• Light as a wave: Diffraction and interference patterns (like those seen with a diffraction grating) only make sense if light is a wave.
• Light as a particle: The photoelectric effect only makes sense if light arrives in discrete packets (photons). Increasing light intensity below the threshold frequency ejects zero electrons, no matter how bright the source.
• Electrons as waves: Electron diffraction experiments show that a beam of electrons produces an interference pattern, just like light waves do. The double-slit experiment demonstrates this dramatically for both photons and electrons.

De Broglie connected these ideas quantitatively. Any particle with momentum $p$ has an associated wavelength:

$\lambda = \frac{h}{p} = \frac{h}{mv}$

where $h$ is Planck's constant ($6.626 \times 10^{-34}$ J·s). For macroscopic objects the wavelength is negligibly small, but for electrons it's on the order of atomic dimensions, which is why wave behavior matters at the atomic scale.

Mathematical Foundations of Quantum Mechanics

The Schrödinger equation is the central equation of quantum mechanics. For stationary (time-independent) states:

$\hat{H}\psi = E\psi$

Here $\hat{H}$ is the Hamiltonian operator (which contains kinetic and potential energy terms), $\psi$ is the wave function, and $E$ is the energy of that state. Solving this equation for a given system yields the allowed wave functions and their corresponding energies.

A few key points about $\psi$:

• The wave function $\psi$ itself doesn't have a direct physical meaning you can measure.
• $|\psi|^2$ is the probability density. Integrating $|\psi|^2$ over a region of space gives the probability of finding the electron there.
• $\psi$ must be single-valued, continuous, and normalizable (total probability integrates to 1).

The Heisenberg uncertainty principle sets a fundamental limit on how precisely you can simultaneously know a particle's position and momentum:

$\Delta x \cdot \Delta p \geq \frac{h}{4\pi}$

This isn't about imperfect instruments. It's a built-in feature of quantum systems. The more precisely you pin down position ($\Delta x$), the less precisely you can know momentum ($\Delta p$), and vice versa.

Quantization of Energy and Atomic Structure

In classical physics, energy can take any value. In quantum mechanics, the energy of a bound electron is quantized, meaning only specific discrete values are allowed. This is a direct consequence of the boundary conditions imposed on $\psi$.

The Bohr model introduced this idea for hydrogen: electrons occupy fixed energy levels labeled by the principal quantum number $n$. While the Bohr model is too simple for multi-electron atoms, it correctly predicts hydrogen's emission spectrum and gives the right energy expression:

$E_n = \frac{-13.6 \text{ eV}}{n^2}$

When an electron transitions between levels, a photon is emitted or absorbed with energy:

$E = h\nu = \frac{hc}{\lambda}$

This is why atomic emission spectra consist of discrete lines rather than a continuous rainbow. Each line corresponds to a specific electronic transition.

Atomic Orbitals and Electron Distribution

An atomic orbital is a one-electron wave function obtained by solving the Schrödinger equation for an atom. Each orbital has a characteristic energy, size, shape, and orientation determined by its quantum numbers.

Quantum Numbers and Orbital Characteristics

Four quantum numbers fully specify the state of an electron in an atom. They're not independent of each other; each one is constrained by the values of the others.

1. Pick a value of $n$ (say $n = 3$).
2. $l$ can range from 0 to $(n-1)$, so $l$ = 0, 1, or 2 (that's the 3s, 3p, and 3d subshells).
3. For each $l$, $m_l$ ranges from $-l$ to $+l$. So the 3d subshell ($l = 2$) has $m_l$ = $-2, -1, 0, +1, +2$, giving five orbitals.
4. Each orbital holds at most 2 electrons with opposite spins ($m_s = +\frac{1}{2}$ and $-\frac{1}{2}$).

The total number of orbitals in a shell is $n^2$, and the maximum number of electrons in a shell is $2n^2$.

Characteristics and Types of Atomic Orbitals

The angular momentum quantum number $l$ determines orbital shape:

• s orbitals ($l = 0$): Spherically symmetric. One per shell.
• p orbitals ($l = 1$): Dumbbell-shaped, oriented along the x, y, and z axes. Three per shell (starting at $n = 2$).
• d orbitals ($l = 2$): More complex shapes, including cloverleaf patterns and the distinctive $d_{z^2}$ orbital with a donut around a central lobe. Five per shell (starting at $n = 3$).
• f orbitals ($l = 3$): Even more complex multi-lobed shapes. Seven per shell (starting at $n = 4$).

As $n$ increases, orbitals of the same type get larger and higher in energy. A 3s orbital is bigger and more diffuse than a 2s orbital.

Electron Probability and Spatial Distribution

Since $|\psi|^2$ gives probability density, you can think about electron distribution in two useful ways:

• Radial probability distribution plots the probability of finding the electron at a given distance $r$ from the nucleus (integrated over all angles). For a 1s orbital, this peaks at the Bohr radius ($a_0 = 52.9$ pm). For higher $n$, the most probable distance shifts farther from the nucleus.
• Angular probability distribution captures the directional dependence of the orbital. This is what gives p, d, and f orbitals their characteristic shapes.

Nodes are regions where $|\psi|^2 = 0$ (zero probability of finding the electron):

• Radial nodes are spherical surfaces at specific distances from the nucleus. The number of radial nodes = $n - l - 1$.
• Angular nodes are planes (or cones) passing through the nucleus. The number of angular nodes = $l$.
• Total nodes = $n - 1$.

For example, a 3p orbital has $n - l - 1 = 3 - 1 - 1 = 1$ radial node and $l = 1$ angular node, for 2 total nodes.

Electron Configuration

Electron configuration describes how electrons are distributed among the orbitals of an atom. Getting this right is essential because it directly determines an element's chemical properties and its position in the periodic table.

Principles Governing Electron Arrangement

Three rules control how electrons fill orbitals:

1. Aufbau principle: Electrons fill orbitals starting from the lowest energy and working up. The general 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, and 6s fills before 4f. This happens because of electron-electron repulsion and shielding effects in multi-electron atoms, which alter the orbital energies compared to hydrogen.

1. Pauli exclusion principle: No two electrons in the same atom can share all four quantum numbers. In practice, this means each orbital holds a maximum of 2 electrons, and those two must have opposite spins ($m_s = +\frac{1}{2}$ and $-\frac{1}{2}$).

2. Hund's rule: When filling a set of degenerate orbitals (orbitals with the same energy, like the three 2p orbitals), electrons occupy them singly with parallel spins before any pairing occurs. This minimizes electron-electron repulsion.

Electron Configuration Notation and Representations

• Full notation lists every occupied subshell: $1s^2\, 2s^2\, 2p^6\, 3s^2\, 3p^6\, 4s^2\, 3d^{10}\, 4p^2$ (germanium, Z = 32).
• Noble gas shorthand replaces the inner core with the preceding noble gas in brackets: $[\text{Ar}]\, 4s^2\, 3d^{10}\, 4p^2$.
• Orbital box diagrams use boxes (or lines) for each orbital and arrows for electrons, making spin assignments and Hund's rule visible.

Valence electrons are those in the outermost shell (highest $n$) and, for transition metals, the outermost d electrons. These are the electrons involved in bonding. Core electrons are everything else, shielded beneath the valence shell.

Exceptions and Special Cases in Electron Configuration

The Aufbau order doesn't always hold. The most important exceptions to know:

• Chromium (Z = 24): Expected $[\text{Ar}]\, 4s^2\, 3d^4$, but the actual configuration is $[\text{Ar}]\, 4s^1\, 3d^5$. A half-filled d subshell provides extra stabilization through favorable exchange energy.
• Copper (Z = 29): Expected $[\text{Ar}]\, 4s^2\, 3d^9$, but the actual configuration is $[\text{Ar}]\, 4s^1\, 3d^{10}$. A fully filled d subshell is similarly stabilized.

These exceptions arise because the energy difference between 4s and 3d is small, and the exchange energy gained from half-filled or fully filled subshells tips the balance. Similar exceptions appear in the heavier transition metals, lanthanides, and actinides, where f orbital filling introduces additional irregularities.

A few other points worth keeping straight:

• Ions: When transition metals lose electrons to form cations, the electrons come from the highest $n$ shell first. For $\text{Fe}^{2+}$, you remove the two 4s electrons from $[\text{Ar}]\, 4s^2\, 3d^6$, giving $[\text{Ar}]\, 3d^6$, not $[\text{Ar}]\, 4s^2\, 3d^4$.
• Excited states occur when an electron absorbs energy and jumps to a higher orbital. These are temporary and not the ground-state configuration.