Zumdahl’s Chapter 7

1. Zumdahl’s Chapter 7 Atomic Structure Atomic Periodicity

2. 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 Chapter Contents

3. Electromagnetic Radiation • Oscillating E & M fields forever. • Wavelength, , distancebetween successive peaks. • Period, τ, time between peaks. • Speed, c =  / τ =   •  = frequency (cycles per second) •  = c /  and  = c / 

4. 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

5. 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?

6. 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)

7. 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 = – (2² Z² e4 m / h²) n–² • En = – RH Z² / n² YES!! • But but but WHY quantize pθ ?!?

8. 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!

9. 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

10. 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!

11. 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!

12. Angular Momentum Quantum Number, l • l= 0, 1, 2, 3, 4, … , (n – 1) • s p d f g … chemist shorthand • lmeasures # 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!

13. Magnetic Quantum Number, ml • mlis the component of l along (up or down) the Z axis in space. • ml= – l, – l + 1, … , –1, 0, 1, … , l–1, l • Because the component can’t exceed its vector. • E only depends upon mlwhen a magnetic field is applied.

14. 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 +

15. Spherical Node Cross-section of 2s

16. 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!

17. 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.

18. P.A.M. DiracWolfgang 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 msmust differ, • without which NO CHEMISTRY.

19. 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.

20. 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:

21. Filling Sequence Mnemonic And that’s as far as the known elements go.

22. 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:

23. 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.

24. 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?

25. 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.

26. 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.

27. 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 moretightly!

28. 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

29. 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.

30. 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.