Quantum Numbers and Atomic Structure Refining Bohr’s Model

What are Quantum Numbers?  Bohr defined the principal energy levels (n = 1,2,3,4…)  experimental evidence indicated the need for changes to this simple system  quantum numbers are quantized values used to describe electrons in an atom  there are four quantum numbers represented by the letters n (Bohr’s number), l, m l and m s

The Principal Quantum Number, n (Bohr, 1913)  based on Bohr’s observations of line spectra for different elements  ‘n’ relates to the main energy of an electron  allowable values: n = 1, 2, 3, 4, …  electrons with higher ‘n’ values have more energy

The Secondary Quantum Number, l (Sommerfeld, 1915)  based on the observation (Michelson, 1891) that lines on line spectra are actually groups of multiple, thin lines  ‘ l ’ relates to the shape of the electrons’ orbits  allowable values: l = 0 to l = n - 1  i.e. for n = 4: l = 0, 1, 2, or 3  the ‘ l ’ values 0, 1, 2, and 3 correspond to the shapes we will call s, p, d and f, respectively

The Magnetic Quantum Number, m l (Sommerfeld and Debye, 1915)  based on the observation (Zeeman, 1897) that single lines on line spectra split into new lines near a strong magnet  ‘m l ’ relates to the direction/orientation of the electrons’ orbits  allowable values: m l = - l to + l  i.e. for l = 2: m l = -2, -1, 0, 1, or 2  electrons with the same l value but different m l values have the same energy but different orientations

The Spin Quantum Number, m s (Pauli, 1925)  based on the observation that magnets could further split lines in line spectra, and that some elements exhibit paramagnetism  ‘m s ’ relates to the ‘spin’ of an electron  allowable values: m s = - ½ or + ½  i.e. for any possible set of n, l, and m l values, there are two possible m s values  when two electrons of opposite spin are paired, there is no magnetism observed; an unparied electron is weakly magnetic

Defining Electrons Using Quantum Numbers Let’s look at the energy level n = 2:  Possible l values: 0, 1  For l = 0, m l = 0  For l = 1, m l = -1, 0 or 1  For every value of m l, there are two electrons (m s = ½ and m s = - ½) So, there would be 8 electrons found in principal energy level 2 and they would have the following designations…

Electrons in energy level 2: Electronn l mlml msms 120 (or s)0½ ½ 321 (or p)½ 421 (or p)- ½ 521 (or p)0½ ½ 721 (or p)1½ ½

Orbits vs. Orbitals  initially, electrons were thought to travel in orbits (2D, travels around nucleus at fixed distance in a circular path, 2n 2 electrons per orbit)  quantum theory describes electrons as existing in orbitals (3D region, distance from nucleus varies, no path, 2 electrons per orbital)

For our purposes:  primary energy level (n) = ‘shell’  energy sublevel ( l ) = ‘subshell’  orbitals are named as a combination of the n and l values  e.g. an electron may exist in a ‘2p’ orbital (n = 2, l = 1 or p)  shapes of these orbitals will be discussed soon

Energy-Level Diagrams  now we can be more specific  for every ‘n,’ energy increases from s  p  d  f  quantum number restrictions state that there can only be:  one s orbital (= 2 electrons) for any value of n  three p orbitals (= 6 electrons) for n = 2,3,4, …  five d orbitals (= 10 electrons) for n = 3,4,5, …  seven f orbitals (=14 electrons) for n = 4,5,6, …

Relative Energies of Electron Orbitals Ref:

When Placing Electrons in Orbitals…  aufbau principle: fill lower-energy orbitals first  Hund’s rule: within the same energy level, give each orbital one electron before pairing up electrons  Pauli exclusion principle: two electrons within the same orbital must have opposite spins

Aufbau (‘building up’) Diagram this diagram will help you remember the proper order for filling orbitals 7s7p7d7f 6s6p6d6f 5s5p5d5f 4s4p4d4f 3s3p3d 2s2p 1s

Energy-Level Diagram for Vanadium  vanadium has 23 electrons  read on pages 189 – 190 to learn how to draw energy- level diagrams for ions

The Following is Just Beautiful…  The quantum theory of the atom agrees completely with the periodic table, which had been around for 30 years and was developed without any knowledge of electron arrangements…. Wait for it…

Relationship between the first two quantum numbers and the periodic table:

Referring to quantum theory and the periodic table of the elements: “The unity of these concepts is a triumph of scientific achievement that is unparalleled in the past of present.” - Text, pg. 185 Read more on pp. 194 – 195 in your text!

Electron Configurations  More concise than energy-level diagrams but provide same information e.g. for vanadium: V: 1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 3d 3 Try chlorine right now… Cl: 1s 2 2s 2 2p 6 3s 2 3p 5

Shorthand Electron Configurations  use noble gases as a starting point e.g. for vanadium: V: [Ar] 4s 2 3d 3 for chlorine: Cl: [Ne] 3s 2 3p 5

The Power of What You Now Know  You have seen that the periodic table is explained for you as never before  Charges of ions can be explained e.g. leadPb: 6s 2 4f 14 5d 10 6p 2 Pb 2+ ion: remove two electrons from 6p Pb 4+ ion: remove two electrons from 6p and two electrons from 6s  Magnetism is explained (pp )