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Section 2: Quantum Theory and Electronic Structure of Atoms - Vocabulary Flashcards

2.1 From Classical Physics to Quantum Theory

  • Planck’s proposal: energy is emitted or absorbed in discrete units (quanta).
  • Quantum of energy for EM radiation: E = h\nu = \frac{hc}{\lambda}
  • Implication: matter and radiation exhibit quantization; classical physics could not fully describe these processes.
  • Contrasts with Classical Physics: Classical view allowed continuous energy exchange; Planck’s view requires fixed energy quanta.
  • Relevance: foundation for quantum theory, governing electronic structure and transitions.

Properties of Waves

  • A wave transmits energy; properties: Amplitude, Frequency (\nu), Wavelength (\lambda), Speed (u).
  • Fundamental relation: u = \lambda \nu
  • For light, speed: c = 3.00 \times 10^{8}\ \mathrm{m\,s^{-1}}

Electromagnetic Radiation

  • All EM radiation described by wave-like electric and magnetic fields (e.g., visible light, X-rays).
  • Characterized by frequency or wavelength.

2.1 The Electromagnetic Spectrum

  • Visible band: \lambda{\text{visible}} \approx 380\text{–}750\ \text{nm}, \nu{\text{visible}} \approx 4.0 \times 10^{14} \text{–} 7.5 \times 10^{14}\ \text{Hz}.
  • Black-body radiation: hot objects emit EM radiation; intensity depends on temperature.

2.1 Practice Exercises

  • FM station at 103.4 MHz: Wavelength \lambda \approx 2.90\ \text{m} (2.90 \times 10^{9}\ \text{nm} or 2.90\times 10^{4}\ \text{Å}).
  • UV light for pathogen (\lambda = 254\ \text{nm}): Frequency \nu \approx 1.18 \times 10^{15}\ \mathrm{Hz}.

2.1 Planck’s Quantum Theory

  • Planck proposed EM energy is quantized into photons: E = h\nu = \frac{hc}{\lambda}.
  • Consequences: A quantum is the fundamental unit of EM energy; emitted/absorbed energy depends on photon energy.
  • Example: Energy of a photon from red light (\lambda = 652\ \text{nm}): E \approx 3.05 \times 10^{-19}\ \mathrm{J}.

2.1 Planck’s Quantum Theory – Bond Breaking

  • Problem: Determine max wavelength to break single O–O (142 kJ/mol) and O=O (498 kJ/mol) bonds.
  • Assumptions: 1 photon/bond; 1 mole = N_A particles.
  • Answers: For O–O, \lambda{\max} = 843\ \text{nm}; for O=O, \lambda{\max} = 240\ \text{nm}.

2.2 Atomic Spectroscopy and The Bohr Model

  • Emission spectra: Atoms emit EM radiation when heated; separated by prism into characteristic line spectra.
  • Hydrogen emission spectrum: discrete lines, not continuous.
  • Bohr’s model postulates:
  1. Only orbits of certain radii are permitted.
  2. Electron in permitted orbit has quantized angular momentum: L = n\hbar\quad (n = 1,2,3,…).
  3. Energy emitted/absorbed only when electron changes allowed energy levels.
  • Energies for hydrogen (Bohr): E_n = -\frac{2.18 \times 10^{-18}}{n^2}\ \mathrm{J}.
  • Transitions and photon energy: \Delta E = Ef - Ei = 2.18 \times 10^{-18}\left(\frac{1}{nf^2} - \frac{1}{ni^2}\right)\ \mathrm{J}.
  • Wavelengths for transitions (Rydberg): \frac{1}{\lambda} = R{\infty}\left(\frac{1}{nf^2} - \frac{1}{ni^2}\right), where R{\infty} \approx 1.097 \times 10^{7}\ \mathrm{m^{-1}}.
  • Practical note: Bohr model explains hydrogen but not multi-electron atoms; led to quantum mechanics.

2.3 The Wavelength Nature of Matter

  • De Broglie hypothesis: matter has a wavelength: \lambda = \frac{h}{p} = \frac{h}{mv}.
  • This wave-particle duality is foundational to quantum mechanics.

2.4 Quantum Mechanics and The Atom

  • Electron treated as a wave with allowed energy and spatial distribution.
  • Key ideas:
    • Electron wave fits as a standing wave around the nucleus in certain configurations.
    • Heisenberg Uncertainty Principle: cannot simultaneously specify exact position and momentum of an electron.
    • Time-independent Schrödinger equation: \hat{H}\psi = E\psi; wavefunction (\psi) encodes electron probability density.
  • Electron exists as a standing wave around the nucleus within an orbital.
  • Quantum numbers define orbitals: n, l, ml, and spin ms.

2.4 Quantum Numbers and Orbitals

  • Principal quantum number: n = 1, 2, 3, \dots (energy level, average distance).
  • Azimuthal (angular momentum) quantum number: l = 0, 1, \dots, n-1 (subshell type: s, p, d, f).
  • Magnetic quantum number: m_l = -l, \dots, +l.
  • Spin magnetic quantum number: m_s = -\tfrac{1}{2}, +\tfrac{1}{2}.
  • Orbital designations:
    • s: (l=0), m_l = 0 (1 orbital)
    • p: (l=1), m_l = -1, 0, +1 (3 orbitals)
    • d: (l=2), m_l = -2, \dots, +2 (5 orbitals)
    • f: (l=3), m_l = -3, \dots, +3 (7 orbitals)
  • Orbital energy ordering: Hydrogen-like depends on n; multi-electron depends on both n and l.

2.4 The Electron Spin Quantum Number (ms)

  • Describes intrinsic angular momentum (spin) of electrons.
  • Spin states: m_s = -\tfrac{1}{2} \quad \text{or} \quad +\tfrac{1}{2}.
  • Experiment (Stern–Gerlach): hydrogen atoms split into two lines, reflecting two spin states.
  • Contributes to electronic structure and magnetic properties.

2.5 The Shape of Atomic Orbitals

  • Orbitals are regions of highest probability (\psi^2) for finding an electron.
  • S orbitals (l=0): spherical, radial nodes increase with n.
  • P orbitals (l=1): dumbbell-shaped, three orientations (x, y, z axes), one nodal plane.
  • D orbitals (l=2): five orbitals, clover/lemon shapes, two nodal planes.
  • F orbitals (l=3): seven orbitals, more complex shapes.
  • Orbitals in subshell: s: 1, p: 3, d: 5, f: 7.
  • Number of orbitals in a shell: n^2.
  • Radial nodes: n - l - 1.

2.5 The Energies of Orbitals (2.6 context)

  • One-electron (H) system: energy depends only on n (E_n = -\frac{2.18\times 10^{-18}}{n^2}\ \mathrm{J}).
  • Multi-electron atoms: energy depends on both n and l due to electron interactions (e.g., 2s vs 2p).

2.6 Electron Configurations

  • Purpose: specify electron arrangement in ground state.
  • Notation: orbitals with superscripts (e.g., 1s^2, 2p^6).
  • Pauli Exclusion Principle: no two electrons in same atom have same four quantum numbers; max 2 electrons per orbital with opposite spins.
  • Example: Helium: 1s^2; Hydrogen: 1s^1.
  • Electron pairing and magnetism: Paired electrons have opposite spins; unpaired electrons lead to paramagnetism; diamagnetic substances have all electrons paired.
  • Hund’s Rule: Electrons occupy degenerate orbitals singly with parallel spins before pairing.
  • Aufbau Principle: electrons fill lowest-energy orbitals first.
  • Noble-gas shorthand: core configuration of nearest noble gas + valence electrons.
  • Practical filling order: 1s \to 2s \to 2p \to 3s \to 3p \to 4s \to 3d \to 4p \to 5s \to \dots
  • Anomalies: Cr ([Ar] 3d^5 4s^1) and Cu ([Ar] 3d^{10} 4s^1) due to subshell energy differences.
  • Ground-state configurations: noble-gas core + valence electrons.

2.6 Building-Up and Related Rules (Recap)

  • Rules:
    • Shell n contains subshells l = 0, \dots, n-1.
    • Subshell l contains 2l + 1 orbitals.
    • Each orbital holds max 2 electrons (Pauli).
    • Max electrons in a principal level: 2n^2.
    • Aufbau principle: practical method for ground-state configurations.
    • Periodic table guides filling order.

2.7 Emission/Absorption and Energy-Level Diagrams

  • Energy-level diagrams show allowed energy levels; transitions correspond to photon emission/absorption matching energy gaps.

Additional Notes and Formulas (Summary of Key Items)

  • Planck’s relation: E = h\nu = \frac{hc}{\lambda}
  • Speed of light: c = 3.00 \times 10^{8}\ \mathrm{m\,s^{-1}}
  • Wave relation: u = \lambda \nu
  • De Broglie relation: \lambda = \frac{h}{p} = \frac{h}{mv}
  • Heisenberg Uncertainty Principle: \Delta x\,\Delta p \ge \frac{\hbar}{2}
  • Schrödinger equation: \hat{H}\psi = E\psi
  • Hydrogen energy levels (Bohr): E_n = -\frac{2.18\times10^{-18}}{n^2}\ \mathrm{J}
  • Hydrogen energy difference: \Delta E = 2.18\times10^{-18}\left(\frac{1}{nf^2} - \frac{1}{ni^2}\right)\ \mathrm{J}
  • Rydberg relation: \frac{1}{\lambda} = R{\infty}\left(\frac{1}{nf^2} - \frac{1}{n_i^2}\right)
  • Orbitals in subshell: s: 1, p: 3, d: 5, f: 7; in shell: n^2; max electrons in shell: 2n^2.
  • Quantum numbers: n = 1, 2, 3, \dots; l = 0, 1, \dots, n-1; ml = -l, \dots, +l; ms = -\tfrac{1}{2}, +\tfrac{1}{2}.

Quick Problems Reference

  • Wavelength of 103.4 MHz FM transmission: \lambda \approx 2.90\ \text{m}
  • UV wavelength 254 nm: frequency \nu \approx 1.18 \times 10^{15}\ \text{Hz}
  • Photon energy for 652 nm: E \approx 3.05 \times 10^{-19}\ \mathrm{J}
  • Bond-breaking wavelengths: \lambda{\max}(O-O) = 843\ \text{nm}, \lambda{\max}(O=O) = 240\ \text{nm}.

Connections to Foundational Principles and Real-World Relevance

  • Quantum theory explains chemical bonding, spectroscopy, and electronic structure (reactions, materials, technologies).
  • Wave-particle duality leads to models for atomic/molecular orbitals.
  • Understanding spin and electron configurations explains periodic trends and elemental behavior.
  • Energy quantization and spectral lines link chemistry to astronomy/astrophysics.

Ethical/Philosophical/Practical Implications

  • Quantum theory challenges classical intuition, introducing uncertainty and probabilistic descriptions.
  • Practical applications in electronics, photonics, medical imaging.
  • Informs material design and environmental considerations.

References to Figures/Examples (Conceptual)

  • Emission spectra and line spectra illustrate quantized transitions.
  • Hydrogen energy levels and spectral lines visualize electronic states.
  • Orbital shapes correspond to angular momentum quantum numbers.
  • Aufbau, Pauli, and Hund’s rules guide electron configurations and predict molecular properties.