What’s coming up??? • Oct 25 The atmosphere, part 1 Ch. 8 • Oct 27 Midterm … No lecture • Oct 29 The atmosphere, part 2 Ch. 8 • Nov 1 Light, blackbodies, Bohr Ch. 9 • Nov 3,5 Postulates of QM, p-in-a-box Ch. 9 • Nov8,10Hydrogen and multi – e atoms Ch. 9 • Nov 12,15 Multi-electron atoms Ch.9,10 • Nov 17 Periodic properties Ch. 10 • Nov 19 Periodic properties Ch. 10 • Nov 22 Valence-bond; Lewis structures Ch. 11 • Nov 24 Hybrid orbitals; VSEPR Ch. 11, 12 • Nov 26 VSEPR Ch. 12 • Nov 29 MO theory Ch. 12 • Dec 1 MO theory Ch. 12 • Dec 2 Review for exam
We can split the hydrogen wavefunction into two: Y(x,y,z) Y(r,q,j) = R(r)xY(q,j) Depends on angular variables Depends on r only
The solutions have the same features we have seen already: • Energy is quantized • En = -RZ2 / n2 = - 2.178 x 10-18 Z2 / n2 J [ n = 1,2,3 …] • Wavefunctions have shapes which depend on the quantum numbers • There are (n-1) nodes in the wavefunctions
Because we have 3 spatial dimensions, we end up with 3 quantum numbers: n, l, ml • n = 1,2,3, …; l = 0,1,2 … (n-1); ml = -l, -l+1, …0…l-1, l • n is the principal quantum number – gives energy and level • l is the orbital angular momentum quantum number – it gives the shape of the wavefunction • ml is the magnetic quantum number – it distinguishes the various degenerate wavefunctions with the same n and l
En = -RZ2 / n2 • = - 2.178 x 10-18 Z2 / n2 J [ n = 1,2,3 …] … degenerate
3/2 æ ö 1 1 -r/a 0 ç ÷ Y = e 100 ç ÷ p a è ø 0 Probability Distribution for the 1s wavefunction: Maximum probability at nucleus
A more interesting way to look at things is by using the radial probability distribution, which gives probabilities of finding the electron within an annulus at distance r (think of onion skins) max. away from nucleus
90% boundary: Inside this lies 90% of the probability nodes
P-orbitals Node at nucleus
The result (after a lot of math!) Node at s = 2!! Nodes at f, q = 0 !!
A Comparison of the Radial Probability Distributions of the 2s and 2p Orbitals
A Cross Section of the Electron Probability Distribution for a 3p Orbital Spatial nodes and Angular nodes!
Nodes at f, q = 0 !! Nodes at s = 0 and 4
Another quantum number! Electrons are influenced by a magnetic field as though they were spinning charges. They are not really, but we think of them as having “spin up” or “spin down” levels. These are labeled by the 4th quantum number: ms, which can take 2 values.
This 2-valued electron spin can be shown in an experiment In silver (and many other atoms) there is one more “spin up” electron than “spin down” or vice versa. This means that an atom of silver can interact with a magnetic field and be deflected up or down, depending on which type of spin is in excess.
3s 3p 3d 2s 2p 1s E In multi-electron atoms things change, because of the influence of electrons which are already there Remember the energies are <0 The degenerate energy levels are changed somewhat to become
4d 5s 4p 3d 4s 3p E 3s 2p 2s 1s THE MULTI-ELECTRON ATOM ENERGY LEVEL DIAGRAM Remember the energies are < 0
THE PAULI PRINCIPLE No two electrons in the same atom can have the same set of four quantum numbers (n, l, ml , ms). An orbital is described by three quantum numbers, Then each electron in a given orbital must have a different ms HOW MANY ELECTRONS IN AN ORBITAL? each orbital may contain a maximum of two electrons, and they must have opposite spins.
ELECTRONIC CONFIGURATIONS THE BUILDING-UP PRINCIPLE. GROUND STATE lowest energy electronic configuration assign electrons to orbitals one at a time Electrons go into the available orbital of lowest energy. Electrons are placed in orbitals according to the Pauli Principle. A maximum of two electrons per orbital.
4d 5s 4p 3d 4s 3p E 3s 2p 2s 1s THE AUFBAU (BUILDING-UP) PRINCIPLE: electrons are added to hydrogen-like atomic orbitals in order of increasing energy The electron configuration of any atom or ion….... can be represented by an orbital diagram
ORBITAL DIAGRAM Hydrogen has its one electron in the 1s orbital: 1s 2s 2p H: 1s1 Helium has two electrons: both occupy the 1s orbital Pauli principle with opposite spins: 1s 2s 2p He: 1s2 1s 2s 2p He:
ORBITAL DIAGRAM Hydrogen has its one electron in the 1s orbital: 1s 2s 2p H: 1s1 Helium has two electrons: both occupy the 1s orbital Pauli principle with opposite spins: 1s 2s 2p He: 1s2 helium ground state Helium can also exist in an excited state such as: 1s 2s 2p He: 1s12s1 Now onto the next atoms
Lithium has three electrons, so it must use the 2s orbital: Beryllium has four electrons, which fill both the 1s and 2s orbitals: Boron’s five electrons fill the 1s and 2s orbitals, and begin to fill the 2p orbitals. Since all three are degenerate, the order in which they are filled does not matter. 1s 2s 2p Li: 1s22s1 1s 2s 2p Be: 1s2 2s2 1s 2s 2p B: 1s22s22p1
CARBON Z=6 A CHOICE 1s 2s 2p C: 1s22s22p2 OR 1s 2s 2p C: 1s22s22p2 How can we decide?????
HUND’S RULE FOR THE GROUND STATE ELECTRONS OCCUPY DEGENERATE ORBITALS SEPARATELY THE SPINS ARE PARALLEL SO FOR CARBON THE GROUND STATE IS 1s 2s 2p C: 1s22s22p2
4d 5s 4p 3d 4s 3p E 3s 2p 2s 1s ENERGY LEVEL DIAGRAM FOR A MULTI-ELECTRON ATOM BROMINE ELECTRONIC CONFIGURATION [Ar] 4s23d104p5
The idea of penetration explains why the 3d orbitals lie higher in energy than the 4s.
H He 1s1 1s2 Li Be B C N O F Ne 2s1 2s2 2p1 2p2 2p3 2p4 2p5 2p6 Na Mg Al Si P S Cl Ar 3s1 3s2 3p1 3p2 3p3 3p4 3p5 3p6 The valence electron configuration of the elements in the periodic table repeat periodically! Every element in a group has the same valence electron configuration!