Lecture 18: Polyelectronic Atoms • Reading: Zumdahl 12.10-12.13 • Outline: – Spin – The Aufbau Principle – Filling up orbitals and the Periodic Table H-atom wavefunctions • The Coulombic potential can be generalized: e- Ze 2 V (r) r r Z P+ • Z = atomic number (= 1 for hydrogen) H-atom wavefunctions • If we solve the Schrodinger equation using this potential, we find that the energy levels are quantized: 2 Z 2 me4 Z E n 2 2 2 2.178x1018 J 2 n 80 h n • n is the principle quantum number, and ranges from 1 to infinity. Orbitals Quantum Numbers and Orbitals n l Orbital 1 2 0 0 1 0 1 2 1s 2s 2p 3s 3p 3d 3 ml 0 0 -1, 0, 1 0 -1, 0, 1 -2, -1, 0, 1, 2 # of Orb. 1 1 3 1 3 5 Spin • Further experiments demonstrated the need for one more quantum number. • Specifically, some particles (electrons in particular) demonstrated inherent angular momentum. Spin (cont.) ms = 1/2 ms = -1/2 • The new quantum number is ms (analagous to ml). • For the electron, ms has two values: +1/2 and -1/2 The Aufbau Principal • For polyelectronic atoms, a direct solution of the Schrodinger Eq. is not possible. • When we construct polyelectronic atoms, we use the hydrogen-atom orbital nomenclature to discuss in which orbitals the electrons reside. • This is an approximation (and it is surprising how well it actually works). The Aufbau Principal (cont.) • When placing electrons into orbitals in the construction of polyelectronic atoms, we use the Aufbau Principle. • This principle states that in addition to adding protons and neutrons to the nucleus, one simply adds electrons to the hydrogen-like atomic orbitals • Pauli exclusion principle: No two electrons may have the same quantum numbers. Therefore, only two electrons can reside in an orbital (differentiated by ms). The Aufbau Principal (cont.) • Finally, orbitals are filled starting from the lowest energy. • Example: Hydrogen 1s1 1s 2s 2p • Example: Helium (Z = 2) 1s2 1s 2s 2p The Aufbau Principal (cont.) • Orbital configurations … The Aufbau Principal (cont.) • Lithium (Z = 3) 1s22s1 1s 2s • Berillium (Z = 4) 2p 1s22s2 1s 2s 2p • Boron (Z = 5) 1s22s22p1 1s 2s 2p The Aufbau Principal (cont.) • Carbon (Z = 6) 1s22s22p2 1s 2s 2p Hund’s Rule: Lowest energy configuration is the one in which the maximum number of unpaired electrons are distributed amongst a set of degenerate orbitals. • Nitrogen (Z = 7) 1s22s22p3 1s 2s 2p The Aufbau Principal (cont.) • Oxygen (Z = 8) 1s22s22p4 1s • Fluorine (Z = 9) 2s 2p 1s22s22p5 1s 2s 2p • Neon (Z = 10) 1s22s22p6 1s 2s 2p full The Aufbau Principal (cont.) • Sodium (Z = 11) 1s22s22p63s1 Ne [Ne]3s1 3s • Argon (Z = 18) [Ne] 3s23p6 Ne 3s 3p The Aufbau Principal (cont.) • We now have the orbital configurations for the first 18 elements. • Elements in same column have the same # of valence electrons! The Aufbau Principal (cont.) The Aufbau Principal (cont.) • What is the radial distribution for different orbitals? The Aufbau Principal (cont.) • Similar to Sodium, we begin the next row of the periodic table by adding electrons to the 4s orbital. • Why not 3d before 4s? • 3d is closer to the nucleus • 4s allows for closer approach; therefore, is energetically preferred. The Aufbau Principal (cont.) • Elements Z=19 and Z= 20: Z= 19, Potassium: 1s22s22p63s23p64s1 = [Ar]4s1 Z= 20, Calcuim: 1s22s22p63s23p64s2 = [Ar]4s2 • Elements Z=21to Z=30 have occupied d orbitals: Z= 21, Scandium: 1s22s22p63s23p64s23d1 = [Ar] 4s23d1 Z = 24, Chromium: [Ar] 4s13d5 exception Z= 30, Zinc: 1s22s22p63s23p64s23d10 = [Ar] 4s23d10 The Aufbau Principal (cont.) • This orbital filling scheme gives rise to the modern periodic table. The Aufbau Principal (cont.) • After Lanthanum ([Xe]6s25d1), we start filling 4f. The Aufbau Principal (cont.) • After Actinium ([Rn]7s26d1), we start filling 5f. The Aufbau Principal (cont.) • Heading on column given total number of valence electrons. The Aufbau Principal (cont.) Summary • Electrons go into hydrogen-like orbitals to construct polyelectronic atoms.