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, ml and ms

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, ml

(Sommerfeld and Debye, 1915)

based on the observation (Zeeman, 1897) that single lines on line spectra split into new lines near a strong magnet

‘ml ’ relates to the direction/orientation of the electrons’ orbits

allowable values: ml = - l to + l i.e. for l = 2: ml = -2, -1, 0, 1, or 2

electrons with the same l value but different ml values have the same energy but different orientations

The Spin Quantum Number, ms(Pauli, 1925)

based on the observation that magnets could further split lines in line spectra, and that some elements exhibit paramagnetism

‘ms ’ relates to the ‘spin’ of an electron allowable values: ms = - ½ or + ½

i.e. for any possible set of n, l, and ml values, there are two possible ms 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, ml = 0 For l = 1, ml = -1, 0 or 1 For every value of ml, there are two

electrons (ms = ½ and ms = - 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:

Electron n l ml ms

1 2 0 (or s) 0 ½

2 2 0 (or s) 0 - ½

3 2 1 (or p) -1 ½

4 2 1 (or p) -1 - ½

5 2 1 (or p) 0 ½

6 2 1 (or p) 0 - ½

7 2 1 (or p) 1 ½

8 2 1 (or p) 1 - ½

Orbits vs. Orbitals

initially, electrons were thought to travel in orbits (2D, travels around nucleus at fixed distance in a circular path, 2n2 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: http://www.chemistry.mcmaster.ca/esam/Chapter_4/section_3.html

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

7s 7p 7d 7f6s 6p 6d 6f5s 5p 5d 5f4s 4p 4d 4f3s 3p 3d2s 2p1s

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: 1s2 2s2 2p6 3s2 3p6 4s2 3d3

Try chlorine right now… Cl: 1s2 2s2 2p6 3s2 3p5

Shorthand Electron Configurations

use noble gases as a starting point

e.g. for vanadium: V: [Ar] 4s2 3d3

for chlorine: Cl: [Ne] 3s2 3p5

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 explainede.g. lead Pb: 6s2 4f14 5d10 6p2

Pb2+ ion: remove two electrons from 6p Pb4+ ion: remove two electrons from 6p and two electrons from 6s

Magnetism is explained (pp. 195-196)