CHE 151: BASIC INORGANIC
CHEMISTRY
LECTURE 1 & 2
Dr. Eric S. Agorku
Senior Lecturer,
Inorganic/Bioinorganic, Coordination Chemistry, Nanotechnology
COURSE OUTLINE
• Atomic structure; qualitative wave mechanics
• Periodic table and periodicity, reactive parameters
• Chemical bonding
– Ionic
– Covalent & Dative (coordinate)
– Multiple bond
– Hybridization and shapes of molecules
– VBT
– Resonance
– Metallic
– van der Waal’s bonding
2
COURSE OUTLINE
• Forces within molecules
– Bond strength
– Bond energy
– Polarity
– Continuity of bonds
• Valence bond theory, resonance, multiple bond, shapes of
molecules, hybridization of non-transition elements
• Crystal structures
• The principal types of unit cell space lattice
• Introduction to coordination chemistry
3
QUIZES AND EXAMINATIONS
• 1st Quiz:
Form of exam: Written/Multiple choice (30 marks)
• Mid-semester:
Form of exam: Written/Multiple choice (30 marks)
• End of Semester:
Form of exam: Written/Multiple choice (100 marks)
4
Reading Material
• J.D. Lee, Concise Inorganic Chemistry, 5th edition.
• G.L. Miessler, P.J. Fischer & D.A. Tarr, 5th edition, Pearson
New International Edition.
• Shriver and Atkins, Inorganic Chemistry, 4th edition.
• J.E. Huheey, E.A. & R.L. Keiter, Inorganic Chemistry, 4th
ed., HarperCollins, N.Y.
• F.A. Cotton & G. Wilkinson, Advanced Inorganic Chemistry,
5th ed., Wiley, N.Y.
INTRODUCTION
Chemistry: The branch of science that deals with the
study of the physical properties and interactions of
matter.
Inorganic chemistry: Deals with the synthesis and
behaviour of inorganic and organometallic
compounds. This field covers
all chemical compounds except the myriad organic
compounds (carbon based compounds, usually
containing C-H bonds), which are the subjects of
organic chemistry.
Atom : An atom is the smallest particle of an element
that retains the properties of that element.
6
Introduction
– In 1661 Robert Boyle defined an element as a
simple substance which could not be broken down
into simpler substances.
– Later, we now define an element as a pure
substance that cannot be broken down into simpler
things by either chemical or physical methods.
– Since elements always combine in fixed ratios, this
lends support to the idea of elements being made of
discrete particles.
History of Atomic Theory
Democritus
– 460-371 B.C.
– ancient Greek philosopher
– believed all matter consisted of
extremely small particles that could
not be divided
– atoms, from Greek word atomos,
means “uncut” or “indivisible”
Aristotle
– believed all matter came from only
four elements—earth, air, fire and
water
Who Was Right?
• Greek society was slave based
• No experiments
– It was all a thought game
• Settled disagreements by argument
• Aristotle was more famous so he won
• His ideas carried through to the middle ages
John Dalton (Late 1700’s)
•
•
•
School teacher in England.
Based his conclusions on experimentation
and observations.
Combined ideas of elements with that of
atoms.
Dalton’s Atomic Theory
1.
2.
3.
4.
5.
All matter is made up of very tiny particles called
atoms.
Atoms of the same element are identical, those of
different atoms are different.
Individual atoms of an element may not all have
the same mass. However, the atoms of an element
have a definite average mass that is characteristic
of the element.
Atoms of different elements have different masses.
Atoms are not subdivided, created or destroyed in
a chemical reaction. No new atoms are created or
destroyed.
Parts of Atoms
• Most of Dalton’s theory is accepted today.
• Except the part about atoms being indivisible.
ATOMIC STRUCTURE
• All atoms consist of a central nucleus surrounded by one or
more orbital electrons.
• It was inconsistent with classical physics which predicted that
the electrons in such an atom should spiral around the nucleus
with concomitant emission of radiation.
• The internal structure of an atom forms the basis for
understanding its chemical behaviour.
• Although many sub-atomic particles are known, we will
consider only electrons, protons and neutrons for our study.
• Details of the atomic structure were formulated from series of
classic experiments during the latter part of the 19th and
beginning of 20th centuries.
13
The Electron
• Discovered by Sir William Crooks in 1979 when he
discovered a beam or streams of negative particles which he
called cathode rays when high voltage electricity was passed
through different gases introduced into sealed glass tube at low
pressures-cathode ray experiment.
• Crookes investigated the rays and found their properties as
follows:
➢ They normally travel in straight line.
➢ They possess considerable momentum.
➢ They are deflected by magnetic and electric fields, the
direction of deflection showing that they are negatively
charged.
➢ They can cause many substances to fluoresce.
14
The Electron
• A cathode ray passed
between two charged
plates is
deflected toward the
positively charged plate.
– The ray is also
deflected by a
magnetic field.
– By measuring the
deflection by both, J.J.
Thomson was able to
calculate the ratio of
charge to mass.
• He was able to measure the deflection because the detecting
screen was coated with zinc sulfide, a substance that produces a
visible light when struck by a charged particle.
15
The Electron
• J. J. Thompson (1897) extended these investigations and determined
the charge-to-mass ratio.
• The same value was obtained irrespective of the nature of the metal
used.
• Thus the cathode rays were considered to be composed of fundamental
constituents of all materials.
• He concluded that electrons must be present in all matter.
16
Thompson’s Model
• Found the electron
– 1 unit of negative charge
– Mass 1/2000 of hydrogen atom
– Later refined by Millikan to 1/1840
Millikan Oil Drop Experiment
▪ In 1906 Robert Millikan passed
mineral oil through a vaporized oil into
an apparatus where he could observe
the drops with a magnifier and make
measurement on them as they drifted
downward.
e
=
m=
e/m
1.60 x 10-19 C
-31 kg
9.11
x
10
=
1.7588 x 108 c/g
✓ He found that the least
charge on any of the
droplets was 1.60 x
10-19 coulombs and
that larger droplets
always had a charge
that
was
some
multiple of this value.
✓ Knowing Thompson’s
work of charge to
mass ratio and the
charge
on
an
individual electron, it
was
possible
to
calculate the mass of
the electron.
The Proton
Eugen Goldstein designed a cathode ray experiment to
demonstrate that positive rays were produced simultaneously
with cathode rays.
He noticed that not
only were cathode
rays emitted from
the cathode rays
but there were
other rays which
originated
from
the anode.
19
The Proton
• The charge on the canal ray particle was found to be equal to that
of the electron, but with opposite sign.
Properties
➢ They travel in a straight line in a direction opposite to the
cathode
➢ They are deflected by electric as well as magnetic field in a way
indicating that they are positively charged.
➢ The e/m of positive particles varies with the nature of the gas
placed in the discharge tube.
➢ They possess mass many times that of an electron.
➢ They cause fluorescence in ZnS.
20
The Proton
• J. J. Thompson (1910) later measured the e/m ratio for the
positive rays and found that the ratio for these rays was not
constant but it varied with the residual gas in the tube.
• The masses of the particle were found to be lowest when the tube
was filled with hydrogen. They were termed protons. Protons are
now known to be the fundamental components in the nuclei of all
atoms (mass of proton = 1.0072 amu)
• Assuming that the magnitude of the charge on the positive
particle is identical with that of the negative charge on the
electron, it is possible to calculate the mass of the positive
particle obtained from hydrogen.
e
=
m=
e/m
1.60 x 10-19 C
= 1.67 x 10-27 kg
4
9.579 x 10 c/g
21
The Neutron
The neutron has a mass of 1.0087 amu which is slightly higher than that of the proton.
2
4
He
+
9
4
Be
12
6
C
+
1
0
n
22
The Neutron
➢ Following the discovery of electrons and protons and the determination of
e/m ratio, J. J. Thompson (1898) proposed that the atom be considered a
sphere of positive electricity with the electrons embedded like jellybean in
a ball of cotton.
➢ Since the nuclear charge must be counter-balanced to make the atom
neutral, Rutherford further assumed that significant number of electrons
must surround the nucleus.
➢ Nuclear model suggested by Rutherford is something which scientists have
found very useful in understanding chemistry.
23
24
The Atomic Number
• About the same era that Rutherford was led to postulate the existence of the
nucleus, H.G.J. Moseley was measuring energies of x-rays emitted by various
elements. Moseley examined the spectrum produced when x-rays were directed
at a metal target. He found that the frequencies (υ) of observed lines obeyed the
relationship
υ = a(z-b)2
where a and b are constants. z was a number, different from each metal, found to
depend upon the position of the metal in the periodic table.
• It increased by a unit from one element to the next. z was called atomic number.
• The atomic number, a fundamental property of the atom is the number of
protons in the nucleus of an atom.
25
Subatomic Particles
• The charge of the proton and the electron are equal in
magnitude but opposite in sign.
• The neutron has no charge.
The Electronic Structure
• This refers to the number and distribution of electrons about a
central nucleus. The nucleus can be considered to consist of the
proper number of protons and neutrons depending on the mass
number and atomic number of the isotope in question.
• Apart from electrostatic repulsions between nuclei, all the major
reactions involve the electrons.
• The electronic structure is a tool used to describe structures and
reactivity of ions and molecules.
• Once electronic structure is known, properties such as energies,
I.E, size and magnetic properties can be determined.
27
The Electronic Structure
• Much of the experimental work on the electronic structure of
atoms that was done prior to 1913 involved measuring frequencies
of electromagnetic radiation that were observed or emitted by
atoms.
• It was found to be characteristic of atoms. The exact pattern of
frequencies is characteristic of each particular substance.
• Although the emission and absorption spectra for most of the
elements were known precisely, a convincing explanation was not
available even for the simplest case-the hydrogen atom.
28
The Electromagnetic Spectrum
•
•
•
•
Light consists of tiny particles called photons.
Isaac Newton (1642-1727) believed light consisted of particles.
By 1900 most scientists believed that light behaved as a wave.
Solids, liquids and gases become incandescent at high temperatures.
Analysis of the radiation emitted by bodies using a spectrophotometer
results in the production of a continuous band of colours.
• If the emitted light is examined, the resulting pattern is not continuous but
consist of series of narrow lines of colours.
• The various colours correspond to the light of definite wavelengths (or
lines) and are called a line spectrum. The line spectrum of an element is
characteristic of that element in such a way that it is used to identify it.
The Electromagnetic Spectrum
• The electromagnetic spectrum represents the range of
energy from low energy, low frequency radio waves with
long wavelengths up to high energy, high frequency gamma
waves with small wavelengths.
The Electromagnetic Spectrum
• Visible light is a small portion of this spectrum. This is the
only part of this energy range that our eyes can detect.
What we see is a rainbow of colours.
RedOrangeYellowGreenBlueIndigoViolet
ROY G BIV
• The product of wavelength and
frequency always equals the speed
of light.
c = λν
• Why does this make sense?
• NOTE:
c is a constant value= 3.00 x 108 m/s
PROBLEMS
• Calculate the wavelength of yellow light emitted from a sodium
lamp if the frequency is
5.10 x 1014 Hz (5.10 x 1014 s-1)
List the known info List the unknown
c = 3.00 x 1010 cm/s
wavelength (λ) = ? cm
Frequency (v) = 5.10 x 1014 s-1
c = λv
λ=c
v
λ = 3.00 x 1010 cm/s = 5.88 x 10-5 cm
5.10 x 1014 s-1
The Hydrogen Atom
• The hydrogen atom is the simplest atom in that it possess a single
extra-electron. The basis of chemical theory is dependent upon a
thorough understanding of this atom.
• It has been found that the emission spectra of this atom consists of a
series of lines or spectroscopic emissions. Within each series the
lines become increasingly closely spaced as λ decreases until they
converge at a limiting value ie. a continuum is reached.
• Such discrimination of emission frequencies lead to the concept of
the existence of discrete energy levels within the atom which are
permitted for electron occupation (quantization of electron energies).
• Frequency is generally expressed as wavenumber (ῡ),
where ῡ = 1/λ m-1.
The Hydrogen Spectra
• When atoms of an element is excited in an electric discharge or by an
electric arc, energy in the form of radiation are emitted. This
radiation can be analyzed by means of a spectrograph into a series of
lines called Atomic Spectrum. The atomic spectrum of H is shown
below:
1 - 1
ῡ = 1 =R
22 ni2
λ
• The lines shown are observed in the visible region and are called the
Balmer series after their discoverer.
The Hydrogen Spectra
• Where R is the Rydberg constant, λ the wavelength of the radiation
and ni and nf have whole number values.
• Several other series of lines were later observed in different regions
of the spectrum. Similar equations were found to hold for the lines in
other series in the hydrogen spectrum. Several series of lines are
observed, all of which fit the formula:
ῡ= 1 = R
λ
1 - 1
nf2 ni2
37
Humphries series (IR)
Pfund series (IR)
Brackett series (IR)
38
Niels Bohr Model (1885-1862)
• To explain these irregularities Bohr suggested that the electron in
an atom existed in definite energy levels ie. The energy states of H
atoms are quantized. Thus electrons move between these energy
levels emit or absorb energy corresponding to the particular
frequency which appear in the spectrum.
• In so doing, he ushered in the modern theory of atomic structure.
His theory provided a link between atomic spectra and Plank’s
constant.
• According to Plank’s quantum theory, energy is not continuous but
discrete (in packets) called quanta of magnitude h/2π.
• Bohr’s theory was founded on plank’s theory on assumption that
for each discrete orbit, the angular momentum of the electron must
be quantized.
Bohr Model
• For an electron of mass, m moving with velocity v in an orbit of
radius r
mv2
Centrifugal force =aaaa
r
• If the charge is e, the number of charges on the nucleus z, and
permittivity of a vacuum εo
ze2
coulombic attractive force = 4π ε r2
o
mv2
r
=
ze2
4π εor2
2
ze
v2 =
4π εomr
For quantization of energy, Bohr showed that
mvr = nh/2π
1
2
Bohr Model
where n is an integer, m and v are mass and velocity of the electron
respectively, r the radius of the orbit and h is Plank’s constant
(6.626x10-34Js).
nh
From eqn 2
v=
2πmr
v2
n2h2
=
4π2m2r2
3
Comparing eqn 1 and 3
n2h2
4π2m2r2
ze2
= 4π ε mr
o
n2h2εo
r=
πmze2
For hydrogen the charge on the nucleus z = 1, and if
n = 1 this gives a value r = 12 x 0.0529nm
n=2
r = 22 x 0.0529nm
n=3
r = 32 x 0.0529nm
4
NB: if
v2 = ze2/nm
Bohr Model
- The requirement that mvr can take only those values that are
multiples of h/2π means that only certain values of r are allowed.
- The atom will only radiate energy when the electron jumps from
one orbit to another. The K.E of an electron is -½mv2. Rearranging
eqn 1
2
E=
-½mv2
=-
ze
8π εor
5
- Substituting for r using eqn 4
E=-
mz2e4
8n2h2εo2
The most exciting support for Bohr’s theory was the collection of constants other
than the quantum number. n is equal numerically to the values of R which is the
Rydberg’s constant determined impirically ie
E = -R/n2
7
Bohr Model
- Thus for an electron to jump from an initial orbit ni to final orbit nf,
the change in E is
ΔE = -
ΔE =
-
mz2e4
8ni2h2εo2
mz2e4
8h2εo2
2e4
mz
- 8nf2h2εo2
1
n f2
1
n i2
8
Therefore if an electron is excited to an orbit with higher energy (n>2) and returns
to the ground state (n = 1) discrete energies equal to
1
2
1
1
n i2
are emitted. Thus the Lyman series of the spectroscopic lines is observed.
Summary of Bohr Model
• Electrons are moving with high velocity around the nucleus in
specified paths called “ orbits or shell ”. As along as the electron
is in a particular orbit its energy is constant. Therefore these
orbits are called “stationary orbit or stationary shell ”.
• Bohr proposed that ‘n’ can take values 1,2,3,4,…etc &
named it is the principal quantum number.
Summary of Bohr Model
▪ Each stationary orbit corresponds to a definite energy. The
stationary orbits are designated by K,L,M,N,O,… . The orbit
close to the nucleus has less energy compared to the orbit away
from the nucleus.
n=3
n=2
n=1
+
n=4
Problems
1. Calculate the wavenumber for all transition from the outer levels to
level 2.
R = 10967800 m-1.
Ans
1 1
ῡ = 1λ = R
22 - 32
ῡ =R
5
36
NB: Calculate for ni = 4, 5, 6
2. Calculate the radius of Bohr’s first orbit for hydrogen atom and the
energy of the electron in the orbit.
Ans
= 0.529x10-8 cm-1
= 0.529 Å
= 0.529x10-10 m
Problems
3. Calculate the wavelength and energy of radiation emitted for the
electronic transition from infinite to stationary state of hydrogen atom.
Given R=1.09678x107 m-1, h=6.6256x10-34Js and c=3x108 ms-1.
Ans
1
λ
1 1
nf2 - ni2
= R
1
λ = R =
c
E = hυ = h
λ
=
1
1.09678x107
=
2.17x10-18 J
= R
=
6.6256x10-34 x
1
12
1
- (∞)
2
9.11x10-8 m
3.0x108
9.11x10-8
= R
Problems
4. Calculate the velocity (cm/s) of an electron placed in the third orbit
of the hydrogen atom. Also calculate the number of revolutions per
second that this electron makes around the nucleus. Given radius of
third orbit = 32 x 0.529 x 10-8 = 4.761 x 10-8 cm.
Ans
mvr = nh/2π
v = nh/2πmr
3 x 6.624 x 10-27
2 x 3.14 x (9.108 x 10-28) x (4.761 x 10-8)
=
= 0.729x10-8 cm/s
Time taken for one revolution = 2πr/v
Number of revolutions per second = 1 = v
2πr
v
=
0.729 x 108
2 x 3.14 x (4.761 x 10-8)
2πr
= 2.4 x 1014 revolutions/sec
Problems
5a. Calculate the
i. Energy
ii. Frequency
iii. Wavelength
of the radiation emitted by the transition from the fifth to the second quantum level in
a hydrogen atom
b.(i) In which spectral region can a line corresponding to this transition be detected?
c.(ii) Calculate the ionization energy of hydrogen in joules per atom and in joules per
moles of atoms.
6. Calculate the wavelength of the first line and the series limit for Lyman series for
hydrogen (R = 10967800 m-1).
7. The Balmer series of spectral lines from hydrogen appear in the visible region.
What is the lower energy level that these electronic transitions start from, and what
transitions correspond to the spectral lines 379 nm and 430 nm respectively.
8. The wavelength of a certain line in Balmer series is observed to be 4.341 Å. To
what value of ‘n’ does this correspond?
9. Calculate the ratio of the velocity of light and the velocity of electron in the first
orbit of hydrogen. [ m = 9.08x10-28 g, r = 0.529x10-8 cm]
10. Calculate the shortest and longest wavelength in the hydrogen spectrum of Lyman
series.
Limitations of the Bohr’s Atomic Model
1. It does not explain the spectra of multi-electron atoms.
2. When a high resolving power spectroscope is used, it is observed that a
spectral line in the hydrogen spectrum is not a simple line but a
collection of several lines which are very close to another. This is known
as fine spectrum. Bohr’s theory does not explain the fine spectra of even
hydrogen atom.
3. It does not explain the splitting of spectral lines into a group of finer
lines under the influence of magnetic field (Zeeman effect) and electric
field (Stark effect).
4. According to Bohr’s theory the electron move in circular orbits round
the nucleus, but modern researches have shown that the motion of the
electron is not limited to a single plane but takes place in three
dimensional space. In other words, atomic model is not flat, as suggested
by Bohr.
5. Bohr’s theory is not in agreement with Heisenberg’s uncertainty
principle.
Sommerfeld’s Atomic Model
•
Summerfeld introduced the concept of sub-energy shells. Energies of subshells differ slightly from one another.
•
Atomic spectra display fine structure due to splitting of spectral lines. An
attempt to account for the fine structure, Arnold Sommerfeld proposed
elliptical orbits instead of circular orbits proposed by Bohr.
•
The atomic spectra lines in the presence of applied magnetic field split into
several lines-Zeeman effect.
•
Sommerfeld theory is also sometimes referred to as Bohr-Sommerfeld theory.
Limitations.
• The theory cannot give the correct
number of lines observed in fine
structure.
• Moreover, it gives no information
about the relativistic intensities of the
‘fine lines’.
• Further, the exact definition of
position and momentum is contrary to
the uncertainty principle.
Particle and Wave Nature of Electron
▪ With the failure of the Bohr model it was found that the property of an
electron in an atom had to be described in a wave-mechanical terms.
▪ Louis victor de Broglie (1924) suggested that all matter could exhibit
wave-like properties. The wave-particle duality of photon is expressed
concisely by combining Plank’s equation which expresses energy in terms
of wave properties.
E = hv = hc/λ
………………………………….1
where λ is wavelength.
• Einstein’s equation for the energy equivalence of mass is
E = mc2
…………………………………………..2
Combining eqn 1 & 2
mc2 = hv = hc/λ
λ = h/mc = h/p (where p is momentum of photon)
λ = h/p or h = λp
………... ………………………...3
• Since h is constant, momentum of a moving electron is inversely
proportional to its wavelength
The Heisenberg Uncertainty Principle
The Heisenberg uncertainty principle states that it is
impossible to know both the momentum and the position of a
particle at the same time.
• This limitation is critical when dealing with small particles
such as electrons.
• But it does not matter for ordinary-sized objects such as cars
or airplanes.
• To locate an electron, you might strike it with a photon.
• The electron has such a small mass that striking it with a
photon affects its motion in a way that cannot be predicted
accurately.
• The very act of measuring the position of the electron changes
its momentum, making its momentum uncertain.
Heisenberg Uncertainty Principle
Let’s find an electron
Photon changes the momentum of electron
x  
p  h/ (smaller , bigger p)
xp > h/4
x – uncertainty of position
p - uncertainty of momentum
Et > h/4
E - uncertainty of energy
t - uncertainty of time
Summary
 Light is made up of photons, but in macroscopic situations
it is often fine to treat it as a wave.
 Photons carry both energy & momentum.
 Matter also exhibits wave properties. For an object of mass
m,
and velocity, v, the object has a wavelength, λ = h / mv
 One can probe ‘see’ the fine details of matter by using
high energy particles (they have a small wavelength !)
For what period of time is the uncertainty of the energy of an
electron 2.5 x 10-19 J?
Et > h/4
(2.5 x 10-19 J)t > h/4
t = 2.1 x 10-16 s
Problems
• Calculate the uncertainty in the position of a particle when the
uncertainty in momentum is (a) 1 x 10-3 g cm s-1 (b) zero.
• The uncertainty in velocity of a cricket ball of mass 150 g if the
uncertainty in its position is of the order of 1 Å.
The Schrödinger Equation
• An elegant solution to the wave-particle duality was provided
by Schrödinger in 1926.
• He proposed the electrons in atoms should be considered not
as particles but as waves and that they could therefore be
described by a suitable wave equation.
• The important conceptual breakthrough here was that the
presence of only certain allowed electron energies is a direct
and natural consequence of a wave treatment and no ad hoc
assumptions are necessary.
58
The Schrödinger Equation
• Definition: It is a differential equation which
forms the basis of the quantum-mechanical
description of matter in terms of the wave-like
properties of particles in a field. Its solution
is related to the probability density of a particle in space and time.
• Erwin Rudolf Josef Alexander Schrödinger in1926 proposed
the wave equation which bears his name. its purpose was to
describe the behaviour of a subatomic particle in the same way
that microscopic particles are described by classical mechanics.
∂2 ψ
∂2x2
+
∂2ψ + ∂2ψ + 8πm (E-V)ψ= 0
h2
∂2y2
∂2 z2
The Schrödinger Equation
• Describes the wave properties in terms of its position, mass, total
energy and potential energy.
∂2 ψ
∂2x2
2ψ
2ψ
8πm (E-V)ψ= 0
∂
∂
+
+
+
2
2
2
2
h2
∂y
∂z
Describes
K.E
x, y, z = Cartesian coordinate
h = Plank’s constant
V = P.E
m = mass
E = total energy
Ψ = wavefunction
Describes
P.E
Time Dependent
Wave Equation!
Time Independent
The Schrödinger Equation
The general form of Schrödinger’s equation is
Hψ = Eψ
• The operator H (called the Hamiltonian operator) prescribes a series
of mathematical operations that are performed on the wave function
(ψ).
• Solution to the wave equation is the wave function which is a
mathematical expression which describes or defines the subatomic
particle (electron) in terms of its wave properties. The wavefunction
ψ has properties analogous to the amplitude of a wave. Its square
(ψ2) is proportional to the probability of finding a particle at the
coordinates x, y, z.
• The probability of finding the electron is maximum when dp/dr = 0.
The Schrödinger Equation
• The Schrödinger wave equation is used to describe the
behavior of electrons in atoms,
– i.e. we treat electrons as waves such that only certain
solutions to the wave equation, or energies for the
electron, are possible and this quantization results directly
from boundary conditions.
• Moreover, we obtain not one solution but a series of
many possible solutions each with certain energy and
each described by a certain set of quantum numbers.
• Each one of these solutions or wavefunctions (ψ)
describes a possible state of the electron in the atom
and this is called an orbital.
64
PHYSICAL SIGNIFICANCE OF ψ AND ψ2
• The wavefunction, ψ, has no physical significance, ψ only
represents the amplitude of the electron wave.
• Of more physical significance is the square of this function
(ψ2) which refers to the electron intensity (density) or more
precisely a probability of finding an electron associated with a
specific energy.
• The space is called an atomic orbital
65
PHYSICAL SIGNIFICANCE OF ψ AND ψ2
• Thus ψ2 at any particular point is the probability of finding the
electron at that point.
• High values of ψ2 mean high probability, low values mean low
probability;
• ψ2 equal to zero means zero probability.
Acceptable solutions to the wave equation, that is solutions which are
physically possible, must have certain properties:
• Ψ must be continuous.
• Ψ must be finite.
• Ψ must be single valued.
• The probability of finding the electron (ψ2) over space from positive
infinity to negative infinity must be equal to one.
The probability of finding an electron at a point x, y, z is ψ2, so
-∞
𝑓ψ2𝑑𝑥𝑑𝑦𝑑𝑧 = 1
+∞
66
Quantum Numbers
• Boundary conditions and the spherical, three-dimensional
nature of the atom give rise to three quantum numbers and
these are given the symbols n, l and ml.
• These can take only certain allowed values and a solution
exists to the Schrödinger equation for certain allowed sets of
these three numbers.
67
Quantum Numbers
• The names and symbols for these quantum numbers and the
values which they can take are given below as follows:
➢ n is the principal quantum number and this determines the
radial part of the wavefunction and determines the energy
corresponding to an orbital. It give the radial distance from the
nucleus. n does not depend on l and ml. The number n can take
integral values 1, 2, 3, 4….∞ but we will be concerned only
with the first few. The energy of the electron in the nth shell,
En is given by
2π2µz2e4
En = n2h2
and the radial distance of the nth shell, rn from the nucleus is
n2h2
rn = - 4π2µze2
68
Quantum Numbers
➢ l is the subsidiary or angular momentum quantum number.
This determines the shape of the orbital and can take values
0,1, 2, 3…n-1, i.e. the possible values of l are dependent on n.
In fact, l determines the type or shape of the orbital and these
are usually referred to by letters.
l=0
l=1
l=2
l=3
s orbital
p orbital
d orbital
f orbital
69
Quantum Numbers
• Orbital are then labeled according to their value of n
and the letter associated with l:
•
•
•
•
•
•
n=1 l=0
1s
n=2 l=0
2s
n=2 l=1
2p
n=3 l=0
3s
n=3 l=1
3p
n=3 l=2
3d
Note that orbitals such as 1p and 2d are not allowed
according to these rules.
70
Quantum Numbers
• ml is the magnetic quantum number and it takes values
-l, -l+1,…0…, l-1, l, i.e. 2l+1 values for a given value
of l.
• Thus for l=0, ml =0 and so there is only one type of s
orbital for any given value of n, i.e. one 1s, one 2s etc.
• For l=1, ml=-1, 0, +1, i.e. three types of p orbital.For
l=2, ml=-2, -1, 0, +1, +2, i.e. five types of d orbital.
• This quantum number specifies the orientation of the
orbital.
71
Quantum Numbers
• When we look at these orbitals or wave functions in
more detail we will see that they contain nodes and there
are a few simple rules concerning nodes that are worth
remembering.
• The total number of nodes in any orbital is given by n-1.
• Some of these are in the radial part of the wave function
and some are in the angular part.
72
Quantum Numbers
• In the case of the hydrogen atom, the energy of an
electron or orbital depends only on the value of n and
so, for example, the 2s and 2p orbitals have the same
energy and are said to be degenerate.
• The energy of the orbitals increases as n increases. The
separation in energy is not constant but that the levels
become closer together as n increases. Moreover, the
energies are given negative values with n=∞ defined as
zero energy.
73
Principal
quantum
Number
n
Subsidiary Magnetic
quantum
Quantum
number
numbers
l
m
Symbol
1
0
0
1s (one orbital)
2
0
0
2s (one orbital)
2
1
-1, 0, +1
3
0
0
3
1
-1, 0, +1
3p (three orbitals)
3
2
-2, -1, 0, +1, +2
3d (five orbitals)
4
0
0
4
1
-1, 0, +1
4p (three orbitals)
4
2
-2, -1, 0, +1, +2
4d (five orbitals)
4
3
2p (three orbitals)
3s (one orbital)
4s (one orbital)
-3, -2, -1, 0, +1, +2, +3 4f (seven orbitals)
74
The angular part of the wavefunction
• The angular part of the wavefunction reveals how the
wavefunction varies as a function of angle from the
origin of some suitably chosen coordinate systems and
thus determines the shape of the orbital.
• It is dependent on the quantum number l and we can
label the types of orbitals according to this quantum
number or more usually with the letters s, p, d, f, etc.
• The number of nodes in the angular part of the
wavefunction for a given orbital is equal to l.
• Thus, s orbitals (l = 0) have no angular nodes. Moreover,
since there is no angular dependence, the orbital is
spherical and this is true for all s orbitals; 1s, 2s, 3s, etc.
75
The angular part of the wavefunction
• For p orbitals, l=1 and there is one angular node.
• This is planar and divides the orbital into two lobes
of opposite sign.
• There are three possible orientations for orbitals of
this type (three values of ml) which lie along the
axes x, y and z, and these are usually designated as
px, py and pz.
• All p orbitals have this shape and there are always
three for any given value of n, i.e. three 2p, three
3p etc.
76
The angular part of the wavefunction
• For d orbitals, l=2 and there are therefore two nodes
associated with these orbitals.
• These are perpendicular and each orbital has four lobes.
• There are five d orbitals which are given the labels dxy,
dxz, dyz, dx2-y2 and dz2.
Figure: diagrams of angular part of the wavefunction for selected orbitals
77
The radial part of the wavefunction
• The radial part of the wavefunction tells us
how the wavefunction varies with distance, r,
from the nucleus, i.e. the effective size of the
orbital.
• Atomic orbitals depend on an exponential
function 𝑒 −𝐵𝑟 where B is some constant and r
is the distance from the nucleus.
• This means that ψ falls away exponentially at
large r values.
78
The radial part of the wavefunction
• For a 1s orbital, a standard expression is
ψ=A𝑒 −𝐵𝑟 and if we plot this function as a graph,
we obtain a curve shown in figure below.
• We can see that there is a maximum value of ψ at
r = 0 or at the nucleus.
79
The radial part of the wavefunction
• The probability of finding an electron on a
particular surface rather than at a point along
a line from the nucleus is of interest at this
stage.
• Considering the 1s orbital, we can imagine a
spherical surface expanding from the nucleus
and it will be useful to know the probability of
finding an electron at some point on this
surface as a function of distance from the
nucleus.
80
The radial part of the wavefunction
• The greater the distance from the nucleus, the
more points on a sphere and so the function
we want will be proportional to the surface
area of the sphere or 4𝜋𝑟 2 .
• We can therefore plot a probability that an
electron is at a certain distance r according to
the function 4𝜋𝑟 2 . 𝜓 2 .
• This is called the radial probability function
(RPF).
81
The radial part of the wavefunction
• The graphs of RPFs of 1s, 2s, 2p, 3s, 3p and 3d
orbitals are shown in Figure 2.
• When r = 0, the RPF is also 0 and so for all
orbitals, the function is zero at the centre of
the nucleus.
• The graph has a maximum, i.e. there is a
distance at which we are most likely to find an
electron, ro.
82
The radial part of the wavefunction
Figure 2 Graphs of RPFs of 1s, 2s, 2p, 3s, 3p and 3d orbitals
83
Periodicity
84
Periodicity
Effective nuclear charge
• Those electrons in the outmost or valence shell
are especially important because they are the
ones that can engage in the sharing and exchange
that is responsible for chemical reactions;
– how tightly they are bound to the atom determines
much of the chemistry of the element.
• The degree of binding is the result of two
opposing forces: the attraction between the
electron and the nucleus, and the repulsions
between the electron in question and all the
other electrons in the atom.
85
Periodicity
Effective nuclear charge
• All that matters is the net force, the difference
between the nuclear attraction and the
totality of the electron-electron repulsions.
• We can simplify the shell model even further
by imagining that the valence shell electrons
are the only electrons in the atom, and that
the nuclear charge has whatever value would
be required to bind these electrons as tightly
as is observed experimentally.
86
Periodicity
Effective nuclear charge
• Because the number of electrons in this model
is less than the atomic number Z, the required
nuclear charge will also be smaller and is
known as the effective nuclear charge.
• Effective nuclear charge is essentially the
positive charge that a valence electron "sees".
87
Periodicity
Effective nuclear charge
• Part of the difference between Z and Zeffective is
due to other electrons in the valence shell, but
this is usually only a minor contributor because
these electrons tend to act as if they are spread
out in a diffuse spherical shell of larger radius.
• The main actors here are the electrons in the
much more compact inner shells which surround
the nucleus and exert what is often called a
shielding or "screening" effect on the valence
electrons.
88
Periodicity
Effective nuclear charge
• The effective nuclear charge is useful in
understanding many aspects of periodicity
• In order to utilize and appreciate the concept
fully, a semi-quantitative scale is needed.
• There have been a number of attempts to do
this, but probably the most useful, in terms of
its simplicity, is a scheme known as the
Slater’s rules.
89
Periodicity
Effective nuclear charge
• The object of these rules is to estimate, for a
particular electron, the strength of the shielding
effect of the other electrons present, and from
this to calculate a shielding constant, S.
• This value can then be used to calculate the
effective nuclear charge, Z*, according to the
following equation, where Z is the actual nuclear
charge:
Z* = Z – S
90
Periodicity
Effective nuclear charge
• To calculate the shielding constant for a ns or np
electron, the following rules apply:
1. Electrons with a higher n contribute zero, i.e. no
shielding
2. Electrons with the same value of n contribute 0.35,
i.e. not very good shielding
3. Electrons with a value of n one less than our chosen
electron contribute 0.85, i.e. rather better shielding;
4. Electrons with lower values of n contribute 1.00, i.e.
complete shielding.
91
Periodicity
Atomic size
• The concept of "size" is somewhat ambiguous
when applied to the scale of atoms and
molecules.
• The reason for this is apparent when you
recall that an atom has no definite boundary;
– there is a finite (but very small) probability of
finding the electron of a hydrogen atom,
• for example, 1 cm, or even 1 km from the nucleus.
92
Periodicity
Atomic size
• It is not possible to specify a definite value for the
radius of an isolated atom;
• The best we can do is to define a spherical shell
within whose radius some arbitrary percentage of
the electron density can be found.
• When an atom is combined with other atoms in a
solid element or compound, an effective radius
can be determined by observing the distances
between adjacent rows of atoms in these solids.
93
Periodicity
Atomic size
• This is most commonly carried out by X-ray
scattering experiments.
• Because of the different ways in which atoms can
aggregate together, several different kinds of
atomic radii can be defined.
• A rough idea of the size of a metallic atom can be
obtained simply by measuring the density of a
sample of the metal.
• This tells us the number of atoms per unit volume
of the solid
94
Periodicity
Atomic size
• The atoms are assumed to be spheres of
radius r in contact with each other, each of
which sits in a cubic box of edge length 2r.
• Although the radius of an atom or ion cannot
be measured directly, in most cases it can be
inferred from measurements of the distance
between adjacent nuclei in a crystalline solid.
95
Periodicity
Atomic size
• This is most commonly carried out by X-ray
scattering experiments.
• Because such solids fall into several different
classes, several kinds of atomic radius are
defined.
• Many atoms have several different radii;
– for example, sodium forms a metallic solid and thus
has a metallic radius,
– it forms a gaseous molecule Na2 in the vapor phase
(covalent radius)
– it forms ionic solids such as NaCl (ionic radius).
96
Periodicity
Atomic size
• Metallic radius is half the distance between
nuclei in a metallic crystal.
• Covalent radius is half the distance between like
atoms that are bonded together in a molecule.
• van der Waals radius is the effective radius of
adjacent atoms which are not chemically bonded
in a solid, but are presumably in "contact".
– An example would be the distance between the iodine
atoms of adjacent I2molecules in crystalline iodine
97
Periodicity
98
Periodicity
• Ionic radius is the effective radius of ions in solids such
as NaCl.
• It is easy enough to measure the distance between
adjacent rows of Na+ and Cl– ions in such a crystal, but
there is no unambiguous way to decide what portions
of this distance are attributable to each ion.
• The best one can do is to make estimates based on
studies of several different ionic solids (LiI, KI, NaI, for
example) that contain one ion in common.
• Many such estimates have been made, and they turn
out to be remarkably consistent.
99
Periodicity
Ionic radius
• The lithium ion is sufficiently small that in LiI,
the iodide ions are in contact, so I-I distances
are twice the ionic radius of I–.
• This is not true for KI, but in this solid,
adjacent potassium and iodide ions are in
contact, allowing estimation of the K+ ionic
radius.
100
Periodicity
Ionic radius
• Many atoms have several different radii;
– for example, sodium forms a metallic solid and
thus has a metallic radius, it forms a gaseous
molecule Na2 in the vapor phase (covalent radius),
and of course it forms ionic solids as mentioned
above.
101
Periodicity
Periodic trends in atomic size
• We would expect the size of an atom to depend mainly on
the principal quantum number of the highest occupied
orbital; in other words, on the "number of occupied
electron shells".
• Since each row in the periodic table corresponds to an
increment in n, atomic radius increases as we move down a
column.
• The other important factor is the nuclear charge; the higher
the atomic number, the more strongly will the electrons be
drawn toward the nucleus, and the smaller the atom.
• This effect is responsible for the contraction we observe as
we move across the periodic table from left to right.
102
Periodic trends in atomic size
Ionic radii
• A positive ion is always smaller than the neutral
atom, owing to the diminished electron-electron
repulsion.
• If a second electron is lost, the ion gets even
smaller;
– for example, the ionic radius of Fe2+ is 76 pm, while
that of Fe3+ is 65 pm.
• If formation of the ion involves complete
emptying of the outer shell, then the decrease in
radius is especially great.
103
Periodic trends in atomic size
Ionic radii
• The hydrogen ion H+ is in a class by itself; having
no electron cloud at all, its radius is that of the
bare proton, or about 0.1 pm— a contraction of
99.999%!
• Because the unit positive charge is concentrated
into such a small volume of space, the charge
density of the hydrogen ion is extremely high;
• It interacts very strongly with other matter,
including water molecules, and in aqueous
solution it exists only as the hydrozonium
ion H3O+.
104
Periodic trends in atomic size
Ionic radii
• Negative ions are always larger than the parent ion;
the addition of one or more electrons to an existing
shell increases electron-electron repulsion which
results in a general expansion of the atom.
• An isoelectronic series is a sequence of species all
having the same number of electrons (and thus the
same amount of electron-electron repulsion) but
differing in nuclear charge.
• Of course, only one member of such a sequence can be
a neutral atom (eg. neon in the series Na+,F-,Ne)
• The effect of increasing nuclear charge on the radius is
clearly seen.
105
Periodic trends in ion formation
• Chemical reactions are based largely on the
interactions between the most loosely bound
electrons in atoms,
• it is not surprising that the tendency of an
atom to gain, lose or share electrons is one of
its fundamental chemical properties.
106
Periodic trends in ion formation
Ionization energy
• This term always refers to the formation
of positive ions.
• In order to remove an electron from an atom,
work must be done to overcome the electrostatic
attraction between the electron and the nucleus;
• this work is called the ionization energy of the
atom and corresponds to the exothermic process
107
Periodic trends in ion formation
Ionization energy
• This term always refers to the formation
of positive ions.
• In order to remove an electron from an atom, work
must be done to overcome the electrostatic attraction
between the electron and the nucleus;
• this work is called the ionization energy of the atom
and corresponds to the exothermic process
M(g) → M+(g) + e–
• in which M(g) stands for any isolated (gaseous) atom.
108
Periodic trends in ion formation
Ionization energy
• An atom has as many ionization energies as it has
electrons.
• Electrons are always removed from the highestenergy occupied orbital.
• An examination of the successive ionization
energies of the first ten elements provides
experimental confirmation that the binding of the
two innermost electrons (1s orbital) is
significantly different from that of the n=2
electrons.
109
Periodic trends in ion formation
Ionization energy
• Successive ionization energies of an atom
increase rapidly as reduced electron-electron
repulsion causes the electron shells to contract,
thus binding the electrons even more tightly to
the nucleus.
• Ionization energies increase with the nuclear
charge Z as we move across the periodic table.
• They decrease as we move down the table
because in each period, the electron is being
removed from a shell one step farther from the
nucleus than in the atom immediately above it.
110
Periodic trends in ion formation
Ionization energy
• This results in the familiar zig-zag lines when the first
ionization energies are plotted as a function of Z.
111
Ionization energy
Points to note
• The noble gases have the highest IE's of any
element in the period.
• This has nothing to do with any mysterious
"special stability" of the s2p6 electron
configuration; it is simply a matter of the high
nuclear charge acting on more contracted
orbitals.
• IE's (as well as many other properties) tend not to
vary greatly amongst the d-block elements.
112
Ionization energy
Points to note
• This reflects the fact that as the more
compact d orbitals are being filled, they exert a
screening effect that partly offsets that increasing
nuclear charge on the outermost s orbitals of higher
principal quantum number.
• Each of the Group 13 elements has a lower first-IE than
that of the element preceding it.
• The reversal of the IE trend in this group is often
attributed to the more easy removal of the single
outer-shell p electron compared to that of electrons
contained in filled (and thus spin-paired) s- and dorbitals in the preceding elements.
113
Electron affinity
• Formation of a negative ion occurs when an
electron from some external source enters the
atom and become incorporated into the
lowest energy orbital that possesses a
vacancy.
• Because the entering electron is attracted to
the positive nucleus, the formation of
negative ions is usually exothermic.
• The energy given off is the electron affinity of
the atom.
114
Electron affinity
• For some atoms, the electron affinity appears to
be slightly negative, suggesting that electronelectron repulsion is the dominant factor in these
instances.
• In general, electron affinities tend to be much
smaller than ionization energies, suggesting that
they are controlled by opposing factors having
similar magnitudes.
• These two factors are, as before, the nuclear
charge and electron-electron repulsion.
• But the latter, only a minor actor in positive ion
formation, is now much more significant.
115
Electron affinity
• One reason for this is that the electrons
contained in the inner shells of the atom exert
a collective negative charge that partially
cancels the charge of the nucleus,
• thus exerting a so-called shielding effect which
diminishes the tendency for negative ions to
form.
• Because of these opposing effects, the
periodic trends in electron affinities are not as
clear as are those of ionization energies.
116
Electronegativity
• When two elements are joined in a chemical
bond, the element that attracts the shared
electrons more strongly is more electronegative.
• Elements with low electronegativities (the
metallic elements) are said to be electropositive.
• It is important to understand that
electronegativities are properties of atoms that
are chemically bound to each other;
• there is no way of measuring the
electronegativity of an isolated atom.
117
Electronegativity
• Moreover, the same atom can exhibit different
electronegativities in different chemical
environments.
• The "electronegativity of an element" is only a
general guide to its chemical behavior rather
than an exact specification of its behavior in a
particular compound.
• Nevertheless, electronegativity is eminently
useful in summarizing the chemical behavior
of an element.
118