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CHAPTER 1: ATOMIC STRUCTURE
An atom is the smallest unit quantity of an element that can exist on its own or
can combine chemically with other atoms of the same or another element.
Composed of protons, neutrons and electrons.
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ATOMIC NUMBER, MASS NUMBER AND ISOTOPES
NUCLIDES, ATOMIC NUMBER AND MASS NUMBER
A nuclide is a particular type of atom and has a characteristic atomic
number, Z.
The mass number, A, of a nuclide is the number of protons and neutrons in
the nucleus.
mass number
𝐴
𝑍E
element symbol
atomic number
RELATIVE ATOMIC MASS
12
6C atom.
to 126C = 12.0000.
Atomic mass unit is defined as 1/12 of the mass of a
Relative atomic masses (Ar) are all masses relative
ISOTOPES
Same number of protons and electrons but different mass numbers.
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EXAMPLE
Calculate the value of Ar for naturally occurring chlorine if the distribution of
35
37
isotopes is 75.77% 35
Cl and
17Cl and 24.23% 17Cl . Accurate masses for
37
Cl are 34.97 and 36.97.
SOLUTION
The relative atomic masses of chlorine is the weighted mean of the mass
numbers of the two isotopes.
75.77
24.23
Ar =
× 34.97 +
× 36.97 = 35.45
100
100
EXERCISE FOR THE IDLE MIND
If Ar for Cl is 35.45, what is the ratio of the two isotopes present in a sample
of Cl atoms containing naturally occurring Cl?
Calculate the value of Ar for naturally occurring copper if the distribution of
isotopes is 69.2% 63Cu and 30.8% 65Cu; accurate masses are 62.93 and
64.93.
CHEM210/Chapter 1/2014/03
QUANTUM THEORY
Development of quantum theory took place in two steps; in the older
theories, the electron was treated as a particle. In more recent models, the
electron is treated as a wave, hence wave mechanics.
A quantum of energy is the smallest quantity of energy that can be emitted
(or absorbed) in the form of electromagnetic radiation (Planck-1901).
The energy, E, is given by:
hc
∆E = hν =
λ
where h = Planck’s constant = 6.626 × 10-34 Js
One of the important applications of early quantum theory was the RutherfordBohr model of the hydrogen atom.
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Balmer (1885) – wavelength of the spectral lines of hydrogen obeyed the
equation:
𝜈 =
1
1
1
= 𝑅 2 − 2
𝜆
2
𝑛
where R is the Rydberg constant for hydrogen, ν is the wavenumber in cm-1
and n is an integer 3, 4, 5, …
The various series in atomic H emission spectrum
Series
Lyman
Balmer
Paschen
Brackett
Pfund
n″
1
2
3
4
5
n′
2,3,4, ….
3,4,5, ….
4,5,6, ….
5,6,7, ….
6,7,8, ….
Region
UV
Visible
IR
IR
IR
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BOHR’S THEORY OF THE ATOMIC SPECTRUM OF HYDROGEN
Bohr (1913) stated two postulates for an electron in an atom:
Stationary states exist in which the energy of the electron is constant and
such states are characterized by circular orbits about the nucleus in which
the electron has an angular momentum, mvr give by the equation:
ℎ
𝑚𝑣𝑟 = 𝑛
2𝜋
where m = mass of electron, v = velocity of electron, r = radius of the orbit
and h = Planck constant
Energy is absorbed or emitted only when an electron moves from one
stationary state to another and the energy change is given by:
∆𝐸 = 𝐸𝑛2 − 𝐸𝑛1 = ℎν
where n1 and n2 are the principal quantum numbers.
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If we apply the Bohr model to the hydrogen atom, the radius of each allowed
circular orbit can be determined from the equation:
𝜀𝑜 ℎ2 𝑛2
𝑟𝑛 =
𝜋𝑚𝑒 𝑒 2
where εo = permittivity of a vacuum = 8.854  10-12 F m-1
For n = 1, the radius for the first orbit of the H atom is 5.293  10-13 m or 52.93
pm.
An increase in the principal quantum number from n = 1 to n = ∞ corresponds
to the ionization of the atom and the ionization energy, IE, quoted per mole of
atoms.
H(g)  H+(g) + e𝐼𝐸 = 𝐸∞ − 𝐸1 =
ℎ𝑐
1
1
= ℎ𝑐𝑅 2 − 2
λ
1
∞
= 2.179  10-18 J = 1312 kJ mol-1
CHEM210/Chapter 1/2014/07
WAVE MECHANICS
WAVE NATURE OF ELECTRONS
de Broglie (1924) – if light composed of particles and showed wave-like
properties, the same should be true for electrons and other particles.
Proposed wave-particle duality and stated that classical mechanics with the
idea of wave-like properties could be combined to show that a particle with
momentum, mv possesses an associated wave of wavelength, λ.
ℎ
λ=
𝑚𝑣
THE UNCERTAINTY PRINCIPLE
If an electron has wave-like properties, it becomes impossible to know both
the momentum and position of the electron at the same instant in time.
To overcome this problem, we use the probability of finding the electron in a
given volume of space and this is determined from the function Ψ2, where Ψ
is the wavefunction.
CHEM210/Chapter 1/2014/08
SCHRӦDINGER WAVE EQUATION
The Schrӧdinger equation can be solved exactly only for a species containing a
4
nucleus and only one electron, e.g. 1H, 2He+ i.e. hydrogen-like system.
Equation may be represented in several forms, but the following equation is
appropriate for motion in the x direction.
𝑑2𝛹
𝑑𝑥 2
+
8𝜋2
ℎ2
𝐸 −𝑉 Ψ=0
where m = mass, E = total energy, V = potential energy of the particle.
In reality, electrons move in 3-dimensional space, and an appropriate form of
the equation is given by:
𝜕2 𝛹
𝜕𝑥 2
+
𝜕2 𝛹
𝜕𝑦 2
+
𝜕2 𝛹
𝜕𝑧 2
+
8𝜋2
ℎ2
𝐸 −𝑉 Ψ=0
Results of the wave equation:
• Wavefunction, Ψ is a solution of the Schrӧdinger equation and describes
the behaviour of an electron in an atomic orbital.
CHEM210/Chapter 1/2014/09
• Can find energy values associated with particular wavefunctions.
• Quantization of energy levels arises naturally from the Schrӧdinger
equation.
ATOMIC ORBITALS
THE QUANTUM NUMBERS n, l and ml
The principal quantum number, n, is a positive integer with values lying
between the limits 1 ≤ n ≤ ∞; arise when the radial part of the wavefunction is
solved.
Two more quantum numbers, l and ml , appear when the angular part of the
wavefunction is solved.
The quantum number l is called the orbital quantum number and has allowed
values of 0, 1, 2, ……. , (n-1). Its value determines the shape of the atomic
orbital and the orbital angular momentum of the electron.
The value of the magnetic quantum number, ml , gives information about the
directionality of an atomic orbital and has integral values between +l and –l.
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SHELLS, SUBSHELLS AND ORBITALS
All orbitals with the same value of n have the same energy and are said to be
degenerate.
Therefore, n defines a series of shells of the atom or sets of orbitals with the
same value of n, hence with the same energy and approximately the same
radial extent.
Shells with n = 1, 2, 3, …. are commonly referred to as K, L, M, …. shells.
Orbitals belonging to each shell are classified into subshells distinguished by a
quantum number l.
For a given value of n, the quantum number l can have the values l = 0, 1, ….. ,
n – 1, e.g. the shell consists of just one subshell with l = 0, the shell with n = 2
consists of two subshells, one with l = 0 and the other with l = 1.
Value of l
Subshell designation
0
s
1
p
2
d
3
f
4
g
…..
…..
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ELECTRON SPIN
Two more quantum numbers required to specify the spatial distribution of an
electron in an atom and are related to spin.
Spin is described by two quantum numbers, s and ms.
Spin magnetic quantum number, ms takes only two values, +½ and -½
NODES
Orbitals are best expressed in terms of spherical polar coordinates.
The positions where the wavefunction passes through zero are called nodes.
Ψ = 𝑅 𝑟 + 𝑌(𝜃, Ф)
There are two types of nodes, radial nodes occur where the radial component
of the wavefunction passes through zero and angular nodes occur where the
angular component of the wavefunction passes through zero.
An orbital with quantum numbers n and l, in general has n – l - 1 radial nodes.
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Relationship between quantum numbers and atomic orbitals.
n
l
ml
No. of orbitals
AO designation
1
0
0
1
1s
2
0
0
1
2s
1
-1, 0, 1
3
2px, 2py, 2pz
0
0
1
3s
1
-1, 0, 1
3
3px, 3py, 3pz
2
-2,-1,0,1,2
5
3dxy, 3dyz, 3dzx,
3
3dx2- y2, 3dz2
An orbital is fully occupied when it contains two electrons which are spinpaired; one electron has a value of ms = +½ and the other, ms = -½.
Atomic orbitals are regions of space where the probability of finding an
electron about an atom is highest.
CHEM210/Chapter 1/2014/19
The s Orbitals
s orbitals are spherically symmetric.
As n increases, the s orbitals get larger and the number of nodes increase.
A node is a region in space where the probability of finding an electron is zero.
At a node, Ψ2= 0 and for an s orbital, the number of nodes is (n-1).
1s
2s
3s
4s
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Height of graph indicates
density of dots as we
move from origin
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The p Orbitals
There are three p-orbitals px, py, and pz which lie along the x-, y- and z- axes of
a Cartesian system.
Correspond to allowed values of ml of -1, 0, and +1.
The orbitals are dumbbell shaped and as n increases, the p orbitals get larger.
All p orbitals have a node at the nucleus.
pz
px
2p
py
pz
px
py
3p
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