Chapter 3 Notes: Electronic Structure and Periodic Properties of Elements

Energy and Energy Changes

• Energy is the capacity to do work or transfer heat; forms include kinetic and potential energy.

• Kinetic energy (Ek) is the energy of motion; a key form for chemists is thermal energy (random motion of atoms/molecules).

• Ek = \frac{1}{2} m u^2, where m is mass and u is velocity.

• Potential energy is stored energy due to position. Two chemist-relevant forms:

  • Chemical energy: energy stored in the structural units of substances.

  • Electrostatic energy: potential energy from interactions of charged particles; Eel = \frac{Q1 Q_2}{d}, with Q1 and Q2 charges separated by distance d.

• Kinetic and potential energy are interchangeable; the total energy of the universe is conserved (Law of Conservation of Energy): energy cannot be created or destroyed, only transformed.

• Example pathways illustrating energy conversion across scales:

  • Diver: Ep (macroscale) converts to Ek (macroscale); Ek,macroscale converts to Ek,nanoscale (motion of water, heat).

  • Water in a dammed lake → water rushing through turbines → electrical energy; moving baseball; diver on a cliff; gallon of gas.

Units of Energy

• SI unit of energy: the joule (J).

• A joule is the energy of a 2-kg mass moving at 1 m/s: $E_k = \frac{1}{2} m u^2 = \frac{1}{2} (2 \text{ kg}) (1 \text{ m s}^{-1})^2 = 1 \text{ J}.$

• Another definition: 1 J = 1 N·m (energy when a force of 1 N acts over 1 m).

• Large amounts are written in kilojoules: 1 kJ = 1000 J.

• Calorie (cal): originally defined as the energy needed to heat 1 g of water by 1°C; now defined as 1 cal = 4.184 J (exactly).

• Dietary Calorie (Cal): the big-C Calorie; 1 Cal = 1000 cal = 1 kcal.

• Radiant energy across the electromagnetic spectrum expands beyond visible light; the energy unit remains the joule.

Visible Light and the Electromagnetic Spectrum

• Electromagnetic radiation travels as waves; key properties:

  • Wavelength (\u039b): distance between successive peaks.

  • Frequency (\u03bd): number of waves passing a point per second (s^{-1}).

  • Amplitude: vertical distance from midline to peak or trough.

• Electromagnetic waves have both electric and magnetic field components; they share the same frequency and wavelength.

• Relationship between speed, wavelength, and frequency: c = \u03bb \u03bd = 2.998 × 10^8 m s^{-1} (in vacuum).

• The speed of light, wavelength, and frequency are linked by: $c = \lambda \nu.$

• The electromagnetic spectrum includes visible light as only a small portion; higher-frequency radiation has shorter wavelengths, and vice versa.

The Double-Slit Experiment and Wave-Particle Nature

• When light passes through two slits, an interference pattern forms, demonstrating wave-like behavior.

  • Constructive interference arises when waves are in phase; destructive interference when out of phase.

• This experiment supports the wave nature of light and underpins wave-particle duality.

• The speed of light in vacuum is a constant: $c = 2.998 \times 10^8 \text{ m s}^{-1}.$

• Core relations:

  • $\lambda = \frac{c}{\nu},\quad \nu = \frac{c}{\lambda}.$

Practice: Basic Radiant-Energy Questions

• Given wavelength or frequency, compute the other quantities using: $\lambda = \frac{c}{\nu}, \quad E = h \nu = \frac{h c}{\lambda}.$

Quantum of Energy: Planck and Photons

• Classical physics described energy as continuous; Planck proposed energy is quantized into discrete packets (quanta).

• Fundamental relation: $E = h \nu = \frac{h c}{\lambda},$ where h is Planck's constant: $h = 6.626 \times 10^{-34} \text{ J s}.$

• The photon concept: light behaves as stream of particles (photons) with energy E = h ν.

• The speed of light relation: $c = \lambda \nu.$

• Quick calculations:

  • Calculate photon energy for a given wavelength: use $E = \frac{h c}{\lambda}.$

  • Alternate expression: E = h \nu = \frac{h c}{\lambda}.$n

• Practice problems include:

  • Energy for photons of various wavelengths (e.g., 501 nm, 50.1 nm).

  • Energy-related questions for photons with given frequencies.

  • Wavelengths corresponding to specified energies or frequencies.

  • For a 337.1 nm nitrogen laser, energy per photon: use E = hc/\lambda.

Bohr’s Theory of the Hydrogen Atom and Atomic Line Spectra

• Emission spectrum: energized atoms emit light at discrete wavelengths (line spectra) unique to each element; continuous spectra occur for hot solids or sunlight.

• Bohr model: electron transitions between energy levels produce photons; energy levels are quantized.

• Key equation for hydrogen: En = -2.18 × 10^{-18} J / n^2, where n is a positive integer (n = 1 is ground state).

• As n increases, En becomes less negative (less tightly bound).

• Emission/absorption: energy absorbed moves electron from ground (n = 1) to excited states (n > 1); emission occurs when electron drops to lower energy levels.

• Radiant energy change during a transition (emission): \Delta E = -2.18 \times 10^{-18}\text{ J} \left( \frac{1}{nf^2} - \frac{1}{ni^2} \right) where ni > nf.

• Hydrogen spectral lines are described by the Rydberg formula for wavelength:

  • \frac{1}{\lambda} = R\infty \left( \frac{1}{nf^2} - \frac{1}{n_i^2} \right), with R∞ = 1.09737317 × 10^7 m^{-1}.

• Bohr emission series (transitions to n_f = 1, 2, 3, 4): Lyman (UV), Balmer (visible/UV), Paschen (IR), Brackett (IR).

• Ground state is n = 1; excited states have n > 1. The energy becomes more negative as n decreases toward 1.

• Practice problems include:

  • Wavelengths/emitted energy for specific transitions (e.g., n = 4 → 3; n = 4 → 2).

  • Energies and wavelengths for various transitions in hydrogen; converting to photon energy or to kJ per mole.

Wave Properties of Matter: de Broglie and Uncertainty

• Louis de Broglie proposed that particles can exhibit wave-like properties; electrons behave like standing waves with allowed wavelengths.

• de Broglie relation: \lambda = \frac{h}{m u},$where m is mass and u is velocity.</p></li><li><p>Wave-particle duality connects matter waves to momentum.</p></li><li><p>Heisenberg Uncertainty Principle:$\Delta x \; \Delta p \ge \frac{\,h\,}{4\pi}.$Here note Δp = m Δu for a particle of mass m.</p></li><li><p>Example calculation: an electron with velocity 4.99×10^6 m/s (±5$n\in {1,2,3,\ldots}$, designates size and energy level.</p></li><li><p>Angular momentum quantum number:$l \in {0,1,2,\ldots, n-1}$, designates orbital shape.</p></li><li><p>Magnetic quantum number:$m_l \in {-l, -l+1, \ldots, 0, \ldots, +l}$, designates orientation.</p></li><li><p>Electron spin quantum number:$m_s \in {+\tfrac{1}{2}, -\tfrac{1}{2}}$, designates spin direction.</p></li></ul></li><li><p>Orbital shapes common: s (l = 0, spherical), p (l = 1, dumbbell), d (l = 2), f (l = 3).</p></li><li><p>Table of allowed values: for each n, list possible l (0 to n-1) and the corresponding m_l values; orbital labels are determined by l (e.g., s: l=0; p: l=1; d: l=2; f: l=3).</p></li><li><p>Summary: quantum numbers define shells (n), subshells (n, l), and orbitals (n, l, m_l); ms specifies spin.</p></li></ul><h3 id="0a09929b-b975-41c5-b445-dea336626f63" data-toc-id="0a09929b-b975-41c5-b445-dea336626f63" collapsed="false" seolevelmigrated="true">Electron Configurations, Aufbau, Pauli, and Hund</h3><ul><li><p>Aufbau Principle: electrons occupy the lowest-energy orbitals first.</p></li><li><p>Pauli Exclusion Principle: no two electrons can share the same set of four quantum numbers (n, l, m<em>l, m</em>s).</p></li><li><p>Hund’s Rule: for degenerate orbitals (same energy), electrons occupy separate orbitals with parallel spins before pairing.</p></li><li><p>Visual representations include energy diagrams with arrows indicating electron spin.</p></li><li><p>Examples (ground-state configurations):</p><ul><li><p>Hydrogen: 1s^1</p></li><li><p>Helium: 1s^2 (ground state)</p></li><li><p>Lithium: [He] 2s^1</p></li><li><p>Beryllium: [He] 2s^2</p></li><li><p>Boron: [He] 2s^2 2p^1</p></li><li><p>Carbon: [He] 2s^2 2p^2</p></li><li><p>Nitrogen: [He] 2s^2 2p^3</p></li><li><p>Oxygen: [He] 2s^2 2p^4</p></li><li><p>Fluorine: [He] 2s^2 2p^5</p></li><li><p>Neon: [He] 2s^2 2p^6 (all paired)</p></li></ul></li><li><p>Paramagnetic vs Diamagnetic:</p><ul><li><p>Paramagnetic: atoms with unpaired electrons; attracted to magnetic fields.</p></li><li><p>Diamagnetic: all electrons paired; weakly repelled by magnetic fields.</p></li></ul></li><li><p>Examples: Ne is diamagnetic; O and F are paramagnetic due to unpaired electrons.</p></li><li><p>Multielectron atoms: energy splitting causes subshells to have different energies (e.g., 3s, 3p, 3d) due to electron-electron interactions.</p></li><li><p>Ground-state configurations of common elements shown using the Aufbau pattern and orbitals.</p></li></ul><h3 id="6775307c-cf72-46ba-b8a8-403c47e39180" data-toc-id="6775307c-cf72-46ba-b8a8-403c47e39180" collapsed="false" seolevelmigrated="true">Periodic Table and Electron Configurations of Representative Elements</h3><ul><li><p>Electron configurations can be simplified using noble gas cores (noble gas shorthand):</p><ul><li><p>Potassium: [Ar] 4s^1 (actual: 1s^2 2s^2 2p^6 3s^2 3p^6 4s^1)</p></li><li><p>Argon: [Ar] (core configuration only as noble gas)</p></li></ul></li><li><p>Group 1A (alkali metals) and Group 2A (alkaline earth metals) follow predictable valence configurations:</p><ul><li><p>Group 1A: [noble gas] ns^1</p></li><li><p>Group 2A: [noble gas] ns^2</p></li></ul></li><li><p>Transition metals show exceptions where d-subshell stability leads to deviations (e.g., Cu: [Ar] 4s^1 3d^{10} rather than [Ar] 4s^2 3d^9; Cr: [Ar] 4s^1 3d^5 rather than [Ar] 4s^2 3d^4).</p></li><li><p>Diamagnetic vs paramagnetic trends continue with multi-electron configurations.</p></li><li><p>Practice tasks include writing configurations for Fe, Ar, Si, S, and others; identifying allowed quantum numbers for orbitals (e.g., 2p has n = 2, l = 1, m_l = -1, 0, +1).</p></li></ul><h3 id="1c18d20e-b64f-4100-a115-54c2f58b05fb" data-toc-id="1c18d20e-b64f-4100-a115-54c2f58b05fb" collapsed="false" seolevelmigrated="true">Ions, Isoelectronic Series, and Atomic Details</h3><ul><li><p>Ions: for main-group elements, remove or add electrons from the atom’s configuration to form cations or anions. Isoelectronic species share the same electron configuration as a noble gas.</p><ul><li><p>Examples: Na: [Ne] 3s^1; Na^+: [Ne] 3s^0 (1s^2 2s^2 2p^6 for the core into Ne configuration)</p></li><li><p>Cl: [Ne] 3s^2 3p^5; Cl^-: [Ne] 3s^2 3p^6 (isoelectronic with Ar)</p></li></ul></li><li><p>d-block ions: electrons are removed primarily from the highest-n shell first, often from 4s before 3d for Fe (Fe^2+: [Ar] 3d^6; Fe^3+: [Ar] 3d^5).</p></li><li><p>Practice tasks include writing configurations for ions like N^3-, Ba^2+, Be^{2+}, Cr^{3+}, Zn^{2+}, Mn^{2+}.</p></li><li><p>Valence electrons are the outermost electrons; core electrons are those in filled shells or noble gas cores.</p></li></ul><h3 id="78c55c46-1640-47f7-81f5-c6521ef92bd4" data-toc-id="78c55c46-1640-47f7-81f5-c6521ef92bd4" collapsed="false" seolevelmigrated="true">Atomic Radius, Zeff, and Periodic Trends</h3><ul><li><p>Atomic radius definitions:</p><ul><li><p>Metallic radius (for metals): half the distance between nuclei of two adjacent identical metal atoms.</p></li><li><p>Covalent radius (for nonmetals): half the distance between nuclei connected by a bond.</p></li></ul></li><li><p>Effective nuclear charge (Zeff): the net positive charge experienced by an electron; shields from other electrons reduce the full nuclear charge.</p></li><li><p>Common approximation: Zeff ≈ Z − σ, where σ is the shielding constant.</p></li><li><p>Trends:</p><ul><li><p>Zeff increases from left to right across a period (core electrons remain the same; Z increases).</p></li><li><p>Zeff changes little down a group (core electrons shield similarly as Z increases).</p></li><li><p>Atomic radius increases down a group (n increases; outer shell farther from nucleus).</p></li><li><p>Atomic radius decreases across a period due to increasing Zeff pulling valence electrons closer.</p></li></ul></li><li><p>Example: Helium removal energy differs because of shielding; Zeff grows across a period.</p></li></ul><h3 id="c66016d5-658c-4aca-bdbe-8fe0c507f5ca" data-toc-id="c66016d5-658c-4aca-bdbe-8fe0c507f5ca" collapsed="false" seolevelmigrated="true">Ionization Energy and Electron Affinity</h3><ul><li><p>Ionization energy (IE): energy required to remove an electron from a gaseous atom to form a cation.</p><ul><li><p>Example: IE1(Na) ≈ 496 kJ/mol; IE2(Na) ≈ 4562 kJ/mol (second ionization much harder because it removes a core-like electron).</p></li><li><p>Generally, IE1 increases from left to right across a period; harder to remove electrons with higher Zeff.</p></li></ul></li><li><p>Within a shell, higher-l electrons are easier to remove; pairing effects also influence IE (paired electrons experience repulsion).</p></li><li><p>Ionization energy trends: IE1, IE2, … increase as electrons are successively removed; core electrons require significantly higher energy.</p></li><li><p>Electron affinity (EA): energy released when an atom in the gas phase accepts an electron; trend across a period generally increases with Zeff.</p><ul><li><p>Some edges: adding to an s orbital vs p orbital; within a subshell, it is easier to add to an empty orbital than to an orbital that already contains electrons.</p></li></ul></li><li><p>Enthral: First electron affinity often positive (energy released after electron attachment); successive electron affinities (EA2, EA3, …) are typically negative (additional electrons are repelled by existing negative charge).</p></li><li><p>Examples: O(g) + e^- → O^-(g) with EA1 ≈ 141 kJ/mol; O^-(g) + e^- → O^{2-}(g) with EA2 ≈ −741 kJ/mol.</p></li></ul><h3 id="8cfd6919-5c02-4566-a5cd-d4c1b66e052c" data-toc-id="8cfd6919-5c02-4566-a5cd-d4c1b66e052c" collapsed="false" seolevelmigrated="true">Ion Size: Ionic Radius and Isoelectronic Series</h3><ul><li><p>Ionic radius: radius of a cation or anion; trends differ from neutral atoms.</p><ul><li><p>Cations: radius decreases due to loss of electrons and reduced repulsion.</p></li><li><p>Anions: radius increases due to added electron-electron repulsion.</p></li></ul></li><li><p>Isoelectronic series: species with identical electron configurations but different nuclear charges. As nuclear charge increases, ionic radius decreases.</p><ul><li><p>Examples: O^{2-}, F^-, Ne all share 1s^2 2s^2 2p^6; radii decrease with more protons (higher Z).</p></li></ul></li><li><p>Practice exercises: identify isoelectronic series within a group (e.g., K^+, Ne, Ar, Kr, P^{3-}, S^{2-}, Cl^-), and arrange by increasing radius.</p></li></ul><h3 id="aa9d551d-2d0c-48d6-a178-ba80517432b6" data-toc-id="aa9d551d-2d0c-48d6-a178-ba80517432b6" collapsed="false" seolevelmigrated="true">Periodic Table: Classification and Relevance of Electron Configurations</h3><ul><li><p>Periodic Table organization: elements arranged by atomic number (not atomic mass); vertical groups and horizontal periods.</p></li><li><p>Major groups and blocks:</p><ul><li><p>Group 1A: Alkali metals; [noble gas] ns^1 valence configuration.</p></li><li><p>Group 2A: Alkaline earth metals; [noble gas] ns^2.</p></li><li><p>8A (noble gases): filled p subshells; highly inert.</p></li><li><p>Transition metals: groups 1B, 3B–8B; d-block; some have d-shell stability effects.</p></li><li><p>Lanthanides and actinides: f-block transition elements.</p></li></ul></li><li><p>Valence electrons are those in the outermost principal quantum number; crucial in bonding.</p></li><li><p>Exceptions to the Aufbau principle often arise due to d-subshell stability (e.g., Cu, Cr).</p></li></ul><h3 id="7698da01-9325-4e0a-8a24-fd2682881161" data-toc-id="7698da01-9325-4e0a-8a24-fd2682881161" collapsed="false" seolevelmigrated="true">Practical Aspects and Compound Types</h3><ul><li><p>Metals, nonmetals, and metalloids definitions:</p><ul><li><p>Metals: good conductors, usually solids, ductile, malleable.</p></li><li><p>Nonmetals: poor conductors; exist in all states; graphite is an exception to conductivity.</p></li><li><p>Metalloids: intermediate properties; semiconductors.</p></li></ul></li><li><p>Ionic compounds: comprised of ions (usually a metal and a nonmetal); ionic bonds arise from electrostatic attraction between cations and anions.</p></li><li><p>Covalent (molecular) compounds: composed of neutral molecules; electrons are shared.</p></li><li><p>Acids and hydrates:</p><ul><li><p>Acids release H+ ions in water (binary and oxyacids).</p></li><li><p>Hydrates are compounds containing H2O as part of their crystal structure.</p></li></ul></li></ul><h3 id="e67f2cf5-b41a-4d25-a630-eb7494c8fa68" data-toc-id="e67f2cf5-b41a-4d25-a630-eb7494c8fa68" collapsed="false" seolevelmigrated="true">Quick Summary of Key Equations and Concepts</h3><ul><li><p>Kinetic energy:$E_k = \frac{1}{2} m v^2$</p></li><li><p>Electrostatic energy:$E{el} = \frac{Q1 Q_2}{d}$</p></li><li><p>Speed of light and light relations:$c = \lambda \nu,\quad c = 2.998 \times 10^8\ \text{m s}^{-1}$</p></li><li><p>Planck relation and photon energy:$E = h \nu = \frac{h c}{\lambda}$</p></li><li><p>Bohr energy levels for hydrogen:$E_n = -\frac{2.18 \times 10^{-18}}{n^2}\ \text{J}$</p></li><li><p>Change in energy during a transition:$\Delta E = -2.18 \times 10^{-18}\ \text{J} \left( \frac{1}{nf^2} - \frac{1}{ni^2} \right)$</p></li><li><p>Rydberg formula:$\frac{1}{\lambda} = R\infty \left( \frac{1}{nf^2} - \frac{1}{ni^2} \right)$with$R\infty = 1.09737317 \times 10^7 \text{ m}^{-1}$</p></li><li><p>de Broglie wavelength:$\lambda = \frac{h}{m v}$</p></li><li><p>Uncertainty principle:$\Delta x \Delta p \ge \frac{h}{4\pi}$$

• Quantum numbers: n, l, ml, ms; orbitals labeled s, p, d, f; shapes and orientations summarized by l and m_l.

• Zeff and radius trends: Zeff increases left-to-right; atomic radius decreases left-to-right and increases top-to-bottom.

• Ionization energy (IE) and electron affinity (EA): IE generally increases across a period; EA generally increases across a period but with orbital-type nuances; extra electron additions can be energy-absorbing or energy-releasing depending on subshell occupancy.

• Isoelectronic series: species with identical electron configurations but different nuclear charges; radius decreases with increasing nuclear charge.

• Periodic table classification: Groups 1A–8A, noble gases in 8A, transition metals in B groups, lanthanides/actinides in f-block; valence electrons govern bonding.