V5History and Physics of the Atomic Model: Wave Mechanics and Quantum States

General and Inorganic Chemistry: Chapter 5 - Atomic Models

Review of Previous Material

• Mass Loss and Binding Energy:     * The mass loss per nucleon ($u$) is plotted against the mass number ($A$).     * Iron ($Fe$) represents the point of maximum stability with the lowest mass per nucleon.     * Binding energy per nucleon ($MeV$) increases rapidly for light nuclei and stays relatively high, peaking near iron, before gradually decreasing for heavier elements like Uranium ($^{238}U$).

• Types of Radioactivity:     * Radioactive preparations in lead capsules can emit three types of radiation, distinguishable by their behavior in an electric field:         * Alpha ($\alpha$)-rays: Positively charged particles.         * Beta ($\beta$)-rays: Negatively charged particles (electrons).         * Gamma ($\gamma$)-rays: Neutral electromagnetic radiation (no deflection).

• Electromagnetic Spectrum:     * Ranges from high energy, small wavelength, and large frequency (Gamma rays, X-rays, Ultraviolet) to low energy, large wavelength, and small frequency (Infrared, Microwaves, Radio waves).     * Visible Light Range: Approximately $400\,nm$ (violet) to $800\,nm$ (red).

• Hydrogen Line Spectrum:     * Occurs when light from a hydrogen source (burner/probe) is passed through a slit and prism.     * Series of spectral lines based on electron transitions to specific energy levels ($n$):         * Lyman Series: Transitions to $n=1$ (Ultraviolet).         * Balmer Series: Transitions to $n=2$ (Visible light).         * Paschen Series: Transitions to $n=3$ (Infrared).         * Brackett Series: Transitions to $n=4$.         * Pfund Series: Transitions to $n=5$.

• Niels Bohr (1913):     * Proposed the Bohr model of the atom.     * Limitation: Interpreting spectra of higher (more complex) elements becomes increasingly difficult with the Bohr model.

The Wave Character of the Electron: de Broglie Relation

• Diffraction (Beugung):     * Definition: The deflection of waves at an obstacle, leading to a change in the propagation direction of waves.     * Diffraction occurs through the creation of new waves along a wavefront.

• Louis Victor Pierre Raymond de Broglie (1892-1987):     * Awarded the Nobel Prize in Physics (1929).     * Wave-Particle Dualism (1924): Moving particles must possess wave properties. Diffraction and interference (superposition of waves) also occur with electrons.

• Mathematical Derivation:     * Known for photons: description as wave or particle (Rest mass = 0).     * Planck Equation: $E = h \cdot \nu$ where $\nu = c / \lambda$.     * Einstein Equation: $E = m \cdot c^2$.     * Combining these for electromagnetic radiation: $\lambda = \frac{h}{m \cdot c}$.     * Substituted for a particle of mass $m$ and velocity $v$: $\lambda = \frac{h}{m \cdot v}$.     * Impuls (Momentum): $p = m \cdot v$, thus $\lambda = \frac{h}{p}$.

• Matter Waves:     * Every particle can be assigned a wave character.     * The wavelength is determined by mass and speed.     * For ordinary/macroscopic objects, wavelengths are so extremely small that wave properties are undetectable.

• Davisson-Germer Experiment (1927):     * Demonstrated that an aluminum foil diffracts an electron beam (energy $600\,eV$, $\lambda = 50\,pm$) in exactly the same way it diffracts X-rays ($energetic electromagnetic radiation$, $\lambda = 71\,pm$).     * Confirmed that the wavelength of an electron beam is correctly represented by the de Broglie relation.     * Stability Condition: An electron is only in a stable state if the electron-wave is time-invariant (a standing wave) on the Bohr circular orbit.     * Condition for standing wave: $2 \cdot r_n \cdot \pi = n\lambda$ (where $n = 1, 2, 3, \dots$).

Heisenbergsche Unschärferelation (Heisenberg Uncertainty Principle)

• Werner Heisenberg (1901-1976):     * Awarded the Nobel Prize in Physics (1932).     * Core Principle: It is impossible to simultaneously determine the position ($x$) and momentum ($mv$) of matter waves.     * Equation: $\Delta x \cdot \Delta (m \cdot v) \geq \frac{h}{4\pi}$.

• Implications for Atomic Models:     * Bohr's model treated the electron as a moving particle on a calculated path, which requires knowing velocity and location simultaneously.     * To locate a small electron, light with a very short wavelength (high energy) is required. This light hits the electron and changes its momentum drastically.     * Therefore, a definitive statement about electron orbits (paths) cannot be made.

• Solution - Wave Mechanics:     * Abandon the classical particle image. Describe the electron as a wave using the de Broglie relation and Erwin Schrödinger's approach, in accordance with Heisenberg's principle.

The Electron as a Standing Wave

• 1D-Case (Vibrating String):     * The electron behaves like a standing wave (Electron in a box).     * Amplitude: Given by the wave function $\Psi$ at location $x$.     * Wellenfunktion (Wave function): $\Psi_n = \sin\left(\frac{\pi \cdot n \cdot x}{L}\right)$ where $n = 1, 2, 3, \dots$ and $x = \text{box length}$.     * Nodes (Knotenpunkte): There are $n-1$ nodes where the amplitude is zero. No charge density or probability of presence exists at these nodes.     * Energy: Higher $n$ results in higher energy ($E \propto n^2$).

• Probability and Charge Density:     * In light, intensity is proportional to the square of the amplitude.     * For electrons, $\Psi^2$ is proportional to the Ladungsdichte (Charge Density).     * The charge of the moving electron is distributed in space as a Ladungswolke (Charge Cloud).     * Aufenthaltswahrscheinlichkeit (Probability of Presence): Derived from $\Psi^2$. Integrating over partial volumes between nodal planes gives the total probability.     * Example $n=3$: The electron is divided into 3 areas; this is a better description as a wave than as a particle.

• Generalization to 2D and 3D:     * 2D Case (Vibrating Drumhead): Requires two quantum numbers, $n$ and $l$.     * 3D Case: Requires three quantum numbers: $n$, $l$, and $m$.     * Nodal lines in 2D become Knotenfl\u00e4chen (Nodal Surfaces) in 3D.     * For an electron inside a hollow sphere, there are spherical ($l=0$) and planar nodal surfaces.     * The wave function $\Psi$ has opposite signs on different sides of the nodal surfaces.

The Schrödinger Equation

• Erwin Schrödinger (1887-1961):     * Awarded the Nobel Prize in Physics (1933).     * Basis of wave mechanics using a differential equation.

• The Equation:     * $\hat{H} \cdot \Psi = E \cdot \Psi$     * Variables:         * $\hat{H}$: Hamilton Operator.         * $E$: Electron energy eigenvalues.         * $\Psi$: Wellenfunktion (Wave function), a function of spatial coordinates $(x, y, z)$.     * Solutions are called Eigenfunctions.

• Hydrogen Atom Application:     * Can be solved exactly for the Hydrogen (H) atom.     * Separation of variables in polar coordinates $(r, \theta, \phi)$: $\Psi_{n,l,m_l} = [N] \cdot [R_{n,l}(r)] \cdot [\chi_{l,m_l}(\theta, \phi)]$         * $[N]$: Normalization constant.         * $R_{n,l}(r)$: Radial function (extent of the charge cloud).         * $\chi_{l,m_l}(\theta, \phi)$: Angular function (shape and spatial orientation).

• Orbitals:     * H-atom eigenfunctions are specifically called Orbitals.     * They are defined by three quantum numbers ($n, l, m_l$).

Spatial Distribution and Radial Probability

• Radial Probability Density:     * Given by $4\pi r^2 \Psi^2$.     * Represents the probability of finding the electron at a specific distance $r$ from the nucleus across all directions.     * The maximum of this function for the 1s state of H is at $a_0 = 53\,pm$, which is identical to the Bohr Radius.     * The curve is asymptotic to the x-axis, meaning the electron can be at any finite distance from the nucleus (never zero probability).

The Four Quantum Numbers

• 1. Hauptquantenzahl (Principal Quantum Number) $n$:     * Determines the energy and size of an orbital.     * Values: $n = 1, 2, 3, \dots$     * Energy Level Equation: $E_n = -\frac{m \cdot e^4}{8 \cdot \epsilon_0^2 \cdot h^2 \cdot n^2}$.     * As $n$ increases, energy levels get closer together.     * Energy levels are designated as Shells: $n=1 (K)$, $2 (L)$, $3 (M)$, $4 (N)$, $5 (O)$.     * Ionisierungsenergie (Ionization Energy): Energy needed to remove an electron to infinity. For H ground state, it is $+13.6\,eV$ ($H \rightarrow H^+ + e^-$).

• 2. Nebenquantenzahl (Secondary/Azimuthal Quantum Number) $l$:     * Also known as the Bahndrehimpulsquantenzahl.     * Determines the shape (Gestalt) of the orbital.     * Values: $l = 0 \dots (n-1)$.     * Letter Symbols (from spectral line descriptions):         * $l=0: s\text{ (sharp)}$         * $l=1: p\text{ (principal)}$         * $l=2: d\text{ (diffuse)}$         * $l=3: f\text{ (fundamental)}$

• 3. Magnetquantenzahl (Magnetic Quantum Number) $m_l$:     * Determines the spatial orientation of the orbital relative to an external magnetic field (Zeeman Effect).     * Values: $m_l = -l, \dots, 0, \dots, +l$.     * Number of orbitals per subshell: $2l + 1$.         * $l=0 (s)$: 1 orbital.         * $l=1 (p)$: 3 orbitals ($p_x, p_y, p_z$).         * $l=2 (d)$: 5 orbitals ($d_{xy}, d_{xz}, d_{yz}, d_{z^2}, d_{x^2-y^2}$).         * $l=3 (f)$: 7 orbitals.

• 4. Spinquantenzahl (Spin Quantum Number) $m_s$:     * Required for full description of an electron.     * Defines the orientation of the electron spin (intrinsic angular momentum) relative to a magnetic field.     * Values: $m_s = +\frac{1}{2}, -\frac{1}{2}$.     * Two electrons with opposite spins have their magnetic moments canceled out.

Overview of Atomic Orbitals and Quantum States

• Quantum States for Hydrogen (up to n=4):     * $n=1 (K)$: 1 subshell (1s), 1 orbital, 2 states total.     * $n=2 (L)$: 2 subshells (2s, 2p), 4 orbitals ($1+3$), 8 states total.     * $n=3 (M)$: 3 subshells (3s, 3p, 3d), 9 orbitals ($1+3+5$), 18 states total.     * $n=4 (N)$: 4 subshells (4s, 4p, 4d, 4f), 16 orbitals ($1+3+5+7$), 32 states total.     * General formula: Total states per shell = $2n^2$.

• Orbital Symmetries:     * s-orbitals: Constant angular function, therefore kugelsymmetrisch (spherically symmetric).     * p and d orbitals: Dependent on angular functions, creating lobes and directional preference.     * Degeneracy: Without an external magnetic field, the three p-orbitals (or five d-orbitals) have the same energy; they are said to be entartet (degenerate).

• Visual Representations:     * Contour Diagrams: Lines enclosing 99% of the total charge. Colored lines indicate nodal surfaces.     * Grenzfl\u00e4chendarstellung (Boundary Surface): Displays the region of highest probability. Signs ($+, -$) in diagrams represent the mathematical sign of the wave function $\Psi$, not electrical charge.

Summary of Orbitals

• Orbitals are areas described by an electron's wave function.

• They represent allowed energy states of electrons.

• Spatial extent and shape result from the solution of the Schrödinger equation.

• Orbitals are not "real" paths like Bohr's orbits, but models of the most probable electron density.

Brogile Beziehung

• beschreibt, dass Partikel wie Elektronen sowohl Wellen- als auch Teilcheneigenschaften aufweisen, was eine fundamentale Grundlage für das Verständnis der Quantenmechanik darstellt.

  • Planck-Gleichung: E = h ▪ ν → ν = c / λ c = Lichtgeschwindigkeit

    → E = h ▪ c/λ

    mit Einstein-Gleichung E = mc2 ergibt sich:

    λ = h/m▪c für das Photon bzw. die el.-magn. Strahlung

    λ = h/m▪c → λ = h/m▪v mv = Impuls

    λ = h/m▪v → Wellenlänge eines fliegenden Teilchens

• Stehende Welle ist eine Welle, die nicht durch den Raum wandert, sondern an bestimmten Punkten, den sogenannten Knoten, ihre Amplitude stets null hat. Diese Art der Welle entsteht durch die Überlagerung zweier entgegengesetzt laufender Wellen. Stehende Welle im 3D Fall

• drei Quantenzahlen benötigt: n, l, und m

• Schrödingergleichung beschreibt die zeitliche Entwicklung des quantenmechanischen Zustands eines Systems und ist fundamental für die Beschreibung von Wellenfunktionen in der Quantenmechanik.

• Aufenhaltswahrscheinlichkeit gibt an, mit welcher Wahrscheinlichkeit sich ein Teilchen an einem bestimmten Ort im Raum befindet. Diese Wahrscheinlichkeit wird durch das Quadrat der Betrags der Wellenfunktion beschrieben, was eine zentrale Rolle in der Quantenmechanik spielt.

• Hauptquantenzahl (n): Bestimmt das Energieniveau eines Elektrons in einem Atom und gibt an, in welcher Schale sich das Elektron befindet. (bestimmt die Größe des Orbitals)

  • K (n=1) L(n=2) M(n=3) N(n=4) O(n=5) Schale

• Nebenquantenzahl (l): Definiert die Form形狀 des Orbitals und beschreibt die Anzahl der Unterzustände innerhalb eines Energieniveaus, wobei l Werte von 0 bis n-1 annehmen kann.

  • s p d f Zustand

  • Zustand: Diese Bezeichnungen stehen für die verschiedenen Orbitalformen, die mit unterschiedlichen Nebenquantenzahlen verbunden sind:

    • s-Zustand (l = 0): Kugelsymmetrische Orbitalform

    • p-Zustand (l = 1): Hantelförmig 啞鈴狀

    • d-Zustand (l = 2): Komplexere Formen, die zwischen den Achsen liegen

    • f-Zustand (l = 3): Sehr komplexe Orbitale, die in der Regel für schwere Elemente relevant sind.

• Magnetquantenzahl (m_l): Gibt die Orientierung des Orbitals im Raum an und kann Werte von -l bis +l annehmen. Diese Quantenzahl ist entscheidend für das Verständnis der räumlichen Verteilung von Elektronen in Atomen.

  • 0 ein s Zustand

  • -1 0 +1 drei p Zustand

  • -2 -1 0 +1 +2 fünf d Zustände: Diese Orbitale sind noch komplexer und können mehrere Elektronen beherbergen, die sich in unterschiedlichen Energieniveaus befinden. Dies hat Auswirkungen auf die chemischen Eigenschaften der Elemente.

  • 3 -2 -1 0 +1 +2 +3 sieben f Zustände: Diese Orbitale spielen eine wichtige Rolle in der Chemie schwerer Elemente, da sie dazu beitragen, die Vielfalt und Komplexität der chemischen Bindungen zu erklären.

• Spinquantenzahl beschreibt die intrinsische Drehimpulsquantisierung eines Elektrons und kann die Werte +1/2 oder -1/2 annehmen. Diese Quantenzahl ist entscheidend für das Verständnis der Elektronenkonfiguration und der Paarungsregel in Atomen.