Textbook on applied chemistry

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Textbook on
Applied Chemistry

Achyutananda Acharya
College of Engineering and Technology
(A Constituent College of Biju Patnaik
University of Technology, Odisha)
Bhubaneswar

Biswajit Samantaray
College of Engineering and Technology
(A Constituent College of Biju Patnaik
University of Technology, Odisha)
Bhubaneswar


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Achyutananda Acharya dedicates this book to his grandmother Late Ahalya Devi
and his father Late Maheswar Acharya
and
Biswajit Samantaray dedicates this book to his mother Late Sandhyarani Mohapatra

Copyright © 2017 Pearson India Education Services Pvt. Ltd
Published by Pearson India Education Services Pvt. Ltd, CIN: 72200TN2005PTC057128, formerly
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eISBN 978-93-325-8763-2
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CONTENTS

Preface

About the Authors

ix
xi

1. Quantum Theory and its Postulates
1.1

1.1 Introduction 
1.1

1.2 Black-Body Radiation
1.1

1.2.1 The Photoelectric Effect
1.4

1.3 Wave-Particle Duality of Light 
1.7

1.4 The Heisenberg Uncertainty Principle
1.9

1.5 Origin of Quantum Mechanics
1.9

1.6 The Schrödinger Equation
1.10

1.6.1 Salient Features of the Schrödinger Equation
1.13

1.6.2 Validation of De Broglie Relation Using the Schrödinger Equation
1.13

1.6.3 The Born Interpretation of the Wave Function
1.14

1.6.4 Wave Function must be Acceptable
1.15
1.7 Use of Operators in Quantum Mechanics
1.16

1.8 Postulates of Quantum Mechanics
1.27

1.9 Particle in a One-dimensional Box
1.32

1.10 Selection Rule for Pure Rotational Spectra ∆J = ±1
1.37

1.11 Selection Rule for Pure Vibrational Spectra ∆ν = ±11.39

1.12 Review Questions
1.40

Solved Problems
1.40
2. UV Spectroscopy

2.1Absorption of Different Electromagnetic radiations by Organic Molecules

2.2 Ultraviolet and Visible Spectroscopy

2.2.1 Types of Electron Transitions

2.3Principles of Absorption Spectroscopy: Beer’s and Lambert’s Law

2.3.1Limitations

2.4 Some Important Terms and Definitions

2.5Applications of Electronic Spectroscopy in Predicting
Absorption Maxima of Organic Molecules

2.6 Absorption by Compounds with C=O Bonds

2.7 Aromatic Compounds

2.1
2.3
2.4
2.5
2.7
2.7
2.9
2.10
2.11
2.13


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iv  Contents




2.8 Review Questions
Solved Problems
Short Answer Questions

2.16
2.16
2.21

3. Rotational Spectroscopy

3.1Introduction

3.2 Energy of a Diatomic Molecule as a Rigid Rotor

3.3 Intensities of Spectral Lines in Rotational Spectrum

3.4 Effect of Isotopic Substitution

3.5 Effect of Centrifugal Distortion

3.6 Polyatomic Molecules

3.6.1 Linear Molecules

3.6.2 Symmetric Top Molecule

3.6.3 Asymmetric Top Molecule

3.7 Stark Effect

3.8 Applications of Microwave Spectroscopy

3.9 Review Questions

Solved Problems

3.1
3.1
3.2
3.6
3.7
3.8
3.10
3.10
3.12
3.14
3.14
3.15
3.15
3.15

4. Vibrational Spectroscopy

4.1Introduction

4.2 Energy of a Vibrating Diatomic Molecule

4.3 Simple Harmonic Oscillator

4.4 Energy of Vibrating Rotor

4.5 Failure of Born-Oppenheimer Approximation

4.6 Vibrations of Polyatomic Molecules

4.6.1 Fundamental Vibrations

4.6.2 Linear Molecules

4.6.3 Polyatomic (Symmetric Top) Molecules

4.7 Applications of Infrared Spectra

4.8 Review Questions

Solved Problems

4.1
4.1
4.1
4.3
4.7
4.8
4.9
4.9
4.11
4.12
4.13
4.14
4.14

5. Phase Rule
5.1

5.1Introduction
5.1

5.2 Phase Rule
5.1

5.3 Meaning of Terms Used
5.1

5.4 Phase Table 
5.2

5.5 Other Examples of Phase
5.2

5.6 Component (C)5.3

5.6.1 Example With Explanation
5.4

5.7 Degree of Freedom on Variance (F)5.4

5.8Components Approach to the Phase Rule (Derivation of Phase Rule)
5.5

5.8.1 Usefullnes of Phase Rule
5.6

5.9 Phase Diagram
5.7

5.10 One Component System
5.8

5.10.1 General Characteristics of One Component System
5.9


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Contents













v

5.11 Water System
5.9
5.11.1 Phase Diagram of Water
5.10

5.12 Metastable System
5.13
5.13 Sulphur System
5.13
5.13.1 Polymorphs/Allotropes of Sulphur
5.13
5.13.2 Phase Diagram of Sulphur5.13
5.14 Phase Rule: Study of Two-Component Systems
5.15
5.15 Simple Eutectic Systems
5.17
5.16 Bismuth–Cadmium System (Bi–Cd)5.20
5.17 Review Questions
5.22
Multiple-choice Questions
5.22
Solved Problems
5.28

6. Basics of Organometallic Compounds
6.1

6.1 Introduction6.1

6.2Nomenclature of Organometallic Compounds of the Transition Elements
6.3

6.3 Naming of Organic Ligands
6.3

6.3.1 Naming of Organic Ligands with One Metal-Carbon Single Bond
6.3

6.3.2Naming of Ligands with Several Metal-Carbon Single Bonds
from One Ligand
6.5

6.3.3 Naming of Ligands with Metal-Carbon Multiple Bonds
6.6

6.4 Use of Notation Kappa (κ), Eta (η) and Mu (µ)6.7

6.4.1 Kappa (κ) Notation
6.7

6.4.2 ETA (η) Notation
6.9

6.4.3 Mu (µ) Notation
6.11

6.5 Nomenclature of Metallocene
6.12

6.6Nomenclature of Organometallic Compounds of the Main Group Elements
6.13

6.6.1 Organometallic Compounds of Groups 1 and 2
6.13

6.6.2 Organometallic Compounds of Groups 12–16
6.13

6.7 Effective Atomic Number (EAN) Rule or 18-Electron Rule
6.15

6.7.1 Electron Counts for Common Ligands
6.16

6.7.2 Determination of Oxidation State
6.17

6.7.3 EAN Rule (18e count) for Organometallic Complexes
6.18

6.8 Counting of Electrons in Organometallic Clusters
6.21

6.9 Catalysis Using Organometallic compounds
6.24

6.9.1Hydroformylation
6.25

6.9.2 Catalytic Hydrogenation of Alkenes
6.26

6.9.3 Alkene Isomerization
6.27

6.10 General Characteristics
6.28

6.11 Characteristics of Lanthanoids
6.29

6.12Applications of Some Important Organometallic Compounds as Catalyst
6.29

6.12.1 Grignard’s Reagent
6.29

6.12.2 Ziegler–Natta Catalyst
6.30

6.12.3 Olefin Metathesis
6.31

6.12.4 Palladium Catalyst 
6.32

6.13 Review Questions
6.32

Solved Problems6.32


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vi  Contents

7.FUEL7.1

7.1Introduction
7.1

7.1.1 Classification of Fuel
7.1

7.2 Characteristics of Good Fuel
7.2

7.2.1 High Calorific Value
7.2

7.2.2 Moderate Ignition Temperature
7.3

7.3Determination of Calorific Value Using Bomb Calorimeter
7.4

7.3.1 Solid Fuels and their Characteristics
7.5

7.4Ash
7.7

7.5Moisture
7.7

7.6 Characteristics of Flame

7.8

7.6.1 Combustion Characteristics
7.8

7.7 Coals and their Characteristics
7.8

7.7.1 Analysis of Coal
7.8

7.8 Ultimate Analysis of Coal
7.9

7.9 Manufactured Solid Fuels and their Characteristics
7.9

7.10 Charcoal and its Characteristics
7.9

7.11 Coke and its Characteristics
7.10

7.12 Briquettes and their Characteristics
7.10

7.13 Bagasse and its Characteristics
7.10

7.14 Liquid Fuel

7.10

7.15 Petroleum and its Characteristics
7.11

7.15.1 Classification of Petroleum
7.11

7.16 Paraffinic Base Type Crude Petroleum
7.12

7.17 Asphalitc Base Type Crude Petroleum
7.12

7.18 Petroleum Formation
7.12

7.18.1 Properties of Petroleum
7.13

7.19 Petroleum Utilization
7.14

7.20 Petroleum Refining
7.15

7.21 Petroleum Products
7.16

7.22 Manufactured Liquid Fuels and their Characteristics

7.17

7.23 Gasoline or Petrol and its Characteristics
7.17

7.24 Diesel Fuel and its Characteristics
7.17

7.25 Kerosene Oil and its Characteristics
7.18

7.26 Heavy Oil and its Characteristics
7.18

7.27Cracking
7.18

7.28 Thermal Cracking
7.19

7.28.1 Mechanism of Thermal Cracking
7.19

7.29 Reforming of petrol
7.22

7.29.1 Reforming Reactions
7.22

7.30 Mechanism of Knocking

7.23

7.31 Adverse Effects of Gasoline Knock
7.24

7.32Knocking in IC Engines can be Minimized Through the Following Measures
7.24

7.33 Octane Number
7.24

7.34 Anti-knocking Agents 
7.25

7.35 Unleaded Petrol
7.25

7.35.1 Advantages of Unleaded Petrol 
7.25

7.36 Catalytic Converter Contains Rhodium Catalyst
7.26


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Contents

vii

7.37
7.38
7.39
7.40
7.41
7.42
7.43

Cetane Number

7.26
Synthetic Petrol
7.27
Bergius Process
7.27
Power Alcohol
7.29
Gaseous Fuel
7.30
Water Gas
7.31
Producer Gas
7.32
7.43.1Manufacture
7.32
7.44Reactions that Take Place in Different Zones of the Fuel Bed
7.33
7.45 Manufactured Gases and their Characteristics
7.34
7.45.1 Coal Gas and its Characteristics
7.34
7.45.2 Blast Furnace Gas and its Characteristics
7.34
7.45.3 Oil Gas and its Characteristics 7.34
7.45.4 Kerosene Oil and its Characteristics7.35
7.46 CNG (Compressed Natural Gas)
7.36
7.47 Composition and Calculation
7.38
7.47.1 Calculation of Air Required for Combustion

7.38
7.48 Review Questions
7.40
Solved Problems
7.40
Short Answer Questions
7.62

8.

Corrosion8.1
8.1Introduction
8.1
8.1.1Definitions
8.1
8.2 Chemical or Dry Corrosion
8.2
8.3 Oxidation Corrosion 
8.3
8.4Mechanism
8.3
8.5 Nature of the Oxide Film Formed on the Surface
8.4
8.6 Pilling Bedworth Rule
8.4
8.7Rate of Diffusion of Metal Ion and Oxide Ion through the Layer Formed
8.5
8.8 Wet or Electrochemical Corrosion
8.6
8.9 Electrochemical Theory of Corrosion
8.6
8.9.1 Mechanism of Electrochemical Corrosion
8.7
8.10 Types of Electrochemical Corrosion
8.10

8.11Concentration Cell Corrosion (Differential Aeration Corrosion)
8.11
8.11.1 Important Characteristic about Differential Aeration Corrosion
8.13
8.12 Water-line Corrosion
8.13
8.12.1Prevention
8.14
8.13 Pitting Corrosion
8.14
8.13.1 Pitting Corrosion may be Caused by
8.15
8.14 Stress Corrosion
8.16
8.14.1 Favourable Conditions for Stress Corrosion
8.16
8.15Causes and Methods of Prevention of Caustic Embrittlement
8.17
8.15.1 Prevention of Caustic Embrittlement
8.18
8.16 Galvanic Series
8.18
8.17 Factors Influencing Corrosion
8.19
8.17.1 Nature of the Metal
8.19


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viii Contents









8.18Protection from Corrosion (Preventive Measures for Corrosion Control)
8.19 Method of Application of Metal Coatings
8.20Paints
8.20.1 Constituents of Paints and their Function
8.21 Use of Inhibitors
8.22 Review Questions
Multiple-choice Questions
Short Answer Questions

8.21
8.26
8.28
8.29
8.30
8.31
8.31
8.36

IndexI.1

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PREFACE
This book is primarily focussed on the undergraduate students of engineering and science. It has been
believed that education in Chemistry will train us to think in terms of molecules and their interactions.
The interaction with the electromagnetic radiation is chosen, which is governed by some rules, and
these are well explained in quantum theory. Further, quantum mechanics is accepted as one of the
important areas of chemistry now-a-days. We start with this topic discussing on their basic concepts,
underlying principles and postulates. Today, many of the rules used in physical and organic chemistry
are a consequence of end result of the quantum theory. The spectra are understood only with the concepts
and understanding on quantum mechanics. The principles of UV-visible, rotational (microwave) and
vibrational (infrared) spectra are discussed and described for small molecules. This will give insight to
understand the nature of molecules, prediction of their structures, bond lengths, moment of inertia and
force constants.
Existence of substances in different phases based on their thermodynamical properties is one of the
valuable areas in Chemistry. Based on these principles, a chapter is included on phase rule. However,
addition of the relevant thermodynamical aspects may be taken up in the subsequent edition.
Students are already aware of organic and inorganic compounds along with some idea on transition
metal complexes. They are now exposed to another class of compounds called “Organometallic”
compounds, which are extensively used as a catalyst in many organic and inorganic reactions.
Discussion is limited to their nomenclature of simple organometallic compounds based on IUPAC
recommendations and some of their applications.
Fuel is an important class of material which is being used in our everyday life. Thus, knowledge
of its composition, various types and overall its combustion process are necessary for which a chapter
highlighting these aspects is included. An unwanted chemical process, which we face in our everyday
life, is corrosion. A chapter on it describing the principles of corrosion, types of corrosion and their
prevention is included.
Considering the limitations of this book, one of the important aspects of Chemistry, i.e. chemical
kinetics, is not included at this point of time. However, in subsequent editions, a chapter on this topic

will be included.
We also solicit the views, suggestions and errors, if any, from any readers including students and
teachers and these will be included in the revised editions appropriately.


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x  Preface

ACKNOWLEDGEMENTS
We are deeply indebted to our revered teacher, Prof. A. C. Dash, for his immense support, thoughtful
guidance, continuous encouragement and deep involvement for pursuing research, writing papers and
book. We would like to thank our parents and teachers for their blessings and support to this valuable
contribution. We would also like to thank Prof. P. K. Patra, Principal, College of Engineering and
Technology, Bhubaneswar for his encouragement to pursue research and write articles and books. We
are also thankful to all our colleagues, relatives and well wishers for their good wishes for publication
of this book.
One of the authors, Achyutananda Acharya, would like to give special thanks to his wife, Smita, for
inspiring him to write a book and without her support it would not have been possible to write this book.
He would also like to thank his affectionate son Litun who helped him in sparing computer peripherals.
The other author, Biswajit Samantaray, would like to give special thanks to his wife, Laxmipriya, for
her valuable support to write his second book. He is very fortunate that his lovely daughter, Khusi, did
not tear the hand written draft at the initial stage of writing this book.
We are very thankful to M. Balakrishnan, King D. Charles Fenny, Ranjan Mishra, and Niraj Mishra,
Pearson India Education Services Pvt. Ltd, who had approached us in the month of May 2016 in getting
this book done. We also thank them for their valuable suggestions, editing the manuscript and also
shown keen interest to publish this book.
Achyutananda Acharya
Biswajit Samantaray

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ABOUT THE AUTHORS
Achyutananda Acharya started his employment as a lecturer in Chemistry. He was awarded SERC
Visiting Fellowship and BOYSCAST Fellowship sponsored by Department of Science and Technology,
Govt. of India. His research work is focussed on kinetics and mechanism of transition metal complexes
in aqueous, aquo-organic and aquo-micellar media. He has also perused his research work in the frontier areas of science and technology, i.e. Advanced Drug Discovery. He has vast experience in teaching
UG and PG programmes, along with research experience in the field of chemical sciences and also has
administrative experience in the field of technical education.
Biswajit Samantaray is currently placed as a lecturer in Chemistry with more than 14 years of teaching and academic/administrative experience. He is imparting teaching to B.Tech. and M.Sc. students.
He is also pursuing the research work in the area of Solution Chemistry. He has authored book on
Engineering Chemistry, which is well accepted by academics. He has attended many seminars, conferences and faculty development programmes.


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1

QUANTUM THEORY AND
ITS POSTULATES

1.1 INTRODUCTION
During seventeenth century laws of classical mechanics were introduced by Isaac Newton. These laws
are very successful in explaining the motion of macroscopic bodies (which are visible to naked eye)
such as planets and everyday objects such as pendulums, projectiles and so on. These laws are useful
to derive the relationship between the concepts of velocity, momentum, acceleration, force, work and

energy. However, towards the end of nineteenth century and particularly after the discovery of subatomic particles, the experimental observations gathered through the laws of classical mechanics failed
when applied to small and subatomic particles such as atoms, nuclei and electrons. Quantum mechanics
emerged due to failure of classical mechanics in case of small particles. Now-a-days quantum mechanical hypotheses are applied to understand the fundamental description of matter, spectroscopic studies
for determination of structure of molecules, electronic structure of atoms and moreover the concept of
chemical reactions. In general, quantum mechanical concepts are applied to every aspect of chemistry
for detailed understanding. Thus, understanding the basics of quantum mechanics is an essential necessity in chemistry.
This chapter begins with an example of failure of classical mechanics in explaining black-body
radiation. And then discusses on wave-particle duality, Heisenberg’s uncertainty principle followed
by introduction of the Schrödinger wave function and its fundamental properties, postulates and
applications.

1.2  BLACK-BODY RADIATION
Studies on electromagnetic radiation are the origin of quantum mechanics. A hot body emits electromagnetic radiation. For example, when an iron rod is heated, it turns dull red and progressively it
becomes deep red as the temperature increases. On further heating, the radiation becomes white and
then turns blue as the temperature continues to increase. This displays that there is a continuous shift
of the colour of the heated body from red through white to blue as the body is heated to higher temperatures. This change of colour can be expressed in terms of frequency, i.e., it changes from a lower
frequency (red colour) to a higher frequency (blue colour) as the temperature increases. A black body,
an ideal body, is an object capable of emitting and absorbing all frequencies of radiation uniformly.
The radiation emitted by a black body is called black-body radiation.


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1.2 Textbook on Applied Chemistry
Figure 1.1 shows variation of energy density with wavelength of radiations of a black body with
varying temperatures.
2000 K

ρ / J m−4

5000

1750 K

3000

1000

1250 K

0
Visible
region

2000

4000
λ /nm

6000

Figure 1.1  Energy density versus l at different temperatures of black-body radiation
The peak shifts towards smaller λ as temperature is increased, i.e., towards visible region means
colour shifts towards blue. In 1893, Wilhelm Wien explained these experimental results and expressed
mathematically as


1
T lmax = c2 , (1.1)
5

where c 2 is the second radiation constant which is equal to 1.44 cm-K. This equation is called
Wien’s displacement law. Using equation (1.1), we predict λmax ∼ 2,900 nm at 1,000 K. In 1979,
Josef Stefan proposed that the total energy density is directly proportional to the fourth power of
temperature, i.e.,


U = aT 4 ,(1.2)

where U is the total energy density, which is total energy per unit volume in the electromagnetic field
and a is the proportionality constant. Equation (1.2) is also alternatively expressed as
M = σT 4

(1.3)

where σ is called the Stefan–Boltzmann constant and is equal to 5.67 × 10−8 Wm−2K−4 and M is called
as excitance which is expressed as the power emitted per unit area. Equations (1.1)–(1.3) are valid for
a particular range of temperature and wavelength but miserably fail to explain the behaviour of blackbody radiation with respect to wide variation of temperature and wavelength.
To overcome these failures, Lord Rayleigh first proposed that the electromagnetic field is a collection of harmonic oscillators. Each oscillator is associated with the same frequency of light and therefore
 c
wavelength λ = . Based on the study on black-body radiation, Lord Rayleigh and James Jean for ν
mulated that


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Quantum Theory and Its Postulates

1.3

dU = fdl,(1.4)

8π kT
f=
where
(1.5)
l4
The factor f is referred as the energy per unit volume per unit wavelength and k is called the
Boltzman’s constant. Equation (1.4) is very successful at large wavelengths and low frequencies but
fails at lower wavelengths. In equation (1.5), as λ decreases, f increases without going through maxima,
which is contradictory to the experimental results of black-body radiation. Similarly, at very low wavelength, the electromagnetic oscillators are strongly excited even at room temperature. Thus, a large
amount of energy is exhibited in the high-frequency region of the electromagnetic spectrum, which is
called Ultraviolet Catastrophe. Hence, all the objects should emit electromagnetic radiation or glow
in the dark and there should be no darkness. It is contrary to fundamental properties of nature, i.e., we
experience day and night in our everyday life.
In 1900, Max Planck made the revolutionary assumption that energies (E) of electromagnetic
oscillators are discrete (not same for all), varied arbitrarily and are proportional to an integral multiple
of the frequency. This is mathematically expressed as
E = nhv, n = 0, 1, 2, … ,(1.6)
where E is the energy of an oscillator, v is the frequency, n is an integer and h is a proportionality
constant. The limitation of energy to discrete values is called the quantization of energy, otherwise
called as energy is quantized. The energies of the oscillators are 0, hv, 2hv, … , considering that it
consists of 0, 1, 2, … particles, each particle having an energy hv. These particles are called photons.
Planck’s hypothesis is explained on the basis that atoms in the walls of black body are in thermal
motion, which excites the oscillators of the electromagnetic field. According to classical mechanics
(Rayleigh-Jeans law), all these oscillators share equally in the energy supplied by the walls; thus, the
highest frequencies are excited. On contrary, according to quantum theory (Planck’s concept), oscillators are excited only when they can acquire an energy of at least hv. This is too large for the walls

of the black body to supply in the case of high-frequency oscillators; thus, they are unexcited. All
the oscillators are not excited thereby indicating that the oscillators are quantized or have discrete
values of energy.
Using quantization of energy and statistical thermodynamics concepts, Planck derived the equation
for energy density

dU = fdl, (1.7)
where f =

8πhc
l

5

(

1
hc
e lkT

)

.

−1

Planck showed that equation (1.7) fitted the experimental data well for all frequencies and temperatures, if h has the value 6.626 × 10−34Js. This constant is called Planck’s constant.
hc

For short wavelength,

hc
is large, thereby e lkT → ∞; therefore, f → 0 as l → 0 and v → ∞
lkT


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1.4 Textbook on Applied Chemistry
8πhc  1 


l 5  ∞−1


8πhc 1
0
=
×
∞
l5
f=

(1.8)

So, the energy density tends to zero at high frequencies, in agreement with the experimental
observation.
hc

Similarly, for long wavelengths,

hc
 1, on expansion of e lkT :
lkT
hc



e lkT = 1 +

hc
+
lkT

1+

hc
(1.9)
lkT

and putting this value in equation (1.7), it reduces to Rayleigh–Jeans law as in equation (1.5)
f=

=


8πhc 
1



l 5 1 + hc −1


lkT

8πhc lkT 8π kT
⋅
=
,
hc
l5
l4

(1.10)

which is applicable at large wavelength and low frequencies, i.e.,


f→

8π kT
l4

as l → ∞ and v → 0.

The Planck’s distribution also explained Stefan’s and Wien’s law as discussed above. The concept
of quantization of energy well explained the experimental results of the black-body radiation. This led
the evolution of quantum mechanics.

1.2.1  The Photoelectric Effect

The concept of quantization is also applicable to photoelectric effect. The photoelectric effect is a
phenomenon in which the ejection of electrons takes place from metals when they are exposed to
electromagnetic radiation. The following characteristics are observed for this effect.
1.Electrons are ejected only when the frequency of radiation exceeds a threshold value characteristic of the metal but independent on its intensity.
2. The kinetic energy of the ejected electrons is expressed as


1
 KE = me v 2  ,(1.11)


2


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Quantum Theory and Its Postulates

1.5

where me and v denote the mass of electron and its velocity, respectively. The kinetic energy is
linearly proportional to the frequency of the incident radiation but independent of its intensity.
3. Electrons are ejected even at low intensities if the frequency exceeds its threshold value.
These observations indicated that electrons are ejected when they are collided with a particlelike projectile having enough energy to remove it out from the metal. Let us consider that the
projectile is a photon of energy hv, where v is the frequency of incident radiation. Based on the
law of conservation of energy and above observations, the kinetic energy of the electron should
obey
1

me v 2 = hv −φ ,
2

(1.12)

where φ is the work function of the metal, and v is the frequency of incident radiation. Each
metal has a unique work function, φ or threshold value.
If hv < φ, then ejection of electrons cannot occur because of insufficient energy of the photon.
This can be interpreted mathematically that kinetic energy in equation (1.12) has negative value, which
is absurd. This indicates that electrons cannot be ejected when hv < φ , which satisfies the first observation. The kinetic energy is linearly proportional to frequency of incident radiation as represented in
equation (1.12). The kinetic energy versus v plot is a straight line having slope h. Electrons are ejected
only when hv > φ even at low intensities of the radiation, which is in agreement with the third observation. The observations of photoelectric effect provided evidence for quantization (h, Planck’s constant).
Further, the collision between photon and electron is elastic in nature, which means that during this
collision process transfer of energy from photon to electron takes place with conservation of energy.
The occurrence of elastic collision between photon and electron, a subatomic particle, indicated that the
photons or electromagnetic radiation or wave behave matter-like or particle-like properties.
Example 1.1
When potassium is irradiated with light, the kinetic energy of the ejected electrons is 2.935 × 10 −19 J
for λ = 300.0 nm and 1.280 × 10 −19 J for λ = 400 nm. Calculate (a) the Plank’s constant, (b) threshold
frequency and (c) work function of the potassium metal.
Solution  (a) Let us say for two different wavelengths: λ1 = 300.0 nm and corresponding kinetic
energy is KE1 = 2.935 × 10 −19 J and similarly λ2 = 400 nm and corresponding kinetic energy is KE2 =
1.280 × 10 −19 J
Using equation (1.12),



KE1 = hv1 − φ
=

hc
− φ
l1

(1.1a)

Similarly,



KE2 = hv2 − φ
=

hc
− φ
l2

(1.1b)


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1.6 Textbook on Applied Chemistry
For the same metal, the work function φ is constant. Subtracting equation (1.1b) from (1.1a), it gives
KE1 − KE2 =



hc

−φ −
l1

 hc

 − φ

 l

2



1
1
= hc  − 
 l1 l2 





1
1

= h × 2.998 × 108 m⋅s−1 
−
 300.0 ×10−9 m 400.0 ×10−9 m 

= 2.935 × 10−19 J – 1.280 × 10−19 J



= h (2.498 × 1014 s−1)



=h=



= 6.625 × 10−34 J⋅s

1.665×10−19 J
2.498×1014 s−1

(b)Putting h value in equation (1.1a)
KE1 = hv1 – φ




=



KE1 =

hc
–φ
l1
hc
– hv0
l1


 hc
 − KE1 

 l1

v0 = 


h
 (6.625×10−34 J⋅ s × 2.998×108 m⋅ s−1 ) 

 − 2.935×10−19 J




300.0 ×10−9 m


= 
−34
6.625×10

J⋅ s



= 5.564 × 1014 Hz


(c)




φ = hv0
= 6.625 × 10−34 J⋅s × 5.564 × 1014 Hz (s−1)
= 3.686 × 10−19 J

■


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Quantum Theory and Its Postulates

1.7

1.3  WAVE-PARTICLE DUALITY OF LIGHT
It was proposed that light displayed definite wavelike character and also it consisted of a stream of
photons. Appearance of a spectrum on dispersion of white light through a prism supported the wavelike

character of light. Similarly, the photoelectric effect is an experiment which supports the particle-like
concept of light. Since light exhibits wavelike in some experiments and particle-like in others, this
disparity is referred to wave-particle duality of light. In 1924, Louis de Broglie, a French scientist,
proposed that if light can show wave-particle duality, then matter, which appears particle-like might
also show wavelike properties under certain conditions. Einstein, based on theory of relativity showed
that, the wavelength, λ, and its momentum p of a photon are related by
h

l=
(∵ p = mv)
p

(1.13)

h
Or
p=
(for photons)
l
Also E = mc 2 or

E
hv
= mc = p,  
= mc;
c
c

(1.14)

where c is the velocity of light.
From equations (1.13) and (1.14),
hv
h

= p = ; lv = c,
c
l
where h is the Planck’s constant. On this basis, de Broglie proposed that light and matter both obey
these relations. Since momentum (p) is mv, this equation predicts that a particle of mass m moving with
a velocity v will have a de Broglie wavelength expressed as
h

l=
.
mv
For macroscopic particles like cricket ball, the mass is very large, so λ is too small to be completely undetectable; thus, the results are of no practical applications. This indicates that the equation is not applicable to macroscopic particles. This is applicable only to microscopic particles like
electrons, protons, atoms and molecules moving with high velocities. Example 1.2 explains the above
conclusion.
Example 1.2
Consider a cricket ball of 400 g and an electron moving with same speed say 1.5 × 104 cm/s. Calculate
l for both the particles.
Solution  For cricket ball, mass = 400 g = 0.4 kg and its speed = 1.5 × 104 cm/s
l =

h
  (1 J = 1 kg⋅m2⋅s−2)
mv

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1.8 Textbook on Applied Chemistry

=

6.62 ×10−34 J ⋅ s
0.4 kg ×1.5×102 ms−1

= 1.10 ×10−35 m



= 1.10 ×10−26 nm (undetectable)
Similarly for electron (mass of electron = 9.1 × 10-31 kg)
l=

h
mv

6.62 ×10−34 J ⋅ s
=

9.1×10−31 kg ×1.5×102 ms−1
= 4.849×10−6 m
= 4, 849 nm (detectable)
For a cricket ball, l is negligible or undetectable, whereas for electron it is detectable. So explains
well that de Broglie equation is not applicable to macroscopic particle whereas it is applicable to subatomic particle.
■
The duality characters of both matter and light are observed experimentally. In a typical experiment, a beam of X-rays are scattered through a very thin aluminium foil. The X-rays scatter from the

foil in rings of different diameters. When a beam of electrons scattered through the thin aluminium foil,
similar diffraction pattern was also observed (see Figure 1.2). Thus, the similarity of the two diffraction pattern shows that both X-rays (wave) and electrons (particle) do indeed behave analogously. The
particle nature of light is also evidenced through Einstein’s photoelectric phenomenon (as light behaves
as projectile hitting electron on metal surface). The wavelike property of electrons is used in electron
microscope to determine the atomic and molecular structures of chemical compounds.

(a)

(b)

Figure 1.2  Diffraction pattern of aluminium foil: (a) X-rays; (b) electrons


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Quantum Theory and Its Postulates

1.9

1.4 THE HEISENBERG UNCERTAINTY PRINCIPLE
Indeed it is true that a wave does not have a definite location at a single point in space; a wave spreads
throughout the space for which the location of the particle could not be determined. Similarly, the precise position of a particle could not be specified if it has a definite momentum (mv). In other words, the
wave spreads everywhere thereby the particle may be found anywhere in the whole space. Based on the
wave-corpuscular dualism, Werner Heisenberg in 1927, proposed that
“It is impossible to specify or determine simultaneously, with arbitrary precision, both the
momentum and position of a particle.”
This is known as Uncertainty Principle and quantitatively expressed as
 

h
∆p ⋅∆x ≥

 =  ,
2π
2π

(1.15)

where ∆p and ∆x denote uncertainty in momentum and position, respectively. When the position of the
particle is known exactly, then ∆x = 0; thus, ∆p must be infinite, which means that linear momentum
cannot be determined accurately. Similarly, if ∆p = 0, then ∆x = ∞, which implies that the position
cannot be determined precisely at the same time.
Example 1.3
Calculate the mass of the particle whose uncertainty in position and velocity are 1.52 × 10−9 m and
6.34 × 10−22 ms−1, respectively.
Solution

∆x ×∆p = ∆x × m ⋅∆v =

h
4π

h
Or
∆x ×∆v =
4π m
h

m=

4π ×∆x ×∆v
6.626 ×10−34 Js
m=

4 ×3.142 ×1.52 ×10−9 m ×6.34 ×10−22 ms−1
6.626 ×10−34 kg ⋅ m 2 s−2 ⋅ s
m=
4 ×3.142 ×1.52 ×10−9 m ×6.34 ×10−2 ms−1 

= 5.47×10−5 kg

■

1.5  ORIGIN OF QUANTUM MECHANICS
Classical Mechanics failed to explain the experimental results of black-body radiation, heat capacity of solids and appearance of spectra. These failures established that the basic concepts of classical


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1.10 
Textbook on Applied Chemistry
mechanics are untenable in case of microscopic and sub-microscopic particles. The failures of classical
mechanics gave birth to a new concept referred to as quantum mechanics.
In classical mechanics, it is considered that the particles are moving on trajectories. Based on de
Broglie’s wave-particle duality, the position of a particle is distributed through space like the amplitude
of a wave. The concept of wave function, ψ (psi), is introduced in the place of the trajectory, which is
used in quantum mechanics. In other words, variables such as position (x), momentum (mv) and energy
(E) of a particle took “probabilistic” rather than fully precise “deterministic” approach.

1.6  THE SCHRÖDINGER EQUATION

In the previous section, the concept of de Broglie’s theorem is discussed. One of the important conclusions is that matter has wavelike character in addition to particle-like character. Since matter does possess wavelike properties, there must be some wave equation that governs its motion.
For simplicity, let us consider the classical one-dimensional wave equation for vibration in a
stretched string:
∂2u
1 ∂2u

=
,
∂x 2 v 2 ∂t 2

(1.16)

where u(x, t) is the displacement or amplitude of the string (at position x and time t) when one-dimensional wave equation describes the motion. This is explained in Example 1.4.
Example 1.4
The one-dimensional wave equation describes the motion of a vibrating string.
Solution  Consider a uniform vibrating string stretched between two fixed points as shown in Figure 1.3.

u, (x, t)
x
l

0

Figure 1.3  Vibrating string whose ends are fixed at O and l and has
amplitude of vibration u(x, t) at position x and time t
The amplitude of the string is defined as the maximum displacement from its equilibrium horizontal position. Let u(x, t) is the displacement of the string which satisfies the equation


∂ 2 u ( x, t )
∂x 2

=

1 ∂ 2 u ( x, t )
v2

∂t 2

,

where v is the speed with which a disturbance moves along the vibrating string. This is the classical
wave equation. It is a partial differential equation because u(x, t) occurs in partial derivatives. The
variables x and t are said to be independent variable, whereas u(x, t) is said to be a dependent variable


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Quantum Theory and Its Postulates

1.11

as it depends on x and t. This equation is a linear partial differential equation because u(x, t) and its
derivatives appear only to the first power and there are no cross terms. The amplitude u(x, t) satisfies
the boundary conditions, i.e.,
u (0, t ) = 0 at x = 0 and u(l, t) = 0 (for all t).

This equation can be solved by the method of separation of variables.

■

Equation (1.16) can be solved by the method of separation of variables. The term u(x, t) can be
described as the product of a function of x and harmonic and sinusoidal function of time “t”.
This can be expressed as
u(x, t) = ψ(x)cos wt,

(1.17)

where ψ(x) is the spatial factor of the amplitude u(x, t). Substituting equation (1.17) in equation (1.16),
we obtain


d2ψ
dx 2

+

w2
v2

ψ( x ) = 0 

(1.18)

This is explained in Example 1.5. Putting the expression for w (= 2πv) and v (= vλ), equation (1.18)
becomes
d 2 ψ 4π 2

+
ψ( x ) = 0 
dx 2
l2

(1.19)

Example 1.5
As the above,

u(x, t) = ψ(x) cos wt.

Solution  LHS of equation (1.16)
du( x, t )
dψ ( x)
= cos wt ⋅

dx
dx
d 2 u( x , t )
d 2ψ ( x)
= cos wt


dx 2
dx 2

(1.5a)

R.H.S of equation (1.16)
du ( x, t )
= ψ( x ) ⋅ w ⋅ (− sin wt )

dt
d 2 u ( x, t )

= −ψ( x ) ⋅ w ⋅ w ⋅ cos wt
dt 2

= −ψ( x )w2 cos wt 

(1.5b)


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1.12 
Textbook on Applied Chemistry
Putting the values of equations (1.5a) and (1.5b) in L.H.S and R.H.S of equation (1.16), we obtain
cos wt

d 2ψ ( x)
dx

2

=−

1

v

2

(−ψ( x)w2 ⋅ cos wt )

w2

=−
ψ( x ) cos wt 
(1.5c)
v2
Dividing cos wt on both sides and then transposing it, the expression at equation (1.16) is obtained
as
d 2 ψ( x )



dx 2

w2

+

v2

ψ ( x ) = 0 ,

(1.5d)



■
Total energy of particle is the sum of its kinetic energy and potential energy (V(x)), i.e.,

1 2
mv + V ( x )
2

p2
=
+ V ( x ),
2m
E=


1 2
p2 

 KE = 2 mv = 2 m 


(1.20)

where p is the momentum of the particle (mv) and V(x) is its potential energy. Rearranging equation
(1.20),
1

p = {2 m [ E − V ( x ) ]} 2 

(1.21)

Incorporating p in the de Broglie’s formula, we obtain
l=



h
=
p

h

{2m [ E − V ( x )]}

1
2



(1.22)

Substituting λ in equation (1.19), we find


or

d 2ψ ( x)

dx 2

+

d 2ψ ( x)
dx

2

8π 2 m
h2
+

2m


2

[ E − V ( x)]ψ ( x ) = 0 

(1.23)


h
[ E − V ( x )] ψ ( x ) = 0    = 


2π 

(1.24)

On rearrangement of equation (1.24), it gives


−

2 d 2ψ ( x)
+ V ( x )ψ ( x ) = E ψ ( x ) 
2m dx 2

(1.25)