Quantum-Mechanical Model of the Atom & Periodic Properties: Study Notes

Quantum-Mechanical Model of the Atom

Introduction to Quantum Mechanics

The quantum-mechanical model is the current scientific framework for understanding the behavior, location, and energy of electrons in atoms and molecules. Quantum mechanics is foundational to chemistry, explaining atomic trends, chemical bonding, and the properties of materials.

• Explains periodic trends: Differences between metals and non-metals, ion formation, atomic size, and bonding.

• Basis for technology: Lasers, computers, and other applications rely on quantum principles.

Electromagnetic Radiation: Wave Nature of Light

Electromagnetic radiation, or "light," encompasses a broad spectrum including radio waves, visible light, and X-rays. The study of light is deeply connected to the behavior of electrons.

• Wave properties: Light is described by its wavelength ($\lambda$) and frequency ($\nu$).

• Relationship: $c = \lambda \nu$, where $c = 3.00 \times 10^8$ m/s (speed of light).

• Frequency units: Hertz (Hz) or cycles per second ($s^{-1}$).

• Color and brightness: Different wavelengths correspond to different colors; amplitude relates to brightness.

Electromagnetic Spectrum

The electromagnetic spectrum arranges radiation by frequency and wavelength, from low-energy radio waves to high-energy gamma rays. Visible light occupies a small portion, ranging from about 400 nm (violet) to 750 nm (red).

Particle Nature of Light: Photons and Quantization

Einstein's photoelectric effect experiment showed that light energy is delivered in discrete packets called photons. This means light exhibits both wave and particle properties—a concept known as wave-particle duality.

• Energy of a photon: $E_n = h\nu$ or $E_n = \dfrac{hc}{\lambda}$

• Planck's constant: $h = 6.626 \times 10^{-34}$ J·s

• Total energy for multiple photons: $E_{total} = nh\nu$ or $E_{total} = n\dfrac{hc}{\lambda}$

Practice Example

• Calculate the frequency of red light ($\lambda = 715$ nm): $\nu = \dfrac{c}{\lambda}$

• Find the energy of a single photon: $E = h\nu$

• Energy of a mole of photons: $E_{total} = N_A h\nu$ (where $N_A$ is Avogadro's number)

Atomic Emission Spectra

Atoms and molecules emit light when they absorb energy and then release it. Passing this light through a prism reveals an emission spectrum—a unique pattern of wavelengths for each element.

• Quantized energy: Only specific energy changes are allowed, unique to each atom.

• Electron transitions: Electrons absorb energy and move to higher energy levels (excitation), then relax to lower levels, emitting photons.

• Each spectral line: Corresponds to a specific electron transition.

Diagram: Electron Transitions

• Excitation: Electron moves from lower to higher $n$ (energy level).

• Relaxation: Electron falls from higher to lower $n$, emitting a photon.

Bohr Model of the Atom

Niels Bohr proposed that atomic energy is quantized, with electrons occupying specific orbits at set distances from the nucleus. This model explained hydrogen's emission spectrum but not those of larger atoms.

• Energy of transition: $E_{photon} = 2.18 \times 10^{-18} \left[ \dfrac{1}{n_{final}^2} - \dfrac{1}{n_{initial}^2} \right]$

• Application: Calculate energy and wavelength for electron transitions (e.g., $n=4$ to $n=2$ in hydrogen).

Wave Behavior of Electrons: Quantum Mechanics Begins

Electrons, traditionally considered particles, also exhibit wave-like properties (de Broglie hypothesis). Heisenberg's uncertainty principle states that both position and velocity of an electron cannot be known simultaneously; only probabilities can be described.

• Probability maps: Statistical equations describe regions (orbitals) where electrons are likely to be found.

Schrödinger’s Equation and Quantum Numbers

Schrödinger developed a mathematical model for electron probability distributions, summarized by four quantum numbers:

• Principal quantum number ($n$): Indicates shell and energy level; $n \geq 1$.

• Angular momentum quantum number ($\ell$): Indicates subshell and orbital shape; $\ell = 0$ to $n-1$.

• Magnetic quantum number ($m_\ell$): Indicates orbital orientation; $m_\ell = -\ell$ to $+\ell$.

• Spin quantum number ($m_s$): Indicates electron spin; $m_s = +\frac{1}{2}$ or $-\frac{1}{2}$.

Quantum Number Details

• $n$ (shell): Larger $n$ means larger, higher-energy orbital.

• $\ell$ (subshell): $\ell=0$ (s, spherical), $\ell=1$ (p, dumbbell), $\ell=2$ (d, cloverleaf), $\ell=3$ (f, complex).

• $m_\ell$ (orbital): Number of orbitals per subshell is $2\ell+1$.

• $m_s$ (spin): Two possible values, differentiating electrons in the same orbital.

Practice Examples

• For $n=3$, possible $\ell$ values: 0, 1, 2 (s, p, d).

• For $\ell=2$, possible $m_\ell$ values: -2, -1, 0, +1, +2.

Orbital Structure and Visualization

Each principal energy level ($n$) contains $n$ subshells ($\ell$ values), and each subshell contains $2\ell+1$ orbitals. Each orbital can hold two electrons (with opposite spins).

Principal Level (n)	Sublevel ($\ell$)	Number of Orbitals
1	0 (s)	1
2	0 (s), 1 (p)	1 (s), 3 (p)
3	0 (s), 1 (p), 2 (d)	1 (s), 3 (p), 5 (d)
4	0 (s), 1 (p), 2 (d), 3 (f)	1 (s), 3 (p), 5 (d), 7 (f)

Orbital Shapes

• s orbital ($\ell=0$): Spherical, maximum probability near nucleus.

• p orbital ($\ell=1$): Dumbbell-shaped, three orientations ($m_\ell=-1,0,+1$).

• d orbital ($\ell=2$): Cloverleaf-shaped, five orientations.

• f orbital ($\ell=3$): Complex shapes, seven orientations.

Summary: Nature of the Electron

• Electrons exhibit both particle and wave properties.

• Energy levels and transitions are quantized, explaining emission spectra.

• Bohr model works for hydrogen; quantum mechanics is needed for larger atoms.

• Quantum numbers define electron energy and position in atoms.

Practice Problems

• List allowed values of $n$ (shells): $n=1,2,3,...$

• Which $n$ value is lowest in energy? ($n=1$) Closest to nucleus? ($n=1$)

• For $n=1$ to $n=4$, list allowed $\ell$ values and letter designations.

• For each subshell, indicate possible $m_\ell$ values.

• How many $m_\ell$ values for 3d? (Five: -2, -1, 0, +1, +2)

• Is $\ell=5$ possible when $n=5$? (No, $\ell$ must be less than $n$)

• For $n=6$, possible $\ell$ values: 0, 1, 2, 3, 4, 5; highest $\ell$ (5) has 11 orbitals.

• For $n=5$, $\ell=2$ (5d), possible $m_\ell$: -2, -1, 0, +1, +2; five orbitals, each with $n=5$, $\ell=2$, $m_\ell$ as above.

• Electron capacity for $n=5$: $2n^2 = 50$ electrons.

Periodic Properties of the Elements

Quantum Mechanics and the Periodic Table

The arrangement of elements in the periodic table reflects repeating patterns in properties, explained by quantum mechanics. The position and energy of valence electrons determine chemical reactivity.

• Periodic trends: Atomic number increases, electron configuration changes, leading to periodicity in properties.

Electron Configurations and Orbital Diagrams

Electron configuration is a shorthand notation for the arrangement of electrons in atomic orbitals. Orbital diagrams visually represent electron filling, including spin.

• Aufbau Principle: Electrons fill lowest energy orbitals first.

• Pauli Exclusion Principle: No two electrons in an atom have the same set of four quantum numbers; each orbital holds two electrons with opposite spins.

• Hund's Rule: Electrons occupy empty orbitals singly before pairing up, maximizing total spin.

Electron Configuration Notation

• Write sublevels in order of filling, with electron count as superscript (e.g., $1s^2 2s^2 2p^6$).

• Abbreviated form uses previous noble gas in brackets (e.g., [Ne]$3s^2 3p^5$).

Valence Electrons and Periodic Table Blocks

• Valence electrons: Electrons in the highest principal energy shell; determine chemical properties.

• Core electrons: Electrons in lower energy shells.

• Periodic table blocks: s, p, d, f blocks correspond to sublevel filling; period number matches principal energy level for s and p electrons.

Irregular Electron Configurations (Transition Metals)

• Some transition metals have electron configurations that differ from expected order due to small energy differences between s and d sublevels.

• These must be determined experimentally.

Element	Expected	Experimental
Cr	[Ar]4s23d4	[Ar]4s13d5
Cu	[Ar]4s23d9	[Ar]4s13d10
Pd	[Kr]5s24d8	[Kr]5s04d10

Properties and Electron Configuration

• Elements in the same column have similar chemical properties due to similar valence electron configurations.

• Alkali metals lose one electron to form cations with noble gas configurations; halogens gain one electron to form anions.

Electron Configurations of Ions

• When atoms form ions, electrons are added or removed from the highest energy orbitals.

• Example: Al atom ($1s^2 2s^2 2p^6 3s^2 3p^1$) vs. Al3+ ion ($1s^2 2s^2 2p^6$).

Shielding and Effective Nuclear Charge

Valence electrons experience reduced attraction to the nucleus due to repulsion from inner electrons (shielding). The net positive charge felt by an electron is the effective nuclear charge ($Z_{eff}$).

• $Z_{eff} \approx Z - \text{core electrons}$

Atomic and Ionic Radius Trends

• Atomic radius: Decreases across a period (increased $Z_{eff}$ pulls electrons closer); increases down a group (higher $n$).

• Ionic radius: Cations are smaller than neutral atoms; anions are larger due to electron repulsion.

Ionization Energy Trends

• Ionization energy (IE): Energy required to remove an electron.

• IE decreases down a group (higher $n$, electrons farther from nucleus).

• IE increases across a period (higher $Z_{eff}$, electrons more tightly bound).

Summary: Periodic Trends

• Atomic/ion size and ionization energy can be predicted using electron configuration, $Z_{eff}$, and periodic table position.

• Key factors: principal energy level ($n$), nuclear charge, shielding, electron repulsion.

Practice Problems

• Determine which atom/ion is larger or smaller based on $n$, number of protons, and electron repulsion.

• Predict ionization energy trends and explain using quantum mechanical principles.