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Chapter 7
The Electronic Structure of Atoms
Dr. Nabil EL-Halabi
In 1900 Max Plank discovered that atoms and molecules emit energy only in certain discrete
quantities, or quanta.
Planks quantum theory turned physics upside down. To understand quantum theory, we must know
something about the nature of waves.
A wave can be thought as a vibrating disturbance by which energy is transmitted.
Wavelength () is the distance between identical points on successive waves.
Frequency () is the number of waves that pass through a particular point in 1 second.
Amplitude is the vertical distance from the midline of a wave to the peak or trough.
u =  x 
u = the speed of the wave
 (lambda) is usually expressed in units of meters, centimeters, or nanometers.
 is measured in hertz (Hz), Hz = 1 cycle/s.
Electromagnetic Radiation:
Electromagnetic radiation is the emission and transmission of energy in the form of
electromagnetic waves.
All types of radiant energy, also called electromagnetic radiation moves through a vacuum at a
speed of about 3.00 x 108 m/s, which is called the speed of light.
For all electromagnetic radiation,
 x =c
Figure 7.2 shows various types of electromagnetic radiations, which differ from one another in
wavelength and frequency.
The shortest waves, with the highest frequency, the more energetic the radiation.
The ultraviolet radiation, X rays, and rays are high-energy radiation.
Example 7.1

Plank’s Quantum Theory
In 1900 Max Planck said that atoms and molecules could emit (or absorb) energy only in discrete
quantities. The smallest quantity of energy that can be emitted (or absorbed) in the form of
electromagnetic radiation is called a quantum.
The energy E of a single quantum of energy is given by E = h
h is called Plank’s constant = 6.63 x 10-34 Js.
 is the frequency of radiation.
The Photoelectric Effect:
 In 1905, Albert Einstein extended Planck’s idea.
 He suggested that a beam of light is a stream of particles.
 These particles of light are called photons.
 Photons are the quanta of electromagnetic energy with energy E = h
 Einstein also stated that the change in the photon’s energy was equal to the ejected electron’s
energy. Therefore, the photon’s energy equaled the electron’s kinetic energy added to the
electron’s binding energy
h = KE + BE
KE = h - BE
KE is kinetic energy of the ejected electron and BE is the binding energy of the electron in the
metal.
The more intense the light, the greater the number of electrons emitted by the target metal; the
higher the frequency of the light, the greater the kinetic energy of the emitted electrons.
Example 7.2
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Bohr’s Theory of the Hydrogen Atom:
Emission Spectra:
The emission spectra of atoms in the gas phase do not show a continuous spread of wavelengths
from red to violet; rather, the atoms emit light only at specific wavelengths. Such spectra are called
line spectra because the radiation is identified by the appearance of bright lines in the spectra.
A line spectrum is produced, when an element in the gas state is heated or an electric current is
passed through it. Figure 7.4.
Each element produces a unique line spectrum.
Classical physics could not explain line spectra. Solids produce continuous spectra.
Bohr’s Model of the Atom
 Electron travels around the nucleus in orbits of fixed size, it can only have specific (quantized)
energy values.
 Light is emitted as electron moves from one energy level to a lower energy level.
Ground state (ground level): refers to the lowest energy state of a system.
Excited state (excited level): the higher in energy than the ground state.
 The energy of the electron is En = - RH(1/n2) RH is the Rydberg constant = 2.18 x 10-18 J,
n is an integer called the principal quantum number, n = 1, 2, 3, . ..
 The size of the orbit also increases with increasing n.
Let us now apply equation En = - RH(1/n2), to the emission process in a hydrogen atom. During
emission, the electron drops from an excited state (Ei) to a lower energy state (Ef).
E = Ef - Ei
E = - RH /nf2 - (- RH /ni2)
E = RH (1/ni2 - 1/nf2)
E = h = RH (1/ni2 - 1/nf2)
When a photon is emitted, ni > nf , E is negative (energy is lost to the surroundings).
When energy is absorbed, ni < nf , E is positive.
The following table lists the series of transitions in the hydrogen spectrum; they are named after
their discoverers.
The Various Series in Atomic Hydrogen Emission Spectrum
Series
nf
ni
Spectrum Region
Lyman
1
2, 3, 4
Ultraviolet
Balmer
2
3, 4, 5
Visible and Ultraviolet
Paschen
3
4, 5, 6
Infrared
Brackett
4
5, 6, 7
Infrared
The Balmer series is particularly easy to study because a
number of its lines in the visible range.
Example 7.3
The Dual Nature of the Electron:
Bohr assumed that the angular momentum of the electron is quantized.
mvr = n(h/2) where n = 1, 2, 3, 4…
Then using the laws of classical physics he derived an equation for the allowed energy levels in the
H atom.
En = - 2.18 x 10-18 (1/n2) J n=1,2,3...
Problems with the Bohr model
1- Why does the electron in a Bohr atom have only certain allowed orbits and energies?
2- Bohr’s approach only worked for the hydrogen atom.
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We know that Light has both:
1- wave nature
2- particle nature
In 1924 Louis de Broglie proposed that electrons behave like waves as well as particles.
 According to de Broglie an electron bound to a nucleus behaves like a standing wave.
(The waves are described as standing, or stationary, because they do not travel along the string.
Some points on the string, called nodes, do not move at all, that is, the amplitude of the wave at
these points is zero).
 An allowed energy state is one in which an integral number of waves will fit around the
circumference of the orbit.
2r = n
De Broglie re related by the expression:  = h/mu
Example 7.4
Bohr’s and de Broglie’s approach did not work for atoms other than hydrogen.
Quantum Mechanics:
In the 1920’s Heisenberg, Schrödinger, and Dirac developed the modern theory of the atom, which
we call Quantum Mechanics.
The Heisenberg Uncertainty Principle: It is impossible to know simultaneously both the
momentum and position of a particle with certainty.
Schrödinger developed a differential equation, which treated the electron as both a wave and a
particle. For the H atom it gave the same energies as Bohr. But, it gives quite a different picture of
the atom. It was successfully applied to other atoms.When the Schrödinger equation is solved for
the H atom we get the exact energy states that the electron can occupy. Associated with each energy
state is a wave function , psi.
The square of  gives the probability that the electron will be found in a particular region. This is
also called an orbital.
Orbital: the region in space where the electron is most likely to be found.
Note: Quantum Mechanics tells us nothing about the path of the electron. Only probability
information is given.
When the Schrödinger equation is solved, the energies and orbitals are found to depend on several
quantum numbers.
The Principal Quantum Number (n) = Energy Level
n
1
2
3
4
5
6
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Name of the level
K L M N O P Q
It tells the principle energy level of the electron and its average distance from the nucleus in a
particular orbital.
The Angular Momentum Quantum Number (l): = Subshells
l = 0, 1, 2, 3, 4, …(n-1)
l
0
1
2
3
4
5
Name of orbital
s
p
d
f
g
h
It tells the sublevel of the electron and the shape of the orbital, and it depends on the value of the
principal quantum number n.
l has possible integral values from 0 to (n – 1).
If n = 1, there is only one possible integral value l; that is, l = n – 1 = 1 – 1 = 0
If n = 2, there are two values of l, given by 0 and 1.
The shell with n = 2 is composed of two subshells, l = 0 and 1. These subshells are called the 2s and
2p subshells where 2 denotes the value of n, and s and p denote the values of l.
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The Magnetic Quantum Number (ml):
It tells the number of orbitals and their orientation in space.For a certain vale of l, there are (2l + 1)
integral values of ml as follows:
ml = - l, (- l + 1), …0,... (+l – 1), + l
If l = 0, then ml = 0.
If l = 1, then there are [(2 x 1) + 1], or three values of ml, -1, 0, 1.
If l = 2, then there are [(2 x 2) + 1], or five values of ml, -2, -1, 0, 1, 2.
The Spin Quantum Number (ms):
A spinning charge generates a magnetic field, thus the electron produces a tiny magnetic field since
it can spin in two possible motions, one clockwise and the other counterclockwise.
When an electron is placed in an external magnetic field its own magnetic field is quantized. It can
line up with the external field or oppose the external field.
The electron spin quantum number may have values of +1/2 or –1/2.
Atomic Orbitals:
The following table shows the relation between quantum numbers and atomic orbitals.
n
1
2
3
l
0
0
1
0
1
2
Relation Between Quantum Numbers and Atomic Orbitals.
ml
Number of Orbitals
Atomic Orbital Designations
0
1
1s
0
1
2s
-1, 0, 1
3
2px, 2py, 2pz
0
1
3s
-1, 0, 1
3
3px, 3py, 3pz
-2, -1, 0, 1, 2
5
3dxy, 3dxz, 3dyz, 3dz2, 3dx2 – y2
The number of orbitals associated with the principle quantum number n = n2
s orbitals:
p orbitals
d orbitals
Examples 7.5, 7.6.
The Energies of Orbitals:
Energy only depends on principal quantum number n in the case of a single electron atom (H atom)
figure 7.18, and it depends on the value of n and l in the case of many–electron atoms figure 7.19.
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Electron Configuration: how the electrons are distributed among the various atomic orbitals.
Figure 7.20 shows the ways buy which atomic subshells are filled in many-electron atoms.
The four quantum numbers n, l, ml, and ms enable us to label completely an electron in any orbital
in an atom. The four quantum numbers for an electron in 2s orbital are:
n = 2, l = 0, ml = 0, ms = +1/2 or –1/2.
Example 7.7.
The Pauli Exclusion Principle: no two electrons in an atom can have the same four quantum
numbers.
It 2 electrons in an atom have the same n, l, and ml values, then they must have different values of
ms.
In other words, only 2 electrons may occupy the same atomic orbitals, and these electrons must
have opposite spins.
The maximum number of electrons permitted in any shell is equal to 2n2.
Paramagnetic substances are those that are attracted by a magnet.
Any atom with an odd number of electrons in its electron configuration must be paramagnetic.
Diamagnetic substances are slightly repelled by a magnet.
The shielding Effect in Many-Electron Atoms:
For the same quantum number n, the penetrating power decreases as the angular momentum
quantum number l increases, or s > p >d > f
Consider the following example:
Be 1s22s2
B 1s2 2s 2 2p 1
Less energy is needed to remove a 2p electron in B than a 2s electron in Be because a 2p electron is
not held quite as strongly by the nucleus.
Hund’s rule: The most stable arrangement of electrons in subshells is the one with the greatest
number of parallel spins.
Examples 7.8, 7.9.
The Building-Up Principle:
The Aufbau Principle: dictates that as protons are added one buy one to the nucleus to build up the
elements, electrons are similarly added to the atomic orbitals.
The electron configuration of all elements except H and He are represented by a nobel gas core,
which shows in bracts the nobel gas element that most nearly precedes the element being
considered.
Be [He] 2s1,
Al [Ne] 3s2 3p1,
K [Ar] 4s1
Transition metals: have incompletely filled d subshells or readily give rise to cations that have
incompletely filled d subshells.
Cr [Ar] 4s1 3d5
Cu [Ar] 4s1 3d10
Lanthanides:(rare earth series) have incompletely filled 4f subshells or readily give rise to cations
that have incompletely filled 4f subshells.
Actinides: most of these elements are not found in nature but have been synthesized.
Example 7.10
Selected Problems: 32, 34, 40, 54, 56, 58, 64, 77, 78, 80, 83, 84.
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