Atoms first chapter 3.7 11

Chemistry: Atoms First
Julia Burdge & Jason Overby
Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Chapter 3
Quantum Theory and
the Electronic
Structure of Atoms
Kent L. McCorkle
Cosumnes River College
Sacramento, CA

Quantum Theory and the Electronic Structure ofQuantum Theory and the Electronic Structure of
AtomsAtoms
3
3.7 Quantum Numbers
Principal Quantum Number (n)
Angular Momentum Quantum Number (l)
Magnetic Quantum Number (ml)
Electron Spin Quantum Number (ms)
3.8 Atomic Orbitals
s Orbitals
p Orbitals
d Orbitals and other High-Energy Orbitals
Energies of Orbitals
3.9 Electron Configuration
Energies of Atomic Orbitals in Many-Electron Systems
The Pauli Exclusion Principle
Aufbau Principle
Hund’s Rule
General Rules for Writing Electron Configurations
3.10 Electron Configurations and the Periodic Table

Quantum MechanicsQuantum Mechanics
Erwin Schrödinger derived a complex mathematical formula to
incorporate the wave and particle characteristics of electrons.
Wave behavior is described with the wave function ψ.
The probability of finding an
electron in a certain area of
space is proportional to ψ2
and
is called electron density.

Quantum MechanicsQuantum Mechanics
The Schrödinger equation specifies
possible energy states an electron can
occupy in a hydrogen atom.
The energy states and wave functions
are characterized by a set of quantum
numbers.
Instead of referring to orbits as in the
Bohr model, quantum numbers and
wave functions describe atomic orbitals.

Quantum NumbersQuantum Numbers
Quantum numbers are required to describe the distribution of
electron density in an atom.
There are three quantum numbers necessary to describe an atomic
orbital.
The principal quantum number (n) – designates size
The angular moment quantum number (l) – describes shape
The magnetic quantum number (ml) – specifies orientation
3.7

The principal quantum number (n) designates the size of the
orbital.
Larger values of n correspond to larger orbitals.
The allowed values of n are integral numbers: 1, 2, 3 and so forth.
The value of n corresponds to the value of n in Bohr’s model of the
hydrogen atom.
A collection of orbitals with the same value of n is frequently
called a shell.

The angular moment quantum number (l) describes the shape of
the orbital.
The values of l are integers that depend on the value of the
principal quantum number
The allowed values of l range from 0 to n – 1.
Example: If n = 2, l can be 0 or 1.
A collection of orbitals with the same value of n and l is referred to
as a subshell.
l 0 1 2 3
Orbital designation s p d f

The magnetic quantum number (ml) describes the orientation of
the orbital in space.
The values of ml are integers that depend on the value of the
angular moment quantum number:
– l,…0,…+l

Quantum numbers designate shells, subshells, and orbitals.

Worked Example 3.8
Strategy Recall that the possible values of ml depend on the value of l, not on
the value of n.
What are the possible values for the magnetic quantum number (ml) when the
principal quantum number (n) is 3 and the angular quantum number (l) is 1?
Solution The possible values of ml are -1, 0, and +1.
Setup The possible values of ml are – l,…0,…+l.
Think About It Consult Table 3.2 to make sure your answer is correct. Table
3.2 confirms that it is the value of l, not the value of n, that determines the
possible values of ml.

The electron spin quantum number (ms ) is used to specify an
electron’s spin.
There are two possible directions of
spin.
Allowed values of ms are +½ and −½.

A beam of atoms is split by a magnetic field.
Statistically, half of the electrons spin clockwise, the other half
spin counterclockwise.

To summarize quantum numbers:
principal (n) – size
angular (l) – shape
magnetic (ml) – orientation
electron spin (ms) direction of spin
Required to describe an atomic orbital
Required to describe an
electron in an atomic
orbital
2px
principal (n = 2)
angular momentum (l = 1)
related to the magnetic
quantum number (ml )

Atomic OrbitalsAtomic Orbitals
All s orbitals are spherical in shape but differ in size:
1s < 2s < 3s
2s
angular momentum
quantum number (l = 0)
ml = 0; only 1 orientation
possible
principal quantum
number (n = 2)
3.8

The p orbitals:
Three orientations:
l = 1 (as required for a p orbital)
ml = –1, 0, +1

The d orbitals:
Five orientations:
l = 2 (as required for a d orbital)
ml = –2, –1, 0, +1, +2

Energies of OrbitalsEnergies of Orbitals
The energies of orbitals in the hydrogen atom depend only on the
principal quantum number.
2nd
shell (n = 2)
3d subshell (n = 3; l = 2)
2p subshell (n = 2; l = 1)
3rd
shell (n = 3)
2s subshell
(n = 2; l = 0)
3p subshell (n = 3; l = 1)3s subshell (n = 3; l = 0)

Worked Example 3.9
Strategy Consider the significance of the number and the letter in the 4d
designation and determine the values of n and l. There are multiple values for ml,
which will have to be deduced from the value of l.
List the values of n, l, and ml for each of the orbitals in a 4d subshell.
Solution 4d
Possible ml are -2, -1, 0, +1, +2.
Setup The integer at the beginning of the orbital designation is the principal
quantum number (n). The letter in an orbital designation gives the value of the
angular momentum quantum number (l). The magnetic quantum number (ml) can
have integral values of – l,…0,…+l.
principal quantum
number, n = 4
angular momentum
quantum number, l = 2
Think About It Consult the following figure to verify your
answers.

Electron ConfigurationsElectron Configurations
The electron configuration describes how the electrons are
distributed in the various atomic orbitals.
In a ground state hydrogen atom, the electron is found in the 1s
orbital.
1s1
principal (n = 1)
angular momentum (l = 0)
number of electrons in
the orbital or subshell
1s
2s 2p 2p 2p
The use of an up arrow indicates an electron
with ms = + ½
Ground state electron
configuration of
hydrogen
3.9

If hydrogen’s electron is found in a higher energy orbital, the atom
is in an excited state.
2s1
1s
2s 2p 2p 2p
A possible excited state electron
configuration of hydrogen

In a multi-electron atoms, the energies of the atomic orbitals are split.
Splitting of energy levels refers to
the splitting of a shell (n=3) into
subshells of different energies
(3s, 3p, 3d)

According to the Pauli exclusion principle, no two electrons in an
atom can have the same four quantum numbers.
1s2
1s
2s
2p 2p 2p
The ground state electron
configuration of helium
Quantum number
Principal (n)
Angular moment (l)
Magnetic (ml)
Electron spin (ms)
1
0
0
+ ½
1
0
0
‒ ½
describes the 1s orbital
describes the electrons in the 1s orbital

The Aufbau principle states that electrons are added to the lowest
energy orbitals first before moving to higher energy orbitals.
1s2
2s1
1s
2s
2p 2p 2p
configuration of Li
The 1s orbital can only accommodate 2
electrons (Pauli exclusion principle)
The third electron must go in the
next available orbital with the
lowest possible energy.
Li has a total of 3 electrons

1s
2s
2p 2p 2p
1s2
2s2
configuration of Be
Be has a total of 4 electrons

1s
2s
2p 2p 2p
configuration of B
1s2
2s2
2p1
B has a total of 5 electrons

According to Hund’s rule, the most stable arrangement of
electrons is the one in which the number of electrons with the same
spin is maximized.
1s2
2s2
2p2
1s
2s
2p 2p 2p
configuration of C
The 2p orbitals are of equal energy, or degenerate.
Put 1 electron in each before pairing (Hund’s rule).
C has a total of 6 electrons

spin is maximized.
1s2
2s2
2p3
1s
2s
2p 2p 2p
configuration of N
The 2p orbitals are of equal energy, or degenerate.
Put 1 electron in each before pairing (Hund’s rule).
N has a total of 7 electrons

spin is maximized.
1s2
2s2
2p4
1s
2s
2p 2p 2p
configuration of O
O has a total of 8 electrons
Once all the 2p orbitals are singly occupied, additional
electrons will have to pair with those already in the
orbitals.

spin is maximized.
1s2
2s2
2p5
1s
2s
2p 2p 2p
configuration of F
F has a total of 9 electrons
When there are one or more unpaired electrons, as
in the case of oxygen and fluorine, the atom is
called paramagnetic.

spin is maximized.
1s2
2s2
2p6
1s
2s
2p 2p 2p
configuration of Ne
Ne has a total of 10 electrons
When all of the electrons in an atom are paired, as
in neon, it is called diamagnetic.

General rules for writing electron
configurations:
1) Electrons will reside in the available
orbitals of the lowest possible energy.
2) Each orbital can accommodate a
maximum of two electrons.
3) Electrons will not pair in degenerate
orbitals if an empty orbital is available.
4) Orbitals will fill in the order indicated
in the figure.

Worked Example 3.10
Write the electron configuration and give the orbital diagram of a calcium (Ca)
atom (Z = 20).
Solution
Ca 1s2
2s2
2p6
3s2
3p6
4s2
Setup Because Z = 20, Ca has 20 electrons. They will
fill in according to the diagram at right. Each s subshell
can contain a maximum of two electrons, whereas each p
subshell can contain a maximum of six electrons.
1s2
2s2
2p6
3s2
3p6
4s2
Think About It Look at the figure again to make sure you have filled
the orbitals in the right order and that the sum of electrons is 20.
Remember that the 4s orbital fills before the 3d orbitals.

Electron Configurations and the Periodic TableElectron Configurations and the Periodic Table
The electron configurations of all elements except hydrogen and
helium can be represented using a noble gas core.
The electron configuration of potassium (Z = 19) is
1s2
2s2
2p6
3s2
3p6
4s1
.
Because 1s2
2s2
2p6
3s2
3p6
is the electron configuration of argon, we
can simplify potassium’s to [Ar]4s1
.
1s2
2s2
2p6
3s2
3p6
4s1
The ground state electron configuration of K:
[Ar] [Ar]4s1
3.10
1s2
2s2
2p6
3s2
3p6
4s1

Elements in Group 3B through Group 1B are the transition metals.

Following lanthanum (La), there is a gap where the lanthanide
(rare earth) series belongs.

After actinum (Ac) comes the actinide series.

There are several notable exceptions to the order of electron filling
for some of the transition metals.
Chromium (Z = 24) is [Ar]4s1
3d5
and not [Ar]4s2
3d4
as expected.
Copper (Z = 29) is [Ar]4s1
3d10
and not [Ar]4s2
3d9
as expected.
The reason for these anomalies is the slightly greater stability of d
subshells that are either half-filled (d5
) or completely filled (d10
).
4s 3d 3d 3d 3d 3d
[Ar]Cr
Greater stability with half-filled
3d subshell

There are several notable exceptions to the order of electron filling
for some of the transition metals.
Chromium (Z = 24) is [Ar]4s1
3d5
and not [Ar]4s2
3d4
as expected.
Copper (Z = 29) is [Ar]4s1
3d10
and not [Ar]4s2
3d9
as expected.
The reason for these anomalies is the slightly greater stability of d
subshells that are either half-filled (d5
) or completely filled (d10
).
4s 3d 3d 3d 3d 3d
[Ar]Cu
Greater stability with filled 3d
subshell

Worked Example 3.11
Write the electron configuration for an arsenic atom (Z = 33) in the ground state.
Solution
As [Ar]4s2
3d10
4p3
Setup The noble gas core for As is [Ar], where Z = 18
for Ar.
The order of filling beyond the noble gas core is 4s, 3d,
and 4p. Fifteen electrons go into these subshells because
there are 33 – 18 = 15 electrons in As beyond its noble gas
core.
2
2
2
2
6
6
3
10
Think About It Arsenic is a p-block element; therefore, we should
expect its outermost electrons to reside in a p subshell.

Atoms first chapter 3.7 11

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Atoms first chapter 3.7 11