22.1 The Structure of the Atom

Atomic structure is the foundation of modern physics and connects directly to how matter behaves at its smallest scale. From Rutherford's gold foil experiment to the quantum mechanical model, each step in our understanding explains why atoms emit light at specific wavelengths, why electrons occupy discrete energy levels, and how atomic interactions give rise to chemical properties.

Atomic Structure

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Rutherford's Gold Foil Experiment

Rutherford's team fired positively charged alpha particles at a thin sheet of gold foil. Based on the "plum pudding" model (which assumed positive charge was spread evenly throughout the atom), they expected nearly all particles to pass straight through with only slight deflection.

What they actually observed:

• Most alpha particles passed through undeflected, suggesting the atom is mostly empty space.
• Some particles deflected at large angles, indicating they came close to a concentrated region of positive charge.
• A very small number bounced almost straight back toward the source, meaning they hit something extremely dense and positively charged head-on.

These results led to the nuclear model: the atom's positive charge and nearly all its mass are packed into a tiny, dense nucleus, while electrons orbit at relatively large distances. The nucleus is roughly 100,000 times smaller than the atom itself, which is why most alpha particles never came near it.

Atomic Spectra

When atoms gain energy (from heat, electricity, etc.), their electrons jump to higher energy levels. When those electrons drop back down, they release photons at specific wavelengths. This produces two complementary types of spectra:

• Emission spectrum: Bright colored lines on a dark background. Each line corresponds to a specific electron transition from a higher to a lower energy level. Every element produces a unique pattern of lines (hydrogen's visible series, for example, includes red at 656 nm and blue-violet at 434 nm).
• Absorption spectrum: Dark lines on a continuous bright background. When white light passes through a cool gas, atoms absorb photons at the exact same wavelengths they would emit. The missing wavelengths show up as dark lines.

Both types of spectra serve as evidence that energy levels in atoms are discrete (quantized), not continuous. Astronomers use these spectral fingerprints to identify the composition of stars and distant gases.

Bohr Model vs. Quantum Model

The Bohr model (1913) pictures electrons traveling in fixed, circular orbits around the nucleus at specific radii. Each orbit corresponds to a definite energy level, and electrons emit or absorb photons when they jump between orbits. It works well for hydrogen but struggles with multi-electron atoms.

The quantum mechanical model replaces fixed orbits with orbitals, which are probability distributions showing where an electron is likely to be found. Electrons are described by four quantum numbers (covered below), and energy levels include sublevels (s, p, d, f) that the Bohr model can't account for.

What they share: both models predict discrete energy levels and correctly explain emission and absorption spectra for hydrogen.

Where they differ: the quantum model handles complex atoms, predicts orbital shapes, and accounts for effects like electron-electron repulsion. The Bohr model is simpler and useful as a starting point, but it's fundamentally limited.

Quantum Mechanics and Atomic Transitions

Electron Energy Transitions

When an electron moves between energy levels, the energy change is:

$\Delta E = E_f - E_i$

where $E_f$ is the final energy level and $E_i$ is the initial level. If $\Delta E$ is negative, the electron dropped to a lower level and a photon was emitted. If $\Delta E$ is positive, the electron absorbed energy and jumped up.

For hydrogen-like atoms (one electron), the Bohr model gives the energy of level $n$ as:

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

where $n$ is the principal quantum number (1, 2, 3, ...). The negative sign means the electron is bound to the nucleus; $n = 1$ is the most tightly bound (ground state) at $-13.6$ eV, while higher $n$ values are closer to zero (less bound).

For multi-electron atoms, this formula needs correction terms that account for electron-electron repulsion and relativistic effects, but for an honors course, the hydrogen formula is your primary tool.

Example: An electron drops from $n = 3$ to $n = 2$ in hydrogen.

1. Calculate $E_3 = -\frac{13.6}{9} = -1.51$ eV

2. Calculate $E_2 = -\frac{13.6}{4} = -3.40$ eV

3. Find $\Delta E = E_2 - E_3 = -3.40 - (-1.51) = -1.89$ eV

4. The emitted photon carries 1.89 eV of energy (the negative sign just tells you energy was released).

Photon Characteristics in Transitions

The energy of the emitted or absorbed photon relates directly to its frequency and wavelength:

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

where:

• $h = 6.626 \times 10^{-34}$ J·s (Planck's constant)
• $\nu$ = frequency (Hz)
• $c = 2.998 \times 10^8$ m/s (speed of light)
• $\lambda$ = wavelength (m)

To find frequency from an energy change: $\nu = \frac{\Delta E}{h}$

To find wavelength from an energy change: $\lambda = \frac{hc}{\Delta E}$

Note: if you're working in eV, convert to joules first ($1 \text{ eV} = 1.602 \times 10^{-19}$ J) before plugging into these equations, since $h$ is in J·s.

Quantum Mechanical Model Features

In the quantum model, each electron in an atom is described by four quantum numbers:

1. Principal quantum number ($n$): Determines the main energy level and the overall size of the orbital. $n = 1, 2, 3, ...$ Higher $n$ means higher energy and larger orbital.

2. Angular momentum quantum number ($l$): Determines the shape of the orbital. $l$ ranges from 0 to $n - 1$. The values correspond to orbital types: $l = 0$ (s, spherical), $l = 1$ (p, dumbbell), $l = 2$ (d), $l = 3$ (f).

3. Magnetic quantum number ($m_l$): Determines the orientation of the orbital in space. $m_l$ ranges from $-l$ to $+l$. For example, a p orbital ($l = 1$) has three orientations: $m_l = -1, 0, +1$.

4. Spin quantum number ($m_s$): Describes the intrinsic spin of the electron, either $+\frac{1}{2}$ (spin up) or $-\frac{1}{2}$ (spin down).

Three rules govern how electrons fill orbitals:

• Pauli exclusion principle: No two electrons in the same atom can share all four quantum numbers. This means each orbital holds at most two electrons (with opposite spins).
• Aufbau principle: Electrons fill orbitals starting from the lowest available energy. The filling order is 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, and so on.
• Hund's rule: When filling orbitals of equal energy (like the three 2p orbitals), electrons occupy them singly with parallel spins before any orbital gets a second electron.

Wave Function and Electron Behavior

The wave function ($\psi$) is a mathematical function that describes the quantum state of an electron. You find it by solving the Schrödinger equation for a given atom, which yields the allowed energy states and their corresponding wave functions.

The wave function itself doesn't have a direct physical meaning you can measure. What matters is $|\psi|^2$, the probability density. This quantity tells you the likelihood of finding the electron in a particular small region of space. Where $|\psi|^2$ is large, you're more likely to detect the electron; where it's small, you're less likely.

Atomic orbitals are the three-dimensional shapes you get by mapping out these probability distributions. An s orbital is spherical, a p orbital has two lobes, and d and f orbitals have increasingly complex shapes.

Electron configuration describes how electrons are arranged across all the orbitals in an atom, following the aufbau principle, Pauli exclusion principle, and Hund's rule. For example, oxygen (8 electrons) has the configuration 1s² 2s² 2p⁴, meaning its 2p subshell has four electrons spread across three orbitals according to Hund's rule.