#### Transcript Zumdahl`s Chapter 7

Zumdahl’s Chapter 7 Atomic Structure Atomic Periodicity Chapter Contents • EM Radiation • EM Quantization • H Spectrum – – – – Niels Bohr L. deBroglie W. Heisenberg E. Schrödinger • Quantum Nos. • Orbital Shapes – Degeneracies • W. Pauli (spin) • Multielectrons • Periodic Table – Aufbau – Property Trends – Groups Electromagnetic Radiation • Oscillating E & M fields forever. • Wavelength, , distance between successive peaks. • Period, τ, time between peaks. • Speed, c = / τ = – = frequency (cycles per second) – = c / and = c / Electromagnetic Quantization • E = h = hc / • Equipartition Theorem demands kT worth of thermal energy to all light and matter modes. Leads to energy. • Vibration overtones in matter are truncated by indivisible atoms. • in light energies overcome by Planck with QUANTIZED energies. • h = Planck’s constant = 6.6x10–34 Js Hydrogen Atom Spectrum • White light is all colors (all ); diffraction in prisms or raindrops gives continuous spectra. • Atomic excitation gives instead discrete colors (few ). • In H atom, E light = R (n2–2 – n1–2) • Why so simple? Niels Bohr • • • • • • • F centripetal = m r ² F attraction = – Z e² / r² Balance: Z e² = m r³ ² E = K+V = ½ m r² ² – Z e² / r E = – ½ Z e² / r on substitution E = – (½ Z e² / r) (Z e² / m r³ ²) E = – Z² e4 m / (2 m² r4 ²) Quantized Momenta • • • • • • E = – Z² e4 m / (2 m² r4 ²) pθ = m r² is angular momentum E = – Z² e4 m / (2 pθ²) pθ = n h / 2 Bohr’s postulate En = – (2² Z² e4 m / h²) n–² En = – RH Z² / n² YES!! – But but but WHY quantize pθ ?!? Louis deBroglie • Just as light moves as a wave but lives and dies as a discrete energy packet (“photon”), • Matter too has wave properties only dominant for light masses: • = h / p = h / mv = h / (m r² ) • And n = 2 or wave kills itself! Werner Heisenberg • Waves must S P A N . • Attempts to narrow the wave, reduce , increasing p = h / • So a minimum uncertainty x in position MUST exist, and • ( x) ( px) ½ h / 2 – Heisenberg’s Uncertainty Principle Edwin Schrödinger • Bohr’s orbits are infinitesimally thin trajectories. Being off them must be infinitely uncertain! • Need a full 3-D wave, , not 1-D. • Schrödinger’s Wave Equation solves H energy in 3-d and finds: • En = – RH Z² / n² also and more! Quantum Numbers (n, l ,ml ,ms) • Principle Quantum Number, n – n = 1, 2, 3, 4, 5, … , – Governs the number of nodes in the electron’s matter wave, ! – # of nodes (where =0) is n – 1 – For “hydrogenic” ions (and H itself) electron energy depends only on n. – Nodes can be spherical or angular! Angular Momentum Quantum Number, l •l • = 0, 1, 2, 3, 4, … , (n – 1) s p d f g … shorthand –l chemist measures # of nodes that are angular; so it must stop at n–1. – Increasing angular nodes squeezes waves, so E usually depends on l – Z component of l is also quantized! Magnetic Quantum Number, ml • ml is the component of l along (up or down) the Z axis in space. – ml = – l, – l + 1, … , –1, 0, 1, … , l–1, – Because the component can’t exceed its vector. – E only depends upon ml when a magnetic field is applied. l 3-D Shapes of Orbitals • Governed by n, l, and ml • l = 0 is spherical – – – – n = 1 means no nodes: (n – 1) = # of nodes, all spherical. n = 2 means one spherical node: Wavefunction, , falls off in intensity to zero at large distance from + Cross-section of 2s Spherical Node Angular Orbitals • l = 1 implies one angular node – Cleave space with an x=0 plane – But y=0 or z=0 work as well, so there are three or 2l+1 suborbitals. – The ml sequence always gives 2l+1 – ml differentiates directions in space for chemical bonding! Degeneracies • In the absence of applied magnetic field, all suborbitals of a given l have the same energy. • This identity of energies is called “degenerate.” • Even nearly degenerate orbitals may be mixed to give new ones. P.A.M. Dirac Wolfgang Pauli • Dirac applied Einstein’s fixed c to Schrödinger’s Equation and found new quantum number, ms. – ms is electron spin number and takes on only two values, ½. • Pauli Principle says only two electrons can occupy any orbital, and their ms must differ, – without which NO CHEMISTRY. Multielectronic Atoms • Beyond H, repulsions BETWEEN electrons compete with nuclear attraction & complicate spectra. – Hund’s Rule: if electrons have the choice between degenerate orbitals, they choose NOT to double occupy them. • It minimizes electronic repulsion. Repulsive Consequences • Energies are now a function of l, the angular quantum number. • The “filling sequence” shows the new energy order: – 1s<2s<2p<3s<3p<4s<3d<4p<5s<4d – <5p<6s<4f<5d<6p<7s<5f<6d etc. – Periodic Table exemplifies it, but a simple pattern emerges: Filling Sequence Mnemonic 1s 2s 3s 4s 5s 6s 7s 8s 2p 3p 4p 5p 6p 7p 8p 3d 4d 5d 6d 7d 8d 4f 5f 6f 7f 8f And that’s as far as the known elements go. 5g 6g 7g 8g 6h 7h 8h 7i 8i 8k Periodic Table • Aufbau (filling sequence) follows that table: – 1s² 2s² 2p6 3s² 3p6 4s² 3d10 4p6 5s² – 4d10 5p6 6s² 4f14 5d10 6p6 7s² 6d10 – and the latest elements among 7p6 • Irregularities occur where ½ filled suborbitals acquire greater stability than a predecessor: Aufbau Hiccups • Vanadium: [Ar] 4s² 3d3 suggests • Chromium: [Ar] 4s² 3d4 is next, BUT IT ISN’T SO! • Chromium: [Ar] 4s1 3d5 lowers its energy by borrowing a 4s to complete a ½–filled d suborbital. • Manganese: [Ar] 4s² 3d5 follows. Periodic Properties • Rows are called “periods” on the Periodic Table. – Columns are called “groups.” • Progression along rows implies adding new electrons & protons. – They get added 1:1 for neutrality. – But new repulsions keep pace with new attractions. Which wins? Effective Potential • Protons exert attraction only toward the atom’s center. • Electrons exert repulsion from all over their wavefunctions. – “Core” electrons are located very close to the nucleus where they repel outer electrons as effectively as their number of protons attract. Effective Charge = – Protons – Effective e • So nucleus’s effective charge is Z – (# of all core electrons) less the effect of “outer” electrons. • While core are 100% effective, “valence” electrons are LESS by virtue of spanning a greater fraction of the atom. Only their inner portion is 100% effective. Inefficiency Wins • Across a row, added electrons are valence, not core. So they repel one another less than the added protons attract them. • Effective potential INCREASES across the row, binding the subsequent electrons ever more tightly! Trends on Periods • Increased electron binding along rows (to the right), generally results in: – – – – Increasing ionization potentials Decreasing atomic sizes Growing electron affinities Increasing electronegativies Groups • Elements in the same column have the same number and type of valence electrons, differing only by n. • Because increasing n by 1 puts one more node in , dimensions of increase, vaulting electrons outside their predecessors. Group Trends • Dropping down a group (column) increases efficiency of core and distance of valence from center. Both conspire to weaken the nucleus’s grasp. – Atomic size increases. – Ionization potential decreases. – Electronegativity decreases.