Chemistry 1000 Lecture 8: Multielectron atoms

Marc R. Roussel

Spin

I Spin (with associated quantum number s) is a type of angularmomentum attached to a particle.

I Every particle of the same kind (e.g. every electron) has thesame value of s.

I Two types of particles:

Fermions: s is a half integer

Examples: electrons, protons, neutrons(s = 1

2)

Bosons: s is an integer

Example: photons (s = 1)

Spin magnetic quantum number

I All types of angular momentum obey similar rules.

I There is a spin magnetic quantum number ms which gives thez component of the spin angular momentum vector:

Sz = ms~

I The value of ms can be −s,−s + 1, . . . , s.

I For electrons, s = 12 so ms can only take the values −1

2 or 12 .

Stern-Gerlach experiment

How do we know that electrons (e.g.) have spin?

Source: Theresa Knott, Wikimedia Commons,

http://en.wikipedia.org/wiki/File:Stern-Gerlach_experiment.PNG

Pauli exclusion principle

No two fermions can share a set of quantum numbers.

Multielectron atoms

I Electrons occupy orbitals similar (in shape and angularmomentum) to those of hydrogen.

I Same orbital names used, e.g. 1s, 2px , etc.I The number of orbitals of each type is still set by the number

of possible values of m`, so e.g. there are three 2p orbitals.

I In multielectron atoms, the orbital energies depend on both nand `.Orbitals corresponding to the same values of n and ` have thesame energy, e.g. the five 3d orbitals all have the same energy.

Rule of thumb: Orbital energy increases with n + `.If two orbitals have the same value of n + `, the onewith the smaller n is lower in energy.

Yes, but why?

I We talk about orbital energies as if there was such a thing,but a better picture (if still imperfect) is that electrons inorbitals have certain energies.

“Orbital energy” or (slightly better) “energy of an occupiedorbital” can then be thought of as shorthands.

I Physically, the energy of an occupied orbital depends on justtwo things:

1. Attraction of electrons for nucleusThis part is simple: closer to nucleus = more attraction

2. Repulsion between electronsIt turns out this part isn’t too complicated either.

A theorem from electrostatics

I The force due to a charged spherical (hollow) shell turns outto have a simple form:

total charge Q

0

r

F =qQ

q4πε r 2

No force

in here

So what does that have to do with orbital energies?

A

B

I If occupied orbital A is mostly closer to thenucleus than occupied orbital B, then

I Orbital B mostly doesn’t affect the energy oforbital A.

I Orbital A screens part of the nuclear chargefrom B. This raises the energy of orbital B.

closer to nucleus

more electron densityfarther from the nucleus

electron density

I if we compare the 2s and 2p orbitals (forexample), we see that the 2s orbital hasmore electron density closer to thenucleus, so

I a filled 2s orbital screens a filled 2porbital more effectively than vice versa.

I The 2p orbital would therefore be higherin energy.

Examples of orbital ordering

I Which is lower in energy, the 3d or 4s orbital?

For the 3d orbitals, n + ` = 3 + 2 = 5, while for the 4s orbital,n + ` = 4 + 0 = 4.

The 4s is therefore lower in energy.

I Which is lower in energy, the 2p or 3s orbital?

For the 2p orbitals, n + ` = 3, while for the 3s orbital,n + ` = 3 + 0 = 3. The n + ` test therefore doesn’t help.

The tie-breaker is the value of n, so the 2p orbitals are lowerin energy.

A simple way to remember the orbital ordering

4s 4p 4d 4f

5s 5p 5d 5f 5g

6s 6p 6d 6f ...

7s ...

1s

2s 2p

3s 3p 3d

Ground-state electron configurations of atoms

I An electron configuration is a way of arranging the electronsof an atom in its orbitals.

I Configurations are denoted by showing the number ofelectrons in an orbital type as a superscript, e.g. 1s22p1 woulddenote an atom with 2 electrons in its 1s orbital, and one inthe 2p orbital.

I The ground state configuration is the lowest-energyconfiguration.

Rules for determining the ground state configuration(Aufbau rules)

1. An orbital can hold exactly one electron with each value of ms

(Pauli principle)

2. Electrons are added to the lowest energy orbital available.

3. Electrons are spread over degenerate orbitals when possible.

4. Maximize the number of parallel spins when possible (Hund’srule).

5. Exceptions:(n − 1)d5 ns1 configurations are generally lower in energy than(n − 1)d4 ns2 configurations.(n − 1)d10 ns1 configurations are generally lower in energythan (n − 1)d9 ns2 configurations.

6. There are other exceptions (not to be learned), particularly inthe transition metals (after the first row), lanthanides andactinides.

Orbital energy diagrams

I Not usually drawn to scaleI Electrons represented by up or down arrows (representing the

two possible values of ms)

Example: Ground-state electronic configuration of lithium:

1s

2s

Example: Ground-state electronic configuration of boron:

2p

1s

2s

Orbital box diagrams

I Compact version of orbital energy diagram with each orbitalrepresented as a box arranged along a line

Example: Ground-state electronic configuration of lithium:

�� �1s 2s

Equivalent line notation: 1s2 2s1

Example: Ground-state electronic configuration of carbon:

�� �� � �1s 2s 2p

Equivalent line notation: 1s22s22p2

Valence vs core electrons

I All atoms beyond helium have a noble gas core, i.e. a set ofelectrons with the configuration of the previous noble gas.

Example: Carbon has a 1s2 core, equivalent to theground-state configuration of helium.

These core electrons are chemically inert.

I Chemical reactivity is associated with the valence electrons,i.e. the electrons outside the noble gas core.

I A filled d subshell in a p-block element is also inert and wouldnot normally count as valence electrons.

The noble-gas abbreviation for electron configurations

I When writing down electron configurations, it’s a pain to writedown the core electrons, which aren’t that interesting anyway.We use the noble-gas abbreviation to represent theseelectrons.

Example: The electron configuration of carbon can bewritten [He]2s22p2.

I The noble-gas abbreviation can also be used with orbital boxdiagrams.

Electron configurations and the periodic table

I Electron configurations can be read off the periodic table

I Examples: F, Cl, Br

I Examples: Cr, Mo, W

I How the periodic table should look

http://en.wikipedia.org/wiki/Template:

Periodic_table_(32_columns,_compact)

Periodic table blocks

f block

d blocks b

lock p block

Ions

I A closed (full) shell (noble gas configuration) represents aparticularly unreactive electronic configuration.

I Common ions are often those that have gained or lostelectrons to obtain a closed shell.

I Large charges (positive or negative) are energeticallyunfavorable.

Examples:I Na has a ground-state electronic configuration

of 1s2 2s2 2p6 3s1.Na+ has a noble-gas electronic configuration1s2 2s2 2p6.

I Oxygen has a ground-state electronicconfiguration of 1s2 2s2 2p4.O2− also has the noble-gas electronicconfiguration 1s2 2s2 2p6.

Cations

Rule: Remove electrons from shell of largest n first.If there are electrons with the same n, removeelectrons first from the shell of largest `.

Example:

I Pb has the ground-state electronic configuration[Xe]6s2 4f14 5d10 6p2.

I Pb2+ has the ground-state electronicconfiguration [Xe]6s2 4f14 5d10.

I Pb4+ has the electronic configuration[Xe]4f14 5d10.

Exceptions: Al3+ and Ga3+ are the most common ions of thesemetals. Al+ is unknown, and Ga+ is uncommon.

CationsExample: copper

I Copper has the ground-state electronic configuration[Ar]4s1 3d10.

I It has two commonly observed ions, Cu+ and Cu2+.

I Cu+ has the ground-state electronic configuration [Ar]3d10.

I Cu2+ has the ground-state electronic configuration [Ar]3d9.

Why Cu2+ and not, say, Cu3+? In this case, the existence ofthe former and not of the latter is an empirical observation.

CationsExample: molybdenum

I Molybdenum has the ground-state electronic configuration[Kr]5s1 4d5.

I Mo2+, Mo3+, Mo4+ and Mo5+ are all known in aqueoussolution.

I Why not Mo+? This is an empirical observation.It is unusual since we almost always get a common ion byremoving the ns electrons from transition metals.

I Mo2+ has the ground-state electronic configuration [Kr]4d4.

I Mo3+ has the ground-state electronic configuration [Kr]4d3.

I Mo4+ has the ground-state electronic configuration [Kr]4d2.

I Mo5+ has the ground-state electronic configuration [Kr]4d1.

Magnetic properties

A diamagnetic substance is repelled by a magnetic field.=⇒ Occurs when all the electrons are paired.

A paramagnetic substance is attracted by a magnetic field.=⇒ Occurs when there are unpaired electrons.

Ferromagnetism is an extreme form of paramagnetism in whichthe magnetic field generated by the spin in one atomtends to align the magnetic moment in its neighbors.Ferromagnetic materials can therefore acquirepermanent magnetization.

Examples:

I The Cr3+ ion has a ground-state electronicconfiguration of [Ar]3d3 and is paramagnetic.

I All noble gas atoms are diamagnetic.