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CH101 GENERAL
CHEMISTRY I
SPRING 2013
• Textbook: ‘Chemical Principles. The
Quest for Insight’ by P Atkins and L
Jones. Freeman, New York, 2010
(International Edition)
1
F1
CHEMISTRY
The science of matter and the changes it can undergo.
A science that deals with the
composition, structure, and
properties of substances and
with the transformations that
they undergo.
F2
Chemistry: A Science at Three Levels
Symbolic level
The expression of chemical phenomena in terms
of chemical symbols and mathematical equations
Macroscopic level
The level dealing with the properties
of large, visible objects
Microscopic level
An underworld of change
at the level of atoms and molecules
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F3
How Science Is Done
4
F4
The Branches of Chemistry
 Traditional areas
Organic chemistry (carbon compounds)
Inorganic chemistry (all other elements and their compounds)
Physical chemistry (principles of chemistry)
 Specialized areas
Biochemistry (chemistry in living systems)
Analytical chemistry (techniques for identifying substances)
Theoretical (computational) chemistry (mathematical and computational)
Medicinal chemistry (application to the development of pharmaceuticals)
 Interdisciplinary branches
Molecular biology (chemical basis of genes and proteins)
Materials science (structure and composition of materials)
Nanotechnology (matter at the nanometer level)
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Chapter 1.
ATOMS: THE QUANTUM WORLD
INVESTIGATING ATOMS
1.1 The Nuclear Model of the Atom
1.2 The Characteristics of Electromagnetic Radiation
1.3 Atomic Spectra
QUANTUM THEORY
1.4 Radiation, Quanta, and Photons
1.5 The Wave-Particle Duality of Matter
1.6 The Uncertainty Principle
1.7 Wavefunctions and Energy Levels
2012 General Chemistry I
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INVESTIGATING ATOMS (Sections 1.1-1.3)
1.1 The Nuclear Model of the Atom
 Discovering the electron
J.J. Thomson (English
physicist, 1856-1940) in 1897
discovers the electron and
determines the charge to
mass ratio (e/me) as “cathode
rays”. In 1906 he wins the
Nobel Prize.
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Robert Millikan designed an ingenious
apparatus in which he could observe
tiny electrically charged oil droplets.
Fundamental charge,
the smallest increment of charge
e = 1.602×10-19 C
From the value of e/me measured
by Thomson,
me = 9.109×10-31 kg
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Brief Historical Summary
Atomic Theory
J Dalton, 1807
Discovery of
electron ('cathode
rays'), value of e/m e
J J Thomson, 1897
Implies internal
structure of sub
-atomic particles
electron + proton
= neutral
Value of e
(1.602 x 10-19 C)
m e = 9.109 x 10-31 kg
R Millikan and
H A Fletcher, 1906
How are electrons and
protons arranged
in the atom?
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Two Models of the Atom
'Pudding' model
(J J Thomson)
'Nuclear' model
(E Rutherford)
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 Nuclear Model
Ernest Rutherford (1871-1937) and his students in England studied α
emission from newly-discovered radioactive elements.
Experiment by Geiger and Marsden
Nucleus occupy a small volume at the center of the atom
Nucleus contains particles called proton (+e) and neutron (uncharged).
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Nuclear Model of the Atom
In the nuclear model of the atom, all the positive
charge and almost all the mass is concentrated in
the tiny nucleus, and the negatively charged
electrons surround the nucleus. The atomic
number is the number of protons in the nucleus.
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Some Questions Posed by the Nuclear
Model
1. How are the electrons arranged around the
nucleus?
2. Why is the nuclear atom stable? (classical
physics predicts instability)
3. What holds the protons together in the
nucleus?
Atomic spectroscopy (involving the absorption
by, or emission of electromagnetic radiation
from atoms) provided many of the clues needed
to answer questions 1 and 2.
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1.2 The Characteristics of Electromagnetic
Radiation
 Spectroscopy – the analysis of the light emitted or absorbed by
substances
- Light is a form of electromagnetic radiation, which is the periodic
variation of an electric field (and a perpendicular magnetic field).
amplitude: the height of the wave
above the center line
intensity: the square of the amplitude
wavelength, l: peak-to-peak distance
wavelength × frequency = speed of light
ln
=
c
c = 2.9979 x 108 ms-1
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The color of visible light depends on its frequency and wavelength;
long-wavelength radiation has a lower frequency than
short-wavelength radiation.
infrared
l > 800 nm
the radiation of heat
visible light l = 700 nm (red light)
to 400 nm (violet light)
ultraviolet
l < 400 nm
responsible for sunburn
15
Color, Frequency and Wavelength of
Electromagnetic Radiation
16
Self-Test 1.1A
Calculate the wavelengths of the light from traffic signals
as they change. Assume that the lights emit the following
frequencies: green, 5.75 x 1014 Hz; yellow, 5.15 x 1014
Hz; red, 4.27 x 1014 Hz.
Solution: Green light l =
2.998 x 108 m/s
5.75 x 1014 /s
= 5.21 x 10-7 m = 521 nm
Similarly, yellow light is 582 nm and red light is
702 nm wavelength
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1.3 Atomic Spectra
White light
Discharge lamp of
hydrogen (emission
spectrum)
spectral lines
discrete energy
levels
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Johann Rydberg’s general empirical equation
R (Rydberg constant) = 3.29×1015 Hz
an empirical constant
n1 = 1 (Lyman series), ultraviolet region
n1 = 2 (Balmer series), visible region
n1 = 3 (Paschen series), infrared region
For instance, n1 = 2 and n2 = 3,
l = 6.57×10-7 m
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Self-Test 1.2A
Calculate the wavelength of the radiation emitted by a
hydrogen atom for n1 = 2 and n2 = 4. Identify the spectral
line in Fig. 1.10b.
Solution
n =
1 _ 1
R
22
42
c
Now, l =
n
=
16c
3R
=
=
3
R
16
16 x 2.998 x 108 m s-1
3 x 3.29 x 1015 s-1
= 4.86 x 10-7 m or 486 nm
It is the second (green/blue) line in the spectrum.
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Absorption Spectra
When white light passes through a gas, radiation is
absorbed by the atoms at wavelengths that correspond
to particular excitation energies. The result is an atomic
absorption spectrum.
Above is an absorption spectrum of the sun: elements can
be identified from their spectral lines.
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QUANTUM THEORY (Sections 1.4-1.7)
1.4 Radiation, Quanta, and Photons
 This section involves two phenomena that classical physics
was unable to explain (along with atomic line spectra): black
body radiation and the photoelectric effect.
 Black body radiation is the radiation
emitted at different wavelengths by
a heated black body, for a series of
temperatures. Two empirical laws are
associated with it (below) and plots
of emitted radiation energy density
versus wavelength give bell-shaped
curves (right)
Stefan-Boltzmann law:
Total intensity = constant × T4
Wien’s law:
T lmax = constant
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Self-Test 1.3A
In 1965, electromagnetic radiation with a maximum of
1.05 mm (in the microwave region) was discovered to
pervade the universe. What is the temperature of
‘empty’ space?
Solution
Use Wien's law, in the form T = constant/ lmax
T=
2.9 x 10-3 m K
-3
1.05 x 10 m
= 2.8 K
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Black Body Radiation Theories
• Rayleigh-Jeans theory was based on classical
physics. It assumed the black body atoms behave
like mechanical oscillators that absorb and emit
energy continuously.
Radiant energy density =
8kBT
l4
• The Rayleigh-Jeans equation did not agree with
experimental data, except at high wavelengths.
• Classical physics predicts intense UV or higher
energy radiation from hot black bodies!
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Planck’s Quantum Theory
• Max Planck (1900) made the assumption that black
body atoms could absorb and emit energy only in
multiples of a fundamental quantity (a quantum),
whose value is
E = hn (4)
• The constant h became known as Planck’s constant
(= 6.626 x 10-34 Js)
• The Planck equation describing the black body
radiation profile agreed well with experiment.
1
8hc
Radiant energy density =
l5
_ hc
e
l kB T
_
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1
Summary
 Ultraviolet catastrophe:
Classical physics predicted that any hot
body should emit intense ultraviolet
radiation and even X-rays and g-rays!
Assumes continuous exchange of energy.
Quantization of electromagnetic radiation
suggested by Max Planck
E = hn
Assumes energy can be exchanged
only in discrete amounts (quanta)
Radiation of frequency n can be generated only if an oscillator of
that frequency has acquired the minimum energy required to
start oscillating.
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 The Photoelectric effect –
ejection of electrons from
a metal when its surface is
exposed to ultraviolet radiation
Photoelectric effect:
observations
1. No electrons are ejected
< a certain threshold value of frequency,
which depends on the metal.
2. Electrons are ejected immediately
at that particular value.
3. The kinetic energy of ejected
electrons increases linearly with the
frequency of the radiation.
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 Albert Einstein proposed that electromagnetic radiation consists
of particles, called “photons”.
– The energy of a single photon is proportional to the radiation
frequency by E = hn.
work function
 Bohr frequency condition:
relates photon energy to energy
difference between two energy
levels in an atom:
28
Self-Test 1.5A (and part of 1.5B)
The work function of zinc is 3.63 eV. (a) What is the
longest wavelength of electromagnetic radiation that
could eject an electron from zinc?
(b) What is the wavelength of the radiation that ejects
an electron with velocity 785 km s-1 from zinc?
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Solution
(a)  =
hc
l0
hence l0 =
hc

Firstly, the work function should converted
from eV to J (1 eV = 1.602 x 10-19 J)
= 3.63 eV x (1.602 x 10-19 J/1 eV) = 5.82 x 10-19 J
l0 =
6.626 x 10-34 J s x 3.00 x 108 m s-1
5.82 x 10-19 J
= 3.42 x 10-7 m = 342 nm
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(b) Ek =
1
x 9.109 x 10-31 kg x (7.85 x 105 m s-1)2
2
= 2.81 x 10-19 J
From Einstein's equation, hc/l = Ek + 
l =
6.626 x 10-34 J s x 3.00 x 108 m s-1
(2.81 x 10-19 J + 5.82 x 10-19 J)
= 2.30 x 10-7 m = 230 nm
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Summary of 1.4
Black body
radiation
Planck's
quantum
hypothesis
Photoelectric
effect
Particulate nature
of electromagnetic
radiation
Bohr's frequency condition: hn = Eupper - Elower
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1.5 The Wave-Particle Duality of Matter
– Wave behavior of light:
diffraction and interference effects of
superimposed waves
(constructive and destructive)
– Louis de Broglie proposed that all particles
have wavelike properties.
l is the de Broglie wavelength of
an object with linear momentum p = mv
electron diffraction reflected from a crystal
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De Broglie Wavelengths for Moving Objects
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1.6 The Uncertainty Principle
 Complementarity of location (x) and momentum (p)
– uncertainty in x is Dx; uncertainty in p is Dp
 Heisenberg uncertainty principle
where ħ = h / 2 = 1.0546×10-34 J·s
– x and p cannot be determined simultaneously.
35
Example Calculation
If the Bohr radius of the H atom is 0.529 A and we know the
position of the electron to within 1% uncertainty, calculate
the uncertainty in the electron's velocity.
-10
Dx = 0.529 x 10 (m) = 5.29 x 10-13 m
100
From the Heisenberg uncertainty principle, DxDp > h
4
6.626 x 10-34 (J s)
Dp >
or > 9.96 x 10-23 kg m s-1
4 x 3.142 x 5.29 x 10-13 (m)
Because Dp = m Dv, the uncertainty in the velocity is Dv =
9.96 x 10-23 (kg m s-1)
9.110 x 10-31 (kg)
= 1.09 x 108 m s-1: an uncertainty that is the magnitude of
the speed of light!
36
1.7 Wavefunctions and Energy Levels
 Erwin Schrödinger introduced a central
concept of quantum theory.
particle trajectory
wavefunction
– Wavefunction (y, psi): a mathematical function
with values that vary with position
– Born interpretation: probability of finding the
particle in a region is proportional to the value
of y2
– Probability density (y2): the probability
that the particle will be found in a small
region divided by the volume of the region.
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Schrödinger Equation: Wave Equation for Particles
The classic differential equation describing a standing wave in one dimension
is
4 2 y = 0
d2 y
( y = wavefunction; l = wavelength) (1)
+
2
2
l
dx
From de Broglie's hypothesis, l2 = h2/2mK = h2/[2m(E-V)] (K is kinetic and Vis
potential energy)
Substituting for l in (1) gives
[82m(E-V)] y = 0
d2 y
+
2
dx
h2
or
_
h2 d2 y _ y
V = Ey (2)
2
2
8 m dx
h d
This implies that the (electron wave) momentum, p , is 
2 i dx
instead of p = mv = mdx/dt
h
2
1
2 d 2
_ h2 d2
p
Hence
K =
=
=
8p2m dx2
2m 2i
dx
2m
Combining (2) and (3) gives
That is
H y
K
(3)
y + V y = K + V y = E y,
= Ey
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 Schrödinger equation: for a particle of mass m moving in one dimension
in a region where the potential energy is V(x)
H = hamiltonian of the system
- H represents the sum of potential energy and kinetic energy in a system
- Origins of the Schrödinger equation:
If the wavefunction is described as y(x) = A sin
d2y(x)
2
=
l
dx2
2
y(x)
l = h/p
2x
l,
d2y(x)
2 p 2 y(x)
=
h
dx2
2
ħ2 d2y(x)
p
y(x) : kinetic energy
=
2m dx2
2m
V(x): potential energy
Particle in a Box
 The ‘particle in a box’ scenario is the simplest application of the
Schrödinger equation
– Mass m confined between two rigid walls a distance L apart
– y = 0 outside the box and at the walls (boundary condition)
– Potential energy is zero within and infinite outside the box
n = quantum number
The Solutions of Particle in a Box
For the kinetic energy of a particle of mass m,
Whole-number multiples of half-wavelengths
can follow the boundary condition,
When this expression for l is inserted into
the energy formula,
41
More General Approach
From the Schrödinger equation with V(x) = 0 inside the box,
Solution:
k2 = 2mE/ħ2, and it follows
From the boundary conditions of y(0) = 0 and y(L) = 0,
0
42
L
Wavefunction obtained so far is (with just A left
to identify)
yn(x) = A sin nx
L
The normalization condition determines A:
yn
2
L
=
A
sin2
2
0
Hence,
yn(x) =
2

sin n x
L
L
nx
L
dx = 1
A = 2/L
 Energies of a particle of mass m in a one dimensional box of length L,
n = quantum number
Energy of the particle is quantized, and restricted to energy levels.
- Energy quantization stems from the boundary
conditions on the wavefunction.
- Energy separation between two neighboring levels
with quantum numbers n and n+1:
- L (the length of the box) or m (the mass of the
particle) increases, the separation between
neighboring energy levels decreases.
44
 Zero-point energy:
The lowest value of n is 1, and the lowest energy is E1 =
h2/8mL2, not zero.
According to quantum mechanics, a
particle can never be perfectly still
when it is confined between two walls:
it must always possess an energy.
consistent with the uncertainty
principle
– The shapes of the wavefunctions of
a particle in a box
E1 = h2/8mL2, E2 = h2/2mL2
45
EXAMPLE 1.8
Treat a hydrogen atom as a one-dimensional box of length 150 pm,
containing an electron. Predict the wavelength of the radiation
emitted when the electron falls to the lowest energy level from the
next higher energy level.
n = 1, n+1 = 2, m = me, and L = 150 pm
= hn = hc/l
Chapter 1.
ATOMS: THE QUANTUM WORLD
THE HYDROGEN ATOM
1.8 The Principal Quantum Number
1.9 Atomic Orbitals
1.10 Electron Spin
1.11 The Electronic Structure of Hydrogen
2012 General Chemistry I
47
THE HYDROGEN ATOM (Sections 1.8-1.11)
1.8 The Principal Quantum Number
 A particle in a box
An electron held within the atom by
the pull of the nucleus
 For a hydrogen atom, V(r) = coulomb potential energy
 Solutions of the Schrödinger equation lead to the expression
for energy:
n is the principal
quantum number
R (Rydberg constant) = 3.29×1015 Hz, in good agreement with experiment
48
 For other one-electron ions, such as He+, Li2+, and even C5+,
- Z = atomic number, equal to 1 for hydrogen
- n = principal quantum number
- As n increases, energy increases, the atom
becomes less stable, and energy states
become more closely spaced (more dense).
- Ground state of the atom:
the lowest energy state, E = –hR when n = 1.
- Ionization: the bound electron reaches E = 0,
and freedom, and has left the atom. Ionization
energy: the minimum energy needed
to achieve ionization
49
1.9 Atomic Orbitals
The Schrödinger equation for the H atom is often written as below:
2
_ h
2
8 m  x
2
or
2
+
2
y
2
_ h2
+
2
2
z
+ V(x,y,z) y (x,y,z) = Ey (x,y,z)
2
y +
Vy
= Ey
or
Hy = Ey
2m
2
1  r2 
r
r2  r
becomes


sin
+

r2sin  
1
2
+
2
r2sin2  
1
in spherical polar coordinates
50
Spherical Polar Coordinate System
51
 Atomic orbitals: the wavefunctions (y) of electrons in atoms:
they are meaningful solutions of the Schrödinger equation.
– The square of a wavefunction (y2) is the probability density of an
electron at each point.
– Expressing wavefunctions in terms of spherical
polar coordinates (r, , ) is more convenient.
radial
angular
wavefunction:
wavefunction:
depends on two quantum depends on two quantum
numbers; n and l
numbers; l and ml
52
For the ground state of the hydrogen atom (n = 1),
Here, Y is a constant (independent
of angle): the wave-function is the
same in all directions : it is
spherically symmetrical).
This is the 1s orbital. It is the
only wavefunction for n = 1.
Bohr radius= 52.9 pm
y2
Its spherical electron cloud representation
and plot of y2 versus r are shown opposite:
r
53
Hydrogenlike Wavefunctions
54
Boundary Conditions
Boundary conditions are constraints that reality places on
the solutions to a physically relevant equation (e.g. a
quadratic equation and, here, the Schrodinger equation for
the H atom).
The solutions of the Schrodinger equation (wavefunctions,
y*) are thus seen to be ‘well-behaved’:
 Ψ must be smooth, single-valued, and finite everywhere in space
 Ψ must become small at large distances r from the nucleus (proton)
Boundary Condition yields quantum numbers!
* Ψ is just the embodiment of de Broglie’s hypothesis of matter waves)
2012 General Chemistry I
55
 Three quantum numbers (n, l, ml) specify an atomic orbital.
 n, principal quantum number (n = 1, 2, 3,…)
• It is related to the size and energy of the orbital
• It defines shell: AOs with the same
n value
56
 l, orbital angular momentum quantum number (l = 0, 1, 2, …, n-1)
• It is related to the orbital angular momentum of the electron
• It defines subshell: AOs with the same n and l values
n subshells with a principal quantum number n
Value of l
0
1
2
3
Orbital type
s
p
d
f
(Orbital angular momentum =
)
 ml, magnetic quantum number (ml = +l, l-1, l-2, …, 0, …, -l)
• It is related to the orientation of the orbital motion of the electron
• There are 2l+1 different values of ml for a given value of l
e.g. when l = 1, ml = +1, 0, -1
57
A summary of the
three quantum numbers
58
Degeneracy
- “Normally” the energy should depend on all three quantum
numbers.
- Hydrogen atom is special in that the energy depends only on
principal quantum number n.
- Two or more sets of quantum numbers corresponds to the
same energy are referred as “degenerate”
E.g. 200, 211, 210, 21-1 (for n = 2) states have the same
energy.
- Each n: given total energy → n2 possible combinations of
quantum numbers (total number of orbitals) → degeneracy
Shape of the s-Orbitals (l = 0)
 Visualizing AO as a cloud of points,
proportional to probability of finding the
electron in that volume (probability density,
y2, plot versus distance, r)
 Radial distribution function: the
probability that the electron will be
found anywhere in a thin shell of radius
r and thickness dr is given by P(r)dr,
with
For s orbitals, y = RY = R/21/2,
http://www.mpcfaculty.net/ron_rinehart/orbitals.htm
60
27s
 RDFs for H atom
- Smooth with one or more peaks
- Nodes appear at radii of zero probability
- Falls to zero smoothly at large r
2012 General Chemistry IDepartment of Chemistry, KAIST
61
61
- H-atom (The only atom for which the Schrödinger Eq can be solved
exactly  a model for bigger and many-electron atoms)
- 1s
orbital (n = 1, ℓ = 0, m = 0) → R10(r)
and Y00(θ,Ф)
a function of r only
* spherically symmetric
* exponentially decaying
* no nodes
- 2s
orbital (n = 2, ℓ = 0, m = 0)
zero at r = 2a0 = 1.06Å
nodal sphere or radial node
[r<2ao: Ψ>0, positive] [r>2ao: Ψ<0, negative]
62
Self-Test 1.9A
Calculate the ratio of the probability density for
the 1s orbital at r = 2a0 and r = 0.
Solution
y 2(2a0)
Probability density at r = 2a0 =
Probability density at r = 0
From Table 2, y
Hence
y 2(2a0)
y 2(0)
2
=
e -2r/a 0
 a03
-4a 0/a0
=
y 2(0)
e
 a03
e-4
_:
0
e
 a03
1
= e-4 = 0.0183
The probability is just under 2% of that finding
the electron in a region of the same volume
located at the nucleus.
63
 Visualizing AO as a boundary surface, that encloses most of
the cloud
The 95% boundary surface
Surface enclosing volume
where probability of finding
an electron is 95%
 radial wavefunction versus radius,
A radial node exists
where the curve crosses
the x-axis.
64
Shape of the 2p-Orbitals
 Radial function
 Boundary surface
+
_
_
+
+
_
- Two lobes with + and - signs
- Nodal plane (y = 0) separating
the two lobes. Also called
angular node. No radial
node for 2p orbitals.
- l = 1 and ml = +1, 0, -1
triply degenerate in energy
- three p-orbitals: px, py, pz
65
The 2p0 or 2py Orbital
-n = 2, ℓ = 1, m = 0 → 2p0 orbital : R21 Y10
· Ф = 0 → cylindrical symmetry about the z-axis
· R21(r) → r/a0 no radial nodes except at the origin
· cos θ → angular node at θ = 90o, x-y nodal plane
(positive/negative)
· r cos θ → z-axis 2p0 → labeled as 2pz
66
The 2px and 2py orbitals
Here, n = 2, ℓ = 1, ml = ±1 → 2p+1 and 2p-1
Their wave functions contain the complex term e±iφ,
which makes description difficult.
Since e±iφ = cosφ ± isinφ (Euler’s formula), a linear
combination of 2p+1 and 2p-1 gives two real orbitals,
2px and 2py, that complement 2pz
- px and py differ from pz only in the
angular factors (orientations).
Department of Chemistry,
KAIST
Shape of the 3d- and 4f-Orbitals
 3d-orbitals
l = 2 and
ml = +2, +1, 0, -1, -2
+
_
_ +
_
+
_
_
+
+_
_
+
_ +
+
Each has two angular
Nodes. No radial nodes for 3d
orbitals.
 4f-orbitals
l = 3 and
seven ml values
Each has three angular
nodes. No radial nodes for
4f orbitals.
68
+
+
_
A Note on Radial (Spherical) and Angular
Nodes
• Nodes are regions of space (spheres, planes or
cones) where y = 0.
• The total number of nodes possessed by a given
orbital = n –1.
• The number of angular nodes for a given orbital =
l.
• The remainder (n – 1 – l) are radial or spherical
nodes.
• Example 1. 4s orbital (n = 4; l = 0) has zero
angular nodes and 3 radial nodes.
• Example 2. 5d orbital (n = 5; l = 2) has 2 angular
nodes and 2 radial nodes.
69
1.10 Electron Spin
• Schrödinger’s theory, although successful, has
several inadequacies:
1. The time-dependent equation is 2nd order with
respect to space, but only 1st order in time.
2. It ignores relativity.
3. It does not account for electron spin.
• The idea of electron spin was first proposed by
Otto Stern and Walter Gerlach (1920). See next.
• Samuel Goudsmit and George Uhlenbeck
suggested electron spin as the cause of the
doublet at 589.2 and 589.8 nm (the ‘D-line’) in the
emission spectrum of sodium.
70
Electron Spin States
 ms, Spin magnetic quantum number
- An electron has two spin states,
as ↑(up) and ↓(down), or a and b.
- The values of ms, only +1/2 and -1/2
for the electron.
71
Dirac’s Relativistic Electron
Dirac (1928)
· Using the Schrödinger’s idea and Einstein’s (1905) special theory of
relativity → 4 coupled Schrödinger-like equations (Dirac Eq)
· Dirac equations → 4th quantum number, ms
(Electrons are required to have spin: a law of nature, NOT a postulate)
· Dirac predicted → antimatter (antiparticle: negative energy)
Dirac Equation for spin -1/2 particles
(relativistic modification of the Schrödinger wave equation)
Department of Chemistry, KAIST
Summary
Stern and Gerlach
experiments (1920)
Inadequacies
of
..
Schrodinger's theory
Electron
spin
Uhlenbeck and
Goudsmit explanation
of sodium D-line doublet
(1925)
EXPERIMENTAL
Dirac's equation
(combination of
special relativity with
wave mechanics; 1928).
Spin is a fundamental
property of electrons.
Spin magnetic quantum
number, ms = +1/2 and
-1/2
THEORY
73
1.11 The Electronic Structure of Hydrogen
 Ground state, n = 1, and there is only the 1s orbital
The 1s electron has the following quantum numbers,
n = 1, l = 0, ml = 0, ms = +1/2 or -1/2
 Excited states are achieved by absorption of photons
- The first excited state with n = 2, has four orbitals,
2s, 2px, 2py, or 2pz
The average distance of an electron from the nucleus increases.
- The next excited state with n = 3 has nine orbitals,
one 3s, three 3p, and five 3d orbitals with larger distances.
- Eventually, the electron can escape the atom by absorbing
enough energy, and thus ionization of hydrogen occurs.
74
Orbital Energy Diagram for Hydrogen
75
Self-Test 1.10A
The three quantum numbers for an electron in a hydrogen
atom in a certain state are n = 4, l = 2, ml = -1. In what type
of orbital is the electron located?
Solution
l =2
d type, n = 4
4d
76
Chapter 1.
ATOMS: THE QUANTUM WORLD
MANY-ELECTRON ATOMS
1.12 Orbital Energies
1.13 The Building-Up Principle
1.14 Electronic Structure and the Periodic Table
THE PERIODICITY OF ATOMIC PROPERTIES
1.15 Atomic Radius
1.16 Ionic Radius
1.17 Ionization Energy
1.18 Electron Affinity
1.19 The Inert-Pair Effect
1.20 Diagonal Relationships
1.21 The General Properties of the Elements
2012 General Chemistry I
77
MANY-ELECTRON ATOMS (Sections 1.12-1.14)
1.12 Orbital Energies
- In many-electron atoms, Coulomb potential energy equals the
sum of nucleus-electron attractions and electron-electron
repulsions.
- There are no exact solutions of the Schrödinger equation.
- For a helium atom,
r1 = the distance of r2 = the distance of
electron 1 from
electron 2 from
the nucleus
the nucleus
r12 = the distance
between the
two electrons
78
The hydrogen atom: no electron-electron repulsion.
All the orbitals of a given shell are degenerate.
Many-electron atoms: electron-electron repulsions.
The energy of a 2p-orbital > that of a 2s-orbital
 Shielding:
Each electron attracted by the nucleus,
and repelled by the other electrons.
→ shielded from the full nuclear
attraction by the other electrons.
- effective nuclear charge, Zeffe < Ze
the energy of electron,
79
 Penetration:
s-electron – very close to the nucleus
penetrates highly through the inner
shell.
p-electron – penetrates less
than an s-orbital
effectively shielded from the
nucleus.
In a many-electron atom,
because of the effects of
penetration and shielding, the
order of energies of orbitals in a
given shell is s < p < d < f.
80
1.13 The Building-Up Principle
The Building-up (Aufbau) Principle is a set of rules that
allows us to construct ground state electron configurations
of the elements.
1.
2.
3.
Assume electrons ‘occupy’ orbitals in such a way as
to minimize the total energy (lowest energy first).
Assume a maximum of two electrons can ‘occupy’ an
orbital and these must have opposite spins. (Pauli’s
Exclusion Principle: no two electrons in an atom can
have the same set of quantum numbers).
Assume electrons occupy unoccupied degenerate
subshell orbitals first, and with parallel spins (Hund’s
rule).
In building-up electron configurations, in order of energy,
subshell energy overlap must be taken into account (next
slide).
81
Subshell Energy Overlap
• The complex shielding/repulsion effects that occur when
more and more electrons are ‘fed’ into orbitals during the
building-up process leads to subshell energy overlap,
beyond n = 3.
• See Figs. 1.41 and 1.44.
• For example, the 4s subshell is slightly lower in energy than
the 3d subshell and hence fills first.
• Likewise the 5s subshell fills before the 4d or 3f subshells,
for the same reason. See next slide.
82
Alternative Pictorial Representation of Orbital
Energy Order
83
Electron Configurations of the Elements in Period
1 (H and He), where n = 1
 Electronic configuration of an atom is a list of all its occupied orbitals,
with the numbers of electrons that each one contains.
 H, 1s1
Element
Electron
configuration
Orbital, with
electron occupancy
 He, 1s2
- Closed shell: a shell containing the maximum number of electrons
allowed by the exclusion principle
84
Electron Configurations of the Elements in Period 2
(from Li to Ne), the valence shell with n = 2
 Li, 1s22s1 or [He]2s1
- core electrons: electrons in filled orbitals
valence electrons: electrons in the outermost shell
- stable ionic form losing valence electrons: Li+
 Be, 1s22s2 or [He]2s2
- stable ionic form: Be2+
85
 B, 1s22s22p1 or [He]2s22p1
 C, 1s22s22p2 or [He]2s22p2
→
1s22s22px12py1
parallel spins
 C is first element to illustrate Hund’s rule:
If more than one orbital in a subshell is available, add electrons with
parallel spins to different orbitals of that subshell rather than pairing
two electrons in one of the orbitals.
- Excited state: An atom with electrons in energy states higher than
predicted by the building-up principle
In carbon, ground state, [He]2s22p2 → excited state, [He]2s12p3
86
87
All atoms in a given period have the same type of core with the same n.
All atoms in a given group have analogous valence electron configurations
that differ only in the value of n.
Group 18/VIIIA
Group IA
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
88
 n = 3: Na, [He]2s22p63s1 or [Ne]3s1 to Ar, [Ne]3s23p6
 n = 4: from Sc (scandium, Z = 21) to Zn (zinc, Z = 30)
the next 10 electrons enter the 3d-orbitals.
 The (n + l ) rule
Order of filling subshells in neutral atoms is determined by filling those
with the lowest values of (n + l) first. Subshells in a group with the same
value of (n + l) are filled in the order of increasing n.
order: 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < …
- There are exceptions, two of which are: Cr, [Ar]3d54s1 instead of [Ar]3d44s2
and Cu, [Ar]3d104s1 instead of [Ar]3d94s2, because of lower energy of halffilled (d5) and filled (d10) 3d subshell.
 n = 6: 5s-electrons followed by the 4f-electrons
Ce (cerium, [Xe]4f15d16s2)
89
36
Anomalous Configurations
Exceptions to the Aufbau principle
2012 General Chemistry I
90
Self-Test 1.12A
Write the ground-state configuration of a bismuth
atom.
Solution
Bi ( Z = 83) is in group VA (or 15) and has 5
electrons in its valence shell. As it is in period 6,
these will be 6s26p3. The nearest noble gas is Xe (Z
= 54), leaving 24 electrons to be accounted for in
filled 4f and 5d subshells.
The configuration is [Xe]4f145d10 6s26p3.
91
Self-Test 1.12B
Write the ground-state configuration of an arsenic
atom.
Solution
As ( Z = 33) is in group VA (or 15) and has 5
electrons in its valence shell. As it is in period 4,
these will be 4s24p3. Also because As is in period 4,
it will have an argon (Ar, Z = 18) core. The remaining
10 electrons are held in the filled 3d subshell. The
configuration is [Ar]3d10 4s24p3.
92
1.14 Electronic Structure and the Periodic
Table
- H and He, unique properties
93
- Main group: s- and p-blocks (groups IA- VIIIA)
-The Roman numeral group number tells us how many valence-shell
electrons are present
The modern nomenclature is groups 1, 2 and 13-18
Group IA
Group 18/VIIIA
94
-Period 1: H, He ; 1s orbital
- Period 2: Li through Ne ; 8 elements, one 2s and three 2p orbitals
- Period 3: Na through Ar ; 8 elements, 3s and 3p
- Period 4: K through Kr ; 18 elements, 4s, 4p and 3d
- Period 5: Rb through Xe ; 18 elements, 5s, 5p and 4d
- Period 6: Cs through Rn ; 32 elements, 6s, 6p, 5d and 4f
Period 2
Period 3
Period 4
Period 5
Period 6
Period 7
95
THE PERIODICITY OF ATOMIC PROPERTIES
(Sections 1.15-1.21)
The variation of effective nuclear charge throughout the periodic table
plays an important part in the explanation of periodic trends.
See below (Fig. 1.45). Zeff refers to outermost valence electron.
96
1.15 Atomic Radius
 Atomic radius: defined as half the distance between the
centers of neighboring atoms.
-For metals, r in a solid sample is the metallic radius
- For nonmetal, r for diatomic molecules is the covalent radius
- For a noble gas, r in the solidified gas is the Van der Waals radius
 General trends
- r decreases from left to right across a period (effective nuclear
charge increases)
- r increases from top to bottom down a group (change in n and
size of valence shell; effective nuclear charge decreases)
- See Figs. 1.46 and 1.47 (next slides)
97
186
98
99
1.16 Ionic Radius
 Ionic radius: its share of the distance between
neighboring ions in an ionic solid
- Cations are smaller than parent atoms.
i.e. Zn (133 pm) and Zn2+ (83 pm)
- Anions are larger than parent atoms.
i.e. O (66 pm) and O2- (140 pm).
 Isoelectronic atoms and ions
are atoms and ions with the same
number of electrons
e.g. Na+, F-, and Mg2+
radius: Mg2+ < Na+ < Fdue to different nuclear charges
100
Self-Tests 1.13A and 1.13B
Arrange each of the following pairs in order of increasing
ionic radius. (a) Mg2+ and Al3+ (b) O2- and S2Solution
(a) Al lies to the right of Mg, hence r(Al 3+) < r(Mg2+)
(b) O lies above S in group VIA, hence r(O2-) < r(S2-)
Arrange each of the following pairs in order of
increasing ionic radius. (a) Ca2+ and K+ (b) S2- and ClSolution
(a) Ca lies to the right of K, therefore r(Ca2+) < r(K+)
(b) Cl lies to the right of S, therefore r(Cl-) < r(S2-)
In both cases, check agreement with Fig. 1.48
1.17 Ionization Energy
 Ionization Energy, I, is the minimum energy needed to remove
an electron from an atom in the gas phase
The first ionization energy, I1
I1 (746 kJ·mol-1)
The second ionization energy, I2
I2 (1958 kJ·mol-1)
- I1 typically decreases down a group.
(change in n of valence electron;
Zeff increases)
- I1 generally increases across a period.
(Zeff increases)
- Metals: lower left of the periodic table, low ionization energies
- Nonmetals: upper right of the periodic table, high ionization energies
102
The periodic variation of the first ionization
energies of the elements (Fig. 1.51)
103
For a particular
element, second (I2),
third (I3) and higher
ionization energies
are greater than I1,
because of the
increasing ionic
charge.
Very high ionization
energies occur when
an electron is
removed from an
inner shell. See Fig.
1.52 (opposite).
Blue outline denotes
ionization from the
valence shell.
104
1.18 Electron Affinity
 Electron Affinity, Eea
The energy released when an electron is added to a gas-phase atom.
e.g.
- Eea generally decreases down a group.
(change in n of valence electron;
Zeff decreases)
- Eea generally increases across a
period. (Zeff increases)
– Group 17/VIIA, the highest 1st Eea (F-)
strongly negative 2nd Eea (F2-)
– Group 16/VI, positive 1st Eea (O-)
negative 2nd Eea (O2-)
105
Self-Test 1.15B
Account for the large decrease in electron affinity
between fluorine and neon.
Solution
2
2
5
e
_
_
For F (1s 2s 2p )
F (1s22s22p6), an electron
enters the 2p subshell and completes the octet
_
_
e
2 2
6
For Ne (1s 2s 2p )
Ne ([Ne]3s1), the electron
enters the energetically higher 3s subshell
1.19 The Inert-Pair Effect
– Tendency to form ions two units lower in charge
than expected from the group number.
– Due in part to the different energies of
the valence p- and s-electrons.
1.20 Diagonal Relationships
– A similarity in properties between diagonal
neighbors in the main groups of the periodic
table.
– Similarity in atomic radius, ionization energy,
and chemical property.
107
1.21 The General Properties of the Elements
 s-Block elements: low ionization energy, easy to lose electrons,
likely to be a reactive metal
 Left side of p-block (heavier elements): relatively low ionization
energy, metallic/metalloid, but less reactive than those in s-block
 Right side of p-block: high electron affinities, tend to gain electrons,
mostly nonmetals, forming molecular compounds
s-block
right
p-block
left
p-block
108
 d-block: metals, transitional between the s- and p-block elements
called “transition metals”, they have similar properties and
form ions with different oxidation states (Fe2+ and Fe3+). They have
catalytic properties, facilitating subtle changes in organisms.
They form alloys
 Lanthanoids: metals incorporated in superconducting materials
Actinides: radioactive, many do not naturally exist on earth
d-Block
f-Block
109