Preview Physical Chemistry for the JEE and Other Engineering Entrance Examinations by K. Rama Rao, S. V. V. Satyanarayana (2013)

Preview Physical Chemistry for the JEE and Other Engineering Entrance Examinations by K. Rama Rao, S. V. V. Satyanarayana (2013) Preview Physical Chemistry for the JEE and Other Engineering Entrance Examinations by K. Rama Rao, S. V. V. Satyanarayana (2013) Preview Physical Chemistry for the JEE and Other Engineering Entrance Examinations by K. Rama Rao, S. V. V. Satyanarayana (2013) Preview Physical Chemistry for the JEE and Other Engineering Entrance Examinations by K. Rama Rao, S. V. V. Satyanarayana (2013)

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Preface iv

Physical Chemistry is the coordination of Physics and Chemistry, of which Oswald laid the foundation Without the

knowledge of those physical methods, which raised Chemistry from being a mere collection of facts to a science with a

rational basis, it is simply impossible to study Chemistry, as Robert Bunsen pointed out, “A chemist who is no physicist

is almost valueless.” Although certain branches of Chemistry, such as Organic Chemistry continue to retain traces of

individuality, it is not possible to study one branch of chemistry in isolation (It must be noted here that recent studies

conducted on organometallic compounds has significantly blurred the line between Organic Chemistry and Inorganic

Chemistry.) It is true that the study of neither Organic Chemistry nor Inorganic Chemistry is possible without a clear

understanding of Physical Chemistry

This book, targeted at students of classes XI and XII preparing for various competitive examinations, presents a logical

and modern approach to the basic concepts of Physical Chemistry Although the historical approach is often believed to be

the best way of inculcating a research outlook—ideally, the basis of all science teaching—we have purposely retained only

those ideas and obsolete experimental procedures that provide a historical introduction or establish the experimental basis

for current theories Physical Chemistry involves the theoretical interpretation of experimental observations However,

all too frequently, students are unaware of the methods used in making these observations So, throughout the book,

sections have been devoted to brief descriptions of the apparatus employed and discussions of the fundamental principles

concerned in the measurement of various physical properties This will help students appreciate the logic of the various

interpretations

From experience, we know that combining the theory with its numerical application enables students to grasp the

subject more easily Hence, many solved examples and problems for practice have been included in the text, with a clear

emphasis on the questions that have appeared in several competitive examinations The intent is to give students a clear

idea of the level of numerical questions that appear in competitive examinations Solutions to numerical questions are

attached as an aid to the understanding of the principles of Physical Chemistry

Physical Chemistry naturally starts with the basic concepts such as development of structure of matter, element,

compound, mixtures, development of chemical combinations These are dealt with in the chapter ‘Basic Concepts of

Chemistry’ The study of atom is one of the most outstanding advances in Physical Chemistry that brought about great

changes in the presentation of facts of Inorganic as well as Organic Chemistry All the topics usually dealt with under the

heading of Physical Chemistry will be found in this book It has been made up-to-date as much as possible by the inclusion

of recent developments in the theories of several works

K Rama Rao

S V V Satyanarayana

1.1 iNTRODUCTiON

The introduction of atomic theory by John Dalton early in

the nineteenth century marks the inception of a modern era

in chemical thinking The virtue of Dalton’s theory was not

that it was new or original, for theories of atoms are older

than the science of chemistry, but that it represented the

first attempt to place the corpuscular concept of

mat-ter upon a quantitative basis The theory of the atomic

constitution of matter dates back at least 2,500 years to the

scholars of ancient Greece and early Indian philosophers

who were of the view that atoms are fundamental

build-ing blocks of matter Accordbuild-ing to them, the continued

subdivision of matter would ultimately yield atoms which

would not be further divisible The word ‘atom’ has been

derived from the Greek word ‘a-tomio’ which means

‘un-cuttable’ or non-divisible Thus, we might say that as far

as atomic theory is concerned Dalton added nothing new

He simply displayed a unique ability to crystallize and

cor-relate the nebulous notions of the atomic constitution

prev-alent during the early nineteenth century into a few simple

quantitative concepts

1.2 ATOmiC TheORy

The essentials of Dalton’s atomic theory may be

summa-rized in the following postulates:

1 All matter is composed of very small particles called

atoms

2 Atoms are indestructible They cannot be subdivided,

created or destroyed

3 Atoms of the same element are similar to one another

and equal in weight

4 Atoms of different elements have different properties

and different weights

5 Chemical combination results from the union of atoms

in simple numerical proportions

John Dalton was born in England in 1766 His ily was poor, and his formal education stopped when

fam-he was eleven years old He became a school teacfam-her

He was color blind His appearance and manners were awkward, he spoke with difficulty in public As an ex-periment he was clumsy and slow He had few, if any outward marks of genius

From 1808, Dalton published his celebrated New Systems of Chemical Philosophy in a series of publi-

cations, in which he developed his conception of oms as the fundamental building blocks of all matter

at-It ranks among two greatest of all monuments to man intelligence No scientific discovery in history has had a more profund affect on the development of knowledge

hu-Dalton died in 1844 His stature as one of the est scientists of all time continues to grow

great-Thus, to Dalton, the atoms were solid, hard, trable particles as well as separate, unalterable individuals

impene-Dalton’s ideas of the structure of matter were borne out of considerable amount of subsequent experimental evidence

as to the relative masses of substances entering into cal combination Among the experimental results and rela-tionship supporting this atomic theory were Gay-Lussac’s law of combination of gases by volume, Dalton’s law of multiple proportions, Avagadro’s hypothesis that equal volumes of gases under the same conditions contain the same number of molecules, Faraday’s laws relating to elec-trolysis and Berzelius painstaking determination of atomic weights

chemi-modern Atomic Theory

Dalton’s atomic theory assumed that the atoms of elements were indivisible and that no particles smaller than atoms

exist As a result of brilliant era in experimental physics

which began towards the end of the nineteenth century

ex-tended into the 1930s paved the way to our present modern

atomic theory These refinements established that atoms

can be divisible into sub-atomic particles, i.e., electrons,

protons and neutrons—a concept very different from that

to Dalton The major problems before the scientists at that

time were

(i) How the sub-atomic particles are arranged within

the atom and why the atoms are stable?

(ii) Why the atoms of one element differ from the

atoms of another elements in their physical and chemical properties?

(iii) How and why the different atoms combine to

form molecules?

(iv) What is the origin and nature of the characteristics

of electromagnetic radiation absorbed or emitted

by atoms?

1.3 SUb-ATOmiC PARTiCleS

We know that the atom is composed of three basic

sub-atomic particles namely the electron, the proton and the

neutron The characteristics of these particles are given in

Table 1.1

Table 1.1 The three main sub-atomic particles

Particle Symbol Mass Charge

H-atom or 9.109 ×10 –28 g

It is now known that many more sub-atomic particles

exist, e.g., the positron, the neutrino, the meson, the

hy-peron etc, but in chemistry only those listed in Table 1.1

generally need to be considered The discovery of these

particles and the way in which the structure of atom was

worked out are discussed in this chapter

1.3.1 Discovery of electron

The term ‘electron’ was given to the smallest particle that could carry a negative charge equal in magnitude to the charge necessary to deposit one atom of a 1-valent

element by Stoney in 1891 In 1879, Crookes ered that when a high voltage is applied to a gas at low pressure streams of particles, which could communicate momentum, moved from the cathode to the anode It did

discov-not seem to matter what gas was used and there was strong evidence to suppose that the particles were com-mon to all elements in a very high vacuum they could not

be detected The cathode ray discharge tube is shown

in Fig 1.1

Fig 1.1 Cathode ray discharge tube

The properties of these cathode rays are given below:

(i) When a solid metal object is placed in a discharge tube in their path, a sharp shadow is cast on the end of the discharge tube, showing that they travel

in straight lines

(ii) They can be deflected by magnetic and electric fields, the direction of deflection showing that they are negatively charged

(iii) A freely moving paddle wheel, placed in their path, is set in motion showing that they possess momentum, i.e., particle nature

(iv) They cause many substances to fluoresce, e.g., the familiar zinc sulphide coated television tube

(v) They can penetrate thin sheets of metal

J.J Thomson (1897) extended these experiments and determined the velocity of these particles and their charge/

mass ratio as follows

The particles from the cathode were made to pass through a slit in the anode and then through a second slit

They then passed between two aluminium plates spaced about 5 cm apart and eventually fell onto the end of the tube, producing a well-defined spot The position of the spot was noted and the magnetic field was then switched

on, causing the electron beam to move in a circular arc while under the influence of this field (Fig 1.2.)

Fig 1.3 Millikan’s apparatus for determining the value of

the electronic charge

/

e e

e m

e m

= 1.6022 101119C 1

1.758820 10 C kg

-

-×

=

= 9.1094 × 10–31 kg (1.3)Small droplets of oil from an atomiser are blown into a still thermostated airspace between parallel plates, and the rate of fall of one of these droplets under gravity is observed, from which its weight can be calculated The airspace is now ionized with an X-ray beam, enabling the droplets to pick up charge by collision with the ionized air molecules

By applying a potential of several thousand volts across the parallel metal plates, the oil droplet can either be speeded

up or made to rise, depending upon the direction of the tric field Since, the speed of the droplet can be related to its weight, the magnitude of the electric field, and the charge it picks up, the value of the charge can be determined

elec-Thomson proposed that the amount of deviation of the

particles from their path in the presence of electrical and

magnetic field depends on

(i) Greater the magnitude of the charge on the

par-ticle, greater is the interaction with the tric and magnetic fields, and thus greater is the deflection

(ii) Lighter the particle, greater the deflection

(iii) The deflection of electrons from its original path

increase in the voltage across the electrodes, or the strength of the magnetic field

By careful and quantitative determination of the

mag-netic and electric fields on the motion of the cathode rays,

Thomson was able to determine the value of charge to

mass ratio as

e

e m

1.758820 10 C kg

me is the mass of the electron in kg and e is the

magni-tude of the charge on the electron in Coulomb

1.3.2 Charge on the electron

Thomson’s experiments show electrons to be negatively

charged particles Evidence that electrons were discrete

particles was obtained by Millikan by his well known oil

drop experiment during the years 1910-14 By a series of

very careful experiments Millikan was able to determine

the value – electronic charge, and the mass Millikan found

the charge on the electron to be –1.6 × 10–19 C The

pre-sent-day accepted value for the charge on the electron is

1.602 × 10–19 C When this value for ‘e’ is compared with

the most modern value of e/m, the mass of the electron can

be calculated

Fig 1.2 Thomson’s apparatus for determining e/m for the electron

Anode (+)

Cathode (–)

Spot of light

Spot of light

Fluorescent screen

Deflecting plates

Gas at low pressure

when top plate is positive

when plate is not charged

– +

noticed a particle of great penetrating power which was unaffected by magnetic and electric fields It was found to have approximately the same mass as the proton (hydrogen ion) The reaction is represented as

4Be+4He→6C+0n

Where the superscript refers to the atomic mass and the subscript refers to the atomic number (the number of protons in the nucleus) Notice that a new element, carbon, emerges from this reaction

1.4 Atomic models

The discovery that atoms contained electrons caused some consternation Left to themselves, atoms were known to be electrically neutral So, the negative charge of the electrons had to be balanced by an equal amount of positive charge

The puzzle was to work out how the two types of charges were arranged To explain this, different atomic models were proposed Two models proposed by J.J Thomson and Earnest Rutherford are discussed here though they cannot explain about the stability of atoms

1.4.1 thomson model of Atom

“A theory is a fool and not a creed.” J.J ThomsonSir Joseph John Thomson 1856-1940Thomson’s researches on the discharge of electric-ity through gases led to the discovery of the electron and isotopes

In 1898, Sir J.J Thomson proposed that the electrons are embedded in a ball of positive charge (Fig 1.5) This model of the atom was given the name plum pudding or raisin pudding or watermelon According to this model

1.3.3 discovery of Proton

If the conduction of electricity, through gases is due to

particles, which are similar to those involved during

elec-trolysis, it was to be expected that positive as well as

negative ones should be involved, and that they would be

drawn to the cathode By using a discharge tube

contain-ing a perforated cathode, Goldstein (1886) had observed

the formation of rays (shown to the right of the cathode in

Fig 1.4.)

Fig 1.4

J.J Thomson (1910) measured their charge/mass ratio

from which he was able to deduce that the particles were

positive ions, formed by the loss of electrons from the

re-sidual gas in the discharge tube The proton is the smallest

positively charged particle equal in magnitude to that on

the electron and is formed from the hydrogen atom by the

loss of an electron

H → H+ + e–Unlike cathode rays, the characteristics of posi -

tively charged particles depend upon the nature of the

gas present in the cathode ray tube These are positively

charged ions The charge to mass ratio of these parti

-cles is found to be dependent upon the gas from which

these originate Some of the positively charged particles

carry a multiple of the fundamental unit of electrical

charge The behaviour of these particles in the magnetic

or electric field is opposite to that observed for electron

or cathode rays

1.3.4 discovery of Neutron

The neutron proved to be a very elusive particles to track

down and its existence, predicted by Rutherford in 1920,

was first noticed by Chadwick in 1932 Chadwick was

bombarding the element beryllium with α-particles and

Fig 1.5 The Thomson model of atom The positive

charge was imagined as being spread over the entire atom and the electrons were put in this background

positive charge and four units of atomic mass He

conclud-ed that a-particles are helium nuclei as when a-particles combined with two electrons yielded helium gas b-rays are negatively charged particles similar to electrons The g-rays are high energy radiations like X-rays, are neutral in nature and do not consist of particles As regards penetrat-ing power, a-particles have the least followed by b-rays (100 times that of a-particles) and g-rays (1000 times that

in the field of radioactivity With Bohr and others, he elaborated a theory of atomic structure In 1919, he produced the first artificial transmutation of an element (that of nitrogen into oxygen) For many years he was

a vigorous leader in laying the foundations of the est developments in atomic science, which he did not live to see He died in 1937

great-In 1911, Earnest (later Lord) Rutherford

demonstrat-ed a classic experiment for testing the Thomson’s model

Rutherford, Geiger and Harsden studied in detail the effect

of bombarding a thin gold foil by high speed a-particle (positively charged helium particles) A thin parallel beam

of a-particles was directed onto a thin strip of gold and the subsequent path of the particles was determined Since the a-particles were very energetic, it was thought they would

go right through the metal foils To their surprise they observed some unexpected results which are summarized

as follows:

(i) It was observed that 99% of the alpha particles passed straight through the foil and struck the screen at the centre

(ii) A few of the alpha particles deflected from their original path through varying angles

(iii) Hardly one out of the 20, 000 a-particles was bounced back on its path

The results of Rutherford’s experiment is represented more explicitly in Fig 1.6

we can assume that just like the seeds of a watermelon are

embedded within the reddish juicy material, the electrons

are embedded in a ball of positive charge It is important to

note that in Thomson’s model, the mass of the atom is

uni-formly distributed over the atom Though this model could

explain the overall neutrality of the atom, but could not

explain the results of later experiments

the discovery of X-rays and Radioactivity

In 1895, Rontgen noticed when electrons strike a material

in the cathode ray tubes, produces a penetrating

radia-tion emitted from the discharge tubes, and it appeared to

originate from the anode The radiation had the following

properties:

(i) It blackened the wrapped photographic film

(ii) It ionized gases, so allowing them to conduct

electricity

(iii) It made certain substances fluoresce, e.g., zinc

sulphide

Furthermore the radiation was shown to carry no

charge since it could not be deflected by magnetic or

elec-tric fields Since Rontgen did not know the nature of the

radiation, he named them as X-rays and the name is still

carried on The true nature of X-rays was not discovered

until 1912, when it became apparent that its properties

could be explained by assuming to be wave like in

char-acter, i.e., similar to light but is much smaller wavelength

It is now known that X-rays are produced whenever

fast-moving electrons are stopped in their tracks by impinging

on a target, the excess energy appearing mainly in the form

of X-radiation

The year after Rontgen discovered X-rays, Henry

Becquerel observed that uranium salts emitted a

ra-diation with properties similar to those possessed by

Xrays The Curies followed up this work and discov

-ered that the ore pitchblend was more radioactive than

purified uranium oxide; this suggested that something

more intensely radioactive than uranium was

respon-sible for this increased activity and eventually the

Cu-ries succeeded in isolating two new elements called

polonium and radium, which were responsible for this

increased activity

In 1889, Becquerel reported that the radiation from

the element radium could be deflected by a magnetic field

and in the same year, Rutherford noticed that the

radia-tion from uranium was composed of at least two distinct

types Subsequently, it was shown that the radiation from

both sources contained three distinct components and are

named as a – b and g-rays Rutherford found that a-rays

consist of high energy particles carrying two units of

On the basis of above observations and sions, Rutherford proposed the nuclear model of atom as follows:

(i) Rutherford suggested that an atom has a centre of the nucleus, in which positive charge and mass are concentrated and he called this centre as the nucleus

The quantitative results of scattering experiments such

as Rutherford’s indicate that the nucleus of an atom has a diameter of approximately one to six Fermis (1 Fermi =

10–10 m) and atoms have diameters about 100000 times as great as the size of the nucleus, i.e., of the order 10–5 m

(ii) The atom as a whole is largely an empty space and the nucleus is located at the centre of the atom

(iii) The nucleus is surrounded by the electrons which

are revolving round the nucleus in closed paths like the planets.

(iv) Electrons and the nucleus are held together by electrostatic forces of attraction

Rutherford, therefore, contemplated the dynamical stability and imagined the electrons to be whirling about the nucleus, similar to the planetary motion

Drawbacks of the Rutherford model

Following objections were raised against the Rutherford’s model, and were reported:

(i) It did not explain how the protons could be packed to give a stable nucleus

(ii) When an electron revolves round the nucleus, it will radiate out energy, resulting in the loss of energy

This loss of energy will make the electron to move slowly and consequently, it will be moving in a spi-ral path and ultimately falling onto the nucleus So, the atom should be unstable, but the atom is stable

Fig 1.7 An electron that is accelerating, radiates energy

As it loses energy, it spirals onto the nucleus

(iii) If an electron starts losing energy continuously, the observed spectrum would be continuous and have broad bands merging into one another But most

of the atoms give line spectra Thus, Rutherford’s model failed to explain the origin of line spectra

Fig 1.6 Results of Rutherford’s experiment.

+ + + +

(a) One layer of metal atoms each with nucleus

+ +

Nucleus

(b) One atom of the metal with nucleus

Rutherford’s Explanation Rutherford pointed out

that his results are not in agreement with Thomson’s

mod-el He explained the above results by drawing the

follow-ing conclusions:

(i) As most of the alpha particles passed very nearly

straight through the foil, it means that the atom is extra ordinarily hollow with a lot of empty space inside

(ii) The only way to account for the large

deflec-tion is to say that the positive charge and mass

in the gold foil are concentrated in very small gions Although most of the alpha particles can

re-go through without any deflection, occasionally some of them which come closer to the region

of positive charge, they repel each other, and the repulsion may be big enough to cause the α-particles to undergo large deviations from its original path

(iii) Due to the rigid nature of the nucleus, some

a-particles on colliding with the positive charge, turn back on their original paths

neutrons The total number of protons and neutrons is called the mass number The particles present in the nu-cleus are called nucleons.

Mass number (A) = Number of protons (Z) + Number

of neutrons (n)

1.5.1 isobars, isotopes and isotones

In modern methods, the symbols of elements are written

as A[ ]n,

Z X x where the left hand superscript A is the mass number, left hand subscript Z is the atomic number, right hand superscript n is the number of charges and right hand

subscript is the number of atoms

Different atoms having same mass number but with different atomic numbers are called isobars, e.g., 14

6C and

7

14N The atoms having same atomic number but with ferent mass numbers are called isotopes The difference between isotopes is due to the difference in the number

dif-of neutrons present in the nucleus, e.g., 1 2 3

1H, D and T1 1are isotopes of hydrogen namely protium, deuterium and tritium respectively

Isotopes exhibit similar chemical properties be cause they depend on the number of protons in the nu-cleus Neutrons present in the nucleus show very little effect on the chemical properties of an element So, all the isotopes of a given element show same chemical behavior

-Sometimes atoms of different elements tain same number of neutrons Such atoms are called isotones For example, 13

con-6C and 14

7 N Isotones differ

in both atomic number and mass numbers but the difference in atomic number and mass number is the same

1.6 DevelOPmeNTS leADiNg TO The bOhR mODel OF ATOm

During the period of development of new models to prove Rutherford’s model of atom, two new concepts played a major role They are

(i) Dual behaviour of electromagnetic radiation

This means that light has both particle like and wave like properties.

(ii) Atomic spectra The experimental results ing atomic spectra of atoms can only be explained

regard-by assuming quantized (fixed) electronic energy levels in atoms

Let us briefly discuss about these concepts before studying a new model proposed by Niels Bohr known as Bohr Model of Atom

1.5 ATOmiC NUmbeR

In 1913, H.G.J Moseley, a young English Physicist and

one of Rutherford’s brilliant students, examined the

X-ray spectra of 38 elements When Moseley bombarded

different elements with cathode rays, the X-rays were

generated which had different frequencies He suggested

that the frequency of X-rays produced in this manner was

related to the charge presented on the nucleus of an atom

of the element used as anode When he took the

frequen-cies of any particular line in all elements such as Ka or Kb

lines, then the frequencies were related to each other by

the equation

v = a (Z–b) (1.3)=a Z v

-Where, v is the frequency of any particular Ka or Kb etc,

line Z is the atomic number of the element and ‘a’ and ‘b’

are constants for any particular type of line A plot of v

vs Z gives a straight line showing the validity of the above

equation This is shown in Fig 1.8

Fig 1.8 Variation of x – rays frequency with atomic

number

However, no such relationship was obtained when

the frequency was plotted against the atomic mass

Moseley, further found that the nuclear charge

in-creases by one unit in passing from one element to the

next element arranged by Mendeleef in the order of

increasing atomic weight The number of unit positive

charges carried by the nucleus of an atom is termed

as the atomic number of the element Since, the atom

as a whole is neutral, the atomic number is equal to

the number of positive charges present in the nucleus.

Atomic number = Number of protons present in the

nucleus

= Number of electrons present outside the nucleus of

the same atom.

While the positive charge of the nucleus is due to

protons, the mass of the nucleus is due to protons and

Characteristics of the Wave motion

1 Wavelength (l): It is the distance between two est crests or troughs It is denoted by the Greek letter Lambda l and is expressed in Angstrom units (Å) It

near-is also expressed as micron meter (m), mill micron ter (mm), nanometer (nm), picometer (pm) etc These units are related to SI unit (m) as

me-1 Å = me-10–10 m; 1 m = 10–6 m, 1 mm = 10–9 m,

1 nm = 10–9 m; 1 pm = 10–12 m

2 Frequency (n): The number of waves passing through

a given point in a unit time is known as its frequency

It is denoted by the Greek letter ν (nu) The frequency

is inversely proportional to the wavelength Its unit in cycles per second (cps) of Hertz (Hz); 1 cps = 1 Hz A cycle is said to be completed when a wave consisting crest and trough passes through a point

3 Velocity (c): The distance travelled by the wave in one second is called the velocity of wave It is equal to prod-uct of wavelength and frequency of the wave Thus,

c = vl

or v = c

4 Amplitude: It is the height of crest or depth of wave’s

trough and is generally expressed by the letter ‘a’

The amplitude of the wave determines the intensity or brightness of radiation

5 Wave number ( )v : It is equal to the reciprocal of

wavelength In other words, it is defined as the number

of wavelengths per centimeter It is denoted by v and

v c

The important characteristics of electromagnetic radiations are

(i) These consist of electric and magnetic fields that oscillate in the directions perpendicular to the di-rection in which the wave is travelling as shown in

1.6.1 Nature of light and electromagnetic

Radiation

Fig 1.9 Diffraction of the waves while passing through

a slit

Fig 1.10 Propagation of wave

A radiation is a mode of transference of energy of

differ-ent forms Light, X-rays and g-radiations are examples of

the radiant energy The earliest view of light, due to

New-ton, regarded light as made up of particles (commonly

termed as corpuscles of light) The particle nature of light

explained some of the experimental facts such as reflection

and refraction of light However, it failed to explain the

phenomenon of interference and diffraction The

corpuscu-lar theory was therefore, discarded and Huygens proposed

a wave-like character of light With the help of wave theory

of light, Huygens explained the phenomena of interference

and diffraction

We know that when a stone is thrown into water of a

quiet pond, on the surface of water waves are produced

The waves originate from the centre of the disturbance

and propagate in the form of up and down movements

The point of maximum upward displacement is called the

crest and the point of maximum downward displacement

is called trough Thus, waves may be considered as

dis-turbance which originate from some vibrating source and

travel outwards as a continuous sequence of alternating

crests and troughs as shown in Fig 1.10

between 3800 Å to 7600 Å The different colours in the visible light corresponds to radiations of different wave-lengths In addition to visible light there are so many other electromagnetic radiations, such as X-rays, ultraviolet rays, infra-red rays, microwaves and radiowaves.

The arrangement of different types of netic radiations in the order of increasing wavelengths (or decreasing frequencies) is known as electromagnetic

electromag-spectrum

Different regions of electromagnetic spectrum are identified by different names The complete electromag-netic spectrum is shown in Fig 1.12

The various types of electromagnetic radiations have different energies and are being used for different purposes

as listed in Table 1.2

Table 1.2 Some applications of electromagnetic

wavesName Frequency Wavelength Uses

“pictures”, material testing

Radio frequency

transmission

Fig 1.11 The two field components have the same wavelength and frequency

(ii) All electromagnetic waves travel with the same

speed In vacuum, the speed of all types of tromagnetic radiation is 3 00 × 108 ms–1 This speed is called the velocity of light

(iii) These electromagnetic radiations do not

require any medium for propagation For ex ample, light reaches us from the sun through empty space

-Fig 1.11 Electric and magnetic fields association with an

electromagnetic wave

electromagnetic Spectrum

The different electromagnetic radiations have different

wavelengths The visible light in the presence of which

our eyes can see, contains radiations having wavelength

Fig 1.12 Complete electromagnetic spectrum

Cosmic rays

1.6.2 Quantum Theory of Radiation

When an object is heated, its colour gradually changes

For example, a black coal becomes red, orange, blue and finally ‘white hot’ with increasing temperature This shows that red radiation is most intense at a particular tempera-ture, blue radiation is most intense at another temperature and so on It indicates that the intensities of radiations of different wavelengths emitted by a hot body depend upon the temperature The curves representing the distribution

of radiation from a black body at different temperatures are shown in Fig 1.13

Fig 1.13 Energy distribution for black body radiation at

different absolute temperatures

A black body is one which will absorb completely the incident radiation of wavelength The ideal black body does not reflect any energy, but it does radiate energy

Although no actual body is perfectly black, it is possible

to postulate a condition where radiation is completely closed in a space surrounded by thick walls at a constant temperature Thus, we may imagine a cavity inside a solid body containing a small hole in one wall whereby the in-vestigator may study the enclosed radiation without alter-ing its character When a beam of light is passed through the hole, it would be absorbed by the inner walls of the enclosure and their temperature increases As soon as the beam was removed, since the enclosed kept at a constant temperature the excess energy emerges through the hole

en-in the form light This gives a disten-inctive spectrum, ally referred to as the black body spectrum If the ener-

usu-gy emitted is plotted against its frequency or wavelength where n = c / l, the curve shows a maximum, as indicated in Fig 1.13 The graph shows that the energy emitted is

Solved Problem 1

Calculate and compare the energies of two radiations,

one with a wavelength of 8000 Å and the other with 4000

19 2

2.475 104.95 10

-

-×

=

×

or 1 2

12

E

E =

or 2E1 = E2

Solved Problem 2

A radio station is broadcasting a programme at 100 MHz

frequency If the distance between the radio station and the

receiver set is 300 Km, how long would it take the signal to

reach the receiver set from the radio station? Also calculate

wavelength and wave number of these radio waves

Solution:

All electromagnetic waves travel in vacuum or in air

with the same speed of 3 × 10–8 m s–1

Distance 300 1000m 1 10 sVelocity 3 10 m s

-

-×

×Calculate of wavelength (l),

Quantized energy (a)

When a beam of light falls on a clean metal plate in

vacu-um, the plate emits electrons This effect was discovered

by Hertz in 1887, and is known as the photoelectric effect

The metal surface emits electrons by the action of light, that can be demonstrated using negatively charged gold leaf electroscope (Fig 1.15) As the light from carbon arc falls

on the metal plate, the diversion of leaves is reduced slowly

This shows that electrons come out of all metal plates

Extensive studies have revealed the following facts:

(i) Photoelectric effect is instantaneous, i.e., as soon

as the light rays fall on the metal surface, electrons are ejected

(ii) Only light of a certain characteristic frequency is required for expulsion of electrons from a particu-lar metal surface

Fig 1.15 (a) Einstein’s explanation of photoelectric

ef-fect, and (b) Experimental device for photoelectric effect

greatest at the middle wavelengths in the spectrum and

least at the highest and lowest frequencies If curves are

plotted for a series of temperatures, as given in Fig 1.13 It

is found that the maximum moves towards shorter

wave-lengths as the temperature rises

max Planck 1858-1947

A German physicist He received his Ph D in

theoreti-cal physics from university of Munich in 1879 Max

Planck was well known for his Quantum theory which

won the Nobel Prize in physics in 1918 He also made

significant contributions in thermodynamics and other

areas of Physics

The shapes of the curves could not be explained on the

basis of wave theory of radiation Max Planck in 1900

re-solved this discrepancy by postulating the assumption that

the black body radiates energy — not continuously but

dis-continuously in the form of energy packets called quanta

The general quantum theory of electromagnetic radiations

can be stated as

(i) When atoms or molecules absorb or emit the

radi-ant energy, they do so in separate units of waves

These waves are called quanta or photons

(ii) The energy of a quantum or photon is given by

E = hv (1.7) where, v is the frequency of the emitted radiation

and h is the Planck’s constant.

(iii) An atom or molecule emits or absorbs either one

quantum of energy (hv) or any whole number

multiples of this unit

This theory provided the basis for explaining the

pho-toelectric effect, atomic spectra and also helped in

under-standing the modern concepts of atomic and molecular

structure

Quantization of energy

The restriction of any property to discrete values is called

quantization A quantity cannot vary continuously to have

any arbitrary values but can change only

discontinu-ously to have some specific values For example, a ball

moves down a staircase (Fig 1.14a) then the energy of the

ball changes discontinuously and it can have only certain

definite values of energy corresponding to the energies

of various steps Energy of the ball in this case is

quan-tized On the other hand, if the ball moves down a ramp

(Fig 1.14 b), then the energy of the ball changes continuously

and the ball can have any value of energy corresponding to

any point on the ramp Energy in this case in not quantized

19 10

v

h

-

-

-

Solution:

(i) Energy of the photon

14 1 9

3 10 m s

11.824 10 s253.7 10 m

c v

-

\ 7.83 101919J 4.887 eV1.602 10 J

E

-

-×

× (ii) Kinetic energy of the photonWork function = 4.65 eV

KE = hv – Work function

= 4.887 eV – 4.65 eV = 0.237 eV

(iii) The number of electrons emitted are directly

pro-portional to the intensity of the light

(iv) The gas surrounding the metal plate has no effect

on photoelectric transmission

(v) Kinetic energy of photoelectron depends upon the

nature of metal

(vi) For the same metal kinetic energy of

photoelec-trons varies directly with the frequency of light

If the frequency is decreased below a certain value

(Threshold frequency), no electrons are ejected at all

The classical wave theory of light was completely

in-adequate to interpret all these facts Einstein applied the

quantum theory of radiation to explain this phenomenon

An electron in a metal is found by a certain amount of

energy The light of any frequency is not able to cause the

emission of electrons from the metal surface There is

cer-tain minimum frequency called the threshold frequency

which can just cause the ejection of electrons Suppose the

threshold frequency of the light required to eject electrons

from the metal is v0 When photon of light of this frequency

strikes the metal surface, it imparts its entire energy (hv0) to

the electron This enables the electrons to break away from

the atom by overcoming the attractive influence of the

nucleus Thus, each photon can eject one electron If the

frequency of the light is less than n0, the electrons are not

ejected If the frequency of the light (say v) is more than

v0, more energy is supplied to the electron The remaining

energy, which will be the energy hv imparted by incident

photon and the energy used up, i.e., hv0, would be given as

kinetic energy to the emitted electron Hence,

2 0

12

where 1

2mv

2 is the kinetic energy of the emitted

electron and hv0 is called the work function

or 1

2mv

2 = h(v – v0) (1.9)This equation is known as Einstein’s photoelectric

equation If the frequency of light is less than the

thresh-old frequency, there will be no ejection of electrons Values

of photoelectric work functions of some metals are given

Also according to Einstein’s mass energy equation E

= mc2 where m is the mass of photon From the above two

equations, we get

2

hv m c

de-to pass through a prism, it splits up inde-to a continuous band

of seven colors from red to violet as observed in a rainbow

This phenomenon is known as dispersion and the pattern

of colors is called a spectrum The red color light radiation having longest wavelength will be deviated least while the radiation of violet color having shortest wavelength will be deviated more The spectrum of white light that we can see ranges from violet at 7.50 × 1014 Hz to red at 4 × 1014 Hz

In this spectrum one color merges into another color

adjacent to it viz violet merges into blue, blue into green and

so on When electromagnetic radiation interacts with ter, atoms and molecules may absorb energy and reach to a higher energy state While coming to the ground state (more stable lower energy state) the atoms and molecules emit ra-diations in various regions of the electromagnetic spectrum

er energy are said to be excited While coming to the ground (normal) state the excited species emit the absorbed energy The spectrum of such emitted radiation is called the emission spectrum Emis-sion spectra are further classified according to its appearance as continuous, line and band spectra

1.6.4 Compton effect

According to classical electromagnetic wave theory,

mono-chromatic light falling upon matter should be scattered

with-out change in frequency But when X-rays are impinged

on matter of low atomic weight, X-rays of slightly longer

wavelength than those of impinging beam were produced

In other words when a photon collides with an electron, it

may leave with a lower frequency, and the electron thereby

acquires a greater velocity In the collision the energy of the

photon is reduced and the energy of the electron is increased

This is called Compton effect and can be explained as a

result of the impact between two bodies, the photon and the

electron and it is of course additional proof of the

corpus-cular nature of light This impact is elastic, and both kinetic

energy and momentum are conserved After the encounter

each body has a different energy and a different momentum

from the values it had before the contact, but the sum of the

energies and the sum of the momenta are unchanged

1.6.5 Dual Nature of electromagnetic

Radiations

As described, photoelectric effect and Compton effect could be

explained considering that electromagnetic radiations

consist-ing of small packets of energy called quanta or quantum (or

single photon) These packets of energy can be treated as

parti-cles, on the other hand, radiations exhibit phenomena of

inter-ference and diffraction which indicate that they possess wave

nature So, it may be concluded that electromagnetic radiation

possesses the dual nature, i.e., particle nature as well as wave

nature Einstein (1905) even calculated the mass of the photon

associated with a radiation of frequency v as given below:

The energy E of the photon is given as E = hv

Albert einstein 1879-1955

Born in Germany but later shifted to America He was

regarded as one of the two great Physicists the world has

known, the other being lssac Newton He is well known

for his three research papers on special relativity,

Brown-ian motion and photoelectric effect These research papers

were published in 1905 while he was working as a technical

assistant in a Swiss patent office in Berne He received the

Nobel Prize in Physics in 1921 for his explanation of the

photoelectric effect His work has influenced the

develop-ment of physics in significant manner

“To him who is discoverer in the field (or science),

the products of his imagination appear so necessary and

natural that he regards them, and would like to have

them regarded by others, not as creations of thought but

as given realities”— Albert Einstein

(ii) Absorption spectra It is produced when the light from a source emitting a continuous spectrum is first passed through an absorbing substance and then recorded after passing through a prism in a spectroscope In that spectrum, it will be found that certain colours are missing which leave dark lines or bands at their places This type of spec-trum is called the absorption spectrum Similar to emission spectra, absorption spectra are also of three types:

Prism Increasing wavelength

Absorption spectrum

Source of white light

Source of white light

Continuous emission spectrum of white light/sunFig 1.18 Emission and absorption spectra

(a) Continuous absorption spectrum This type

of spectrum arises when the absorbing terial absorbs a continuous range of wave-lengths An interesting example is the one in which red glass absorbs all colors except red and hence, a continuous absorption spectrum will be obtained

(b) Line absorption spectrum In this type, sharp dark lines will be observed when the

(b) Line spectrum Line spectrum is obtained when the light emitting substance is in the atomic state Hence, it is also called as the atomic spec-trum Line spectrum consists of discrete wave-lengths extended throughout the spectrum and are generally obtained from the light sources like mercury, sodium, neon discharge tube, etc

(c) Band spectrum This type of spectrum arises when the emitter in the molecular state is ex-cited Each molecule emits bands which are characteristics of the molecule concerned and that is why we call this as molecular spec-trum also The sources of band spectrum are (i) carbon with a metallic salt in its core, (ii) vacuum tube, etc

In the emission spectra bright lines on black

back-ground will appear

Emission spectrum of Barium

Absorption spectrum of Barium

Fig 1.17 Illustrating Emission and Absorption Spectra

It may be noted that the dark lines in absorption

spec-tra appear exactly at the same place where the coloured

lines appear in the emission spectra

hydrogen Spectrum

As explained above, each element emits its own istic line spectrum which is different from that of any other element Since, hydrogen contains only one electron, its spectrum is the simplest to analyze

character-To get the spectrum of hydrogen, the gas is enclosed in

a discharge tube under low pressure and electric discharge

is passed through it The hydrogen molecules dissociate into atoms and get excited by absorbing the energy: On the principle that what goes up must come down, sooner or later the atoms must loose their energy and fall back to the ground state, by loosing the energy as light The light that

is given out can be measured in a spectrometer and the tern recorded on a photographic paper The pattern is called hydrogen spectrum

pat-The hydrogen spectrum obtained consists of a series

of lines in the visible, ultraviolet and infrared regions

These have been grouped into five series in Fig 1.19 which are named after their discoverers These are given

in Table 1.3 with the year when discovered The Balmer series happened to be the first series of lines in the hy-drogen spectrum This was because the lines were in the visible part of the spectrum and therefore the easiest to observe

Balmer (1885) discovered a relationship between the wave number and the position of the line in the series

absorbing substance is a vapour or a gas The spectrum obtained from sun gives Fraunhofer absorption lines corresponding to vapours of different elements which are supposed to be present on the surface of the sun

(c) Band absorption spectrum When the sorption spectrum is in the form of dark bands, this is known as band absorption spectrum

ab-An interesting example is that of an aqueous solution of KMnO4 giving five absorption bands in the green region

The pattern of lines in the spectrum of an element is

characteristics of that element and is different from those

of all other elements In other words, each element gives a

unique spectrum irrespective of even the form in which it

is present For example, we always get two important lines

589 nm and 589.6 nm in the spectrum of sodium whatever

may be its source It is for this reason that the line spectra

are also regarded as the fingerprints of atoms

Since atoms of different elements give

characteris-tic sets of lines of definite frequencies, emission spectra

can be used in chemical analysis to identify and estimate

the elements present in a sample The elements rubidium,

caesium, thallium, gallium and scandium were discovered

when their minerals were analyzed by spectroscopic

meth-ods The element helium was discovered in the Sun by

spectroscopic method

Brackett Series

(a) Relative location of the Lyman, Balmer, Paschen,

Brackett and Pfund series of hydrogen spectrum.

Solved Problem 7

In hydrogen atom, an electron jumps from fourth orbit to first orbit Find the wave number, wavelength and the en-ergy associated with the emitted radiation

In 1913, he returned to Copenhagen and in 1920 he was named as Director of the Institute of Theoretical Physics

Bohr’s theory of atomic structure, which was sented in 1913, laid a broad foundation for the great atomic progress of recent years

pre-Second only in importance to his celebrated theory are the facts that it was in Bohr’s laboratory that the implications of nuclear fission were first predicted and that he obtained an understanding of nuclear stability that contributed greatly to the spectacular development

of atomic energy

After the first world war, Bohr worked energies for peaceful uses of atomic energy He received the first Atoms for Peace award in 1957 Bohr was awarded the Nobel Prize in Physics in 1922

“ The very word “experiment” refers to a ation where we can tell others what we have done and what we have learned” — Neils Bohr

situ-“Our experiments are questions that we put to Nature.”

— Neils Bohr

In order to explain the line spectra of hydrogen and the overcome to objections leveled against Rutherford’s model of the atom, Neils Bohr (1913) proposed his quantum mechani-cal structure of the atom He made use of quantum theory, according to which energy is lost or gained not gradually, but

in bundles or quanta The concept of nucleus explained by Rutherford was retained in Bohr’s model Through this model does not meet the modern quantum mechanics, it is still in use

to rationalize many points in the atomic structure and spectra

The important postulates of the Bohr Theory are

1 Electrons move in certain fixed orbits associated with

a definite amount of energy The energy of an electron moving in an orbit remains constant as long as it stays

in the same orbit called stationary state or orbit.

v is the wave number and m is an integer having

val-ues 3, 4, 5, 6, etc Ritz (1908) gave a generalization known

as Ritz Combination principle which was nicely

applica-ble to hydrogen spectrum According to this principle, the

wave number in any line in a series can be represented as a

difference of two square terms, one of which is consistent

and the other varies throughout the series Thus,

where, RH is Rydberg constant

All the lines appearing in the hydrogen spectrum are

governed by the above equation For,

An electronic transition from M shell (m = 3) to K shell

(n = 1) takes place in a hydrogen atom Find the wave

num-ber and the wavelength of radiation emitted (R = 1,09,677

v

Based on the above postulates, Bohr calculated the radii of the various orbits and the energies associated with the electrons present in those orbits The frequen-cies of the spectral lines determined experimentally by Lyman, Balmer and others are in excellent agreement with those calculated by Bohr’s theoretical equations.

1.7.1 Bohr’s Theory of the Hydrogen Atom

Bohr pictured the hydrogen atom as a system consisting

of a single electron with a charge designed as e, rotating

in a circular orbit of radius r about the nucleus of charge

Ze with a velocity v.

Tangential velocity of the revolving electron

Calculation of velocity V: The angular momentum of the electron is defined as the product of the velocity of the electron in its orbit, its mass, and the radius of its orbit The

product is symbolized as mvr According to Bohr’s lates, the angular momentum are whole multiples of h / 2p

postu-Therefore, according to this quantum restriction, gular momentum may be restricted as

an-mvr

2

nh mvr=p

\ The velocity of the electron in an orbit, v

2

nh v mr

=

Radius of an Orbit

According to Coulomb’s law, the electrostatic force of

attrac-tion Fe between the charges may be evaluated mathematically as

F e

2 2

e

Ze F r

By definition, the magnitude of the centrifugal force

Fc for an electron of mass m, with a velocity in its orbit of

V and with an orbit radius r is given as

2

c

mv F r

Hence, a certain fixed amount of energy is associated

with each electron in a particular orbit Thus, stationary

orbits are also known as energy levels or energy shells

Bohr gave number 1, 2, 3, 4 etc, (starting from the nucleus

to these energy levels) The various energy levels are now

designated as K, L, M, N etc, and are termed as principal

quantum numbers

2 Energy is emitted or absorbed only when an electron

jumps from one orbit to another, i.e., form one energy

level (or principal quantum number) to another

Since each level is associated with a definite amount

of energy, the farther the energy level from the nucleus, the

greater is the energy associated with it

P O N M

3 Permissible orbits are those for which the angular

mo-mentum is an integral multiple of (h/2 p).

The electrons move in such orbits without any loss of

energy Thus, angular momentum

where, n is an integer 1, 2, 3 me is mass of

elec-tron and r is the radius of orbit The n values corresponds

to the principal quantum number

4 When an electron gets sufficient energy from outside,

an electron from an inner orbit of lower energy

state E1 moves to an outer orbit of higher energy

state E2.

The excited state lasts for about 10–8 s The difference

of energy E2 –E1 is thus radiated out During the emission

or absorption of radiant energy to Planck’s, Einstein

equa-tion E = hv is obeyed Thus the frequency of emitted

where, En is the total energy of the electron in the orbit

designated by the quantum number n.

mZe

=p Since, n = 1 and Z for hydrogen = 1

2

2 24

h r me

=p

2 27

2

nh V mr

=p

h

Zn

n =2π22 4× 22 Since, n = 1, and Z = 1

Energy of electron in the first orbit

2 4 2

2 me E

or KJ mol–1 or electron volt (eV)One erg/molecule = 1.44 × 1013 Kcal / mol

= 6.22 × 1013 KJ / mol

\ E1 = (–2.179 × 10–11) × (1.44 × 1013) = –313.77 Kcal / mol = –1312.19 KJ / molNow 1eV = 1.602 × 10–12 ergs

\ 1 1112

2.179 10 erg

13.6 /atom1.602 10

-

-×

-×

In order that the electron’s orbit may remain stable, it

is necessary to assume that the two forces, electrostatic and

centrifugal, are equal and opposed to each other From this

assumption may be deduced the following mathematical

relationship

mvr

Zer

2 2 2

mZe

=

Consequently, the radius r0 of the smallest orbit (first

orbit) for the hydrogen atom is

2

0 4 2 2

h r me

=

energy of the electron

By definition the kinetic energy of body is equal to 1 2

2mv

The total energy E of the electron is the sum of its kinetic

and potential energies If the potential energy of an electron

is taken as zero when it is at an infinite distance from the

nucleus, the value at a distance r is given by –Ze2 / r This

value may be obtained by integrating Eq (1.24) between

the limits of r and infinity The negative sign indicates that

work must be performed on the electron to transfer it to

infinity Therefore, the total energy of the electron is

2

mv Ze E

and upon combining Eqs (1.30) and (1.31) the

follow-ing expression may be derived

22

Ze E r

Now if we substitute the value r from Eq (1.28) into

(1.32) the total energy of the electron may be stated as

If the term wave number is used for frequency

v v c

All the terms in the fraction 2p2Z2 e4 m/h3c are

constant For hydrogen atom Z is 1 If we evaluate the

fraction from the constant terms, the following value is obtained:

2 2 4

1 3

equa-in Eq (1.15)

1.7.3 limitations of the bohr’s model

Bohr’s atomic model was a very good improvement over Rutherford’s nuclear model and enabled in cal-culation of radii energies of the permissible orbits in the hydrogen atom The calculated values were in good agreement with the experimental values Bohr’s theory could also explain the hydrogen spectra successfully and also the spectra of hydrogen like atoms (He+, Li2+,

Be3+, etc) The theory was, therefore, largely accepted and Bohr was awarded Nobel prize in recognition of

1.7.2 Origin of Spectral lines and the

hydrogen Spectrum

As described by Bohr, energy is radiated when an electron

moves from one orbit of a quantum number of n2 to an

orbit of quantum number n1, where n1 is the inner orbit

Energy is absorbed if the electron moves in opposite

di-rection, i.e., from inner orbit to outer orbit The energy

DE which is emitted may be represented as a difference

in energy of the two electronic states and can be indicated

Spectral lines are produced by the radiation of

pho-tons, and the position of the lines on the spectral scale is

determined by the frequency, or frequencies, of the

pho-tons emitted Transitions to the innermost level n1 from

orbits n2, n3, n4 etc, gives rise to the first, second, third etc,

lines of the Lyman series in the UV region Transitions

from the outer most energy levels gives rise to spectral

lines of higher frequencies Transition of electron to n2

level from outer level gives Balmer series of lines in the

visible region Similar transitions from higher orbits to

the third orbit (n = 3) produces Paschen series in the

infra-red region Other series of lines have been discoveinfra-red for

similar shifts in the far infrared region A sketch

indicat-ing the transitions which produces these spectral lines is

given in Fig 1.21

From the Eq (1.34) the difference between the

energies of an electron in the two orbits n2 and n1 may be

Brackett series

Paschen series

Balmer

series

Lyman series

Fig 1.21 Origin of emission spectrum of hydrogen atom

louis de broglie 1892–1987

He was a French physicist At the beginning he studied history but while working on radio communications in first world war as an assignment, he developed interest

in science He received his Dr Sc from the university of Paris in 1924 He war professor of theoretical physics

at the university of Paris from 1932 to 1962 He was awarded Nobel Prize in physics in 1929

By making use of the Einstein’s (E = mc2) and Planck’s

quantum theory, (E = hv) de Broglie deduced a

fundamen-tal relation called the de Broglie equation:

h mv

This equation gives the relationship between the wavelength of the moving particle and its mass In the

Eq (1.39), l is the wavelength of the wave associated with

an electron of mass ‘m’ moving with velocity v.

Eq (1.39) can be written as

In the E.q (1.42) ‘p’ represents the momentum mv of

particle; and l corresponds to the wave character of matter

and p its particle character Thus, the momentum (p) of a moving particle is inversely proportional to the wavelength

of the waves, associated with it.

The revolutionary postulate of de Broglie received direct experimental verification in 1927 by Davisson and Germer, G.P Thomson and later by Stern They showed that heavier particles (H2, He etc.) showed diffraction pat-terns when reflected from the surface of crystals Particu-larly Davisson and Germer found that electron beam was diffracted when striken on a single crystal of nickel, which proves the wave like character of electrons

de brolie’s Relationship and bohr’s Theory

Application of de Broglie’s relationship to a moving tron around a nucleus puts some restrictions on the size

elec-of the orbits It means that electron is not a mass particle moving in a circular path but instead a standing wave train (non-energy, radiating motion) extending around the nucle-

us in the circular path as shown in Fig 1.22 (a) and 1.22 (b)

For the wave to remain continually in phase, the cumference of the orbit should be an integral multiple of wavelength l,

cir-this work However, Bohr’s model could not explain

the following points:

(i) Bohr’s model could not explain the spectra of

at-oms containing more than one electron Bohr’s model could not explain hydrogen spectrum ob-tained using high resolution spectroscopes Each spectral line, on high resolution was found to con-sist of two closely spaced lines

(ii) It was observed that in the presence of a magnetic

field each spectral line gets splitted up into closely spaced lines This phenomenon, known as Zee-man effect, could not be explained by the Bohr’s model Similarly, the splitting of spectral line un-der the effect of applied electric field, known as Stark effect, could not be explained by the Bohr’s model

(iii) Bohr’s model could not explain the spectra of

atom of elements other than hydrogen

(iv) Bohr’s model could not explain the ability of

atoms to form molecules and the geometry and shapes of molecule

1 8 WAveS AND PARTiCleS

Because of the limitations of Bohr’s atomic model

several scientists tried to develop a more subtle and

general model for atoms During that period two

im-portant proposals were contributed significantly for

the modern quantum mechanical model of the atom

They are

(i) Dual nature, i.e., wave as well as particle nature of

matter

(ii) Heisenberg’s uncertainty principle

1.8.1 Dual Nature of matter

According to Maxwell’s concept, light, radiation

con-sists of waves while Planck’s quantum theory considers

photons as particles Thus, light, which consists of

elec-tromagnetic radiation, is both a wave and particle Based

on this analogy in 1924, the Frenchman, Prince Louis de

Broglie published an exceedingly complicated account

of the wave-particle duality de Broglie stated that any

form of matter such as electron, proton, atom or

mole-cules, etc, has a dual character The waves predicted by

de Broglie are known as matter waves These waves are

quite different from electromagnetic waves in following

two respects:

(i) Matter waves cannot radiate through empty space

like the electromagnetic waves

(ii) Speed of matter waves is different from that of

electromagnetic waves

meaning of y and Its Significance

y is the wave function or the amplitude of the wave The value of amplitude increases and reaches the maximum which is indicated by peak in the curve This is shown by the upward arrow in the figure The value of the ampli-tude decreases after reaching the maximum value This is shown by the downward arrow

Above the X-axis the amplitude is shown as +ve, along the X-axis it is zero and below the X-axis it is nega-tive The intensity of light is proportional to the square of

amplitude (or A2) Therefore, A2 can be taken as a measure

of the intensity of light since light is considered to consist

of photons (corpuscular theory), the density of photons is

considered to be proportional to A2 Thus, as far as light

waves are concerned A2 indicates the density of photons in space or the intensity of light

9.1 10 kg 10 ms

h mv

Cal-Fig 1.22 Diagrammatic representation of electron orbits

one of which is (a) in phase; and the other (b) out of phase

Substituting the value of l in Eq (1.43), we get

2 r nh mv

p =

or

2

nh mvr=

which is the same as Bohr’s postulate for angular

moment of electron From the E.q (1.45) it can be known

that electrons can move only is such orbits for which the

angular momentum must be an integral multiple of h/2π

If the circumference is bigger or smaller than the value

as given by in Fig 1.22 b, the electron wave will be out

of phase Thus, de Broglie relation provides a theoretical

basis for the Bohr’s postulate for angular momentum In

Fig 1.22 (a) the wave is in phase continually

In the summer of 1927 physicists from all over the

world arrived in Brussels at the Solway congress At

this congress de Broglie’s concept on the relationship

between waves and particles was totally rejected For

many years to come, a complete different

representa-tion of this relarepresenta-tionship led the way It was Heisenberg

and Schrodinger who supported and strongly

repre-sented the concept of de Broglie at another congress

and got it accepted

de Broglie’s equation is true for material particles of all

sizes and dimensions However, in the case of small micro

objects like electrons, the wave character is of significance

only In the case of large macro-objects the wave character

is negligible and cannot be measured properly Thus, de

Broglie equation is more useful for small particles

“For a subatomic object like electron, it is impossible

to simultaneously determine its position and velocity at any given instant to an arbitary degree of precision.”

Heisenberg gave mathematical relationship for the certainty principle by relating the uncertainty in position (Dx) with uncertainty in momentum (Dp) as

The ≥ in the Eq (1.46) means that the product of Dx and

Dp can be either greater than or equal to but would be never smaller than h / 4p But h / 4p is constant Therefore, it follows from Eq (1.46) that smaller the uncertainty in locating the exact position (Dx), greater will be the uncertainty in locating

the exact momentum (Dp) of the particle and vice versa.

As Dp is equal to mDv the Eq (1.46) is equivalent to

saying that position and velocity cannot be simultaneously determined to an arbitrary precision

However, in our daily life, these principles have no significance This is because, we come across only large objects The position and velocity of these objects can be determined accurately because in these cases the changes that occur due to the impact of light are negligible The mi-croscopic objects suffer a change in position or velocity as

a result of the impact of light For example, to observe an electron, we have to illuminate it with light or electromag-netic radiation The light must have a wavelength smaller than the wavelength of electron When the photon of such light strikes the electron, the energy of the electron chang-

es In this process, no doubt, we shall be able to calculate the position of the electron, but we would know very little about the velocity of the electron after the collision

1.8.3 Significance of Uncertainty Principle

The most important consequence of the Heisenberg

un-certainty principle is that it rules out existence of nite paths or trajectories of electrons and other similar particles The trajectory of an object is determined by

defi-its location and velocity at various moments At any ticular instant if we know the position of a body and also its velocity and the forces acting on it at that instant we can predict at what position it will present at a particular time This indicates that the position of an object, and its velocity completely determine its trajectory Because, it

par-is not possible simultaneously to determine the position and velocity of an electron at any given instant precisely

it is not possible to talk of the trajectory of an electron

The effect of Heisenberg uncertainty principle is nificant only for motion of microscopic objects and is neg-ligible for that of macroscopic objects This can be seen from these illustrated examples

2 2.8 109.1 10 m s

- -

-× ×

=

×25

2 2 31

2 2.8 10

m s9.1 10

- -

h mv

He received his Ph D in physics from the university

of Munich in 1923 Later he worked with Max Born

at Gottingen and with Niels Bohr at Copenhagen He

spent about 14 years (1927 – 1941) as professor of

physics at the university of Leipzig He lead the

Ger-man team working for atomic bomb during second

world war Once the war ended he became the director

of Max Planck Institute for physics in Gottingen He

was awarded Nobel Prize in physics 1932

“The exact sciences also start from the assumption

that in the end, it will always be possible to understand

nature, even in every new field of experience but we

make no priori assumptions about the meaning of the

word understand

— W HeisenbergAccording to classical mechanics, a moving electron

is considered to be a particle Therefore, its position and

momentum could be determined with a desired accuracy

On the other hand de Broglie also considered a moving

electron to be wave-like Therefore, it becomes impossible

to locate the exact position of the electron on the wave

be-cause it is extending throughout a region of space So, the

following fundamental question arises:

“If an electron is exhibiting the dual nature, i.e., wave

and particle, is it possible to know the exact location of the

electron in space at some given instant.”

The answer to the above question was given by

Heisenberg in 1927 who stated that

Concept of Probability

The consequences of the uncertainty principle are far reaching and probability takes the place of exactness in ve-locity (which is related to kinetic energy) of an electron

The Bohr concept of the atom, which regards the electrons

as rotating in definite orbits around the nucleus, must be abandoned and should be replaced by a theory which con-siders probability of finding the electrons in a particular region of space This means that, it is possible to state the probabilities of the electron to be various distances with respect to the nucleus In the same way, probable values of velocity can also be given It should be clearly understood, however, that a knowledge of probability for an electron

‘moves’ from one location to another

1.8.4 Quantum mechanical model of Atom

Classical mechanics based on Newton’s laws of motion successfully describes the motion of all macroscopic ob-jects since the uncertainties in position and velocity are small enough to be neglected However, it fails in the case

of microscopic objects like electrons, atoms, molecules etc

So, while considering the motion of microscopic objects, the concept of dual behaviour of matter and the uncertainty principle are taken into account The branch of science that takes into account this dual behaviour of matter is called quantum mechanics

As far as the other dynamical variables of an electron are concerned, we can show that the uncertainty principle leads to the following result.

The total energy of an electron in an atom or ecule has a well-defined (sharp) value The probability distribution as well as the sharp values can be calculated from a function designated as y (x) and called the wave function or the psi function y (x) is obtained by solv-ing the Schrodinger equation which is the fundamental equation in quantum mechanics in the same manner that Newton’s equation is fundamental in classical mechanics

mol-important Features of the Quantum mechanical model of Atom

Quantum mechanical model of the atom is the outcome of the application of Schrodinger wave equation to atoms Its important features are

• Quantization of the energy of electron in atoms, i.e., it can have certain discrete values

• The allowed solutions of Schrodinger wave tion tell about the existence of quantized electronic energy levels for electrons in atoms having wave like properties

equa-Solved Problem 11

Calculate the uncertainty (Dv) in the velocity of a cricket

ball of mass 1 kg, if certainty (Dx) in its position is of the

m

∆ =

π∆ x But h = 6.6 × 10–34 kg m2 s–2 s: Dx = 1Å = 10–10 m and

7 6.6 10

m s

4 22 10

- -

A microscope using suitable photons is employed to locate

an electron in an atom within a distance of 0.1 Å What is

the uncertainty involved in the measurement of its velocity?

Solution:

4

h v

x m

D =

pDBut h=6.626 10× - 34kg m s s;2 - 2

= 0.579 × 107 m s–1

= 5.79 × 106 m s–1

Why bohr’s model was a Failure?

Bohr considered the electron as a charged particle

mov-ing in well defined circular orbits around the nucleus So,

the position and the velocity of the electron can be known

exactly at the same time Uncertainty principle shows that

it is impossible to measure these variables simultaneously

because the wave character of electron was not

consid-ered in the Bohr model Calculation of the trajectory of

an electron in an atom or molecule is, therefore, a futile

exercise We can now appreciate the major fault of the

Bohr model In calculating electron orbits precisely, Bohr

was violating this fundamental requirement and hence his

theory was only partially successful So, a model which

can account the wave-particle duality of matter and be

consistent with Heisenberg uncertainty principle was in

quest This came with the advent of quantum machanics

Schrodinger published his ideas in January 1926 This date represents one of the mile-stones in the history of Chem-istry His work formed the basis of all our present ideas

on how atoms bond together The heart of his method was his prediction of the equation that governed the behaviour

of electrons, and the method of solving it Schrodinger’s equation is

where, m = Mass of electron

E = Total energy of the electron (kinetic energy +

po-tential energy)

U = Potential energy

h = Planck’s constant

y = Wave functionThis equation applies to stationary waves as it does not have the time dependence part of the wave function

Schrodinger won the Nobel prize in physics in

1933 The solution of this equation are very complex and you will learn them for different systems in higher classes

For a system (such as an atomic or a molecule whose energy does not change with time), the Schrodinger equa-tion is written as Ĥy = Ey, where Ĥ is a mathematical operator called Hamiltonian Schrodinger gave a recipe

of constructing this operator from the expression for the total energy of the system The total energy of the system takes into account the kinetic energies of all the sub-atomic particles (electrons, nuclei) attractive potential between the electrons and nuclei and repulsive potential among the electrons and nuclei individually Solutions of this equa-

tion gives E and y.

1.8.6 The meaning of Wave Function

The wave function could only be used to provide tion about the probability of finding the electron in a given region of space around the nucleus Max Born, a German physicist proposed that we must give up ideas of the elec-tron orbiting the nucleus at a precise distance In this re-spect Bohr was wrong in thinking that the electron in the ground state of the hydrogen atom was always to be found

informa-at a distance a0 form the nucleus Rather, it was only most probably to be found at this distance The electron had a

smaller probability of being found at a variety of other tances as well

dis-It is important to realise that we should not try to talk about finding the electron at a given point The reason for this is that there is an infinite number of points around the nucleus So, the probability of finding the electron at any one of these points is infinitely small, i.e., zero Then

• As per Heisenberg’s uncertainty principle, both the

position and velocity of an electron cannot be

deter-mined simultaneously and exactly So, the path of the

electron can never be determined accurately Because

of this reason the concept of probability was

intro-duced for finding the electron at different points in an

atom

• The wave function y for an electron in an atom is

the atomic orbital Whenever we say about electron

by a wave function it occupies that orbital Because

several wave functions are possible for an electron,

there also as many atomic orbitals These one

elec-tron orbital wave functions or orbital form the

ba-sis for electronic structure of atoms Every orbital

possess certain energy and can accommodate only

two electrons In atoms having several electrons, the

electrons are filled in the order of increasing

ener-gy In the multielectron atoms each electron has an

orbital wave function characteristic of the orbital it

occupies All the information about the electron in

an atom is stored in its orbital wave function y and

quantum mechanics makes it possible to extract this

information out of y

The probability of finding the electron at a point

within an atom is proportional to y2 at that point

Though y is sometimes negative, y2 is always

posi-tive and is known as the probability density The

values of y2 predicts the different points in a region

within an atom at which the electron will be most

probably found

1.8.5 Schrodinger Wave equation

erwin Schrödinger 1887–1966

Erwin Schödinger was born in Austria He received

his Ph D in theoretical physics from the university

of Vienna in 1910 At the request of Max Planck,

Schrödinger became his successor at the university

of Berlin in 1927 Because of his opposition to

Hit-ler and Nazi policies, he left Berlin and returned to

Austria and in 1936, when Austria was occupied by

Germany, he was forcibly removed from his

pro-fessorship Then he moved to Dublin, Ireland He

shared the 1933 Nobel Prize for physics with P.A.N

Dirac

During 1920 there was a great deal of interest in

wave-particle duality and de Broglie’s matter waves It was the

Austrian physicist Erwin Schrödinger who invented a

method of showing how the properties of waves could

be used to explain the behaviour of electrons in atoms

1.9 QUANTUm NUmbeRS

The mathematical solution of three dimensional

Schro-dinger wave equation gives three values of E for

accept-able values of y These values are related to one

anoth-er through whole numbanoth-ers These values are tanoth-ermed as

Quantum numbers and represented with n, l, m — called

principal, azimuthal and magnetic quantum numbers, respectively These three quantum numbers together with

the fourth called spin quantum number describe fully the

location and energy of an electron Thus, quantum numbers

are the numbers which determine the energy of electron, the angular momentum, shape of the electron orbital, the orientation of the orbital and spin of the electron Thus,

each quantum number is associated with a particular acteristic of the electron These quantum numbers are dis-cussed below briefly

1 Principal Quantum Number (n) The number

allot-ted to Bohr’s original stationary states, visualized as circular orbits is called the principal quantum number

The innermost orbit, i.e., that nearest to the nucleus has a principal quantum number 1, the second orbit has a quantum number of 2, and so on So, the prin-

cipal quantum number denoted by n have value 1, 2,

3, 4, Alternatively, letters are used to characterize the orbits, K, L, M, N, for 1, 2, 3, 4, The choice

of letters originates from Mosely’s work on the X-ray spectra of the elements He called group of lines in the spectra the K, L, M, N, groups

The number of electrons in an atom which can have the same principal quantum number is limited and is giv-

en by 2n2 where n is the principal quantum number

con-cerned Thus,

Principal quantum number (n) 1 2 3 4Letter designation K L M NMaximum number of electrons 2 8 18 32This is the most important quantum number as it de-termines to a large extent the energy of an electron It also determines the average distance of an electron from the

nucleus As the value of n increases, the electron gets

far-ther away from the nucleus and its energy increases The

higher the value of n, the higher is the electronic energy

For hydrogen and hydrogen like species, the energy and size of the orbital are determined by the principal quantum number alone

The principle quantum number also identifies the ber of allowed orbitals within a shell With the increase in

num-the value of ‘n’ num-the number of allowed orbitals increases and are given by n2 All the orbitals of a given value of ‘n’

constitute a single shell of atom

to predict the position of electron, we can imagine

tak-ing out a series of photographs of an atom to give us an

instantaneous picture of the whereabouts of the electron

If we combine all pictures we would end up with a picture

as shown in Fig 1.24 The separate dots have overlapped

to give regions in which the density of dots is very high

and regions where the density of dots is much lower In the

high density regions, we say that there is a high

probabil-ity densprobabil-ity The maximum in the densprobabil-ity comes at exactly

the same distance a0, as Bohr predicted in his work The

circulated symmetry of the probability density is clear

However, we should remember that atoms exist in three

dimensions, so really the diagram should be shown as a

sphere

a0

Fig 1.24 Electron density

It is easier to draw circles rather than spheres, so usually

we draw the density diagram as a circle Also, it is common

practice not to include the shading and to agree that when a

circle is drawn, it provides a boundary surface within which,

say, there is a 95% probability of finding the electron

When Schrodinger equation is solved for hydrogen

atom, the solution gives the possible energy levels the

elec-tron can occupy and the corresponding wave function(s) y

of the electron associated with each energy level

Accepted solutions to the wave functions are called

Eigen wave functions The probability of finding an

elec-tron at a point in space whose coordinates are x, y and

z is given by y2 (x, y, z) This three dimensional region

obtained where the probability to find the electron is about

95%, is called an orbital

Because atomic behavior is so unlike ordinary

ex-perience, it is very difficult to get used to and it appear

peculiar and mysterious to everyone, both to the novice

and to the experienced scientist Even the experts do

not understand it the way they would like to, and it is

perfectly reasonable that they should not, because all of

direct human experience and of human intution applies

to large objects We know how large objects will act, but

things on a small scale just do not act that way So, we

have to learn about them in a sort of abstract or

imagina-tive fashion and not by connection with our direct

expe-rience Feynman Lectures on Physics Vol l Chapter 37.

3 Magnetic Quantum Number (m l) This quantum

num-ber which is denoted by m l refers to the different tions of the electron cloud in a particular subshell These different orientations are called orbitals The number of orbitals in a particular sub-shell within a principal en-ergy level is given by the number of values allowed to

orienta-m l which in turn depends upon on the value of l The possible values of m l range from + l through 0 to – l, thus making a total of (2l + 1) values Thus, in a subshell, the number of orbitals is equal to (2l + 1).

For l = 0 (i.e., s-subshell) m l can have only one

val-ue m l = 0 It means that s-subshell has only one orbital

For l = 1 (i.e., p-subshell) m l can have three values +1,

0 and –1 This implies that p-subshell has three

orbitals

For l = 2 (i.e., d-subshell) m l can have five values +

2, + 1, 0, and –2 It means that d-subshell has five

orbitals

For l = 3 (i.e., f-subshell) m l can have seven ues + 3, + 2, + 1, 0, –1, –2 and –3 It means that

val-f-subshell has seven orbitals.

The number of orbitals in various types of subshells are as given below

No of orbitals (2l + 1) 1 3 5 7 9The relationship between the principal quantum

number (n), angular momentum quantum number (l) and magnetic quantum number (m l) is summed up in Table 1.4

2 Azimuthal Quantum Number (l) It is also known

as orbital angular momentum or subsidiary quantum

number It defines the three dimensional shape of the

sub-shell For each value of the principal quantum

number there are several closely associated orbitals,

so that the principal quantum number represents a

group or shell of orbits In any one shell, having the

same principal quantum number the various

subsidi-ary orbits are denoted as s, p, d, f, sub-shells The

letters originates from the sharp, principal, diffuse and

fundamental series of the lines in spectra.

The number of sub-shells or sub-levels in a principal

shell is equal to the value of n The values of subshells

is represented with l l can have values ranging from

0 to n – 1, i.e., for a given value of n the possible value of

l can be 0, 1, 2 (n–l) For example when n = 1 value of

l is only 0 For n = 2 the possible value of l can be 0 and

1 For n = 3 the possible values are 0, 1 and 2 Sub-shells

corresponding to different values are as follows:

value for l 0 1 2 3 4 5……

Notation for subshell s p d f g h

The permissible values for ‘l’ for a given principal

quantum number and the corresponding sub-shell notation

In a given energy level

4s 4p 4d 4f

0 + 1, 0, –1 + 2, +1, 0, –1, –2 + 3, + 2, + 1, 0, – 1, – 2, – 3

1 3 5 7

16

1.10 ShAPeS OF ORbiTAlS

The nature of the Schrodinger equation is such that the wave function y may be regarded as an amplitude function This corresponds, on a three-dimensional scale, to the function which expresses the amplitude of vibration of a plucked string

As it is the square of the amplitude of the vibrating string which measures the intensity of the wave involved,

so y2 measures the probability of an electron existing at

a point, meaning therefore, that the chance of finding an electron at that point is zero A high value of y2 at a point means that there is a high chance of finding an electron at the point

The value of the wave function y, for an tron in an atom is dependent, in general, both on the

elec-radial distance, r, of the electron from the nucleus of

the atom and on its angular direction away from the nucleus

The nucleus, itself, however, cannot provide any sense

of direction until a set of arbitrary chosen axes are posed If Cartesian axes are chosen as in Fig 1.25 (a) then

superim-a point in spsuperim-ace, P, superim-around the nucleus, N, csuperim-an be defined

in terms of x–, y– and z-coordinates Such axes do not, however, have any absolute directional significance until

an external magnetic field is applied The axes then have a definite direction in relation to the direction of the magnetic field Alternatively, polar coordinates in Fig 1.25 (b) can be used

Fig 1.25 The position of a point P relative to point

N can be expressed in terms of (a) Cartesian coordinates

x, y, z or (b) Polar coordinates r, θ and f.

The mathematical relationship between y and r and the angular direction can be established accurately for a single electron in a hydrogen like atom For more compli-cated atoms containing more than one electron, the cor-responding relationships can only be established approx-imately because of the difficulty involved in solving the mathematical equations

4 Spin Quantum Number (ms) This quantum number

which is denoted by ms does not follow from the wave

mechanical treatment Electron spin was first

postu-lated in 1925 by Uhlenbeck and Goudsmit to account

for the splitting of many single spectral lines into

dou-ble lines when examined under a spectroscope of high

resolving power The electron in its motion about the

nucleus also rotates or spins about its own axis In

other words, an electron has, besides charge and mass,

an intrinsic spin angular quantum number

Spin angular momentum of the electron –a vector

quantity, can have two orientations relative to a chosen

axis The spin quantum number can have only two values

i.e., ↑ and the other indicates anti-clockwise spin, generally

represented by an arrow downwards, i.e., ↓ The electrons

that have different ms values (one 1

2

+ and the other 1

2

- ) are said to have opposite spins An orbital cannot hold

more than two electrons and these two electrons should

have opposite spins

Orbit and Orbital

Orbit and orbitals are different terms Bohr proposed

an orbit as a circular path around the nucleus in which

the electrons moves As per Heisenberg’s

uncertain-ty principle the precise description is not possible

So, its physical existence cannot be demonstrated

experimentally

An orbital is a three dimensional region calculated

from the allowed solutions of the Schrodinger’s

equa-tion It is quantum mechanical concept and refers to one

electron wave function y in an atom It is characterised

by three quantum numbers (n, l and ml) and its value

depends upon the coordinates of the electron y has no

physical meaning but its square y2 at any point in an

atom gives the value of probability density at that point

Probability density (y2) is the probability per unit

vol-ume and the electron in that volvol-ume y2 varies from one

region to another region in the space but its value can be

assumed to be constant within a small volume The total

probability of finding the electron in a given volume can

be calculated by the sum of all products of y2 and the

cor-responding volume elements It is thus possible to get the

probable distribution of an electron in an orbital

can be directly related with probability of finding electron at

a point which is at a distance r from the nucleus.

Now let us consider the space around the nucleus (taken at the centre) to be divided into a large number of

thin concentric spherical shells of thickness dr Consider one of these shells with the inner radius r (Fig 1.27).

r

r + dr O

Fig 1.27 Division of space around the nucleus into small

spherical shells of very small thickness

The volume of such a shell viz, dV, will be given by

The probability of finding electron within the small

ra-dial shell of thickness dr around the nucleus, called rara-dial

probability will therefore be given by

r dV = yn, l × dV (1.47)

= 4pr2 dr y2

n, l (1.48)

The radial probability distribution between r = 0 to r

= r will be given by the summation of all the probability distributions for concentric radial shells from r = 0 to r =

The differences between s–, p– and d-orbitals depend

on the different ways in which y and / or y2 vary with r

and with the angular direction, as explained in the following

sections:

The probability of finding an electron in a given

vol-ume of space is represented by radial probability

distribu-tion curves These curves indicate how the probability of

finding an electron varies with the radial distance from

the nucleus without any reference to its direction from

the nucleus The radial wave function is generally

writ-ten as yn, l Let us now plot the function yn, l against r for

electrons belonging to different orbitals and try to

cor-relate them with the probability density around a point at

a distance r from the nucleus The plots are as shown in

Fig 1.26

From these plots, it is clear that y n, l cannot be related

with probability density around any point at a distance r

from the nucleus because

(i) y n, l is maximum at r = 0 If y n, l represents

prob-ability density the electron will have maximum probability of occurring at the nucleus This can-

not be true as the actual probability of finding an electron at the nucleus is zero

(ii) y n, l has both positive as well as negative values (if

probability curves for 2s and 3s electrons)

Howev-er, probability density cannot have a negative value.

Objection No (ii) can be removed by considering yn, l

to be related to probability density instead of y n, l Now

even if y n, l is negative at some space its square has to be

positive The plots of y n, l versus r for electrons belonging

to different orbitals are as shown in Fig 1.28

As can be seen the curves do not exhibit negative value

for the radial wave functions at any distance But even y n, l

cannot be related with probability density because yn, l is

maximum at r = 0 This means that the probability of

find-ing electron is maximum on the nucleus which again cannot

be true (as already explained) Thus, neither y n, l nor y2n, l

In other words, the radical probability

distribu-tion* of electron may be obtained by plotting the function

4pr2 y2

graphs are radial probability distribution curves

The radial probability distribution curves for 2s, 3s, 3p

and 3d electrons are shown in Fig 1.29.

It may be noted that for 1s orbital, the probability

den-sity is maximum at the nucleus and it decreases sharply as

we move away from it On the other hand, for 2s orbital the

probability density first decreases sharply to zero and again

starts increasing After reaching a small maxima it

decreas-es again and approachdecreas-es zero as the value of r increasdecreas-es

further The region where this probability density function reduces to zero is called nodal surfaces or nodes In gen-

eral it has been found that ns-orbital has (n – 1) nodes, that

is, number of nodes increases with increase of principal

quantum number n In other words, number of nodes for 2s

orbital is one, two for 3s and so on

These probability density variations can be ized in terms of charge cloud diagrams (Fig 1.30) In these

visual-2 2,0

we have already met; the spherically symmetric ones

For p-orbitals there are three possible values of m (+ 1,

0, –1) — the three types of p-orbitals are called px, py

and pz The boundary surface diagrams are known as dial probability distribution curves or radial charge den- sity curves or simply as radial distribution curves Such

ra-curves truly depict the variation of probability density

of electronic charge with respect to r (distance of charge

from the nucleus.)

Radial probability at a distance r is the probability of finding electron at all points in space which are at a distance r from the nucleus and the radial probability distribution is the graph of these probabilities as a function of r.

The radial probability distribution curves for 1s and 2p

electrons are as shown in Fig 1.31The radial probability function 4pr2ψn l2, written for the sake of simplicity as 4pr2 y2 is evidently the product of two factors While the probability factor y2

decreases as r increases This gives rise to curves of the

type shown in Fig 1.31 At r = 0, though the factor y2 is maximum, (Fig 1.28) the factor 4pr2 is zero Similarly,

when r is very large, the factor 4pr2 is no doubt very large but the probability factor y2 is negligible so that the radial probability is exceedingly small as shown in Fig 1.32

Here unlike s-orbitals, the boundary surface diagrams are not spherical Instead each p-orbital consists of two sec-

tions called lobes that are one either side of the plane that passes through the nucleus The probability density func-tion is zero on the plane where the two lobes touch each other This plane is called nodal plane The nodal plane for

px orbital is YZ-plane, for py orbital, the nodal plane is XZ

plane, and for pz orbital nodal plane is XY plane It should

be understood, however, that there is no simple relation

be-tween the values of ml (–1, 0 and + 1) and the x, y and z

directions The size, shape and energy of the three orbitals are identical

diagrams, the density of the dots in the region represents

electron probability density in that region

1.10.1 boundary Surface Diagrams

The shapes of the orbitals in which constant

probabil-ity densprobabil-ity for different orbitals can be represented with

boundary surface diagram The value of probability

density y2 is constant in this boundary surface drawn in

space In principle, many such boundary surfaces may be

possible However, for a given orbital, only that boundary

surface diagram of constant probability density is taken to

be good representation of the shape of the orbital which

encloses a region or volume in which the probability of

finding the electron is very high, say 90%, but cannot be

100% because always there will be some value, however

small it may be, at any finite distance from the nucleus

Boundary surface diagram for an ‘s’ orbital is actually a

sphere centred on the nucleus In two dimensions, this

sphere looks like a circle The probability of finding the

electron at any given distance is equal in all directions It

is also observed that the size of the s-orbital increases in n,

that is, 4s > 3s> 2s> 1s and the electron is located further

away from the nucleus as the principal quantum number

increases

The probability density diagrams for p orbitals are very

different to those of s-orbitals If you look back at

quan-tum numbers you will find that we said that the magnetic

quantum number can tell us the number of orbitals of a

given type The result is that for s-orbitals the magnetic

quantum number, m l, has only one value This means that

there is only one variety of s-orbitals that is the variety

Z

–Z

Y

X –X

–Y Radial node

–Y Radial node

–Y

1s orbital

Fig 1.30 The shapes of various s-orbitals

zero and infinite distance, as the distance increases The

number of nodes are given by n – 2, that is, number of radial nodes is 1 for 3p-orbital, two for 4p-orbital three and

so on The shapes of 2px, 3px and 4px orbits with radial nodes are shown in Fig 1.33

−

−

showing the radial nodes

The orbitals having l value as 2 are known as d-orbitals

The minimum value of principal quantum number must be

3 to have l value 2 as the value of l cannot be greater than

n – 1 There are five d-orbitals corresponding to ml values

(–2, –1, 0, + 1 and + 2) for l = 2 Their boundary surface

diagrams are shown in Fig 1.34

The five d-orbitals are designed as d d d d xy, yz, xz, x2-y2

and d The orbital z2 d has a dumbell shaped curve sym- z2

metric structure about the z-axis having a ring-like collar

in XY plane The dumbell shaped part of the curve has a

positive geometric sign (because whatever be the sign of

z positive or negative, its square is always positive), ring

the collar in the XY plane has a negative geometric sign.

The orbital d has a double dumbell shape The quan- xy tity XY will be positive when both x and y are positive or when both are negative However, XY will be negative when either of x or y is negative Thus, the sign of the curve

is positive in first and third quadrants while it is negative in second and fourth quadrants The shape of d d orbirals xz, yzcan be explained in a similar manner

The orbital d x2-y2is also double dumbell shaped but

its lobes lie on X and Y axes The signs of the lobes on

X-axis will always be positive (whatever be the sign of X,

X2 will always be positive) whereas the sign of the lobes

on Y axis will always be negative ( whatever be the sign

of Y, – Y2 will always be negative) The exact shapes of d

orbitals are obtained by taking into consideration the total wave function Accordingly 3d x2-y2 orbital would be simi-lar in shape to 4d x2-y2 orbital or 5d x2-y2 orbital except for the fact that

1 A 5d orbital would have two, a 4d orbital have one and

a 3d orbital have no radial node.

Like s-orbitals, p-orbitals increase in size and energy

with increase in the principal quantum number and hence

the order of the energy and size of various p-orbitals is 4p >

3p> 2p.Further, like s-orbitals, the probability density

func-tions for p-orbital also pass through value zero, besides at

For any orbital (i) The number of angular nodes (nodal planes) is

equal to l value.

(ii) The number of radial nodes (nodal surfaces or

nodes) is equal to (n–l–1).

(iii) The total number of radial nodes and angular

nodes for any orbital is equal to (n – l – 1) + l =

n – 1.

The orbitals having l value as 3 are known as f-orbitals

The minimum value of principal quantum number must be

4 to have l value 3 as the value of l cannot be greater than

n – 1 There are seven f-orbitals corresponding to m l values

(–3, –2, –1, 0, + 1, + 2, + 3) for l = 3.Their boundary

sur-face diagrams are shown in Fig 1.36

2 A 5d orbital would occupy more space than a 4d

or-bital and a 4d oror-bital would occupy more space than a

3d orbital as shown in Fig 1.35.

Every d-orbital also have two nodal planes passing

through the origin and bisecting the XY plane containing

Z axis These are called angular nodes and are given by l.

three dimensional space showing radial nodes

z

x y

The attractive and repulsive interactions depend upon the shell and shape of the orbital in which the electron is

present The spherical s orbital can shield the electrons from the nuclear attraction more effectively than the p-orbital with

dumbell shape This is because of the spherical nature of the

s-orbital can shield nucleus from all directions equally but the electron in a p-orbital say px can shield the nucleus in only one direction, i.e., x-direction Similarly because of difference in

their shape and more diffused character the d-orbitals have less shielding power than p-orbitals Further because of their

different shapes the electron in spherical orbital spends more

time close to the nucleus when compared to p-orbital and the p-electron spends more time in the vicinity of nucleus when compared to d-electron For this reason the effective nuclear

charge experienced by different sub shell decreases with

in-crease in the azimuthal quantum number (l) From this we can easily understand that s-electron is strongly attracted by the nucleus than the p-electron which in turn will be strongly attracted than the d-electron and so on Thus, the different

sub-shells belonging to same principal quantum number have different energies However, in hydrogen atom, these have the same energy In multielectron atoms, the dependence of

energies of the sub shells on ‘n’ and ‘l’ are quite complicated but in simple way is that of combined value of n and l Lower the value n + l for a sub shell, lower is its energy If the two sub-shells have the same n + l value, the orbital having lower

n value have the lower energy The energies of sub-shells in

the same shell decrease with increase in the atomic number (Z*) For example, the energy of 2s orbital of hydrogen atom

is greater than that of 2s orbital of lithium and that of lithium

is greater than that of sodium and so on, i.e., E2s (H) > E2s (Li)

> E2s (Na) > E2s (K)

1.11 FilliNg OF ORbiTAlS

The distribution of electrons in various orbitals is known

as electronic configuration Having derived the energy

level sequence, it is now a simple matter to write the tronic configurations of atoms by making use of Aufbau principal, Pauli’s exclusion principle and the Hund’s rule

elec-of maximum multiplicity

1 Aufbau Principle

Aufbau is a German term meaning “building up” This ciple is utilized to deduce the electronic structure of poly-electron atoms by building them up, by filling up of orbit-

prin-als with electrons The Aufbau principle states that “in the ground state of the atoms, the orbitals are filled in order of their increasing energies.” In other words, in the ground state

of atom, the orbital with a lower energy is filled up first before the filling of the orbital with a higher energy commences

in any one of the sub shells belonging to the same

prin-cipal quantum number The orbitals having same energy

are called degenerate orbitals In hydrogen atom when

electron is present in 1s orbital, the hydrogen atom is

con-sidered in most stable condition and it is called the ground

state This is because the electron is nearer to the nucleus

But when electron is present in 2s, 2p or higher orbitals in

the hydrogen atom, it is said to be in the excited state

In multielectron atoms, unlike the hydrogen atom,

the energy of an electron depends both on its

princi-pal quantum number (shell), and the azimuthal

quan-tum number (sub shell) This means that different sub

shells viz., s, p, d, f, in a principal quantum number

will have different energies, because of the mutual

repul-sion between the electrons In hydrogen atom there is

only attractive force between nucleus and electron but no

repulsive forces The stability of an electron in a

multi-electron atom is because the total attractive interactions

are more than the repulsive interactions experienced by

every electron with other electrons The attractive force

of nucleus on the electrons increases with increase in

the positive charge (Ze) of nucleus The attractive power

of nucleus on the electrons will be less due to

screening effect of inner electrons The net positive charge ex

-perienced by the electron from the nucleus is known as

effective nuclear charge (Z*)

4p 4s 4d

3p 3s 2p 2s 1s 3d

Fig 1.37 The relative energies of the atomic orbitals as

a function of atomic number

n = 1, l = 0, m = 0, s = 1 0 0 1

2

n= l= m= s=

-The p-sub-shell which is having 3 orbitals (px, py and

pz ) can accommodate 6 electrons, the d-sub shell which ing 5 orbitals can accommodate 10 electrons and the f-sub

hav-shell having 7 orbitals can accommodate 14 electrons This

can be summed up as the maximum number of electrons in the shell with principal quantum number n is equal to 2n2.Here a question arises? Why only two electrons can present in one orbital? It can be answered as fol-lows We may recall from our knowledge of elementa-

ry physics that the motion of an electric charge creates

a magnetic field In an orbital the electron spin may be clockwise or anti-clockwise as shown in Fig 1.39 This results in the formation of two magnets Placing of two electrons together in the same orbital results in consid-erable repulsion due to same negative charge By over-coming the repulsive forces and to keep the two electrons together in the same orbital some energy is required, called pairing energy If the two electrons also have the same spin there will be magnetic repulsion also, causing the requirement of more pairing energy But, if the two electrons are spinning in opposite directions and forming a pair in an orbital, mutually cancel their magnetic moments Now, we can easily understand that for an electron if pairing energy is less than the energy of next higher energy level the electron will be paired in the orbital But if the pairing energy is greater than the energy of next higher energy level the electron goes to the next higher energy level

The increasing order of energy of various orbitals is

1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s,

The above sequence of energy levels can be easily

remembered with the help of the graphical representation

shown in Fig 1.38

2 Pauli’s exclusion Principle

The four quantum numbers define completely the position

of electron in an atom It is thus possible to identify an

electron in an atom completely by stating the values of its

four quantum numbers Wolfgang Pauli, an Austrian

sci-entist, put forward an ingenious principle which controls

the number of electrons to be filled in various orbitals, and

hence, it is named as the exclusion principle It states that

no two electrons in an atom can have the same set of

four quantum numbers

Thus, in the same atom any two electrons may have

three quantum numbers identical but not the fourth which

must be different This means that the two electrons can

have the same value of three quantum numbers, n, l and

ml but must have the opposite spin quantum numbers This

restricts that only two electrons may exist in the same

or-bital and these electrons must have opposite spins Pauli’s

exclusion principle helps in calculating the number of

electrons to be present in any sub shell For example, the

1s sub-shell comprises one orbital and thus the maximum

number of electrons present in 1s sub shell can be two

because of the following two possibilities

n = 1, l = 0, m = 0, s = 1

2+

Fig 1.38 The order in which the atomic

or-bitals are used in building up the electron

con-figuration of many-electron atoms The

orbit-als are used in sequence, from the bottom in

accordance with the Aufbau principle, Hund’s

rule and Pauli’s exclusion principle

4b

(iii) Hund’s rule of maximum multiplicity The tronic configuration of different atoms can be represented in two ways:

(i) nlx or sa pb dc notation (ii) Orbital diagram method

In the nlx method, n represents the principle quantum number, l represents the azimuthal quantum number (sub

shell) and x represents the number of electron in that sub

shell In sa pb dc notation the sub shell is depicted with

a superscript like a, b, c, etc, along with the principal

quantum number before the respective sub shell In the orbital diagram notation each orbital of the sub shell

is represented by a box and the electron is represented

by an arrow (↑) for positive spin and an arrow (↓) for negative spin The advantage of the second notation over the first notation is that it represents all the four quantum numbers Based on these notations, the elec-tronic configurations of atoms of various elements of the Periodic Table in their ground state are as given in Table 1.5

Hydrogen has only one electron (Z = 1) This tron will enter the lowest energy orbital which is 1s This, the solitary electron of hydrogen atom, will occupy the 1s

elec-orbital The electronic configuration of hydrogen atoms is,

therefore, represented as 1s1

The next element, helium, has two electrons (Z = 2)

One of these electrons occupies the 1s orbital as in the

case of the hydrogen atom The second electron can also enter this orbital so as to fill it completely The electronic

configuration of helium is therefore, represented as 1s2 The two electrons occupying this orbital will have the

Fig 1.39 No two electrons in an atom can have the

same set of four quantum numbers

3 hund’s Rule of maximum multiplicity

We know that the orbitals belonging to same sub shell have

same energy and are called degenerate orbitals Hund’s rule

states that the electron pairing in degenerate orbitals of a

given sub shell will not take place unless all the available

orbitals of a given shell contains one electron each.

Since there are three p, five d and seven f orbitals,

pair-ing of electrons will start in p, d and f orbitals with the

en-try of 4th, 6th and 8th electrons, respectively It is also found

that half-filled and fully filled degenerate set of orbitals

acquire extra stability

1.11.1 Electronic Configuration of Atoms

The filling of electrons in different orbitals in an atom are

governed by the above three rules viz.,

(i) Aufbau principle,

(ii) Pauli’s exclusion principle, and

Table 1.5 Electronic Configuration of First 10 Elements