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Atomic Structure
B.Sc Programme 1st Year
Paper no:1 Unit :1
Chapter No and Name: 1,Atomic Structure
Author : Dr. Geetika Bhalla
Department/College : Hindu College University of Delhi
Reviewer: Dr. K. L. Kapoor, Associate Professor(retd.)
Collge/Department: Hindu College,University of Delhi
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Atomic Structure
1.1 Introduction
Though the atom has been in existence since the beginning of creation, it was only in
465 BC that it was first thought of and mentioned by the Greek philosopher Democritus
(460-370 B.C). He believed that matter was indestructible and made up of tiny particles
called atoms. The structure of atom as known to our ancient Hindu philosophers (600500 BC) was made up of exceedingly small, indivisible and eternal particles called
"Paramanu". For the next 2000 years and more the atom was forgotten until in 1808,
Dalton put forward the living hypothesis called Dalton's atomic theory.
Dalton in 1808, proposed that atom was the smallest indivisible particle of matter.
However, later on, the researches carried out by various eminent scientists like J. J.
Thomson, Goldstein, Chadwick, and many others established beyond doubt that the
atom was not the smallest particle but had a complex structure of its own and was made
up of still smaller particles like electrons, protons and neutrons. However, an atom is the
smallest particle, which retains the properties of an element. The arrangement of
particles within the atom was put forward by Rutherford in 1911 on the basis of his
‘scattering experiments’ or nuclear structure of atoms. This model, however, suffered
from a serious drawback that it could not explain the stability of the atom.
Niels Bohr in 1913 proposed a model of the hydrogen atom that not only did away with
the problem of Rutherford’s unstable atom but also satisfactorily explained the line
spectrum of hydrogen element. The Bohr model was the first atomic model based on the
quantization of energy.
Biographic Sketch
Biographic Sketch
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Biographic Sketch
did you know
Amongst the various scientists who proposed different models for structure of an atom,
the Nobel Prize was conferred to Thomson in 1906, Rutherford in 1908, Bohr in 1922,
Heisenberg in 1932 and Schrödinger in 1933. Although the models proposed by some
of these scientists were rejected later but all of them were significant contributions in
the progress of science and were a major breakthrough in their times.
Historical Context
The history of the study of the atomic nature of matter illustrates the thinking process
that went on as science progressed. The models used by various scientists might not
provide an absolute understanding of the atom but only a way of abstracting so that
they can make useful predictions about them.
Scientist
Year
Contributions and Postulates
Democritus
460 B.C.
Developed the idea of tiny particles called
atoms, which could not be subdivided or made
any smaller.
John Dalton
1803
Matter is made up of tiny particles called
atoms. The atom is the smallest particle of
matter that takes part in a chemical reaction.
Atoms are indivisible and cannot be created or
destroyed.
Atoms of the same element are identical in
every respect.
J.J. Thomson
1897
Discovered
electrons
in
Cathode
Ray
experiments.
His model of the atom looked like raisins stuck
on the surface of a lump of pudding.
E. Goldstein
1900
Discovered protons in Anode Ray experiments.
Max Planck
1900
Showed that when you vibrate atoms strong
enough, such as when you heat an object until
it glows, you can measure the energy only in
discrete units. He called these energy packets,
quanta.
Albert Einstein
1905
Explained that light absorption can release
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electrons from atoms, a phenomenon called the
"photoelectric effect."
Published the famous equation E=mc2
R.A. Millikan
1909
Oil drop experiment determined the charge
(e=1.602 x 10-19 coulomb) and the mass (m =
9.11 x 10-28 gram) of an electron.
E. Rutherford
1911
Discovered the nucleus and provided the basis
for the modern atomic structure through his
alpha particle scattering experiment.
He established that the nucleus was: very
dense, very small and positively charged.
He also assumed that the electrons were
located outside the nucleus.
Drawback: The theory of electricity and
magnetism predicted that opposite charges
attract each other and the electrons should
gradually lose energy and spiral inward.
Niels Bohr
1912
Electrons do not spiral into the nucleus.
Electrons can orbit only at certain allowed
distances from the nucleus.
Atoms radiate energy when an electron jumps
from a higher-energy orbit to a lower-energy
orbit. Also, an atom absorbs energy when an
electron gets boosted from a low-energy orbit
to a high-energy orbit.
Drawbacks: The theory only worked roughly in
case of heavier elements.
Could not explain Zeeman effect and Stark
effect.
Gave the concept of stationary orbits.
Louis de Broglie
1924
Showed that if light can exist as both particles
and waves, then atomic particles also behave
like waves; gave the principle of dual behaviour
of matter.
Pauli
1924
Gave a rule governing the behavior of electrons
within the atom that agreed with experiment: If
an electron has a certain set of quantum
numbers, then no other electron in that atom
can have the same set of quantum numbers.
Physicists call this "Pauli's exclusion principle."
Werner Heisenberg
1925
Had a theory of his own called matrix
mechanics, which also explained the behavior
of atoms.
Heisenberg based his theory on mathematical
quantities called matrices that fit with the
conception of electrons as particles.
Schrödinger
1926
Schrödinger's wave mechanics.
He based his theory on waves and used partial
differential equations to solve the system of
equations.
Max Born
1926
Gave statistical interpretation of wave function
'psi'. Born thought they resembled waves of
chance and hence gave the idea of probability
of finding the particle in space (a given volume
element) around the nucleus.
Heisenberg
1927
Formulated an idea, which agreed with tests,
that no experiment can measure the position
and momentum of a quantum particle
simultaneously.
Scientists call this the "Heisenberg uncertainty
principle." This implies that as one measures
the certainty of the position of a particle, the
uncertainty
in
the
momentum
gets
correspondingly larger. Or, with an accurate
momentum measurement, the knowledge
about
the
particle's
position
gets
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correspondingly less.
James Chadwick
1932
Found the weight of atom did not check out
when working with isotopes and thus found the
predicted particles in nucleus with no charge,
called the neutron.
Neutron weighs about the same as a proton.
Neutron has no electrical charge.
1.2 Bohr's Model of The Atom
The main postulates of Bohr’s theory of atom are:
1. An atom consists of a small, heavy positively charged nucleus in the centre and
the electrons revolve around bit in circular paths called orbits.
2. Out of an infinite number of circular orbits possible, the electrons revolve only in
certain permissible orbits having a fixed value of energy.
Figure 1.1: Bohr's model of atom
These orbits (shown in Figure 1.1) are called energy levels or stationary states. The
energies of the different stationary states in case of hydrogen atom are given by
expression,
where, m = mass of the electron, h = Planck’s constant
e = electronic charge and ε0 = permittivity = 8.8542 × 10-12 C2N-1m-2
and n is an integer that can take the values 1, 2, 3, …, etc. for first, second, third…, etc.
energy levels, respectively.
1. Since the electrons revolve only in certain permissible orbits, they can have only
certain definite or discrete values of energy and not any arbitrary value of their
own. This is expressed by saying that the energy of an electron is quantized.
2. Only those orbits are permissible whose angular momentum, mvr, is an integral
multiple of h/2π . Mathematically it can be expressed as:
3.
4. Hence, the angular momentum of an electron in an atom is also quantized, i.e., it
can have certain definite or discrete values and not any arbitrary value of its
own.
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1. While revolving in a particular orbit, the electron does not lose or gain any
energy. Energy is absorbed or emitted only when the electrons jump from one
orbit to another. the amount of energy absorbed or emitted during this transition
is given by:
Did yo know
Bohr was the first to introduce the concept of quantization of angular momentum
through the relation
mvr = nh / 2π
but this concept had no proof at that time and hence became a limitation of Bohr model
of the atom.
v.
– = Δ=
or
Δ=
E2E1E hv
E
where, E2= energy of higher orbit,
E1= energy of lower orbit,
ΔE = difference in energy between higher and lower orbits,
v = frequency of light absorbed or emitted,
λ = wavelength of light absorbed or emitted,
c = velocity of light (3.0 x 108 m s-1) and
h = Planck’s constant.
Bohr was able to explain the stability of atoms as well as the emission spectrum of
hydrogen with these postulates. However, soon it was realized that Bohr’s model of
atom had many limitations and needed to be redefined.
1.2.1 Limitations of The Bohr’s Model of Atom
1. Bohr’s model of atom could not explain the spectra of atoms containing more
than one electron. It failed even for the simple helium atom, which has only two
electrons.
2. Although Bohr’s model was successful in explaining the spectra of hydrogen atom
and hydrogen like systems such as He+, Li2+ etc., it could not account for the fine
structure in the spectrum observed using refined spectroscopic techniques. It is
found that each spectral line on high resolution is in fact a doublet, i.e., two
closely spaced lines.
3. Bohr’s theory was also unable to account for the splitting of the spectral lines in
the presence of a magnetic field (Zeeman Effect) or an electric field (Stark
Effect).
4. Bohr proposed a two dimensional or flat model of the atom which now has
become obsolete as it has been established that atom is a three dimensional
entity.
5. It failed to account for Heisenberg uncertainty principle.
6. It failed to account for the shapes of molecules.
FAQs
What are Zeeman and Stark effects? Were they explained by the Bohr’s theory?
The splitting of spectral lines into many components in the presence of a magnetic field
is called Zeeman effect. Analogously, the splitting of spectral lines into many
components in the presence of an electric field is called Stark effect. These effects could
not be explained within the framework of Bohr's theory and required the presence of
multiple non-circular orbits corresponding to each energy level.
An effort, with some degree of success, was made by Sommerfeld to explain these
phenomena by invoking multiple elliptical orbits corresponding to each energy level. The
Bohr-Sommerfeld theory was ultimately discarded along with Bohr's model for other
reasons.
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FAQs
Were many electron atoms explained by the Bohr’s model?
Bohr's model especially with Sommerfeld's extension explained most details of the
hydrogen atom spectrum. However, it failed to explain the spectrum of many-electron
atoms.
Any electron in such an atom would be influenced by the electric field of other electrons
and by the magnetic field due to the motion of other electrons. The latter was also the
reason for the presence of hyperfine structures within the spectral lines of manyelectron atoms.
Bohr's model had no provision for the incorporation of inter-electronic interactions in
many-electron atoms.
Need for a New Approach to the Atomic Model
In view of the shortcomings of the Bohr model, attempts were made to develop a more
suitable and general model for atom. Two important developments that contributed
significantly in the formulation of such a model were:
(i)
(ii)
Dual behavior of matter
Heisenberg uncertainty principle
1.3 Dual Behaviour of Matter and Radiation
In case of light, some phenomenon, like interference [shown in Fig. 1.2(a)], diffraction,
etc., can be explained if the light is supposed to possess wave character. However,
certain other phenomenon, such as black body radiation and photoelectric effect [shown
in Fig. 1.2(b)] can be explained only if it is believed to be a stream of photons, i.e., it
has particle character (or is corpuscular in nature). Thus, light is said to have a dual
character.
Comment
The quanta of light (radiant energy) are called ‘photons’ (from the Greek ‘phos’ = light;
introduced by G. N. Lewis in 1926). They are defned by the equation:
E = hυ
where h is Planck’s constant (named for the German physicist Max Planck, 1858–1947)
and υ is the frequency of radiation.
(a)
(b)
Figure1.2: (a) Interference pattern shown by a beam of light
(b) photoelectric effect
Louis de-Broglie, the French physicist in 1924 proposed that matter, like radiation,
should also exhibit dual behaviour, i.e., both particle and wave-like properties (the
distinction between the properties of a particle and a wave is given in Table 1.1). This
means that just as a photon has momentum as well as wavelength, electrons should
also have momentum as well as wavelength. de-Broglie, from this analogy, gave the
following relation between wavelength (λ) and momentum (p) of a material particle, i.e.,
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where ‘m’ is the mass of the particle, ‘v’ its velocity, and ‘p’ its momentum.
Did you Know
1.1 de-Broglie’s Wavelength can be Obtained from the Quantum and Einstein Theory
For electromagnetic radiations,
E
or
υ
E = hυ
(according to quantum theory)
and also, E = mc2 (Einstein equation)
therefore, hυ = mc2
or
(
)
where c is the velocity of light
Concept map
Do you know that the electron microscope is based on the wave like property of
electrons?
In an electron microscope, the wavelength of the electrons can be controlled through an
applied voltage and using electric and magnetic fields can readily focus the electron
beam.
Electron microscopes are now used more routinely in chemistry and biology to
investigate atomic and molecular structures.
Table 1.1 Distinction between a particle and a wave
S.No.
Particle
Wave
1.
A particle occupies a well-defined
position in space, i.e., a particle
is
localized
in
space.
For
example, a cricket ball.
A wave is spread out in space. For
example, on throwing a stone in a pond of
water, the waves start moving out in the
form of concentric circles. Thus, wave is
delocalized in space.
2.
When a particular space is
occupied by one particle, the
same space cannot be occupied
simultaneously by any other
particle. Particles do not undergo
interference.
Two or more waves can co-exist in the
same region of space and hence undergo
interference.
3.
When a number of particles are
present in a given region of
space, their total value is equal
to their sum.
When a number of waves are present in a
given region of space, due to interference,
the resultant wave can be larger or smaller
than the individual waves.
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Table 1.2 Characteristics of matter and electromagnetic waves
S.No.
Electromagnetic Waves
Matter Waves
1.
The electromagnetic waves (photons and
radiations) are associated with electric and
magnetic fields perpendicular to each other
and to the direction of propagation.
Matter
waves
(associated
with
material particles, like electron,
proton etc.) are not associated with
electric and magnetic fields.
2.
They do not require any medium for They require medium for their
propagation, i.e., they can pass through propagation, i.e., they cannot pass
vacuum.
through vacuum.
3.
They travel with the same speed as that of The speed of these waves is not the
light.
same as that of light. Moreover, it is
not
constant
for
all
material
particles.
4.
Their wavelength is given by
Their wavelength is given by
The wave character of matter should not be confused with the wave character of
electromagnetic waves. They differ in various aspects as given in Table 1.2.
Although, de-Broglie equation is applicable to all material objects but it has significance
only in case of microscopic particles.
For example, consider a ball of mass 0.1 kg moving with a speed of 60 m s-1 . From deBroglie equation, the wavelength of the associated wave is h/mv = (6.62 x 10-34 )/(0.1 x
60) or 10-34 m. It is apparent that this wavelength is too small for ordinary observation.
On the other hand, an electron with a rest mass equal to 9.11 x10-31 kg, i.e.,
approximately 10-30 kg moving at the same speed would have a wavelength = (6.62 x
10-34 )/(10-30 x 60) =10-5 m, which can be easily measured experimentally.
Davisson and Germer, in 1927, confirmed de-Broglie’s prediction experimentally. They
showed that a crystal diffracts that electron beam in the same way as light radiation is
diffracted, a phenomenon characteristic of waves. This provided a proof of wave nature
of electrons. It needs to be noted that according to de-Broglie, every object in motion
has a wave character. The wavelengths associated with ordinary objects are so short
(because of their large mass) that their wave properties cannot be detected. The
wavelengths associated with electrons and other subatomic particles (with very small
mass) can however be detected experimentally.
Example
A moving particle has 4.55 x 10-25 J of kinetic energy. Calculate its wavelength. (mass =
9.1 x 10-31 kg and h = 6.6 x 10-34 kg m2 s-1)
Solution:
Since, kinetic energy = 1/2 mv2 = 4.55 x 10-25 J (given)
m = 9.1 x 10-31 kg
h = 6.6 x 10-34 kg m2 s-1
Thus, ½ (9.1 x 10-31) v2 = 4.55 x 10-25
or v2 = 106 or v = 103 m s-1
λ = h/mv = 6.6 x 10-34 / (9.1x10-31) x 103
= 7.25 x 10-7 m.
Example
Two particles A and B are in motion. If the wavelength associated with particle A is 5 x
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10-8 m, calculate the wavelength associated with particle B if its momentum is half of A.
Solution:
Using de-Broglie equation,
λA = h/pA and λB= h/pB
so λA/ λB = pB/pA
But pB = ½ pA (given)
or λB = 2 λA = 2 (5 x 10-8) m =10-7 m
Interesting fact
A very interesting fact about the life of two scientists lies in the concept of the
wave-particle duality of matter.
It was J.J. Thomson who first showed in 1895 that electrons are subatomic particles and
his son G.P. Thomson was one of the first who showed in 1926 that electrons could also
behave as waves.
J.J. Thomson, the father, won the Nobel Prize in 1906 for showing particle nature of
electrons and his son won Nobel Prize in 1937 for showing that electrons behave as
waves.
1.4 Heisenberg's Uncertainty Principle
Werner Heisenberg, a German physicist, in 1927, stated the uncertainty principle as a
consequence of dual behaviour of matter and radiation. It states that it is impossible to
determine simultaneously the exact position and velocity of an electron.
The uncertainty principle can best be understood with the help of an example.
Suppose you are asked to measure the thickness of a sheet of paper with an unmarked
meter-stick, obviously, the result obtained would be extremely inaccurate and
meaningless. In order to obtain any accuracy you would have to use an instrument
graduated in units smaller than the thickness of a sheet of paper. Analogously, in order
to determine the position of an electron, we must use a 'metrestick' calibrated in units
smaller than the dimensions of an electron. To observe an electron, we can illuminate it
with light or electromagnetic radiation. The light used must have a wavelength smaller
than the dimensions of an electron, but photon of such light would possess very high
energy. The high momentum photons of such light (p = h/λ ) would change the velocity
of electrons on collision. In this process, although we are able to calculate the position of
the electron accurately, we would know very little about the velocity of the electron after
the collision. On the other hand, if we reduce the energy (h ) of the photons, i.e., we
increase wavelength (λ) of light, the effects of impacts could be considerably diminished.
But at the same time the electron would be less accurately defined owing to the
decreased resolving power of the microscope. Therefore, with light of low frequency, it is
possible to know the velocity of an electron, but not its position.
The uncertainty principle is inherent in the nature of things.
For example, let us try to measure the momentum of an electron moving with a velocity
of say 108 m s-1.
Momentum (mv) of the electron = 10-30 x 108 = 10-22 kg m s-1. The γ photon needed for
looking at the electron has wavelength of the order of 6 x 10-15m.
But
So the momentum of photon,
It is clear that the γ-photon required to detect the electron will hit with a momentum
1000 times greater than its own. So, the very act of observing the electron disturbs it.
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In other words, it can be stated that if the position of the electron is measured
accurately, there is an uncertainty in its velocity or momentum and vice-versa.
Heisenberg proposed a mathematical expression for the uncertainty principle as:
or
where Δx = uncertainty with regard to position, Δv = uncertainty with regard to velocity
and m = mass of the particle.
Example
Calculate the uncertainty in the velocity of an electron if the uncertainty in its position is
1 Å or 100 pm. (h = 6.6 × 10-34 kg M2 s-1)
Solution:
∆
= 10-10 m (given)
m = 9.1 x 10-31 kg
Applying uncertainty principle :
(for sake of calculation we have replaced “
” with “=”)
= (6.6 x 10-34)/(4 x 3.14 x 9.1 x 10-31 x 10-10)
= 5.77 x 105 m s-1
Did you know
The nonexistence of electron in the nucleus can be explained on the basis of
Heisenberg’s uncertainty principle. The approximate dimension of a nucleus of an atom
is of the order of 10–14. Now, if the electron were supposed to be present inside the
nucleus, the maximum uncertainty in its position would be 10–14. Using the uncertainty
expression, the maximum uncertainty in its velocity would be:
or
This uncertainty in the velocity is much higher even than the velocity of light, which is
not possible. Hence, such an electron cannot exist.
FAQs
Is the uncertainty in Heisenberg’s principle due to limitations in our technology
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for making precise measurements?
No, the uncertainty is not due to limitations in our technology for making precise
measurements. The uncertainty is inherent in the nature of subatomic particles and in
the very process of making measurements. For making a measurement, the instrument
must interact with the particle on which the measurement is made. During this
interaction it changes the properties of the particle 'uncontrollably'. If the interaction is
through photons (which have a dual nature) then shorter wavelength photons can
determine the position of the particle more accurately. But such photons will have very
high energy and at the instant of measurement of the position it will grossly change the
momentum of the particle. If, in order to accurately determine the momentum, we use
low energy photons, then the wavelength would become very large and the uncertainty
in position will increase.
Did you know
Heisenberg’s Uncertainty Principle states that the uncertainties in measurement of
position and momentum of a particle are related by the expression,
Δx .Δpx ≥ h / 4π
The uncertainties in measurement of time (Δt) and energy (ΔE) of a particle are related
by the expression,
Δt .ΔE ≥ h / 4π
This relation is very essential in spectroscopic studies because it gives information about
the width of spectral lines. This expression is also one of the forms of the Heisenberg
Uncertainty Principle.
1.4.1 Concept of Probability
According to the uncertainty principle, the position and the momentum of an electron
cannot be known accurately at the same time. So, the Bohr's concept of well-defined
paths or orbits for a moving electron becomes meaningless. There is always some
uncertainty associated with the location the electron. In other words, we can only talk
about probability of finding an electron at a point in space rather than describing it
accurately.
Bohr model of the hydrogen atom not only ignores dual behavior of matter but also
contradicts Heisenberg's uncertainty principle. In view of this inherent weakness in the
Bohr's model, there was no point in extending his model to other atoms. What was
needed was an insight into the structure of the atom that takes into account waveparticle duality of matter and was consistent with Heisenberg uncertainty principle. This
came with the advent of quantum mechanics.
Check your progress
According to Heisenberg’s uncertainty principle, why it is impossible to measure
simultaneously the exact position and momentum of a fast moving electron?
1.5 Quantum Mechanical Model of Atom
In classical mechanics, the ‘state’ of a system is defined by specifying all the forces
acting on the system and all the positions and velocities of the particles. Classical
mechanics is based on Newton’s laws of motion that successfully describes the motion of
all macroscopic objects such as a falling stone, orbiting planets etc., which have
essentially a particle-like behaviour. However, it fails when applied to microscopic
objects like electrons, atoms, molecules etc. This is mainly because of the fact that
classical mechanics ignores the concept of dual behavior of matter and the uncertainty
principle. The branch of science that takes into account this dual behavior of matter is
called quantum mechanics.
Quantum mechanics is a theoretical science that deals with the study of the motion
of the microscopic objects that have both observable wave like and particle like
properties. It specifies the laws of motion that these objects obey. When quantum
mechanics is applied to macroscopic objects (for which wave like properties are
insignificant) the results are the same as those from the classical mechanics.
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Did you know
Quantum Mechanics was developed separately by two groups of scientists:one was Schrödinger, who gave the concept of wave mechanics (in 1926) and obtained
the famous Schrödinger equation by solving partial differential equations and second
was Heisenberg who devised matrix mechanics (in 1925) for solving the system of
equations. But both the approaches were later shown to be equivalent by Dirac and
Neumann (in 1928).
Wave mechanical model of atom is based on the following postulates:
•
•
•
•
•
Heisenberg uncertainty principle governs atomic particles. Thus, the idea of
uncertainty in the position and the velocity overruled Bohr's picture of fixed
circular orbits. The term probability distribution is used to describe position of an
electron.
A stream of electrons is associated with a wave, whose wavelength (λ) is given
by de-Broglie relationship:
Since electron is under the influence of nuclear forces, it can be compared to a
stationary or standing wave formed by a vibrating string fixed between two
points.
Energy values of the electron are quantized as calculated by Bohr.
FAQs
What is the role of quantum mechanics in chemistry, if it is a science dealing
with "laws of motion" of microscopic objects?
Quantum mechanics caters to three fundamental problems in chemistry namely,
structure, bonding and reactivity. Solutions to these problems depend on the detailed
understanding of the behavious of these microscopic particles in matter. Microscopic
particles can never be at rest. An in depth understanding of the behaviour of these
particles is achieved by quantum mechanical studies.
Quantum mechanics was developed independently in 1927 by Werner Heisenberg and
Erwin Schrödinger. Here, however, we shall be discussing the quantum mechanics of
Schrödinger, which is based on the ideas of wave motion. With the wave nature of a
moving electron confirmed, it was expected that its behaviour could be described by a
suitable wave equation just as for waves of light, sound, etc.
The fundamental equation of quantum mechanics is the Schrödinger equation:
where ψ = amplitude of the wave, m = mass of the electron, h = Planck’s constant, E =
total energy of the electron, V = potential energy of the electron and x, y and z are the
coordinates of the electron with nucleus at the origin.
This equation was proposed by Schrödinger, which won him the Nobel Prize in Physics in
1933.
In quantum mechanics, the state of the system is defined by a mathematical function
(ψ) called the state function or wave function. A wave equation gives a complete
description of a system. Just as we can write a volume function for a gas, V = nRT/P, we
can write a wave function for a wave in the form of ψ.
For waves of a vibrating string, we have the simplified relation,
where x = displacement, A = constant and ψ = amplitude of the wave.
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For electron waves, we can write,
where x = distance from the nucleus Differentiating with respect to x
Differentiating again,
From de Broglie’s relation we have,
Replacing λ in equation (1.1), we have,
The total energy of a particle is sum of potential energy and kinetic energy, i.e.,
E = P.E. + K.E.
= V + ½ mv2
E – V = ½ mv2
v2 = 2(E - V)/ m
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The
Schrödinger
equation
can
be
and shortened to
rearranged
to
the
form
Ĥψ = Eψ
where Ĥ is called a Hamiltonian operator and it defines the operation or sequence of
operations to be performed (i.e., taking second differentials, multiplying their sum by –
h2/8π2m and adding the potential energy, V) on the function ψ. The result of carrying
out these operations on function ψ is the same as multiplying ψ by the electronic
energy, E. This form of the Schrödinger wave equation (i.e. Ĥψ= Eψ) is called an
eigenvalue equation.
FAQs
What is an operator?
An operator is basically a mathematical command, which operates on a function that
follows it. For every observable/ measurable property in classical mechanics, there is a
corresponding operator in quantum mechanics. Like d/dx (first derivative of function),
d2/dx2 (second derivative of function), √ (square root of function) etc, are some
examples of operators.
Think
Does the Hamiltonian operator, Ĥ, have units?
The Hamiltonian operator defines the sequence of mathematical operations to be
performed on the wave function, Ψ , which follows it in the Schrödinger
Equation.
ĤΨ = EΨ
i.e.,
The wave function, y, is an eigenfunction and the energy, E is an eigenvalue of the
Hamiltonian operator. This indicates a correspondence between the Hamiltonian
operator and the energy. In fact, the Hamiltonian operator in the above equation is
constructed in a manner such that it contains both the kinetic energy term and the
potential energy term and therefore it is evident that the Hamiltonian operator has the
dimensions of energy (energy units) in the Schrödinger Equation.
Table 1.3 Difference between an orbit and an orbital
S.No
Orbit
Orbital
1.
An orbit is a well-defined circular
path around the nucleus in which
an electron revolves.
It represents the planar motion of
an electron.
The concept of a well-defined orbit
is
against the
Heisenberg's
uncertainty principle and the deBroglie concept.
All orbits are circular.
It represents the region in space around the
nucleus in which the probability of finding the
electron is the maximum.
It represents the three-dimensional motion of
an electron around the nucleus.
This
concept
is
in
accordance
to
the Heisenberg Uncertainty principle and the
de-Broglie relation.
2.
3.
4.
5.
Orbits do not
characteristics.
have
Orbitals have different shapes. For example,
s- orbital is spherically
symmetrical and
p-orbital is dumbbell
shaped.
directional Except the
s-orbital,
all
others
have
directional characteristics and, thus, they
clearly explain the shape of a molecule during
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6.
bonding.
The maximum number of electrons An orbital can accommodate only two
in an orbit is 2n2, where n electrons and that too with opposite spins.
represents the number of orbit.
The solution of the Schrödinger wave equation led to the concept of the most probable
regions where the chance to find the electrons is maximum. This is in contrast to the
well-defined circular paths proposed by Bohr. In accordance with Heisenberg’s
uncertainty principle, the electron cannot exist at a definite point but at certain regions
in space around the nucleus where the probability of finding electron is about 90-95%.
These regions in space are called orbitals.
An orbital may be defined as a region in space, around the nucleus, where the
probability of finding electron is maximum.
Table 1.3.
The difference between an orbital and orbit is given in Table 1.3.
Difference between an Orbit and an Orbtal
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1.5 Quantum Mechanical Model of Atom
Schrödinger equation cannot be derived, i.e., it cannot be proved from fundamental
principles of quantum mechanics. It was the genius of Schrödinger to arrive at equation
(1.4) by intuition and to justify it by solving this equation to give values for ‘E’ in
agreement with the experiments. It may be emphasized again that the above treatment
is in no way a ‘proof’ of the Schrödinger equation, it merely shows that if the deBroglie’s relationship is assumed and if the motion of the electron is analogous to a
system of standing waves, equation (1.4) is the type of the wave equation to be
expected. The Schrödinger equation shows a logical coherence to a vast amount of
experimental observations. It has been found to hold true in both microscopic and
macroscopic systems.
The mathematical task in applying the Schrödinger equation to a particular problem is to
obtain a suitable expression for ψ showing how the wave amplitude varies with distance
along the x, y, z-axes and then to derive solutions of the differential equation.
There will be, in every case, many expressions for ψ satisfying the Schrödinger
equation, all of which are, however, not acceptable. Only those wave functions which
satisfy certain conditions are acceptable and are called eigenfunctions of the system,
while the energy E, corresponding to the these eigenfunctions are called eigenvalues.
For hydrogen atom, the eigenvalues correspond to the discrete sets of energy values
postulated by Bohr theory, i.e., the occurrence of definite energy levels in an atom
follows directly from the wave mechanical concept.
The conditions which a wave function, ψ, must satisfy before it is accepted are:
(i)
ψ must be continous (all real values are continous).
(ii) ψ must be finite (ψ2 must be finite since it represents probability).
(iii) ψ must be single valued, i.e., it must have only one definite value at a particular
point in space.
(iv) The probability of finding the electron over the entire space from –∞ to +∞ must
be equal to one, that is,
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1.5.1 Significance of Ψ and Ψ2
The physical significance of ψ is nebulous. The wave function ψ has no physical
significance except that it represents the wave amplitude. ψ depends on the coordinates
of the particle. In some cases, it may turn out to be a complex function of the form:
ψ = a + ib
where a and b are real functions of the coordinates and i is iota
This complex nature of ψ is necessary if one is to use superposition of wave functions to
describe interference effects in matter waves. The complex conjugate of ψ is:
ψ* = a – ib
therefore,
ψψ* = (a + ib) (a – ib) = a2+ b2
If ψ is real, ψψ* is identical to ψ2.
Neither ψ nor ψ* has any physical significance but ψ ψ* (or ψ2) has. The product of ψ
ψ* will always be real, whereas, ψ2 (ψ ψ) can possibly be imaginary.
In dealing with all forms of wave motions such as light waves, sound waves or matter
waves, the square of the wave amplitude at any point is interpreted as the intensity of
the effect at that point. So, the value of ΨΨ* or Ψ2 at any point around the nucleus
gives measure of the electronic charge density at that point, the density of which varies
from point to point around the nucleus. According to the statistical interpretation, the
electron is still considered as a particle and the value of Ψ2 at any point is taken to
represent the probability of finding the electron at that point at a given instant.
According to uncertainty principle, the position of the electron cannot be determined
with certainty. Thus Ψ2 is interpreted as giving a direct measure of probability.
Consequently, the greater the intensity of wave function at a particular point, greater is
the probability of locating the electron at that point. When Ψ2 is high, electron density is
high, i.e., the probability of finding an electron is high.
Did you know
ψ is known as the “storehouse” of information because the information regarding any
property of the system like energy, radius etc., can be obtained from ψ.
1.5.2 Normal and Orthogonal Wave Functions
The wave function, Ψ, is said to be normalized if the probability of finding an electron
over the entire space (from –
to + ) is equal to unity. Mathematically, it can be
expressed as:
This is quite reasonable, as the probability of finding the electron in the entire universe
has to be 100%. Sometimes Ψ may not be a normalized wave function. In that case, we
multiply the function Ψ with a constant, N such that, the function NΨ is also a solution
to the wave equation. The value of N is so chosen that the new function NΨ be
normalized. Mathematically, the expression can be written as:
where N is called normalization constant.
The wave function Ψ is said to be orthogonal if it is different from all the other wave
functions over the entire space (from –
to + ). Mathematically, it can be expressed
as:
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This is reasonable, as each wave function (that is solution to the wave equation) is
unique and different. Thus, any two wave functions (say, Ψ1 and Ψ2) should be
independent of each other, i.e., they should be orthogonal to one another.
The quantum mechanical study of any system consists of
(i) Writing Schrödinger wave equation for the system.
(ii) Solving Schrödinger wave equation for the meaningful solutions of the wave
functions and the corresponding energies.
(iii) Calculation of all the observable properties of the system from ψ.
Check your progress
Explain how the quantum mechanical model of atom differs from the Bohr's model of
atom.
1.5.3 Quantum Mechanical Treatment of The Hydrogen Atom
Hydrogen atom is the simplest chemical system consisting of one proton and one
electron. Schrödinger equation can be solved exactly for hydrogen-atom but not for
atoms having more than one electron. Assuming that the electron moves at a distance
around the stationary nucleus, then the nucleus can be taken as the origin in a
coordinate system. Schrödinger equation for the hydrogen atom can be written in terms
of the Cartesian coordinates (x, y, z) as given below:
The solutions of the Schrödinger equation for the hydrogen atom lead to important
conclusions. It gives the values of energy, which agrees well with those obtained
experimentally, and also with those given by Bohr for his model of atom. It also gives
the allowed energy levels for electrons, which are same as that derived by Bohr.
The problem here is that of calculating the amplitude of electron waves at various
points in a hydrogen atom. These points can be defined by drawing a set of Cartesian (x,
y, z) axes through an origin at the nucleus of the atom and locating points by x, y, z
coordinates.
However, it turns out that the mathematics involved is much simpler if we use an
alternative way of specifying position namely the polar coordinates (r, θ, φ). Since the
atom has spherical symmetry, it is apparent that in the polar coordinates, the separation
of the variables for the hydrogen system is easier than in Cartesian coordinates. Hence,
Schrödinger equation is transformed into polar coordinates. The coordinate r, measures
radial distance from the origin, θ is a lattitude and φ is a longitude. Since the electron
is moving in three dimensions, three coordinates are sufficient to describe its position at
any time.
Transformation of Cartesian Coordinates to Polar Coordinates
The relation between the two coordinate systems is shown in the fig. 1.3. It can be seen
from the fig. 1.3 that the coordinates x, y, and z of the electron with respect to nucleus
in terms of polar coordinates can be written as:
Transformation of Cartesian Coordinates to Polar Coordinates
By replacing x, y, z coordinates with the polar coordinates in the Schrödinger equation,
we get,
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The above equation is the Schrödinger wave equation for H-atom in terms of polar
coordinates.
A series of wave functions exist that
are solutions to the above equation. A
standard method of solving the
classical wave equation is by the
method of separation of variables.
This can be done if it is assumed that
the actual wave function ψ (r, θ , φ )
is the product of three separate wave
functions each containing one of the
three variables. This can be
represented as:
Figure 1.3: Relationship
between
Cartesian coordinates
(x, y, z) and
polar
coordinates (r, θ , φ).
For convenience, the solutions are given in terms of radial part R(r) and angular part
Θ(θ).ф(φ) or A(θ,φ).
When Schrödinger equation in polar coordinates is solved for the hydrogen atom, it
gives the possible energy states and the corresponding wave functions [ψ(r, θ,
φ)] (called atomic orbitals or hydrogenic orbitals). The quantum mechanical solution of
the hydrogen atom successfully predicts all aspects of the hydrogen atom spectrum and
other phenomena that could not be explained by the Bohr model.
Did you know
The quantum mechanical solution of the hydrogen atom system gives three quantum
numbers n, l and m only. The fourth quantum number, i.e., the spin quantum number
does not come from the quantum mechanical results. The spin of the electron was
introduced based on the concepts of relativistic quantum mechanics by Peschke (1988)
and Morrison (1986) using the theory of relativity.
Comment
The total wavefunction can be written as the product of three wavefunctions each
containing one of the three variables
y (r, q, f) = R(r) . Q(q). F(f) = R(r) . Y(q,f)
Radial Angular part
part
•
•
The angular part is known as denoted by (, ).spherical harmonicsYl,mqf
Spherical Harmonics is independent of principal quantum number ‘n’.
1.6 Quantum Numbers
The application of Schrödinger equation to hydrogen atom and other atoms yields three
quantum numbers n, l and m, characterized as principal quantum number, azimuthal
quantum number and magnetic quantum number, respectively. These three quantum
numbers arise as a natural consequence in the solution of the Schrödinger equation. The
restrictions on the values of these three quantum numbers also come naturally from this
solution.
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The fourth quantum number, s, characterized as the spin quantum number, arises from
the spectral evidence. Using refined spectroscopic techniques, it was found that in highresolution spectrum of H-atom, each spectral line is in fact a doublet, i.e., two closely
spaced lines. This can be accounted for only if we assume that electron in its motion
around the nucleus, not only rotates but also spins about its own axis. So, four quantum
numbers are actually required in order to define an electron completely in an atom.
These quantum numbers describe the whole atomic state.
Principal or Radial quantum number (n)
The principal quantum number, n, is a positive integer with values of 1,2,3…, etc. The
principal quantum number determines the size and to a large extent the energy of the
orbital. ‘n’ specifies location and energy,
For hydrogen and hydrogen-like species (e.g., He+, Li2+), it alone determines the energy
and size of the orbital. The lower the value of n, the more stable will be the orbital. The
principal quantum number also identifies the shell to which the electron belongs. Each
value of n corresponds to a shell, which can be represented by the following letters.
n
=
1
2
3
4
shell
=
K
L
M
N
Azimuthal or Orbital Angular Momentum Quantum Number(l)
A shell consists of one or more subshells or sublevels. Each subshell in a shell is
assigned an azuimuthal quantum number, l. The number of subshells in a principal shell
is equal to the value of n, i.e. for a given value of n, l can have values ranging from 0 to
(n-1). For example, for n =1, the only value of l is 0 and there is only one sub-shell. For
n=2, there are two subshells that have values of l as 0 and 1. For n =3, the values of l
are 0,1 and 2 for three subshells.
Subshells corresponding to different values of l are represented by the following
symbols:
l =
notation =
0,
s,
1,
p,
2,
d,
3,
f,
4,
g,
5
h,
The symbols s, p, d and f stands for the initial letters as in the words sharp, principal,
diffuse and fundamental respectively which had been used to define the spectral lines.
The other symbols, i.e., g, h,…. and so on proceed alphabetically. Table 1.4 shows the
permissible values of l for a given principal quantum number and the corresponding
subshell notations.
Table 1.4 Subshell notations
n
1
2
l
0
0
1
Subshell
1s
2s
2p
0
3s
3
1
3p
4
2
3d
0
4s
1
4p
2
4d
3
4f
The azimuthal quantum number gives an idea about the shape of the orbital. For
example, orbital in a ‘s’ subshell is spherical, orbitals in a ‘p’ subshell are dumbbell
shaped, orbitals in a ‘d’ subshell are double dumbbell shaped and so on.
It also describes the motion of the electron in terms of orbital angular momentum. The
orbital angular momentum (L) of an electron is given by the relation:
For hydrogen atom, energies of the subshells belonging to the same shell are equal,
i.e., ns = np = nd = nf. However, for atoms with more than one electron, the energy
increases slightly with an increase in the value of l, i.e., ns < np < nd < nf.
Magnetic Quantum Number (ml)
Each subshell consists of one or more orbitals and each orbital is identified by a
magnetic orbital quantum number ml, which gives information about the orientation of
the orbital. The number of orbitals in a subshell is given by (2 l + 1). For instance, when
l = 0 (s-subshell) there is only one orbital, when l = 1 (p-subshell) there are three
orbitals, when l = 2 (d-subshell) there are five orbitals and so on. This is also equal to
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the number of values that ml may assume for a given value of l. Since ml determines the
orientation of orbitals, the number of orbitals must be equal to the number of ways in
which they are oriented. For a given value of l, the values of ml are -l,…..0,….,+l. Table
1.5 lists various values of ml for different values of l.
Table 1.5 Values of ml for different values of l
l
0
1
2
Subshell
s
p
d
ml
0
-1, 0, +1
-2, -1, 0, +1, +2
It may be noted that the values of ml are derived from l and that the values of l are
derived from n. For any shell represented by the quantum number ‘n’,
The number of subshells (or values for the quantum number l) = n
The number of orbitals (or values for the quantum number ml) = n2
Orbitals within a given subshell differ in their orientation in space, but not in their
energies, i.e., they are degenerate (having same energies). However, on applying an
external magnetic or electric field, the degeneracy is lifted (orbitals with different
orientations acquire different energies in presence of the magnetic or electric field).
Spin Quantum Number (s) and Magnetic Spin Quantum Number (ms)
The spinning of an electron is untenable from quantum mechanics. It arises from the
spectral evidence. The electron not only revolves around the nucleus but also spins
about its own axis. This spin of the electron is associated with spin angular momentum,
which is characterized by spin quantum number, s (=½ specifically for electrons). In an
analogous way of orbital angular momentum, L, the spin angular momentum can be
obtained by the expression:
where s = spin quantum number and h = Plank’s constant
.
The spinning of an electron around its own axis can be either in a clockwise or anticlockwise direction. A spinning electron generates a magnetic field and thus a magnetic
moment. The orientation of this magnetic moment vector with respect to an external
magnetic field (along z-axis) is characterized by magnetic spin quantum number, ms. An
electron spinning in clockwise or anti-clockwise direction is assigned the value of ms = +
½ or - ½ . It is also represented as
the
(spin up) or
(spin down) respectively.
The component of the spin angular momentum along the z-axis is given by
expression:
A spinning electron acts like a tiny magnet. So, two electrons with opposite spins will
behave as two magnets with opposite poles towards each other and hence there is
attraction.
For every value of the quantum number ml (orbital), there are only two permitted values
of ms, i.e., + ½ and –½ . That is to say that every orbital can have not more than two
electrons and that too having opposite spins.
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Interesting fact
The Mystery of the fourth quantum number (s).
Quantum Mechanics is not just a theoretical science but is a result of an interplay of
theory and experiment. The advent of and the progress made in the field of quantum
mechanics in the first quarter of the 20th century had provided a much better solution to
the problem of hydrogen atom and addressed the anomalies in the Bohr’s model,
however, still many experimental observations were still raising the eyebrows of many
scientist of the time and these observations were the following:
•
•
- On passing a beam of silver atoms through an inhomogeneous magnetic field
they unexpectedly observed a splitting of the beam into two parts.SternGerlach observation
A doublet in the spectra of alkali metals.
In order to address these questions, Wolfgang Pauli postulated that an electron can exist
in two distinct states and introduced a fourth quantum number in a rather ad-hoc
manner.
This fourth quantum number is now called the spin quantum number, s and has a value
½ for electrons. It is interesting to know that Pauli did not give any interpretation to
this fourth quantum number and therefore this fourth quantum number was just a
mystery at that time though it provided explanation to various experimental observation
listed above.
It was finally the attempt of George Uhlenbeck and Samuel Goudsmit in 1925 who
showed that there are two intrinsic states of an electron, which are due to the motion of
the electron about its own axis and identified these two intrinsic states with two intrinsic
angular momentum or spin states.
That is how the concept of spin and thereby the spin quantum number came in to
existence. It did not come from the solution of the Schrodinger wave equation but due
to the explanations for the various experimental observations at that time.
Discovery of electron spin and spin quantum number
As stated earlier, the spin quantum number, s, does not follow directly from the solution
of the Schrödinger wave equation for the hydrogen atom. It arose out of necessity to
explain two types of experimental evidences, which suggested an additional property of
the electron. One was the closely spaced splitting of the hydrogen spectral lines, called
fine structure. The other was the Stern-Gerlach experiment, which showed in 1922 that
a beam of silver atoms directed through an inhomogeneous magnetic field would be
forced into two beams. Both of these experimental situations were consistent with the
possession of an intrinsic angular momentum and a magnetic moment by individual
electrons. Classically, this could occur if the electron was a spinning ball of charge, and
this property was called electron spin. George Uhlenbeck and Samuel Goudsmit, in
1925, proposed the idea that each electron spins with an angular momentum of one half
Planck constant and carries a magnetic moment of one Bohr magneton (µB= 9.27 x 10-24
J/T ).
Figure 1.4: An electron spin angular momentum vector of length
units (one unit =
h/2π) can take only two
orientations with respect to a specifed axis
Spinning of Electron
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The experimental evidences suggest just two possible states for this angular momentum
described by spinning of electron in either clockwise or in anti-clockwise direction. In the
presence of an external magnetic field an electron can have one of the two orientations
corresponding to magnetic spin quantum number, ms = ±½ as shown in Figure 1.4. The
magnetic spin quantum number, ms, is used to define the z-component of the spin
angular momentum.
Exhibit
1.1 Stern–Gerlach Experiment
The direct observation of the electron's intrinsic angular momentum was achieved in
the Stern-Gerlach experiment. In 1922 two physicists named Otto Stern and Walther
Gerlach made an interesting experiment. They took a beam of electrically-neutral
silver atoms and let it pass through a non-uniform magnetic field. The magnetic field
deflected the beam and split it into two parts.
The result of this experiment was totaly unexpected and very surprising as the
electrons in silver atoms had no special orientation in space, therefore the spins of the
outer electrons in these atoms should point into all possible directions in space. On
the contrary, one half of the electrons in the beam were deflected up and the other
half were deflected down. The amount of deflection up or down is exactly the same
magnitude. Whether an individual electron is deflected up or down appears to be
random. There was only one possible explanation for this behaviour: The magnetic
moments and therefore the spins can only have two certain orientations in
space.
Soon, a new version of the Schrödinger wave equation was developed by Paul Dirac,
which was relativistically invariant, (called Dirac's equation). In this equation the
additional quantum number ‘s’ intrinsic to the electron arose naturally during its
solution.
Check your progress
Give the values of the quantum numbers for the electron with the highest energy in
sodium atom?
1.7 Atomic Orbitals and Their Pictorial Representations
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An atomic orbital is a one-electron wave function ψ(r, θ, φ) obtained from the solution
of the Schrödinger wave equation. It is a mathematical function of the three coordinates
of the electron (r, θ, φ) and can be factorized into three separate parts each of which is
a function of only one coordinate:
where R (r) is the radial wave function which gives the dependence of the distance, r, of
the electron from the nucleus and
are the angular wave functions
giving the direction of the electron in terms of the angles θ and φ respectively. On
solving the three wave equations (each involving only one variable) separately, it was
found that the radial wave function depends upon the quantum numbers n and l, the
wave function
depends upon quantum numbers l and ml and is independent of n,
while, the wave function
depends on quantum number ml only. The total wave
function ψ may, therefore, be more explicitly written as:
It would be interesting to know how ψ and |ψ|2 vary as a function of the three
coordinates r, Θ and φ for different orbitals. Such representation of the variations of
ψ or ψ2 in space would however need a four dimensional graph - three dimensions for
the coordinates and the fourth for ψ or |ψ|2 . It is not possible to show such variation in
a single diagram since we can draw only two-dimensional diagrams on paper. We can
get over this difficulty by drawing separate diagrams for: (i) variation of radial function
and (ii) angular function.
1.7.1 The Radial Wave Function, R(r)
The solutions of radial wave functions are in the form of polynomials in ‘r’ and are known
as the associated Laguerae polynomials. The normalized solutions of these wave
equations are quite complex. However, they reduce to relatively simple forms on
introduction of particular values of the parameters, i.e., n parameter and l parameter. (n
= 1, 2, 3……
and l = 0 to n-1).
Mathematical expressions for radial functions of 1s, 2s, and 2p are given in the Table
1.6.
Table 1.6 Radial functions of 1s, 2s and 2p orbitals
S. No.
n
l
Orbital
1.
1
0
1s
2.
2
0
2s
3.
2
1
2p
Radial wave function, R(r)*
*where Z = nuclear charge, e = base of natural logarithm, r = distance from the
nucleus, ao=radius of the Bohr’s orbit=
wave mechanics it is the most probable radii.
= 0.529 Å for 1s in H atom and in
The important aspects of radial functions may be made apparent by grouping constants
(Table 1.7). For a given atom, Z is constant.
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Table 1.7 Radial wavefunctions of 1s, 2s and 2p orbitals
S. No.
n
l
Orbital
1.
1
0
1s
2.
2
0
2s
3.
2
1
2p
Radial Wave function, R(r)*
*where k1s, k2s and k2p are the constants.
All these radial wave functions represent an exponential decay and that for n = 2, the
decay is slower than for n = 1.
1.7.2 Plots of the Radial Wave Function, R(r)
The radial part of the wave function R(r) gives the distribution of the electron with
respect to its distance, r, from the nucleus. The importance of these plots lies in the fact
that they give information about how the radial wave function changes with distance, r,
and about the presence of nodes. As stated earlier, the radial wave function depends
only on the values of quantum numbers n and l. It is mainly goverened by exponential
term, e-zr/naº.
In all cases wave function, R(r), approaches zero as r approaches infinity. However, the
actual shape of the curve is determined by actual wave function that depends on the
value of the quantum numbers n and l. For 1s orbital (n =1; l =0), the wave function
depends only on the exponential function. So, a plot of R(r) versus r is an exponential
curve as shown in fig. 1.5(a). In this plot, at r = 0, R(r) has a maximum value and as r
increases, R(r) decreases exponentially. For a 2s orbital (n =2; l =0), the wave function
R(r), in addition to the exponential term also depends on the term
So the
wave function, R(r), initially decreases as r increases and becomes zero at
, it
passes through a minimum and then increases to an infinitesimally small negative value.
This is shown in fig 1.5(b). R(r) for the 2s orbital becomes zero at a particular value of r
between r = 0 and r =
, at this point, the so called nodal point, R(r) changes sign
from positive to negative.In general, the number of nodes in ns-orbital is given by (n –
1). There is only one node in 2s radial function. The radial plot for a 3s-orbital (n =3; l
=0), is shown in fig.1.5(c). The radial wave function, R(r), for the 3s-orbital becomes
zero at two points between 0 and
therefore, it has two nodes.
(a)
(b)
(c)
Figure 1.5: The plots of the radial wave function, R(r), as a function of distance, r,
of the electron from the nucleus for (a) 1s-orbital (b) 2s-orbital and (c) 3s- orbital
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The radial function for a 2p-orbital (n =2; l =1), depends on the term
in addition
to the exponential term
. So, in the plot of R(r) versus r, the function R(r)
becomes zero at r = 0. It increases initially with r, reaches a maximum value and then
decreases to an infinitesimally small value.
The function R(r) remains positive throughout as shown in fig. 1.6(a), hence there are
no nodes in the 2p radial function. The radial wave function for 3p orbital [fig. 1.6(b)]
increases to a maximum in the beginning and then decreases passes through zero and
decreases further and finally increasing again to an infinitesimally small value. There is
one node in the radial function plot of 3p-orbital. The radial wave function plot for 3dorbital is shown in fig. 1.6(c).
The general formula for calculating the number of nodes in any orbital is (n – l – 1).
(a)
(b)
(c)
Figure 1.6: The plots of the radial wave function, R(r), as a function of distance, r,
of
the electron from the nucleus for (a) 2p-orbital (b) 3p-orbital
(c) 3d-orbital
and
Animation
Plots of the Radial Wave Function, R(r)
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1.7.3 Plots of Radial Probability Density R2(r) or R2
The square of the radial wave function R2 for an orbital gives the radial density. The
radial density gives the probability density of finding the electron at a point along a
particular radius line. To get such a variation, the simplest procedure is to plot R2
against r (fig. 1.7 and 1.8). These plots give useful information about probability density
or relative electron densitys at a point as a function of radius. It may be noted that while
for s-orbitals the maximum electron density is at the nucleus; all other orbitals have
zero electron density at the nucleus.
(a)
(b)
(c)
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Figure 1.7:
The plots of the radial probability density, R2, as a function of distance,
r, of the electron from the nucleus for (a) 1s orbital (b) 2s orbital and (c)
2p orbital
(a)
(b)
(c)
2
Figure 1.8: The plots of the radial probability density, R , as a function of
distance, r, of the electron from the nucleus for (a) 3s orbital (b)
3p orbital and (c) 3d orbital.
Polts of Radial probability Density R(r)2 or R2
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FAQs
What is the role of quantum mechanics in chemistry, if it is a science dealing
with "laws of motion" of microscopic objects?
Quantum mechanics caters to three fundamental problems in chemistry namely,
structure, bonding and reactivity. Solutions to these problems depend on the detailed
understanding of the behavious of these microscopic particles in matter. Microscopic
particles can never be at rest. An in depth understanding of the behaviour of these
particles is achieved by quantum mechanical studies.
1.7.4 Plots of Radial Probability Distribution Function, 4πr2R2(r)
The radial density, R2,for an orbital, as discussed earlier, gives the probability density of
finding the electron at a point at a distance r from the nucleus. Since the atoms have
spherical symmetry, it is more useful to discuss the probability of finding the electron in
a spherical shell between the spheres of radius (r + dr) and r (fig. 1.9). The probability
of finding the electron between the shell with radius r and the shell with radius (r + dr)
is called the radial probability.
Volume of sphere with radius r is given as:
Multiplying by R2
Figure 1.9: Spherical shell of thickness dr
The probability of finding the electron in the spherical shell of thickness dr is equal to
(volume of the shell x probability function). This represents radial probability
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distribution function (
), which gives the probability of finding the electron in
a shell at a distance, r, from the nucleus regardless of direction.
FAQs
What is the difference between ψ2 and 4πr2R2 terms if both represent the
probability of finding an electron around the nucleus?
ψ2 or ψ*ψ represents the probability of finding the electron at a point. The value of ψ2 at
a point represents the probability of finding the electron at that point. ψ depends on
variables (x, y, z) in Cartesian system and (r, θ, f) in polar system.
When we perform the separation of variables of the wave function ψ(r, θ, f) into radial
part R(r) and angular part Y(θ, f), the R2 term gives us the radial probability distribution
of electron around the nucleus because R, the radial part of wave function, is only
dependent on r (distance of electron from the nucleus) and not on the other two polar
coordinates (θ and f).
The radial probability represents the total probability of finding the electron in a
spherical shell situated at a distance r from the nucleus and having a thickness dr.
The radial probability distribution curves, (obtained by plotting radial probability
distribution function,
, versus distance, r) for 1s. 2s and 2p orbitals are
shown in fig. 1.10. An important contrasting feature of the radial probability distribution
curves with regard to radial wave function and radial probability density can be seen in
the plots of s-orbitals. It may be noted that at r = 0, the latter two functions have a
maxima while the former has a zero value. The maxima in the former is very close to r
= 0 but not at r = 0. This is justified as the radial probability distribution function,
, also depends on ‘r’ (in addition to
), so its value is zero at r =
0.
The importance of these curves lies in the fact that they give the true picture of the
probability of finding the electron with respect to the nucleus as it has been established
that an electron can never exist at the nucleus (i.e., at r = 0).
1s orbital
The radial probability distribution function for the 1s orbital, [fig. 1.10.(a)], initially
increases with increase in distance from the nucleus. It reaches a maximum at a
distance very close to the nucleus and then decreases. The maximum in the curve
corresponds to the distance at which the probability of finding the electron is maximum.
This distance is called the radius of maximum probability or most probable
distance. For 1s orbital, it is equal to 52.9 pm, same as Bohr's radius for hydrogen
atom.
Did you know
Bohr was the first to introduce the concept of quantization of angular momentum
through the relation
mvr = nh / 2π
but this concept had no proof at that time and hence became a limitation of Bohr model
of the atom.
(a)
(b)
(c)
Figure 1.10: The plots of the radial probability distribution function, 4π r2drR2, as
a
function of distance, r, of the electron from the nucleus for (a) 1sorbital (b)
2s-orbital and (c) 2p-orbital.
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2s and 2p-orbitals
The radial probability function curve for 2s orbital, [fig. 1.10.(b)], shows two maxima, a
smaller one near the nucleus and a bigger one at a larger distance. In between these
two maxima it passes through a zero value indicating that there is zero probability of
finding the electron at that distance. The region at which the probability of finding the
electron is zero is called a node. The most probable distance for a 2s electron
corresponds to the value of r where we have a bigger maxima.
The distance of maximum probability for a 2p electron, [fig. 1.10.(c)], is slightly less
than that for a 2s electron. However, in contrast to 2p curve, there is a small additional
maxima in the 2s curve, which lies at or around the maxima for a 1s orbital. This
indicates that the electron in 2s orbital spends some of its time near the nucleus. In
other words, the 2s electron penetrates into the inner 1s shell and therefore, is held
more tightly than the 2p electron. That is the reason why 2s electron is more stable and
has lower energy than a 2p electron.
3s, 3p and 3d-orbitals
Radial probability distribution curves for 3s, 3p and 3d-orbitals are shown in fig. 1.11.
(a)
a
(b)
Figure 1.11: The plots of the radial probability distribution function, 4π r2R2, as
function of distance, r, of the electron from the nucleus for
(a) 3s-orbital (b) 3s, 3p and 3d-orbitals.
For 3s orbital, there are three regions of high probability separated by two nodes. The
first two peaks indicate penetration of the electrons.
In general, the number of high probability regions = (n – l)
And the number of radial nodes = (n – l – 1)
For example, in a 3p-orbital, number of peaks = n – l = 3 – 1 = 2
and number of nodes = n – l – 1 = 3 – 1 – 1 = 1
It can be seen in Fig. 1.11(b) that the most probable distance for 3s, 3p and 3d orbitals
decreases in the order 3s > 3p > 3d. That is to say that the 3s orbital is more extended
in space than the 3p orbital which is in turn more extended than 3d orbital. However,
the radial distribution of 3s orbital spreads into the curve for 2s and 1s orbital. Similarly,
the radial distribution of 3p-orbital spreads into the curve for 2p orbital and so on. This
is called penetration of orbitals to inner cores. Hence, an electron in an outer orbital is
not fully screened or shielded by the inner electrons from the nuclear charge. It is only
partially screened from the nuclear charge. The extent of penetration decreases from s
to f orbitals, i.e. s > p > d > f. This variation in the extent of penetration greatly
influences the effective nuclear charge and relative energies of orbitals in multi–electron
atoms (discussed later in this chapter).
Plots of Radial Probability Distribution Function , 4nr2 R(r)2
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1.7.5 Plots of Angular Wave Function Θ(θ)Φ(φ)
The angular part of the wave function determines the shape of the electron cloud and
varies depending upon the type of orbitals involved i.e., s, p, d, f and their orientation in
space.
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As already mentioned, the angular wave function 'Θ(θ)Φ(φ)’ depends only on the
quantum numbers l and ml and is independent of the principal quantum number n. It
therefore means that all s orbitals will have same angular wave function irrespective of
the shell they belong to. Similarly, all px orbitals will have same angular wave function
and so on. The plots of angular wave function with θ and φ) are 3-dimensional plots,
which give an idea about the shapes of the orbitals. Mathematical expressions for
angular wave functions of s- and p-orbitals are given in the table 1.8:
Table 1.8 Angular wavefunctions for s- and p-orbitals
l
ml
Orbital
0
0
s
1
0
pz
1
1
px
1
-1
py
Angular wave functions
The angular wave function for s-orbital is independent of angle θ and φ and therefore,
its plot is spherically symmetrical. This is shown in fig. 1.12. There are no angular nodes
in s-orbital. The angular wave function plots for p-orbitals give two tangent spheres. The
plots for px, py and pz orbitals are identical in shape but are oriented along the x, y and z
axes respectively.
Figure 1.12: Angular wave function plot for 1s orbital
Each p-orbital has an angular node (represented by a plane) as shown in fig. 1.13. The
angular wave function plots for d and f orbitals are four-lobed and six-lobed respectively.
There are two nodal planes in each d-orbital except for dz2 , which has a nodal surface
as shown in the fig. 1.14.
It is necessary to keep in mind that in the angular wave function plots, the distance from
the center is proportional to the numerical values of θ and φ in that direction and is not
the actual distance from the center of the nucleus.
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Figure 1.13: Angular wave function plots for 2p-orbitals
Figure 1.14: Angular wave function plots for 3dxy and 3dz2- orbitals showing
the
angular nodes.
Figure 1.13: Angular wave function plots for 2p-orbitals
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showing the
Figure 1.14: Angular wave function plots for 3dxy and 3dz2- orbitals
angular nodes.
1.7.6 Plots of Angular Probability Density: Shapes of Atomic Orbitals
The actual shapes of the orbitals are obtained by plotting square of the angular wave
function as it represents the probability distribution. On squaring, different orbitals
change in different ways. For an s orbital, the squaring causes no change in shape since
the function everywhere is the same; thus another sphere is obtained. For both p and d
orbitals, however, on squaring, the plot tends to become more elongated as shown for
pz in fig. 1.15. Instead of two tangent spheres, we get two dumbbells symmetrical about
z-axis. Similarly, for px and py orbitals, we get dumbbell-shaped lobes symmetrical about
x-axis and y-axis respectively. For d-orbitals, four dumbbell shaped lobes are obtained
except for dz2 orbital, which consists of two lobes along z-axis with a ring of electron
density around the nucleus in xy-plane as shown in fig. 1.20(a).
Figure 1.15: Angular probability density function for pz , px and py orbital.
Check your Progress
How do electrons cross nodes when they can never be present at the nodes?
A mathematical expression, which describes the probability that an electron of an atom
will be at a certain point in space, is called an atomic orbital. A complete picture of an
orbital can be obtained by summing up both the radial and angular distribution functions
together. These pictures of orbitals are called the plots of total probability density.
Ideally, probability distribution extents till infinity but most of it exists very near to the
nucleus. Chemists tend to think of electron clouds and hence, ψ2 probably gives the best
intuitive picture of an orbital. Although, electron density may be shown either by
shading or by contours of equal electron density, only the latter method is quantitatively
accurate. Figure 1.16 shows the pictorial representation of the electron density in a
hydrogen like 3p-orbital.
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Figure 1.16: Countour diagram for a 3pz- orbital.
1.7.7 Plots of Total Probability Density: Shapes of Atomic Orbitals
We can also construct boundary surfaces or boundary surface diagrams such that
they contain a volume within which there is a 99% probability of finding the electron.
Such a boundary surface for an s-orbital (l = 0) has the shape of a spherical shell
centered on the nucleus [fig. 1.17(b)]. For each value of n, there is one s orbital. As n
increases, there are (n-1) concentric spherical shells like the successive layers in an
onion [fig. 1.17(a)].
An alternative approach is to draw charge cloud diagrams. In this approach, the
probability density |ψ|2 is shown as a collection of dots such that the density of dots in
any region represents the electron probability density in that region. The greater the
density of dots at a region, the greater is the probability of an electron to be found
there. These diagrams for s-orbitals are shown in [fig. 1.17(c)]. Charge cloud diagrams
give a better picture of orbitals as compared to boundary surface diagrams because in
the former plots, density of dots is proportional to electron probability density. Hence,
just by looking at these plots, we can predict where the probability of finding an electron
is maximum. Moreover, the radial nodes can be seen in these plots, which are not visible
in boundary surface diagrams.
Figure 1.17: Total probability density plots for s-orbitals (a) crossection of boundary
surface plots showing radial nodes (b) boundary surface diagrams for 1s, 2s and 3s
orbitals (c) charge cloud diagrams for 1s, 2s and 3s-orbitals.
Boundary surface diagrams for the three 2p-orbitals (l = 1) are shown in fig. 1.18(a). In
these diagrams, the nucleus is at the origin. Each p-orbital consists of two lobes on
either side of the plane that passes through the nucleus. The size, shape and energy of
the three orbitals are identical. They differ, however, in the way the lobes are oriented.
Since the lobes may be considered to lie along the x, y or z axis, they are given the
designations 2px, 2py and 2pz. Like s orbitals, p-orbitals increase in size with increase in
the principal quantum number and hence 4p > 3p > 2p. Figure 1.18(b) shows the
charge cloud diagram for 2py orbital.
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Figure 1.18: Plots for 2p-orbitals (a) boundary surface diagrams
for 2px, 2py and 2pz orbitals (c) charge cloud diagram for 2py
orbital.
Boundary surface diagrams for the three 3p-orbitals (l = 1) are shown in fig. 1.19(a).
There are three 3p-orbitals, as there were three 2p-orbitals. The shapes of the 3porbitals are similar to those of the 2p-orbitals, except that the 3p-orbitals have a
spherical node that cuts each lobe into two distinct sections. Figure 1.19(b) shows the
charge cloud diagram for 3pz orbital.
Did you know
How px, py and pz orbitals get their designations?
p-orbitals are those with
.
When
, there are three possible values of ml.
i.e. ml = 0, +1, -1. Correspondingly there are three p-orbitals.
i.e.
ℓ
1
1
1
ml
0
+1
-1
For pz orbital, ml = 0 (i.e. pz = p0), while px and py orbitals are formed from a linear
combination of +1 and -1 values of ml. It is wrong to designate ml = +1 to px orbital
and ml = -1 to py orbital. In fact, no single value of ml can be assigned to px and py
orbitals.
Let us see how linear combination of +1 and -1 values results in the designation of px
and py orbitals.
Angular dependence of p-orbitals
•
For , = 0
ml
Since z = r cosq, this is oriented along z-axis.
Thus it designates pz orbital since cosq gives the projection of a unit vector on the zaxis. Therefore, ml = 0 is only for pz orbital. pz orbital is independent of f and is
therefore symmetric about z-axis.
•
When , two other possible values of are = +1and = -1.
mlmlml
In the above expression, complex quantity
is appearing and for complex quantity,
no physical picture or interpretation can be assigned. Thus, no one value of magnetic
quantum number can be assigned to px or py orbitals.
Since p+1 (for ml =1) and p-1 (for ml = -1) are degenerate and include complex
quantities, new orbitals without complex quantities can be generated from their linear
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combinations as shown in the following.
Linear combinations are
and
.
•
Substituting the expressions of p+1 and p-1, we get,
---(1)
Since,
and
Therefore,
and
Substituting this in equation (1), we get,
As x = r sinq cos f, the above expression designates px orbital.
•
Substituting the expressions of p+1 and p-1, we get,
As y = r sinq sin f, this represents py orbital.
Similarly, in case of d-orbitals (l = 2), there are five possible values of ml, i.e. ml =
0, +1, -1, +2, -2. Correspondingly there are five d-orbitals. The
orbital has ml
= 0, while dxz and dyz are formed from the linear combinations of 3d+1 and 3d-1 (ml =
+1, -1) and dxy and
(ml = +2, -2), respectively.
are formed from the linear combinations of 3d+2 and 3d-2
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boundary
Figure: 1.19: Total probability density plots for 3p orbitals (a)
surface diagrams for 3px, 3py and 3pz orbitals (c) charge cloud Diagram for
3pz orbital.
Boundary surface diagrams of the d-orbitals (l=2) are shown in fig. 1.20(a). There are
five d orbitals which are designated as dxy, dyz , dxz , dx2 -y2 and dz2. The shape of
3dz2 orbital is different from that of others but all five 3d orbital are equivalent in
energy. The d orbitals for which n is greater than 3 (4d, 5d……) have similar shapes.
Figure 1.20(b) shows the charge cloud diagram for 3dxz , 3dx2-y2 and 3dz2 orbitals.
Figure 1.20: Plots for five d-orbitals (a) boundary surface diagrams for 3dxz , 3dyz ,
3dxy
3dx2-y2 and 3dz2 orbitals (b) charge cloud diagram for 3dxz , 3dx2-y2 and
3dz2 orbitals.
An important feature to be noted here is that all angular nodes except for dz2 orbital are
represented as two-dimensional planes or surfaces while all radial nodes are represented
as spheres. The general formula for calculating the number of nodes is:
Total number of nodes (radial + angular) in an orbital = (n – 1)
Number of radial nodes = (n – l – 1)
Number of angular nodes = l
For instance, in a 4p-orbital, there are in all three nodes out of which two are radial
nodes and one is an angular node.
Check your progress
Explain why only s-orbital is spherically symmetrical.
1.7.8 Sign of Wave Function
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The sign of the wave function can be obtained from its angular part. The wave function,
ψ , can be either positive or negative depending on the values of θ and Φ. The s-orbital
being independent of angle θ and Φ is positive all through. The p-orbitals, on the other
hand, have one positive and one negative lobe as shown in the fig. 1.18(a). Angular
dependence of d-orbitals gives opposite lobes with identical signs as shown in fig.
1.20(a). In dz2 orbitals, two opposite lobes have the positive sign and the ring in the
xy-plane caries the negative sign. These positive and negative sign does not indicate the
electron density in any way; they only represents the sign of the wave function. The true
significance of the sign is reflected only when orbitals combine to form bonds.
Common Misconce
The positive and negative signs in the orbitals
s-orbital
p-orbital
Attention must be paid to a confusing practice. In above figures, it is noted tha
and minus sign appear. The above figures refer to the orbital pictures which giv
probability of finding the electron and thus must be positive everywhere. The sig
(-) refer to the sign of the original wave function, y, in these region of space. T
must also not be confused with charges.
1.8 Relative Energies of Orbitals
For hydrogen like systems (i.e., systems with only one electron), the energies of orbitals
depend only on the quantum number, n and therefore as n increases the energy of the
orbital increases. Thus the relative energies of the orbitals of hydrogen like system have
the following order:
On the other hand, for multi-electron systems (i.e. atoms/molecules with more than one
electron), the relative energies of orbitals not only depends on the principal quantum
number, n, but also on the azimuthal quantum number, l. The following factors are
responsible for the relative energies of orbitals for multi-electron systems:
(i)
(ii)
Effective nuclear charge (depends on ‘n’ – distance from the nucleus)
Penetration effect (depends on ‘l’ – shape of the orbitals)
1.8.1 Effective Nuclear Charge
Effective nuclear charge may be defined as the actual charge felt by the electron. The
electron present in the innermost orbital (i.e., 1s) experiences the full charge of the
nucleus. However, the charge experienced by the electron in the outer orbitals is
influenced by the repulsions from the inner core electrons. As a result of these
repulsions, the net charge experienced by an electron is always less than the actual
nuclear charge. The reduced effect of the nuclear charge by the inner electrons is called
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shielding. Consider for example the lithium atom, which has the ground state electronic
configuration as 1s2 2s1. Lithium has three protons, which gives the nucleus a charge of
+3, but the full attractive force of this charge on the 2s electron is partially offset by
electron – electron repulsion from the inner 1s2 electrons. Consequently, the 1s
electrons shield the outer 2s electron from the nucleus.
The effective nuclear charge (Zeff), experienced by an electron can be expressed as:
Zeff = (Z- σ)
where Z is the actual nuclear charge (atomic number of the element) and σ (sigma) is
called shielding constant. The shielding constant is the sum of shielding contributions
from all the inner electrons and the electrons in the same shell. About 520 kJ of energy
is required to remove the first electron from 1 mol of Li atoms and 7298 kJ of energy to
remove the second electron from 1 mol of Li+ ions. Thus, much more amount of energy
is required to remove second electron as the electron feels the greater effective nuclear
charge. Effective nuclear charge depends on distance of the electron from the nucleus.
The greater the distance, the lesser is Zeff.
1.8.2 Penetration Effect
As already mentioned, the radial probability distribution curves show that the 2s-orbital
also has a maximum at or around the maximum for 1s orbital. Thus the electrons in the
2s-orbital penetrates the lower 1s orbital as shown in fig. 1.21(a). This means that
although 2s electron is shielded by the 1s core yet the 2s electron penetrates it to some
extent and experiences somewhat higher nuclear charge. This penetration exposes the
electrons to more influence of the nucleus and causes them to be more tightly bound
and thus lowering their associated energy states. For example, in a sodium atom, 3s
electron penetrates the inner core more than the 3p and is significantly lower in energy.
This is shown in fig. 1.21(b).
Figure 1.21: (a) Penetration of a 2s-orbital to the inner 1s core (b)
Penetration of 3s and 3p-orbitals to the inner 1s core in a sodium atom.
Check your progress
Explain why only s-orbital is spherically symmetrical.
Therefore, the order of energies depends on the degree of penetration, lesser is the
energy associated with that orbital. As discussed earlier, the degree of penetration
decreases in the order :
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s>p>d>f
Hence, the order of energy is 2s < 2p. similar consideration lead to 3s < 3p < 3d and 4s
< 4p < 4d < 4f. this means that within a given main energy level (n). the energy states
do not remain degenerate but increase in the order ns < np < nd < nf.
1.9 Rules For Filling Electrons in Various Orbitals
In multielectron atoms, the following rules or principles govern the filling of electrons
into various orbitals.
(i) Pauli’s exclusion principle
(ii) Hund’s rule of maximum multiplicity
(iii) Aufbau principle
These are discussed below.
Pauli exclusion principle was established by an Austrian scientist Wolfgang Pauli in 1925.
According to this principle, each electron is described by a unique set of four quantum
numbers. In other words, no two electrons in an atom can have the same set of four
quantum numbers. Thus, even if two electrons have the same value for n, l and ml, they
must possess different spins, i.e., different values of ms, which are either +½ or -½ .
This is illustrated in Fig. 1.22.
Figure 1.22: Filling of electrons in orbitals according to Pauli exclusion principle
Thus, 1s-orbital (n = 1, l = 0, ml = 0) can contain a maximum of two electrons having
opposite spins (+½ and –½). With two electrons of opposite spins, the s-orbital is said
to be ‘closed’. Using similar considerations, a p-subshell can accommodate a maximum
of six electrons, two in each px, py and pz orbitals. A d-orbital can accommodate ten
electrons and so on as shown in Fig. 1.23. Pauli exclusion principle has no fundamental
explanation but is an experimental fact.
Figure 1.23: Accommodation of electrons in various orbitals according to
the Pauli exclusion principle
1.9.2 Hund’s Rule of Maximum Multiplicity
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This rule deals with the filling of electrons into the orbitals belonging to the same
subshell (i.e., orbitals of equal energy, called degenerate orbitals). It states that pairing
of electrons in the orbitals belonging to the same subshell does not take place
until each orbital belonging to that subshell is singly occupied (i.e., has got one
electron each). This is justified as the electrons in two different orbitals of same subshell
experience lesser inter-electronic repulsions as compared to the electrons in the same
orbital. Thus, it leads to lower energy state. This is illustrated in Fig. 1.24.
Figure 1.24: Filling of electrons in orbitals according to Hund’s rule
Since there are three p, five d and seven f orbitals, therefore the pairing of electrons will
start in the p, d and f orbitals with the entry of 4th, 6th, and 8th electron, respectively.
The unpaired electrons present in the orbitals of same subshell should have
parallel spins.
The preference of the electrons for parallel spins within a subshell is due to their
stabilization by exchange energy(discussed later). It can be more appropriately
understood
in
terms
spin
multiplicity,
which
is
given
by
the
formula:
Spin multiplicity = (2S +1)
where S is sum of spin quantum numbers of all the electrons (ΣSi).
Multiplicity is originally derived from the number of lines shown in a spectrum. It is
related to the number of unpaired electrons. Hund’s rule states that the ground state of
an atom will be the one having the greatest spin multiplicity. Spin multiplicity
corresponding to the number of unpaired electrons is given in Table 1.9.
Table 1.9 Spin multiplicity
Number of Unpaired
Electrons
Resultant
Spin, S
Multiplicity,
(2S+1)
Name of the State
0
0
1
Singlet
1
1/2
2
Doublet
2
1
3
Triplet
3
3/2
4
Quartet
4
2
5
Quintet
Consider a p2 state
In the singlet state, the electrons are close together, and consequently, there will be
high interelectronic repulsion energy. In the triplet state this will be smaller owing to the
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greater distance of the electrons. Therefore, triplet state with electrons of the same spin
will be more stable (possess lower energy).
1.9.3 Aufbau Principle
This principle is based on the relative energies of the orbitals as determined by the
effective nuclear charge and the penetration effect. The word ‘Aufbau’ in German, means
‘building up’. The electronic configuration of atoms is built up by adding the electrons in
various orbitals in a stepwise manner. In the ground state of the atoms, the orbitals are
filled in order of their increasing energies. This means that electrons first occupy the
lowest energy orbital available to them and enter into higher energy orbitals only after
the lower energy orbitals are completely filled.
The increasing order of energies of the orbitals and hence the order in which the orbitals
are filled is as follows:
1s < 2s < 2p < 3s < 3p < 4s ≈ 3d < 4p < 5s ≈ 4d < 5p < 6s ≈ 4f ≈ 5d < 6p < 7s…
As the energy of the orbitals depends on the quantum number n and l, the order of
filling the orbitals proceeds according to (n + l) rules. According to this rule, the lower
the value of (n + l) the lower is the energy of an orbital. For example, 3d (n + l = 3 + 2
= 5) is filled after 4s (n + l = 4 + 0 = 4). If two orbitals have the same (n + l) value,
the orbital with lower value of n has the lower energy. For example, 2p (n + l = 2 + 1 =
3) is filled before 3s (n + l = 3 + 0 = 3).
This order of filling the orbitals may be remembered by using the method given in Fig.
1.25. Starting from the top, the direction of the arrows gives the order of filling of
orbitals.
Figure 1.25: Order of filling of orbitals
Aufbau Principale
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1.9.4 Stability of Completely Filled and Half-Filled Subshells
The ground state electronic configuration of an element always corresponds to the state
of the lowest total electronic energy. The electronic configurations of most of the atoms
follow the basic rules as you have studied earlier. However, in certain elements such as
Cu or Cr, where the two subshells (4s and 3d) differ slightly in their energies, an
electron shifts from a subshell of lower energy (4s) to a subshell of higher energy (3d),
provided such a shift results in either completely filled or half-filled subshell. The valence
electron configurations of Cr and Cu, therefore, are 3d54s1 and 3d104s1 respectively and
not 4s23d4 and 4s23d9. Similarly, molybdenum (Mo) and silver (Ag) have electronic
configuration 4d55s1 and 4d105s1 respectively. Thus, generally only one electron jumps
from lower energy orbital to higher energy orbital, e.g., from 4s to 3d. However, in case
of palladium, two electrons are involved (the only case with a difference). The reason for
the tendency of the subshells to be completely filled or exactly half-filled is that it leads
to greater stability.
This stabilization is due to the following two factors:
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Atomic Structure
1. : It is well known that symmetry leads to stability. The completely filled or halffilled subshells has symmetrical distribution of electrons in them and are
therefore more stable. For example, Symmetrical distribution of electrons
Cr – atomic number 24
2. : Two electrons with the same charge tend to keep apart and two electrons with
different
spin
tend
to
come
together.
For
example,
consider
a
configuration,Exchange energyp2
Charge correlation energy (e2/r) dictates that a lower energy situation exists if the two p
electrons have different angular functions (i.e., different ml values). Spin correlation
affords a lower energy for like spins (i.e., same ms values). The chemical significance of
spin correlation of electron motions is that, since electrons are charged and repel one
another, the repulsion is less by an amount called the ‘exchange energy’ for electrons of
like spin than for electrons of different spin.
The exchange energy, Eex, is proportional to the number of pairs of electrons of the
same spin that can be arranged from ‘n’ electrons with parallel spins. The pair in this
context means simply a group of two electrons with same spin quantum number.
This exchange energy provides the basis of Hund’s rule that electrons in the same orbital
have their spins parallel as far as possible. For example,
This difference of 9k in exchange energy is sufficient to compensate for the loss in
orbital energy in moving the electron from 4s-orbital to a 3d-orbital. Hence, half-filled
subshell is more stable. Similarly, stability for completely filled-orbitals can be proved.
Check your progress
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Atomic Structure
Explain the extra stability of fully filled and half-filled subshells.
1.10 Electronic Configurations of Atoms
The distribution of electrons into orbitals of an atom is called its electronic
configuration. If one keeps in mind the basic rules, which govern the filling of different
atomic orbitals, the electronic configurations of first few atoms can be written very
easily. The order may be continued for the higher elements also with minor anomalies
arising in few cases. There are two ways to write the electronic configuration of
elements. One is subshell notation which involves writing the electronic configurations of
the atoms in terms of the subshells being filled. For example, the subshell notation for a
neutral fluorine atom is:
F (Z = 9): 1s2 2s2 2p5
The other way to represent electronic configurations is orbital diagrams. An orbital
diagram gives a little more information about the properties of individual atoms. The
way to draw orbital diagram is to have squares to represent each of the individual
orbitals. The difference between the two electrons in each individual orbital is the
electron spin or the magnetic spin quantum number. They are differentiated by drawing
one with an arrow up and one with an arrow down. For example, the orbital notation for
a fluorine atom is:
The Table 1.10 below gives the ground state electronic configuration of various elements
and Fig. 1.26 shows these configurations in the periodic table.
Table 1.10 Electronic configuration of elements in the ground state
Atomic number
Element
Electronic configuration
1
H
1s1
2
He
1s2
3
Li
[He]2 2s1
4
Be
[He]2 2s2
5
B
[He]2 2s2 2p1
6
C
[He]2 2s2 2p2
7
N
[He]2 2s2 2p3
Table 1.10
(Continued)
Atomic number
Element
Electronic configuration
8
O
[He]2 2s2 2p4
9
F
[He]2 2s2 2p5
10
Ne
[He]2 2s2 2p6
or 1s2, 2s2, 2p6
11
Na
[Ne]10 3s1
12
Mg
[Ne]10 3s2
13
Al
[Ne]10 3s2 3p1
14
Si
[Ne]10 3s2 3p2
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Atomic Structure
Table 1.10
15
P
[Ne]10 3s2 3p3
16
S
[Ne]10 3s2 3p4
17
Cl
[Ne]10 3s2 3p5
18
Ar
[Ne]10 3s2 3p6
or 1s , 2s2, 2p6, 3s2 ,3p6
19
K
[Ar]18 4s1
20
Ca
[Ar]18 4s2
21
Sc
[Ar]18 3d1 4s2
2
(Continued)
Atomic number
Element
Electronic configuration
22
Ti
[Ar]18 3d2 4s2
23
V
[Ar]18 3d3 4s2
*24
Cr
[Ar]18 3d5 4s1
25
Mn
[Ar]18 3d5 4s2
26
Fe
[Ar]18 3d6 4s2
27
Co
[Ar]18 3d7 4s2
28
Ni
[Ar]18 3d8 4s2
*29
Cu
[Ar]18 3d10 4s1
30
Zn
[Ar]18 3d10 4s2
31
Ga
[Ar]18 3d10 4s2 4p1
32
Ge
[Ar]18 3d10 4s2 4p2
33
As
[Ar]18 3d10 4s2 4p3
34
Se
[Ar]18 3d10 4s2 4p4
35
Br
[Ar]18 3d10 4s2 4p5
*Anomalous Configurations
Table 1.10
(Continued)
Atomic number
Element
Electronic configuration
[Ar]18 3d10 4s2 4p6
2
or 1s ,2s2,2p6,3s2,3p6,3d10,4s2,4p6
36
Kr
37
Rb
[Kr]36 5s1
38
Sr
[Kr]36 5s2
39
Y
[Kr]36 4d1 5s2
40
Zr
[Kr]36 4d2 5s2
*41
Nb
[Kr]36 4d4 5s1
*42
Mo
[Kr]36 4d5 5s1
43
Tc
[Kr]36 4d5 5s2
*44
Ru
[Kr]36 4d7 5s1
*45
Rh
[Kr]36 4d8 5s1
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Atomic Structure
*46
Pd
[Kr]36 4d10 5s0
*47
Ag
[Kr]36 4d10 5s1
48
Cd
[Kr]36 4d10 5s2
49
In
[Kr]36 4d10 5s2 5p1
50
Sn
[Kr]36 4d10 5s2 5p2
51
Sb
[Kr]36 4d10 5s2 5p3
*Anomalous Configurations
Table 1.10
(Continued)
Atomic number
Element
Electronic configuration
52
Te
[Kr]36 4d10 5s2 5p4
53
I
[Kr]36 4d10 5s2 5p5
54
Xe
[Kr]36 4d10 5s2 5p5
or 1s ,2s ,2p ,3s2,3p6,3d10,4s2,4p6,4d10,5s2,5p6
55
Cs
[Xe]54 6s1
56
Ba
[Xe]54 6s2
*57
La
[Xe]54 5d1 6s2
*58
Ce
[Xe]54 4f2 5d0 6s2
59
Pr
[Xe]54 4f3 5d0 6s2
60
Nd
[Xe]54 4f4 6s2
61
Pm
[Xe]54 4f5 6s2
62
Sm
[Xe]54 4f6 6s2
63
Eu
[Xe]54 4f7 6s2
Gd
[Xe]54 4f7 5d1 6s2
65
Tb
[Xe]54 4f9 6s2
66
Dy
[Xe]54 4f10 6s2
67
Ho
[Xe]54 4f11 6s2
*64
2
2
6
*Anomalous Configurations
Table 1.10
(Continued)
Atomic number
Element
Electronic configuration
68
Er
[Xe]54 4f12 6s2
69
Tm
[Xe]54 4f13 6s2
70
Yb
[Xe]54 4f14 6s2
71
Lu
[Xe]54 4f14 5d1 6s2
72
Hf
[Xe]54 4f14 5d2 6s2
73
Ta
[Xe]54 4f14 5d3 6s2
74
W
[Xe]54 4f14 5d4 6s2
75
Re
[Xe]54 4f14 5d5 6s2
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76
Os
[Xe]54 4f14 5d6 6s2
*77
Ir
[Xe]54 4f14 5d7 6s2
*78
Pt
[Xe]54 4f14 5d9 6s1
*79
Au
[Xe]54 4f14 5d10 6s1
80
Hg
[Xe]54 4f14 5d10 6s2
81
Tl
[Xe]54 4f14 5d10 6s2 6p1
82
Pb
[Xe]54 4f14 5d10 6s2 6p2
83
Bi
[Xe]54 4f14 5d10 6s2 6p3
*Anomalous Configurations
Table 1.10
(Continued)
Atomic number
Element
Electronic configuration
84
Po
[Xe]54 4f14 5d10 6s2 6p4
85
At
[Xe]54 4f14 5d10 6s2 6p5
86
Rn
[Xe]544f145d106s26p6
or 1s ,2s ,2p6,3s2,3p6,3d10,4s2,4p6, 4d10, 5s2,
5p6, 4f14, 5d10, 6s2, 6p6
87
Fr
[Rn]86 7s1
88
Ra
[Rn]86 7s2
89
Ac
[Rn]86 6d1 7s2
90
Th
[Rn]86 5f2 6d10 7s2
91
Pa
[Rn]86 5f3 6d17s2
92
U
[Rn]86 5f2 6d1 7s2
93
Np
[Rn]86 5f4 6d1 7s2
94
Pu
[Rn]86 5f6 7s2
95
Am
[Rn]86 5f7 7s2
96
Cm
[Rn]86 5f7 6d1 7s2
97
Bk
[Rn]86 5f9 7s2
98
Cf
[Rn]86 5f10 7s2
99
Es
[Rn]86 5f11 7s2
Table 1.10
2
2
(Continued)
Atomic number
Element
Electronic configuration
100
Fm
[Rn]86 5f12 7s2
101
Md
[Rn]86 5f13 7s2
102
No
[Rn]86 5f14 7s2
103
Lr
[Rn]86 5f14 6d1 7s2
Rf
(Rutherfordium)
Db
(Dubnium)
[Rn]86 5f14 6d2 7s2
104
105
106
Sg
[Rn]86 5f14 6d3 7s2
[Rn]86 5f14 6d3 7s2
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(Seaborgium)
107
108
109
110
Bh
(Bohrium)
Hs
(Hassium)
Mt
(Meitnerium)
Uon
(Ununnium)
[Rn]86 5f14 6d4 7s2
[Rn]86 5f14 6d5 7s2
[Rn]86 5f14 6d6 7s2
[Rn]86 5f14 6d8 7s2
1.10 Electronic Configurations of Atoms 1
The hydrogen atom has only one electron, which goes in the orbital with lowest energy,
namely 1s. The electronic configuration of the hydrogen atom is 1s1 ,meaning it has one
electron in the 1s orbital. The second electron in helium (He) can also occupy the 1s
orbital. Its configuration is therefore, 1s2. The third electron of lithium (Li) is not allowed
in the 1s-orbital because of the Pauli exclusion principle. It therefore, takes the next
available choice, namely the 2s orbital. The electronic configuration of lithium is 1s2 2s1.
The 2s-orbital can accommodate one more electron. The configuration of the beryllium
(Be) atom is, therefore, 1s2 2s2.
Figure 1.26: Electronic configuration of elements in the periodic table
In the next six elements-boron (B, 1s2, 2s2 2p1), carbon (C, 1s2 2s2 2p2), nitrogen (N,
1s2 2s2 2p3), oxygen (O, 1s2 2s2 2p4), fluorine (F, 1s2 2s2 2p5) and neon (Ne, 1s2 2s2 2p6
) the 2p-orbitals get progressively filled. This process is completed with the neon atom.
The electronic configurations of the elements from sodium (Na, 1s2 2s2 2p6 3s1) to argon
(Ar, 1s2 2s2 2p6 3s2 3p2) follow exactly the same pattern as the elements from lithium to
neon with the difference that the 3s and 3p-orbitals are getting filled now. In potassium
(K) and calcium (Ca), the 4s orbital, being lower in energy than the 3d-orbitals, is filled
first and is occupied by one and two electrons, respectively.
A new feature shows up beginning with scandium (Sc). The 3d-orbital, being lower
in energy than 4p, is filled first. Consequently, in the next ten elements, scandium (Sc,
3d1 4s2), titanium (Ti, 3d2 4s2), vanadium (V, 3d3 4s2), chromium (Cr, 3d5 4s1),
manganese (Mn, 3d5 4s2), iron (Fe, 3d6 4s2), cobalt(Co, 3d7 4s2), nickel (Ni, 3d8 4s2),
copper (Cu, 3d10 4s1) and zinc (Zn, 3d10 4s2)—the five 3d-orbitals are progressively
occupied. We may be puzzled by the fact that chromium and copper have five and ten
electrons in 3d-orbitals rather than four and nine as their position would have indicated
(with one electron in the 4s orbital). The reason is that fully filled orbitals and half-filled
orbitals have extra stability (i.e., lower energy) as discussed below. Thus, chromium and
copper adopt the d5 and d10 configurations in preference to the d4 and d9 configurations.
Similarly, from yttrium (Y, 4d1 5s2) to cadmium (Cd, 4d10 5s2), 4d-orbitals are
progressively filled. However, in the sixth period, 4f orbitals start appearing. Between
lanthanum (La, 5d1 6s2) and hafnium (Hf, 5d2 6s2), the 4f orbitals are filled and these
elements are called lanthanides. These lanthanides are listed at the bottom of the
periodic table. Similarly, in the seventh period, 5f orbitals are being filled and these
elements are called actinides. They occur between actinium (Ac, 6d1 7s2) and
rutherfordium (Rf, 6d2, 7s2) and are also placed at the bottom of the periodic table.
Some important points that should be remembered while writing the electronic
configurations of atoms are:
(i) To avoid writing of the electronic configurations in a lengthy way, usually the
symbols [He]2, [Ne]10, [Ar]18 etc., are used as the first part of the configuration.
Such a symbol stands for the electronic configuration of that inert gas and is usually
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called the core of the inert gas.
(ii) Although the orbitals of lower energy are filled first but the electronic configuration
are written not in the order in which the orbitals were filled but in the order of
principal quantum numbers. For example, Scandium [Ar]18 3d14s2. In this case,
although 4s is filled before 3d yet it is written after 3d. This is because the principle
quantum number for s-orbital (n = 4) is greater than for d-orbital (n = 3).
(iii) Unless otherwise mentioned, electronic configuration always means the electronic
configuration in the ground state of the element.
(iv) One should always remember that while writing the electronic configuration of an ion
(cation or anion), first the electronic configuration of basic atom is written and then
the electron is added or removed from the system otherwise there is always a
chance of error. For example, for Cr3+, first, Cr 3d5 4s1 then Cr3+ 3d3 4s0, for Cu2+,
first Cu 3d10 4s1 then Cu2+ 3d9 4s0.
One may ask what is the utility of knowing the electron configuration. The modern
approach to chemistry, in fact, depends almost entirely on electron distributions to
understand and explain chemical behaviour.
Check your progress
Explain the extra stability of fully filled and half-filled subshells.
1.10.1 Anomalies in the Electronic Configurations as Predicted by Aufbau Principle
Although the Aufbau principle and the order of filling of orbitals given previously may be
used reliably to determine electronic configurations, yet for elements with high atomic
numbers, some deviations are observed.
1. As discussed above, the anomalous configurations of certain elements given in the
Table 1.11 are due to the extra stabilities associated with half-filled and completely filled
electronic configurations.
Table 1.11 Anomalous configurations of certain elements
S. No. Atomic number Element Expected configuration Actual configuration
1.
24
Cr
3d4 4s2
3d5 4s1
2.
29
Cu
3d9 4s2
3d10 4s1
3.
42
Mo
4d4 5s2
4d5 5s1
4.
47
Ag
3d9 4s2
3d10 4s1
5.
79
Au
5d9 6s2
5d10 6s1
(ii)
2. The predicted electronic configuration of palladium (Pd, Z = 46) is 4d8 5s2, however,
the observed configuration is 4d10 5s0. This is the only case of its kind where two
electrons are misplaced from their expected subshells.
3. The predicted electronic configuration of niobium (Nb, Z = 41) is 4d3 5s2, however,
the observed configuration is 4d4 5s1. In the same period, ruthenium (Ru, Z = 44) and
rhodium (Rh, Z = 45) has the electronic configuration 4d7 5s1 and 4d8 5s1 respectively
instead of the expected 4d6 5s2 and 4d7 5s2. This shows that the preference for half-filled
and fully filled configurations is not the only determining factor but other factors such as
nucleus–electron attraction and electron-electron repulsion also plays some important
role in determining the actual configuration.
4. In the sixth period, 4f and 5d subshells are exceedingly close in energy. At lanthanum
(Z = 57), the last electron does not go to 4f as predicted in Aufbau order but is added to
5d subshell to give it a configuration 4d105s25p65d16s2. However, the next electron for
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cerium (Z = 58) is added to the 4f subshell and the previous electron in 5d in lanthanum
is also shifted to 4f to give it a configuration 4d105s25p64f25d06s2.
5. In the third transition series, Pt (Z = 78) has an observed configuration of
5d 96s1 as compared to the expected configuration of 5d86s2. There is a jump of one
electron from 6s-orbital to 5d-orbital although this jump does not give a completely
filled 5d-orbital.
Despite having certain limitations, Aufbau order of filling the orbitals is quite reliable in
predicting the electronic configurations and the chemical behaviour of the elements.
Common Miscoception
Limitations of Aufbau Principle
While explaining anomalous electronic configuration of chromium (Cr) and copper (Cu),
it is stated that half filled and fully filled orbitals have extra stability, which of course is
true but it must not be over emphasized. For example,
a) General electronic configuration of halogens is ns2 np5 and not ns1np6. Similarly,
ns2np4 and not ns0np6 for halogens; ns2np2 for carbon group and not ns1np3, as one
may predict by over extending the idea of half filled and fully filled orbital associated
with extra stabilization concept.
b) The electronic configuration of tungsten (W), a member of chromium group in
periodic table, is [Xe]6s25d4 and not [Xe]6s15d5 by analogy to chromium electronic
configuration.
c) The electronic configurations of group 10 elements of periodic table (nickel group) are
excellent examples of over emphasis of either Aufbau rule or half filled and fully filled
orbitals extra stability concept.
Element of Group 10 Electronic Configuration
Remarks
Ni
[Ar]4s23d8
As per Aufbau rule
Pd
As per fully filled orbitals
[Kr]5s04d10
extra stability.
Pt
Neither of the above
[Xe]6s15d9
two explanations!
Then how do we understand the electronic configuration of elements (especially heavier
transition and inner transition elements). As per the Aufbau principle, the ordering of
orbitals (sub-shells) energy is determined by the (n + l) rule and the value of n. In
every case in which exception to the Aufbau principle occurs, it is noted that energy
levels involved are exceedingly close together and factors such as exchange energy,
symmetry, pairing energy and electronic repulsions may invert the energy levels at a
particular atomic number (or more precisely for a particular value of effective nuclear
charge, Z*).
This explains why main group elements do not have exceptions to Aufbau principle, as
ns and np orbitals (sub shells) have energy difference high enough which do not allow
change in order of their filling due to other factors.
The electronic configuration adopted by an atom is one with the lowest energy state,
which therefore is determined on the basis of combined effect of all factors and not on
any individual factor. In case of Pt (platinum) the ground state electronic configuration,
i.e. the one with minimum energy has electronic configuration [Xe]6s15d9 instead of
[Xe]6s25d8 as determined by Aufbau principle or [Xe]6s05d10 as determined by fully filled
orbital considerations. It is the combined effect of all factors the (n + l) value, exchange
energy, pairing energy, symmetry and electronic repulsions, which determined the
ground state electronic configuration at a particular atomic number.
Did you know
Some of the atoms are found to possess exceptional electronic configurations, which are
not in accordance with the Aufbau filling of orbitals, but still the atoms are found to be
stable. The exceptional stability of such electronic configurations can be explained on the
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basis of formation of exchange pairs. More the number of exchange pairs of electrons
having the same spin in a particular electronic configuration, higher the exchange
energy released from the system and hence higher is the stability of the system.
One may also note that the formation of exchange pairs would be possible only if
the
orbitals under consideration have comparable energies because if their energy
levels are widely separated then the formation of exchange pairs would not be feasible.
For example, Cr (Z = 24) prefers to exist in 3d5 4s1 (exceptional) electronic state rather
than in the 3d4 4s2 state (which is in accordance with the Aufbau filling). This is because
in the 3d5 4s1 configuration of Cr, 15 exchange pairs of electrons are possible, while in
the 3d4 4s2 state, only 10 such pairs are possible:In 3d5 4s1 state, number of electrons with like spins, n=6. Thus, number of exchange
pairs is given by nC2 = 6C2 = 15.
While in the 3d4 4s2 state, number of electrons with like spins, n=5. Thus, number of
exchange pairs is given by nC2 = 5C2 = 10.
As a result, more exchange energy is liberated in the former case leading to extra
stability. Similarly, Cu (Z = 29) prefers to exist in the 3d10 4s1 state rather than in the
3d9 4s2 state. Thus, the rules for filling of orbitals cannot be generalized for all the
elements of the periodic table as rigid guidelines.
Summary
Atoms are the building blocks of elements. They are the smallest parts of an element
that chemically react like a bulk sample of the element. Towards the end of the
nineteenth century it was proved experimentally that atoms are divisible and consist of
three fundamental particles: electrons, protons and neutrons. The discovery of subatomic particles led to the proposal of various atomic models to explain the structure of
atom.
Quantum mechanical model of the structure of atom is based on the principles of waveparticle duality of matter and Heisenberg uncertainty principle. According to the principle
of wave-particle duality of matter, waves known as matter waves or de Broglie waves
are associated with moving particles. The wavelength of such waves associated with a
particle of mass m moving with velocity v is given by the de Broglie relation λ = h / mv.
Heisenberg uncertainty principle is a consequence of the dual behaviour of matter and
radiation. According to this principle, it is impossible to determine simultaneously both
the position and momentum of a moving particle with certainty.
Quantum mechanics is the theoretical science of microscopic matter. It specifies the
laws of motion that microscopic particles obey. Its central equation is Schrödinger
equation. The solution of the Schrödinger equation of the hydrogen atom gives the
possible energy states of the electron and the corresponding wave functions, also known
as atomic orbitals. These atomic orbitals can also be used with a reasonable accuracy in
multielectron atoms. An atomic orbital is a one electron wave function (ψ) that defines
the distribution of electron density (ψ2) in space. Orbitals can be represented by charge
cloud diagrams or boundary surface diagrams.
The quantized energy states and corresponding wave functions which are characterized
by a set of three quantum numbers (principal quantum number, n, atimuthal quantum
number, l, and magnetic quantum number, ml) arise as a natural consequence in the
solution of the Schrödinger equation.
For hydrogen and hydrogen like systems (such as He+, Li2+ etc.) all the orbitals within a
given shell have same energy while the energy of the orbitals in a multi–electron atom
depends upon the values of n and l. The lower the value of (n + l) for an orbital, the
lower is its energy. If two orbitals have the same (n + l) value, the orbital with lower
value of n has the lower energy. In an atom many such orbitals are possible and
electrons are filled in those orbitals in order of increasing energy in accordance with
Pauli exclusion principle (no two electrons in an atom can have the same set of four
quantum numbers) and Hund’s rule of maximum multiplicity (pairing of electrons in the
orbitals belonging to the same subshell does not take place until each orbital belonging
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to that subshell has got one electron each, i.e., is singly occupied). This forms the basis
of the electronic structure of atoms.
Exercises
1. What is the wavelength of an electron if its kinetic energy is 1.375× 10-25 J?
(Mass of electron = 9.1 × 10-31 kg, h = 6.63 × 10-34 J s).
2. A beam of helium atoms moves with a velocity of 2.0 × 103 m s-1 . Find the
wavelength of particles constituting the beam. (h = 6.63 × 10-34 J s).
3. Calculate the uncertainty in the position of an electron if uncertainty in its
velocity is
(i) 0.001%
(ii) zero (mass of electron = 9.1 × 10-31 kg, velocity of electron =
300 m s-1 ). (Ans. (i) 1.92 ×10-2 m (ii) ∞)
1. Calculate the uncertainty in the velocity of a wagon of mass 2000 kg whose
position is known to an accuracy of ±10 m.
2. Why cannot we overcome the uncertainty predicted by Heisenberg’s principle by
building more precise devices to reduce the error in measurement below the h/4π
limit?
3. Which orbital in each of the following pairs is lower in energy in a many electron
atom?
(i)
(ii)
(iii)
(iv)
2s, 2p
3p, 3d
3s, 4s
4d, 5f
Explain the meaning of the symbol 4d6.
Write electronic configuration of Cu (Z = 29) and Cr (Z = 24).
Explain why 4s electrons are removed before 3d electrons in transition metals?
What are the n, l, m values for 2px and 3pz electrons?
An electron is in a 4f orbital. What possible values for the quantum numbers, n, l,
m and s can it have?
6. Which of the following orbitals are not possible: 1p, 2p, 2s, 3f, and 4d.
7. If n is equal to 3, what are the values of quantum numbers l and m?
8. Write the correct orbital notations for each of the following sets of quantum
numbers.
1.
2.
3.
4.
5.
(i) n = 1, l = 0, m = 0
(ii) n = 2, l = 1, m = -1
(iii) n = 3, l = 2, m = +1
1. Give an expression for time independent Schrödinger wave equation in three
dimensions and explain the various terms in it.
2. Explain the significance of ψ and ψ2.
3. What are normal and orthogonal wave functions.
4. What is the difference between radial probability function and probability?
5. What do you understand by
(i) Radial probability density, R2 and
r2 R2 ? How do they vary with r
(ii) Radial probability function 4
for 1s, 2s and 2p atomic orbitals of hydrogen atom?
1. Draw radial probability distribution curves for n=3, l=0, and n=4, l=2.
2. Draw the shapes of d-orbitals indicating the sign of wave function.
3. What do you understand by a node? How many nodes are present in 3s and 4porbitals?
4. Draw the shapes (boundary surfaces) of the following orbitals:
Institute of Lifelong Learning, University of Delhi
Atomic Structure
5.
(i)
2py
(ii)
3dz2
(iii)
3dx2-y2
1. For an element with atomic number 20, which is the last or highest occupied
subshell of atomic orbitals? 1s, 2s, 2p, 3s, 3p, 3d, 4s, or 5s.
2. Which of the following two electronic configurations is more stable?
(i)
[Ar] 3d5 4s1
(ii)
[Ar] 3d4 4s2
1. The electronic configuration of palladium, Pd (Z = 46), is:
(i)
[Ar] 4d8 5s2
(ii)
[Ar] 4i10 5s0
1. Discuss the similarities and differences between a 1s and a 2s orbital.
References
Web side References
http://www.shodor.org/unchem/basic/atom/index.html(atomic structure, quantum
numbers, concept of orbital, electronic configuration)
http://web.jjay.cuny.edu/~acarpi/NSC/3-atoms.htm(introduction to atomic structure)
http://physics.bu.edu/~duffy/PY106/PeriodicTable.html (quantum mechanical view of
the atom)
http://ritchie.chem.ox.ac.uk/Grant Teaching/The Quantum Theory of%20Atoms%20%20The%20Schrodinger%20equation.pdf(Schrodinger equation)
http://www.webelements.com (interactive periodic table)
http://csep10.phys.utk.edu/astr162/lect/light/bohr.html (Bohr Model)
http://umanitoba.ca/outreach/crystal/resources for
teachers/QuantumMechanical%Model%20of%20the%20Atom%20C12-204.doc(Historical Development of the Quantum Mechanical Model of the Atom)
Book References
P.W.Atkins: Physical Chemistry, Oxford University Press.
Institute of Lifelong Learning, University of Delhi