Notes on Quantum Mechanics by Mr. Scott The Schrodinger equation uses THREE QUANTUM NUMBERS to describe the orbitals and locations of each electron in an atom. A FOURTH Q. N. was later added. Principal quantum number, n, is an integer that describes the average distance of the electron from the nucleus. (n = 1, 2, 3, …) Azimuthal quantum number, , can have integral values between 0 and n-1, and defines the SHAPE OF THE ORBITAL’s electron density. The values of are usually associated with letter designations that are drawn from the descriptions of spectra that were developed in the past: 0 1 2 3 Letter s p d f Magnetic quantum number, m , can have integral values between + and - , and describes the ORIENTATION of the orbital space. s p d f m 0 1 0 -1 2 1 0 -1 -2 3 2 1 0 -1 -2 -3 f32 f32 f3 2 f fff22 22 22 px ppyz ddyz xz ddxy 22 d2 z −−33zr x xr y − 3yr xyz y ()x −z x(z −y )z (x −y ) x − yz 55 5 Spin quantum number, ms , introduced by Wolfgang Pauli in his exclusion principle, requires that two electrons in the same orbital must have opposite polarity or SPIN. These values are + ½ or - ½ . Describing the electronic structure of an atom There are several ways of writing the structures of the electrons of atoms. These are referred to as NOTATION and they all follow two basic rules or principles: Aufbau Principle = electrons occupy the lowest available energy positions first Hund’s Rule = orbitals of equal energy each hold one electron before they get a second electron Increasing Energy Aufbau Diagram 1s (diagonal rule) 2s 2p 3s 3p 3d 4s 4p 4d 4f 5s 5p 5d 5f 6s 6p 6d 6f 7s 7p 7d 7f Orbital Notation Uses a line for each orbital and an arrow for each electron. Each orbital represented is labeled beneath the line. The progression is in the order of energy. Both AUFBAU and HUND are used. Examples: Carbon (z = 6) ground state Sodium (z = 11) ground state ↑↓ ↑↓ ↑ ↑ ↑↓↑ ↓↑ ↓↑ ↓↑ ↓↑ 12s sp2x 2py 12s sp2xy2p2pz3s This notation shows 2 unpaired electrons. This notation shows 1 unpaired electron. Electron Configuration Notation Uses a number-letter designation for each orbital type and a superscript number for the number of electrons that occupy each type of orbital. The progression is in the order of energy. The AUFBAU DIAGRAM is best used to determine the correct order. Carbon (z = 6) ground state Sodium (z = 11) ground state 1s22s22p2 1s22s22p63s1 The superscripts add up to the number of electrons for the atom (or ion). 2 + 2 + 2 = 6 2 + 2 + 6 + 1 = 11 2 6 10 14 The maximum amounts are: s , p , d , f Dot Notation Uses the symbol of the element with dots placed around it in a certain pattern that represent the VALENCE electrons. The valence electrons are those in the highest occupied PRINCIPAL ENERGY LEVEL (n = 1, 2, 3, …). The dot pattern followed is similar to the pattern followed in orbital notation. 3 6 px 41 The dot pattern is: representing p s 72XyXy y 5 8 pz Carbon (z = 6) ground state Sodium (z = 11) ground state i i C : Na i Why just four dots? Why just one dot? Lets reconsider the electron configuration of each of these atoms. 1s22s22p2 1s22s22p63s1 The highest occupied level is n = 2 The highest occupied level is n = 3 2s22p2 = 4 electrons 3s1 = 1 electron so 4 dots are shown so 1 dot is shown .