Chapter 2. Atomic Structure
Part II
Chapter 2. Atomic Structure
Outline
1. Atomic spectra and the Bohr model
2. Quantum mechanics and the electronic structures of hydrogen-like
species
2.1. The wave nature of matter
2.2. Quantum mechanics and atomic structures
2.3. Quantum numbers
2.4. Energy of atomic orbitals
2.5. Shapes of atomic orbitals
3. Polyelectronic atoms (and Electron configuration)
4. Periodic trends in atomic properties
3. Polyelectronic atoms
3.1. Energy levels of atomic orbitals
Polyelectronic atoms: atoms with more than one e(e.g. He).
Problem: The Schrödinger wave equation cannot be
solved exactly, due to e-e correlation problem.
two electrons
Three components must be considered in order to describe the total
energy for He:
i.
Kinetic energy of electrons
ii. Potential energy from the attraction between electrons
and the nucleus.
iii.
Potential energy from the repulsion between electrons.
Since the electron pathways are unknown, the e–e repulsion (item iii
above) cannot be determined easily, making it impossible to solve the
Schrodinger equation exactly.
r1
e-
r12 ?
2+
r2
e-
electron-electron
correlation
3
3. Polyelectronic atoms
z
z
How can we deal
with poly e- atoms?
z
s
s
p
p
Assumptions:
y
x
y
x
•Electrons occupy
hydrogenic
(hydrogen-like)
atomic
x
orbitals
z
s
px
pz
py
etc,….
y
x
pz pz
y
px px
py py
•The energy (and size) of atomic orbitals are different from
d d
those
in
hydrogen
atom.
p
dxz
dyz
d
dxz dxz dyz dyz
py
dz2
pz
2 2 px
dxy
dx
-y changes
What are the
dxy dxy
in the
energies of the atomic orbitals ?
2
2 2-y2 dz2 dz
dx
dx -y
2
4
What are the changes in the energies of the atomic
orbitals?
Energy levels of atomic orbitals
of single-electron atoms:
Orbitals with the same n have the
same energy.
[Orbitals having the same energy
are said to be degenerate].
Energy levels of atomic orbitals
of polyelectronic atoms:
The degeneracy of atomic
orbitals of polyelectronic
atoms partially is removed.
5
Selected Calculated orbital energy (for reference)
Why is the order of orbital energy levels of polyelectronic atoms
different from that of hydrogen-like species?
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3.2. Shielding Effect, Penetration Effect and
Effective Nuclear Charge
• For hydrogen-like atoms (1e-):
Z+
eEn = - 13.6
Z2
2
n
(ev)
• For polyelectronic atoms (more than one e-), the above
equation could not be used, because e- will feel a
positive charge less than Z.
Z
e1
e2
The e2 feels a nuclear charge
smaller than Z.
7
Shielding Effect in Polyelectronic Atoms
• Effective nuclear charge (Z*):
Real charge felt by an electron.
•
Z* is the net result from the
nuclear attraction and the
average repulsions from the
other electrons.
Z
e1
e2
Z*(2) = Z - (1)
where  = shielding parameter
• Shielding (or screening,屏蔽) effect: the ability to decrease in
the attraction between the nucleus and an electron.
In the case above, we say e1 has shielding effect on e2.
8
Shielding Effect in Polyelectronic Atoms
Z= 19
19
K
Zeff
+
 For the outermost electron, Zeff < Z (19) because of the repulsion
from the other inner-core electrons.
9
Shielding Effect in Polyelectronic Atoms
The energy of AOs of polyelectronic atoms could be
evaluated according to the modified equation:
En = - 13.6
Z*2
2
n*
(ev)
Zeff
Z* = Effective nuclear charge, the “actual” charge felt by an electron
n* = Effective principal quantum number
The larger the Z*, the lower the energy.
The lager the n*, the higher the energy
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Shielding Effect in Polyelectronic Atoms
Factors influencing Z*?
The effective nuclear charge felt by an electron may be affect by
(i) relative locations of the electron concerned and other electrons,
(ii) distribution of the electron density (penetration effect).
Z= 19
19
K
+
11
Shielding Effect in Polyelectronic Atoms
• Outer electrons are shielded from nuclear charge by inner
electrons, which reduce electrostatic attraction between
positively-charged protons in nucleus and the outer electrons
Li
Be
3p
B
5p
4p
1s
1s
2s
1s
2s
2s, 2p
• Electrons in the same shell can also have shielding effect on
other electrons.
• For a given shell, electrons in different subshells have a
different shielding effect and experience a different effective
nuclear charge due to penetration effect.
12
Penetrating Effect
Penetrating Effect:
Due to the wave nature, it is possible for
an outer electron to occupy the region
near the nucleus.
An 2s orbital
• An electron’s penetration effect is
determined by the atomic orbital’s
radial probability distribution.
13
Penetrating Effect
As the penetrating effect of an electron increases:
•Probability to find the e- near the nucleus ________
•Effective nuclear charge felt by the electron _________
•Energy of the electron __________
•Shielding effect (to others) of the electron ________
Z+
e-
e-
En = - 13.6
Z*2
2
n*
(ev)
14
Comparison of 2s and 2p orbitals
Order of the Penetrating Effect:
(have high probability to find electron
close to the nucleus).
Order of the Shielding Effect:
Order of the Z*:
Order of the Energy:
15
Comparison of ns, np, nd, and nf
Orbitals in Polyelectronic Atoms
Order of the Penetrating Effect:
(have high probability to find electron
close to the nucleus)
ns np nd nf
16
Effective Nuclear Charges of Electrons in Selected
Atoms (for reference only)
Slater developed some empirical rules to calculate effective nuclear
charges for various atoms [out of the scope]
Note: except for H, Z* is always less than Z.
17
Periodic Trend in Effective Nuclear Charges
Effective Nuclear Charge vs. Atomic Number
9
For outmost electrons only
8
Ar
7
Ne
Effective Nuclear Charge
6
F
5
O
4
C
3
Be
2
Na
1
Li
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
Atomic Number
Across a period, Z* increases
18
Exercises
1. Consider p electrons in O and C. In which atom
would the p electrons feel a larger effective nuclear
charge?
A) O
B) C
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Exercises
2. Consider e- in 4p and 4f orbitals in a polyelectronic
atom.
(a) Which e- has a higher probability to be found close to the
nucleus?
(b) Which e- has a larger screen (shielding) effect?
(c) Which e- feels a large effective nuclear charge?
20
Exercises
3. Predict the relative energy of 4s and 4d orbitals in
(a) Li2+
(b) N6+
(c) Cs
(d) Ni2+
21
3.3. Electron Configurations
• How are electrons distributed in various
atomic orbitals?
• Electron configuration: the distribution
of electrons of an atom in atomic orbitals
Representations:
Letter representation:
Box representation:
H: 1s1
He: 1s2
B: 1s22s22p1
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Three Rules for Writing an Electron Configuration:
(1) The Aufbau Principle: Which orbitals are filled first?
General sequence:
The electrons in an atom will fill in orbitals of lower energy first.
General sequence according to the Aufbau principle:
1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p ……
23
Three Rules for Writing an Electron Configuration:
(2) The Pauli Exclusion Principle:
How many e-’s can occupy an atomic orbital?
No two electrons in an atom can have the same four quantum
numbers.
2e1s
B
A
In other words, an atomic orbital can have a maximum of two
electrons with opposite spins.
For example,
ns
Maximum #e?:
np
24
Three Rules for Writing an Electron Configuration:
(3) Hund's Rule: If there are degenerate orbitals, how are
e-’s filled?
The most stable arrangement of electrons in subshells is the one with
the greatest number of parallel spins.
e.g. 2e-’s in 2p orbitals: three possibilities are shown below:
A
B
C
25
Exercises
1. What is the maximum number of electrons that can
occupy a subshell with a quantum number l?
26
Exercises
2. What is the maximum number of electrons that can
occupy a shell with a quantum number n?
27
Exercises
3. (i) Write the electron configurations for:
(a) B, Z = 5; (b) Ar, Z = 18; (c) Fe, Z = 26.
(ii) Indicate unpaired electrons in each atoms.
(i)
(ii)
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Abbreviated Electron Configurations
He, Z = 2; 1s2
[He]
B, Z = 5; 1s22s22p1
[He]2s22p1
Ar, Z = 18; 1s22s22p63s23p6
[Ar]
Fe, Z = 26; 1s22s22p63s23p64s23d6
[Ar] 4s23d6
Full Electron Configurations
Symbols of nearst
smaller noble gas
The remaining
electrons
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Exceptions in Electron Configurations
BEWARE!!! There are many exceptions in electron configurations. This is
particularly true for transition metals and f-block elements.
For example:
Expected
Cu (Z = 29)
[Ar]4s2 3d9
Observed
[Ar] 4s1 3d10
Mo (Z = 42)
[Kr] 5s2 4d4
[Kr]5s1 4d5
Ag (Z = 47)
[Kr] 5s2 4d9
[Kr]5s1 4d10
The exact reason for the observed configuration is still under debate.
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More Examples of Exceptions
31
You are NOT expected to memorize these exceptional cases !!!
Periodic Table and Electron Configuration
nsx
ns2(n-1)dx
ns2npx
32
The last subshell to be filled with electrons
33
Periodic Table and Electron Configuration
[x]nsx
n = 2-3, [X]ns2npx
n = 4-5, [X]ns2(n-1)d10npx
n = 6-7, [X]ns2(n-2)f14(n-1)d10npx
n = 4-5, [X]ns2(n-1)dx
n = 6-7, [X]ns2(n-2)f14(n-1)dx
35
Periodic Table and Electron Configuration
Write Abbreviated Electron Configurations with the Help of Periodic Table.
e.g. Zinc (Zn), Rhenium (Re)
2
10
Zn: [Ar] 4s 3d
36
Exercises
1. Determine the expected electron configurations for:
(a) S,
(b) Ta.
37
3.4. Valence Electrons
Valence electron: the electrons in the “outermost” orbitals
of an atom.
(valence orbitals = 2s, 2p)
e.g. Ne: 1s22s22p6
(#valence electrons = 8)
Core electrons/orbital
Valence electrons/orbital
The elements in the same group in the periodic table have the
same valence electron configuration.  same # of valence e-.
s block: electrons in ns orbital
p block: electrons in outer ns np orbitals
d block: electrons in ns (n-1)d orbitals
Valence Electrons
e.g.
N:
Pt:
Fe:
s block: electrons in ns orbital;
d block: electrons in ns (n-1)d orbitals
p block: electrons in outer ns np orbitals
3.5. Electron Configurations of Ions
Formation of Cations: Electrons are not necessarily removed in the
same order as we put them in according to the Aufbau Principle.
Electrons leave theouter most shell (orbitals with larger n and l) first.
e.g. Na, 1s2 2s2 2p6 3s1
Na+, 1s2 2s2 2p6
Formation of anion:
Electrons are added in
the same order as we
put them in according to
the Aufbau Principle.
e- in 3s is removed.
Electron Configurations of Ions
3.5. Electron Configurations of Ions
For reference
4. Periodic Variation in Atomic Properties
4.1. Atomic and Ionic Sizes.
Their sizes can be described in terms of their radii.
Three types of commonly used radii
•Covalent radius:
Half internuclear separation of neighboring atoms
of the same element in a molecule.
•Metallic radius:
Half the experimentally determined distance between
the nuclei of nearest neighbor atoms
• Ionic Radius:
Related to the distance between neighboring
cation and anion. (The radius of O2- is taken as
1.40 Å).
Trend in atomic sizes (covalent radius)
From left to right across a period: atom size decreases, because
Down a group: atom sizes increases, because
 In each period, alkali metals have the largest, noble gases the smallest atomic radii.
Variations of Atomic Size with Oxidation State
M-, M, M+, which one is the biggest? Why?
In cation, removal of e- increases Z*
felt by the outer e-’s, thus the
electron cloud contracts.
In anion, the extra e- increases
the e-e repulsion and
decreases Z* felt by outer e-.
All anions are ___________ than their parent atoms.
All cations are ___________ than their parent atoms.
Exercise
Arrange the following in order of increasing sizes:
Se2-, Br-, Sr2+, Rb+.
4.2. Ionization Energy (I)
The energy required to remove an electron from a gaseous
atom or ion.
A(g)  A+(g) + e- (g)
I1
1st ionization energy
A+(g)  A2+(g) + e- (g) I2
2nd ionization energy
o
E
n = infinite
I
Example:
Mg (g) → Mg+ (g) + e–
Mg+
(g) →
Mg2+
(g) +
e–
Mg2+ (g) → Mg3+ (g) + e–
Note:
I1 = 735 kJ/mol I > I by a factor of ~2
2
1
However, I3 > I2 by a factor
I2 = 1445 kJ/mol of ~5
I3 = 7730 kJ/mol Why is this?
48
General Trend in 1st Ionization energy
Across a Period: I1 increases
Down a Group: I1 decreases
Some Zig-zag feature is also noted.
Another View
Across a Period:
I1 increases.
Down a Group:
I1 decreases.
Zig-zag feature
is also noted
Explanations
I is dependent on:
• Effective nuclear charge
• From which orbital the electron is removed
• How the orbitals are occupied
En = - 13.6
Z*2
2
n*
(ev)
Z* = Z-
binding energy
• Down a group: I1 decreases, because ____________
• Across a period: I1 increases, because ___________
Exercises
Explain the following facts:
(1) Why I(Be)  I(Li)?
(2) Why I(Be)  I(B)?
(3) Why I(N)  I(O)?
4.3. Electronegativity
Electronegativity (c): A measure of the ability of atoms in a
molecule to attract electrons to itself.
Cl
Cl
H
Cl
Cl
: Cl
H
: Cl
Cl is more electronegative
than H
Important NOTE:
•Electronegativity cannot be directly measured and must
be calculated from other atomic or molecular properties.
•There are several scales (methods) to calculate
electronegativity. The one often used and cited is the scale
proposed by Linus Pauling, and is represented by χp.
Linus Pauling
1901 –1994, American
Nobel Prize in Chem. 1954
Nobel Prize in Peace 1962
Electronegativity, Pauling scale
Values range from 0.7 (Cs) to 4.0 (F)
Decreasing electronegativity
Periodic Trend in Electronegativity (c) of Main Group
Elements
•
c (non-metal) > c (metal)
• In each period, halogens have the largest, and alkali
metals the smallest electronegativities.
• Group is more important than period.
Periodic Trend in Electronegativity (c) of Main Group
Elements
Decreasing electronegativity
Increasing electronegativity
Across a period: c increases, because ____________.
Down a group: c decreases, because _____________
__________________________.
____
Exercises
1. Arrange the elements oxygen, fluorine, and sulfur
according to increasing in:
Ionization energy:
Atomic size:
Exercises
2. Rank the electronegativity of (a) K, (b) Cl, (c) Si.
End of Chapter 2
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