STRUCTURE OF ATOM

The rich diversity of chemical behaviour of different elements

can be traced to the differences in the internal structure of

atoms of these elements.

UNIT 2

STRUCTURE OF ATOM

After studying this unit you will be

able to

••

• know about the discovery of

electron, proton and neutron and

their characteristics;

••

• describe Thomson, Rutherford

and Bohr atomic models;

••

• understand the important

features of the quantum

mechanical model of atom;

••

• understand nature of

electromagnetic radiation and

Planck’s quantum theory;

••

• explain the photoelectric effect

and describe features of atomic

spectra;

••

• state the de Broglie relation and

Heisenberg uncertainty principle;

••

• define an atomic orbital in terms

of quantum numbers;

••

• state aufbau principle, Pauli

exclusion principle and Hund’s

rule of maximum multiplicity; and

••

• write the electronic configurations

of atoms.

The existence of atoms has been proposed since the time

of early Indian and Greek philosophers (400 B.C.) who

were of the view that atoms are the fundamental building

blocks of matter. According to them, the continued

subdivisions 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

‘uncut-able’ or ‘non-divisible’. These earlier ideas were

mere speculations and there was no way to test them

experimentally. These ideas remained dormant for a very

long time and were revived again by scientists in the

nineteenth century.

The atomic theory of matter was first proposed on a

firm scientific basis by John Dalton, a British school

teacher in 1808. His theory, called Dalton’s atomic

theory, regarded the atom as the ultimate particle of

matter (Unit 1). Dalton’s atomic theory was able to explain

the law of conservation of mass, law of constant

composition and law of multiple proportion very

successfully. However, it failed to explain the results of

many experiments, for example, it was known that

substances like glass or ebonite when rubbed with silk

or fur get electrically charged.

In this unit we start with the experimental

observations made by scientists towards the end of

nineteenth and beginning of twentieth century. These

established that atoms are made of sub-atomic particles,

i.e., electrons, protons and neutrons — a concept very

different from that of Dalton.

Objectives

2020-21

30 CHEMISTRY

2.1 DISCOVERY OF SUB-ATOMIC

PARTICLES

An insight into the structure of atom was

obtained from the experiments on electrical

discharge through gases. Before we discuss

these results we need to keep in mind a basic

rule regarding the behaviour of charged

particles : “Like charges repel each other and

unlike charges attract each other”.

2.1.1 Discovery of Electron

In 1830, Michael Faraday showed that if

electricity is passed through a solution of an

electrolyte, chemical reactions occurred at

the electrodes, which resulted in the

liberation and deposition of matter at the

electrodes. He formulated certain laws which

you will study in class XII. These results

suggested the particulate nature of

electricity.

In mid 1850s many scientists mainly

Faraday began to study electrical discharge

in partially evacuated tubes, known as

cathode ray discharge tubes. It is depicted

in Fig. 2.1. A cathode ray tube is made of

glass containing two thin pieces of metal,

called electrodes, sealed in it. The electrical

discharge through the gases could be

observed only at very low pressures and at

very high voltages. The pressure of different

gases could be adjusted by evacuation of the

glass tubes. When sufficiently high voltage

is applied across the electrodes, current

starts flowing through a stream of particles

moving in the tube from the negative electrode

(cathode) to the positive electrode (anode).

These were called cathode rays or cathode

ray particles. The flow of current from

cathode to anode was further checked by

making a hole in the anode and coating the

tube behind anode with phosphorescent

material zinc sulphide. When these rays, after

passing through anode, strike the zinc

sulphide coating, a bright spot is developed

on the coating [Fig. 2.1(b)].

Fig. 2.1(a) A cathode ray discharge tube

Fig. 2.1(b) A cathode ray discharge tube with

perforated anode

The results of these experiments are

summarised below.

(i) The cathode rays start from cathode and

move towards the anode.

(ii) These rays themselves are not visible but

their behaviour can be observed with the

help of certain kind of materials

(fluorescent or phosphorescent) which

glow when hit by them. Television picture

tubes are cathode ray tubes and

television pictures result due to

fluorescence on the television screen

coated with certain fluorescent or

phosphorescent materials.

(iii) In the absence of electrical or magnetic

field, these rays travel in straight lines

(Fig. 2.2).

(iv) In the presence of electrical or magnetic

field, the behaviour of cathode rays are

similar to that expected from negatively

charged particles, suggesting that the

cathode rays consist of negatively

charged particles, called electrons.

(v) The characteristics of cathode rays

(electrons) do not depend upon the

2020-21

STRUCTURE OF ATOM

material of electrodes and the nature of

the gas present in the cathode ray tube.

Thus, we can conclude that electrons are

basic constituent of all the atoms.

2.1.2 Charge to Mass Ratio of Electron

In 1897, British physicist J.J. Thomson

measured the ratio of electrical charge (e) to

the mass of electron (m

) by using cathode ray

tube and applying electrical and magnetic field

perpendicular to each other as well as to the

path of electrons (Fig. 2.2). When only electric

field is applied, the electrons deviate from their

path and hit the cathode ray tube at point A

(Fig. 2.2). Similarly when only magnetic field

is applied, electron strikes the cathode ray tube

at point C. By carefully balancing the electrical

and magnetic field strength, it is possible to

bring back the electron to the path which is

followed in the absence of electric or magnetic

field and they hit the screen at point B.

Thomson argued that the amount of deviation

of the particles from their path in the presence

of electrical or magnetic field depends upon:

(i) the magnitude of the negative charge on

the particle, greater the magnitude of the

charge on the particle, greater is the

interaction with the electric or magnetic

field and thus greater is the deflection.

(ii) the mass of the particle — lighter the

particle, greater the deflection.

(iii) the strength of the electrical or magnetic

field — the deflection of electrons from its

original path increases with the increase

in the voltage across the electrodes, or the

strength of the magnetic field.

By carrying out accurate measurements on

the amount of deflections observed by the

electrons on the electric field strength or

magnetic field strength, Thomson was able to

determine the value of e/m

as:

= 1.758820 × 10

C kg

–1

(2.1)

Where m

is the mass of the electron in kg and

e is the magnitude of the charge on the electron

in coulomb (C). Since electrons are negatively

charged, the charge on electron is –e.

2.1.3 Charge on the Electron

R.A. Millikan (1868-1953) devised a method

known as oil drop experiment (1906-14), to

determine the charge on the electrons. He found

the charge on the electron to be

– 1.6 × 10

–19

C. The present accepted value of

electrical charge is – 1.602176 × 10

–19

C. The

mass of the electron (m

) was determined by

combining these results with Thomson’s value

of e/m

ratio.

= 9.1094×10

–31

kg (2.2)

Fig. 2.2 The apparatus to determine the charge to the mass ratio of electron

2020-21

32 CHEMISTRY

2.1.4 Discovery of Protons and Neutrons

Electrical discharge carried out in the modified

cathode ray tube led to the discovery of canal

rays carrying positively charged particles. The

characteristics of these positively charged

particles are listed below.

(i) Unlike cathode rays, mass of positively

charged particles depends upon the

nature of gas present in the cathode ray

tube. These are simply the positively

charged gaseous ions.

(ii) The charge to mass ratio of the particles

depends on the gas from which these

originate.

(iii) Some of the positively charged particles

carry a multiple of the fundamental unit

of electrical charge.

(iv) The behaviour of these particles in the

magnetic or electrical field is opposite to

that observed for electron or cathode

rays.

The smallest and lightest positive ion was

obtained from hydrogen and was called

proton. This positively charged particle was

characterised in 1919. Later, a need was felt

for the presence of electrically neutral particle

as one of the constituent of atom. These

particles were discovered by Chadwick (1932)

by bombarding a thin sheet of beryllium by

α-particles. When electrically neutral particles

having a mass slightly greater than that of

protons were emitted. He named these

particles as neutrons. The important

properties of all these fundamental particles

are given in Table 2.1.

2.2 ATOMIC MODELS

Observations obtained fr

om the experiments

mentioned in the previous sections have

suggested that Dalton’s indivisible atom is

composed of sub-atomic particles carrying

positive and negative charges. The major

problems before the scientists after the

discovery of sub-atomic particles were:

•

••

• to account for the stability of atom,

••

•

• to compare the behaviour of elements in

terms of both physical and chemical

properties,

Millikan’s Oil Drop Method

In this method, oil droplets in the form of

mist, produced by the atomiser, were allowed

to enter through a tiny hole in the upper plate

of electrical condenser. The downward motion

of these droplets was viewed through the

telescope, equipped with a micrometer eye

piece. By measuring the rate of fall of these

droplets, Millikan was able to measure the

mass of oil droplets. The air inside the

chamber was ionized by passing a beam of

X-rays through it. The electrical charge on

these oil droplets was acquired by collisions

with gaseous ions. The fall of these charged

oil droplets can be retarded, accelerated or

made stationary depending upon the charge

on the droplets and the polarity and strength

of the voltage applied to the plate. By carefully

measuring the effects of electrical field

strength on the motion of oil dr

oplets,

Millikan concluded that the magnitude of

electrical charge, q, on the droplets is always

an integral multiple of the electrical charge,

e, that is, q = n e, where n = 1, 2, 3... .

Fig. 2.3 The Millikan oil drop apparatus for

measuring charge ‘e’. In chamber, the

forces acting on oil drop are:

gravitational, electrostatic due to

electrical field and a viscous drag force

when the oil drop is moving.

••

• to explain the formation of different kinds

of molecules by the combination of different

atoms and,

••

•

• to understand the origin and nature of the

characteristics of electromagnetic radiation

absorbed or emitted by atoms.

2020-21

STRUCTURE OF ATOM

Table 2.1 Properties of Fundamental Particles

Different atomic models were proposed to

explain the distributions of these charged

particles in an atom. Although some of these

models were not able to explain the stability of

atoms, two of these models, one proposed by

J.J. Thomson and the other proposed by

Ernest Rutherford are discussed below.

2.2.1 Thomson Model of Atom

J. J. Thomson, in 1898, proposed that an atom

possesses a spherical shape (radius

approximately 10

–10

m) in which the positive

charge is uniformly distributed. The electrons

are embedded into it in such a manner as to

give the most stable electrostatic arrangement

(Fig. 2.4). Many different names are given to

this model, for example, plum pudding, raisin

pudding or watermelon. This model can be

In the later half of the nineteenth century

different kinds of rays were discovered,

besides those mentioned earlier. Wilhalm

Röentgen (1845-1923) in 1895 showed

that when electrons strike a material in

the cathode ray tubes, produce rays

which can cause fluorescence in the

fluorescent materials placed outside the

cathode ray tubes. Since Röentgen did not

know the nature of the radiation, he

named them X-rays and the name is still

carried on. It was noticed that X-rays are

produced effectively when electrons strike

the dense metal anode, called targets.

These are not deflected by the electric and

magnetic fields and have a very high

penetrating power through the matter

and that is the reason that these rays are

used to study the interior of the objects.

These rays are of very short wavelengths

(∼

0.1 nm) and possess electro-magnetic

character (Section 2.3.1).

Henri Becqueral (1852-1908)

observed that there are certain elements

which emit radiation on their own and

named this phenomenon as

radioactivity and the elements known

as radioactive elements. This field was

developed by Marie Curie, Piere Curie,

Rutherford and Fredrick Soddy. It was

observed that three kinds of rays i.e., α,

β- and γ-rays are emitted. Rutherford

found that α-rays consists of high energy

particles carrying two units of positive

charge and four unit of atomic mass. He

concluded that α- particles are helium

Fig.2.4 Thomson model of atom

visualised as a pudding or watermelon of

positive charge with plums or seeds (electrons)

embedded into it. An important feature of this

model is that the mass of the atom is assumed

to be uniformly distributed over the atom.

Although this model was able to explain the

overall neutrality of the atom, but was not

consistent with the results of later experiments.

Thomson was awarded Nobel Prize for physics

in 1906, for his theoretical and experimental

investigations on the conduction of electricity

by gases.

Electron e –1.602176×10

–19

–1 9.109382×10

–31

0.00054 0

Proton p +

1.602176×10

–19

+1 1.6726216×10

–27

1.00727 1

Neutron n 0 0 1.674927×10

–27

1.00867 1

Name Symbol Absolute Relative Mass/kg Mass/u Approx.

charge/C charge mass/u

2020-21

34 CHEMISTRY

nuclei as when α- particles combined with

two electrons yielded helium gas. β-rays

are negatively charged particles similar to

electrons. The γ-rays are high energy

radiations like X-rays, are neutral in

nature and do not consist of particles. As

regar

ds penetrating power, α-particles are

the least, followed by β-rays (100 times

that of α–particles) and γ-rays (1000 times

of that α-particles).

2.2.2 Rutherford’s Nuclear Model of Atom

Rutherford and his students (Hans Geiger and

Ernest Marsden) bombarded very thin gold foil

with α–particles. Rutherford’s famous

αα

α–particle scattering experiment is

represented in Fig. 2.5. A stream of high energy

α–particles from a radioactive source was

directed at a thin foil (thickness ∼

100 nm) of

gold metal. The thin gold foil had a circular

fluorescent zinc sulphide screen around it.

Whenever α–particles struck the screen, a tiny

flash of light was produced at that point.

The results of scattering experiment were

quite unexpected. According to Thomson

model of atom, the mass of each gold atom in

the foil should have been spread evenly over

the entire atom, and α– particles had enough

energy to pass directly through such a uniform

distribution of mass. It was expected that the

particles would slow down and change

directions only by a small angles as they passed

through the foil. It was observed that:

(i) most of the α–particles passed through

the gold foil undeflected.

(ii) a small fraction of the α–particles was

deflected by small angles.

(iii) a very few α–particles (∼1 in 20,000)

bounced back, that is, were deflected by

nearly 180

On the basis of the observations,

Rutherford drew the following conclusions

regarding the structure of atom:

(i) Most of the space in the atom is empty as

most of the α–particles passed through

the foil undeflected.

(ii) A few positively charged α–particles were

deflected. The deflection must be due to

enormous repulsive force showing that

the positive charge of the atom is not

spread throughout the atom as Thomson

had presumed. The positive charge has

to be concentrated in a very small volume

that repelled and deflected the positively

charged α–particles.

(iii) Calculations by Rutherford showed that

the volume occupied by the nucleus is

negligibly small as compared to the total

volume of the atom. The radius of the

atom is about 10

–10

m, while that of

nucleus is 10

–15

m. One can appreciate

this difference in size by realising that if

Fig. 2.5 Schematic view of Rutherford’s

scattering experiment. When a beam

of alpha (α) particles is “shot” at a thin

gold foil, most of them pass through

without much effect. Some, however,

are deflected.

A. Rutherford’s scattering experiment

B. Schematic molecular view of the gold foil

2020-21

STRUCTURE OF ATOM

a cricket ball represents a nucleus, then

the radius of atom would be about 5 km.

On the basis of above observations and

conclusions, Rutherford proposed the nuclear

model of atom. According to this model:

(i) The positive charge and most of the mass

of the atom was densely concentrated in

extremely small region. This very small

portion of the atom was called nucleus

by Rutherford.

(ii) The nucleus is surrounded by electrons

that move around the nucleus with a very

high speed in circular paths called orbits.

Thus, Rutherford’s model of atom

resembles the solar system in which the

nucleus plays the role of sun and the

electrons that of revolving planets.

(iii) Electrons and the nucleus are held

together by electrostatic forces of

attraction.

2.2.3 Atomic Number and Mass Number

The presence of positive charge on the

nucleus is due to the protons in the nucleus.

As established earlier, the charge on the

proton is equal but opposite to that of

electron. The number of protons present in

the nucleus is equal to atomic number (Z ).

For example, the number of protons in the

hydrogen nucleus is 1, in sodium atom it is

11, therefore their atomic numbers are 1 and

11 respectively. In order to keep the electrical

neutrality, the number of electrons in an

atom is equal to the number of protons

(atomic number, Z ). For example, number of

electrons in hydrogen atom and sodium atom

are 1 and 11 respectively.

Atomic number (Z) = number of protons in

the nucleus of an atom

= number of electrons

in a nuetral atom (2.3)

While the positive charge of the nucleus

is due to protons, the mass of the nucleus,

due to protons and neutrons. As discussed

earlier protons and neutrons present in the

nucleus are collectively known as nucleons.

The total number of nucleons is termed as

mass number (A) of the atom.

mass number (A) = number of protons (Z)

+ number of

neutrons (n) (2.4)

2.2.4 Isobars and Isotopes

The composition of any atom can be

represented by using the normal element

symbol (X) with super-script on the left hand

side as the atomic mass number (A) and

subscript (Z) on the left hand side as the atomic

number (i.e.,

X).

Isobars are the atoms with same mass

number but different atomic number for

example,

C and

N. On the other hand, atoms

with identical atomic number but different

atomic mass number are known as Isotopes.

In other words (according to equation 2.4), it

is evident that difference between the isotopes

is due to the presence of different number of

neutrons present in the nucleus. For example,

considering of hydrogen atom again, 99.985%

of hydrogen atoms contain only one proton.

This isotope is called protium (

H). Rest of the

percentage of hydrogen atom contains two other

isotopes, the one containing 1 proton and 1

neutron is called deuterium (

D, 0.015%)

and the other one possessing 1 proton and 2

neutrons is called tritium

(

T ). The latter

isotope is found in trace amounts on the earth.

Other examples of commonly occuring

isotopes are: carbon atoms containing 6, 7 and

8 neutrons besides 6 protons (

C, C, C

);

chlorine atoms containing 18 and 20 neutrons

besides 17 protons (

Cl, Cl

Lastly an important point to mention

regarding isotopes is that chemical properties

of atoms are controlled by the number of

electrons, which are determined by the

number of protons in the nucleus. Number of

neutrons present in the nucleus have very little

effect on the chemical properties of an element.

Therefore, all the isotopes of a given element

show same chemical behaviour.

2020-21

36 CHEMISTRY

Problem 2.1

Calculate the number of protons,

neutrons and electrons in

Solution

In this case,

, Z = 35, A = 80, species

is neutral

Number of protons = number of electrons

= Z = 35

Number of neutrons = 80 – 35 = 45,

(equation 2.4)

Problem 2.2

The number of electrons, protons and

neutrons in a species are equal to 18, 16

and 16 respectively. Assign the proper

symbol to the species.

Solution

The atomic number is equal to

number of protons = 16. The element is

sulphur (S).

Atomic mass number = number of

protons + number of neutrons

= 16 + 16 = 32

Species is not neutral as the number of

protons is not equal to electrons. It is

anion (negatively charged) with charge

equal to excess electrons = 18 – 16 = 2.

Symbol is

32 2–

Note : Before using the notation

X, find

out whether the species is a neutral atom,

a cation or an anion. If it is a neutral atom,

equation (2.3) is valid, i.e., number of

protons = number of electrons = atomic

number. If the species is an ion, determine

whether the number of protons are larger

(cation, positive ion) or smaller (anion,

negative ion) than the number of electrons.

Number of neutrons is always given by

A–Z, whether the species is neutral or ion.

2.2.5 Drawbacks of Rutherford Model

As you have learnt above, Rutherford nuclear

model of an atom is like a small scale solar

system with the nucleus playing the role of the

* Classical mechanics is a theoretical science based on Newton’s laws of motion. It specifies the laws of motion of

macroscopic objects.

massive sun and the electrons being similar

to the lighter planets. When classical

mechanics* is applied to the solar system, it

shows that the planets describe well-defined

orbits around the sun. The gravitational force

between the planets is given by the expression

1 2

m m













where m

and m

are the masses, r

is the distance of separation of the masses and

G is the gravitational constant. The theory can

also calculate precisely the planetary orbits and

these are in agreement with the experimental

measurements.

The similarity between the solar system

and nuclear model suggests that electrons

should move around the nucleus in well

defined orbits. Further, the coulomb force

(kq

where q

and q

are the charges, r is

the distance of separation of the charges and

k is the proportionality constant) between

electron and the nucleus is mathematically

similar to the gravitational force. However,

when a body is moving in an orbit, it

undergoes acceleration even if it is moving with

a constant speed in an orbit because of

changing direction. So an electron in the

nuclear model describing planet like orbits is

under acceleration. According to the

electromagnetic theory of Maxwell, charged

particles when accelerated should emit

electromagnetic radiation (This feature does

not exist for planets since they are uncharged).

Therefore, an electron in an orbit will emit

radiation, the energy carried by radiation

comes from electronic motion. The orbit will

thus continue to shrink. Calculations show

that it should take an electron only 10

–8

s to

spiral into the nucleus. But this does not

happen. Thus, the Rutherford model

cannot explain the stability of an atom.

If the motion of an electron is described on the

basis of the classical mechanics and

electromagnetic theory, you may ask that

since the motion of electrons in orbits is

leading to the instability of the atom, then

why not consider electrons as stationary

2020-21

STRUCTURE OF ATOM

around the nucleus. If the electrons were

stationary, electrostatic attraction between

the dense nucleus and the electrons would

pull the electrons toward the nucleus to form

a miniature version of Thomson’s model

of atom.

Another serious drawback of the

Rutherford model is that it says nothing about

distribution of the electrons around the

nucleus and the energies of these electrons.

2.3 DEVELOPMENTS LEADING TO THE

BOHR’S MODEL OF ATOM

Historically, results observed from the studies

of interactions of radiations with matter have

provided immense information regarding the

structure of atoms and molecules. Neils Bohr

utilised these results to improve upon the

model proposed by Rutherford. Two

developments played a major role in the

formulation of Bohr’s model of atom. These

were:

(i) Dual character of the electromagnetic

radiation which means that radiations

possess both wave like and particle like

properties, and

(ii) Experimental results regarding atomic

spectra.

First, we will discuss about the duel nature

of electromagnetic radiations. Experimental

results regarding atomic spectra will be

discussed in Section 2.4.

2.3.1 Wave Nature of Electromagnetic

Radiation

In the mid-nineteenth century, physicists

actively studied absorption and emission of

radiation by heated objects. These are called

thermal radiations. They tried to find out of

what the thermal radiation is made. It is now

a well-known fact that thermal radiations

consist of electromagnetic waves of various

frequencies or wavelengths. It is based on a

number of modern concepts, which were

unknown in the mid-nineteenth century. First

active study of thermal radiation laws occured

in the 1850’s and the theory of electromagnetic

waves and the emission of such waves by

accelerating charged particles was developed

in the early 1870’s by James Clerk Maxwell,

which was experimentally confirmed later by

Heinrich Hertz. Here, we will learn some facts

about electromagnetic radiations.

James Maxwell (1870) was the first to give

a comprehensive explanation about the

interaction between the charged bodies and

the behaviour of electrical and magnetic fields

on macroscopic level. He suggested that when

electrically charged particle moves under

accelaration, alternating electrical and

magnetic fields are produced and transmitted.

These fields are transmitted in the forms of

waves called electromagnetic waves or

electromagnetic radiation.

Light is the form of radiation known from

early days and speculation about its nature

dates back to remote ancient times. In earlier

days (Newton) light was supposed to be made

of particles (corpuscules). It was only in the

19th century when wave nature of light was

established.

Maxwell was again the first to reveal that

light waves are associated with oscillating

electric and magnetic character (Fig. 2.6).

Although electromagnetic wave motion is

complex in nature, we will consider here only

a few simple properties.

(i) The oscillating electric and magnetic fields

produced by oscillating charged particles

Fig.2.6 The electric and magnetic field

components of an electromagnetic wave.

These components have the same

wavelength, frequency, speed and

amplitude, but they vibrate in two

mutually perpendicular planes.

2020-21

38 CHEMISTRY

are perpendicular to each other and both

are perpendicular to the direction of

propagation of the wave. Simplified

picture of electromagnetic wave is shown

in Fig. 2.6.

(ii) Unlike sound waves or waves produced

in water, electromagnetic waves do not

require medium and can move in

vacuum.

(iii) It is now well established that there are

many types of electromagnetic radiations,

which differ from one another in

wavelength (or frequency). These

constitute what is called electromagnetic

spectrum (Fig. 2.7). Different regions of

the spectrum are identified by different

names. Some examples are: radio

frequency region around 10

Hz, used for

broadcasting; microwave region around

Hz used for radar; infrared region

around 10

Hz used for heating;

ultraviolet region around 10

Hz a

component of sun’s radiation. The small

portion around 10

Hz, is what is

ordinarily called visible light. It is only

this part which our eyes can see (or

detect). Special instruments are required

to detect non-visible radiation.

Fig. 2.7 (a) The spectrum of electromagnetic radiation. (b) Visible spectrum. The visible region is only

a small part of the entire spectrum.

(iv) Different kinds of units are used to

represent electromagnetic radiation.

These radiations are characterised by the

properties, namely, frequency (

) and

wavelength (λ).

The SI unit for frequency (

) is hertz

(Hz, s

–1

), after Heinrich Hertz. It is defined as

the number of waves that pass a given point

in one second.

Wavelength should have the units of length

and as you know that the SI units of length is

meter (m). Since electromagnetic radiation

consists of different kinds of waves of much

smaller wavelengths, smaller units are used.

Fig.2.7 shows various types of electro-

magnetic radiations which differ from one

another in wavelengths and frequencies.

In vaccum all types of electromagnetic

radiations, regardless of wavelength, travel at

the same speed, i.e., 3.0 × 10

m s

–1

(2.997925

× 10

m s

–1

, to be precise). This is called speed

of light and is given the symbol ‘c‘. The

frequency (

), wavelength (λ) and velocity of light

c =

λ (2.5)

(a)

(b)

2020-21

STRUCTURE OF ATOM

The other commonly used quantity

specially in spectroscopy, is the wavenumber

(

). It is defined as the number of wavelengths

per unit length. Its units are reciprocal of

wavelength unit, i.e., m

–1

. However commonly

used unit is cm

–1

(not SI unit).

Problem 2.3

The Vividh Bharati station of All India

Radio, Delhi, broadcasts on a frequency

of 1,368 kHz (kilo hertz). Calculate the

wavelength of the electromagnetic

radiation emitted by transmitter. Which

part of the electromagnetic spectrum

does it belong to?

Solution

The wavelength,

, is equal to c/

, where

c is the speed of electromagnetic radiation

in vacuum and

is the frequency.

Substituting the given values, we have

λ =

This is a characteristic radiowave

wavelength.

Problem 2.4

The wavelength range of the visible

spectrum extends from violet (400 nm) to

red (750 nm). Express these wavelengths

in frequencies (Hz). (1nm = 10

–9

Solution

Using equation 2.5, frequency of violet

light

= 7.50 × 10

* Diffraction is the bending of wave around an obstacle.

** Interference is the combination of two waves of the same or different frequencies to give a wave whose distribution at

each point in space is the algebraic or vector sum of disturbances at that point resulting from each interfering wave.

Frequency of red light

= 4.00 × 10

The range of visible spectrum is from

4.0 × 10

to 7.5 × 10

Hz in terms of

frequency units.

Problem 2.5

Calculate (a) wavenumber and (b)

frequency of yellow radiation having

wavelength 5800 Å.

Solution

(a) Calculation of wavenumber (

)

λ=5800Å = 5800 × 10

–8

= 5800 × 10

–10

(b) Calculation of the frequency (

)

2.3.2 Particle Nature of Electromagnetic

Radiation: Planck’s Quantum

Theory

Some of the experimental phenomenon such

as diffraction* and interference** can be

explained by the wave nature of the

electromagnetic radiation. However, following

are some of the observations which could not

be explained with the help of even the

electromagentic theory of 19th century

physics (known as classical physics):

(i) the nature of emission of radiation from

hot bodies (black -body radiation)

(ii) ejection of electrons from metal surface

when radiation strikes it (photoelectric

effect)

(iii) variation of heat capacity of solids as a

function of temperature

2020-21

40 CHEMISTRY

(iv) Line spectra of atoms with special

reference to hydrogen.

These phenomena indicate that the system

can take energy only in discrete amounts. All

possible energies cannot be taken up or

radiated.

It is noteworthy that the first concrete

explanation for the phenomenon of the black

body radiation mentioned above was given by

Max Planck in 1900. Let us first try to

understand this phenomenon, which is given

below:

Hot objects emit electromagnetic radiations

over a wide range of wavelengths. At high

temperatures, an appreciable proportion of

radiation is in the visible region of the

spectrum. As the temperature is raised, a

higher proportion of short wavelength (blue

light) is generated. For example, when an iron

rod is heated in a furnace, it first turns to dull

red and then progressively becomes more and

more red as the temperature increases. As this

is heated further, the radiation emitted becomes

white and then becomes blue as the

temperature becomes very high. This means

that red radiation is most intense at a particular

temperature and the blue radiation is more

intense at another temperature. This means

intensities of radiations of different wavelengths

emitted by hot body depend upon its

temperature. By late 1850’s it was known that

objects made of different material and kept at

different temperatures emit different amount of

radiation. Also, when the surface of an object is

irradiated with light (electromagnetic radiation),

a part of radiant energy is generally reflected

as such, a part is absorbed and a part of it is

transmitted. The reason for incomplete

absorption is that ordinary objects are as a rule

imperfect absorbers of radiation. An ideal body,

which emits and absorbs radiations of all

frequencies uniformly, is called a black body

and the radiation emitted by such a body is

called black body radiation. In practice, no

such body exists. Carbon black approximates

fairly closely to black body. A good physical

approximation to a black body is a cavity with

a tiny hole, which has no other opening. Any

ray entering the hole will be reflected by the

cavity walls and will be eventually absorbed by

the walls. A black body is also a perfect radiator

of radiant energy. Furthermore, a black body

is in thermal equilibrium with its surroundings.

It radiates same amount of energy per unit area

as it absorbs from its surrounding in any given

time. The amount of light emitted (intensity of

radiation) from a black body and its spectral

distribution depends only on its temperature.

At a given temperature, intensity of radiation

emitted increases with the increase of

wavelength, reaches a maximum value at a

given wavelength and then starts decreasing

with further increase of wavelength, as shown

in Fig. 2.8. Also, as the temperature increases,

maxima of the curve shifts to short wavelength.

Several attempts were made to predict the

intensity of radiation as a function of

wavelength.

But the results of the above experiment

could not be explained satisfactorily on the

basis of the wave theory of light. Max Planck

Fig. 2.8(a) Black body

Fig. 2.8 Wavelength-intensity relationship

2020-21

STRUCTURE OF ATOM

arrived at a satisfactory relationship by

making an assumption that absorption and

emmission of radiation arises from oscillator

i.e., atoms in the wall of black body. Their

frequency of oscillation is changed by

interaction with oscilators of electromagnetic

radiation. Planck assumed that radiation

could be sub-divided into discrete chunks of

energy. He suggested that atoms and

molecules could emit or absorb energy only

in discrete quantities and not in a continuous

manner. He gave the name quantum to the

smallest quantity of energy that can be

emitted or absorbed in the form of

electromagnetic radiation. The energy (E ) of a

quantum of radiation is proportional

to its frequency (

) and is expressed by

equation (2.6).

E = h

(2.6)

The proportionality constant, ‘h’ is known

as Planck’s constant and has the value

6.626×10

–34

J s.

With this theory, Planck was able to explain

the distribution of intensity in the radiation

from black body as a function of frequency or

wavelength at different temperatures.

Quantisation has been compared to

standing on a staircase. A person can stand

on any step of a staircase, but it is not possible

for him/her to stand in between the two steps.

The energy can take any one of the values from

the following set, but cannot take on any

values between them.

E = 0, h

, 2h

, 3h

....nh

.....

Photoelectric Effect

In 1887, H. Hertz performed a very interesting

experiment in which electrons (or electric

current) were ejected when certain metals (for

example potassium, rubidium, caesium etc.)

were exposed to a beam of light as shown

in Fig.2.9. The phenomenon is called

Photoelectric effect. The results observed in

this experiment were:

(i) The electrons are ejected from the metal

surface as soon as the beam of light strikes

the surface, i.e., there is no time lag

between the striking of light beam and the

ejection of electrons from the metal surface.

(ii) The number of electrons ejected is

proportional to the intensity or brightness

of light.

(iii) For each metal, there is a characteristic

minimum frequency,

(also known as

threshold frequency) below which

photoelectric effect is not observed. At a

frequency

, the ejected electrons come

out with certain kinetic energy. The kinetic

energies of these electrons increase with

the increase of frequency of the light used.

All the above results could not be explained

on the basis of laws of classical physics.

According to latter, the energy content of the

beam of light depends upon the brightness of

the light. In other words, number of electrons

ejected and kinetic energy associated with

them should depend on the brightness of light.

It has been observed that though the number

of electrons ejected does depend upon the

brightness of light, the kinetic energy of the

Fig.2.9 Equipment for studying the photoelectric

effect. Light of a particular frequency strikes

a clean metal surface inside a vacuum

chamber. Electrons are ejected from the

metal and are counted by a detector that

measures their kinetic energy.

Max Planck

(1858 – 1947)

Max Planck, a German physicist,

received his Ph.D in theoretical

physics from the University of

Munich in 1879. In 1888, he was

appointed Director of the Institute

of Theoretical Physics at the

University of Berlin. Planck was awarded the Nobel

Prize in Physics in 1918 for his quantum theory.

Planck also made significant contributions in

thermodynamics and other areas of physics.

2020-21

42 CHEMISTRY

ejected electrons does not. For example, red

light [

= (4.3 to 4.6) × 10

Hz] of any brightness

(intensity) may shine on a piece of potassium

metal for hours but no photoelectrons are

ejected. But, as soon as even a very weak yellow

light (

= 5.1–5.2 × 10

Hz) shines on the

potassium metal, the photoelectric effect is

observed. The threshold frequency (

) for

potassium metal is 5.0×10

Hz.

Einstein (1905) was able to explain the

photoelectric effect using Planck’s quantum

theory of electromagnetic radiation as a

starting point.

minimum energy required to eject the electron

is h

(also called work function, W

; Table 2.2),

then the difference in energy

– h

) is transferred as the kinetic energy of

the photoelectron. Following the conservation

of energy principle, the kinetic energy of the

ejected electron is given by the equation 2.7.

(2.7)

where m

is the mass of the electron and v is

the velocity associated with the ejected electron.

Lastly, a more intense beam of light consists

of larger number of photons, consequently the

number of electrons ejected is also larger as

compared to that in an experiment in which a

beam of weaker intensity of light is employed.

Dual Behaviour of Electromagnetic

Radiation

The particle nature of light posed a dilemma

for scientists. On the one hand, it could explain

the black body radiation and photoelectric

effect satisfactorily but on the other hand, it

was not consistent with the known wave

behaviour of light which could account for the

phenomena of interference and diffraction. The

only way to resolve the dilemma was to accept

the idea that light possesses both particle and

wave-like properties, i.e., light has dual

behaviour. Depending on the experiment, we

find that light behaves either as a wave or as a

stream of particles. Whenever radiation

interacts with matter, it displays particle like

properties in contrast to the wavelike

properties (interference and diffraction), which

it exhibits when it propagates. This concept

was totally alien to the way the scientists

thought about matter and radiation and it took

them a long time to become convinced of its

validity. It turns out, as you shall see later,

that some microscopic particles like electrons

also exhibit this wave-particle duality.

C:\Chemistry XI\Unit-2\Unit-2(2)-Lay-3(reprint).pmd 27.7.6, 16.10.6 (Reprint)

Albert Einstein, a German

born American physicist, is

regarded by many as one of

the two great physicists the

world has known (the other

is Isaac Newton). His three

research papers (on special

relativity, Brownian motion

and the photoelectric effect)

which he published in 1905,

Albert Einstein

(1879-1955)

while he was employed as a technical

assistant in a Swiss patent office in Berne

have profoundly influenced the development

of physics. He received the Nobel Prize in

Physics in 1921 for his explanation of the

photoelectric effect.

Shining a beam of light on to a metal

surface can, therefore, be viewed as shooting

a beam of particles, the photons. When a

photon of sufficient energy strikes an electron

in the atom of the metal, it transfers its energy

instantaneously to the electron during the

collision and the electron is ejected without

any time lag or delay. Greater the energy

possessed by the photon, greater will be

transfer of energy to the electron and greater

the kinetic energy of the ejected electron. In

other words, kinetic energy of the ejected

electron is proportional to the frequency of the

electromagnetic radiation. Since the striking

photon has energy equal to h

and the

Metal Li Na K Mg Cu Ag

/eV 2.42 2.3 2.25 3.7

4.8

4.3

Table 2.2 Values of Work Function (W

) for a Few Metals

2020-21

STRUCTURE OF ATOM

Problem 2.6

Calculate energy of one mole of photons

of radiation whose frequency is 5 ×10

Hz.

Solution

Energy (E) of one photon is given by the

expression

E = h

h = 6.626 ×10

–34

J s

= 5×10

–1

(given)

E = (6.626 ×10

–34

J s) × (5 ×10

–1

)

= 3.313 ×10

–19

Energy of one mole of photons

= (3.313 ×10

–19

J) × (6.022 × 10

mol

–1

)

= 199.51 kJ mol

–1

Problem 2.7

A 100 watt bulb emits monochromatic

light of wavelength 400 nm. Calculate the

number of photons emitted per second

by the bulb.

Solution

Power of the bulb = 100 watt

= 100 J s

–1

Energy of one photon E = h

= hc/λ

6.626 10 J s 3 10 m s

34 8 1

× × ×

− −

−

400 10

= 4.969 × 10

-19

Number of photons emitted

100

4 969 10

2 012 10

20 1

J s

−

= ×

Problem 2.8

When electromagnetic radiation of

wavelength 300 nm falls on the surface

of sodium, electrons are emitted with a

kinetic energy of 1.68 ×10

J mol

–1

. What

is the minimum energy needed to remove

an electron from sodium? What is the

maximum wavelength that will cause a

photoelectron to be emitted?

Solution

The energy (E) of a 300 nm photon is

given by

= 6.626 × 10

-19

The energy of one mole of photons

= 6.626 ×10

–19

J × 6.022 ×10

mol

–1

= 3.99 × 10

J mol

–1

The minimum energy needed to remove

one mole of electrons from sodium

= (3.99 –1.68) 10

J mol

–1

= 2.31 × 10

J mol

–1

The minimum energy for one electron

This corresponds to the wavelength

∴

× × ×

− −

−

λ=

6.626 10 J s 3.0 10 m s

3.84 10 J

34 8 1

= 517 nm

(This corresponds to green light)

Problem 2.9

The threshold frequency

for a metal is

7.0 ×10

–1

. Calculate the kinetic energy

of an electron emitted when radiation of

frequency

=1.0 ×10

–1

hits the metal.

Solution

According to Einstein’s equation

Kinetic energy = ½ m

=h(

–

)

= (6.626 ×10

–34

J s) (1.0 × 10

–1

– 7.0

×10

–1

)

= (6.626 ×10

–34

J s) (10.0 ×10

–1

– 7.0

×10

–1

)

= (6.626 ×10

–34

J s) × (3.0 ×10

–1

)

= 1.988 ×10

–19

2020-21

44 CHEMISTRY

2.3.3 Evidence for the quantized*

Electronic Energy Levels: Atomic

spectra

The speed of light depends upon the nature of

the medium through which it passes. As a

result, the beam of light is deviated or refracted

from its original path as it passes from one

medium to another. It is observed that when a

ray of white light is passed through a prism,

the wave with shorter wavelength bends more

than the one with a longer wavelength. Since

ordinary white light consists of waves with all

the wavelengths in the visible range, a ray of

white light is spread out into a series of

coloured bands called spectrum. The light of

red colour which has longest wavelength is

deviated the least while the violet light, which

has shortest wavelength is deviated the most.

The spectrum of white light, that we can see,

ranges from violet at 7.50 ×

Hz to red at

4×10

Hz. Such a spectrum is called

continuous spectrum. Continuous because

violet merges into blue, blue into green and so

on. A similar spectrum is produced when a

rainbow forms in the sky. Remember that

visible light is just a small portion of the

electromagnetic radiation (Fig.2.7). When

electromagnetic radiation interacts with matter,

atoms and molecules may absorb energy and

reach to a higher energy state. With higher

energy, these are in an unstable state. For

returning to their normal (more stable, lower

energy states) energy state, the atoms and

molecules emit radiations in various regions

of the electromagnetic spectrum.

Emission and Absorption Spectra

The spectrum of radiation emitted by a

substance that has absorbed energy is called

an emission spectrum. Atoms, molecules or

ions that have absorbed radiation are said to

be “excited”. To produce an emission

spectrum, energy is supplied to a sample by

heating it or irradiating it and the wavelength

(or frequency) of the radiation emitted, as the

sample gives up the absorbed energy, is

recorded.

An absorption spectrum is like the

photographic negative of an emission

spectrum. A continuum of radiation is passed

through a sample which absorbs radiation of

certain wavelengths. The missing wavelength

which corresponds to the radiation absorbed

by the matter, leave dark spaces in the bright

continuous spectrum.

The study of emission or absorption

spectra is referred to as spectroscopy. The

spectrum of the visible light, as discussed

above, was continuous as all wavelengths (red

to violet) of the visible light are represented in

the spectra. The emission spectra of atoms in

the gas phase, on the other hand, do not show

a continuous spread of wavelength from red

to violet, rather they emit light only at specific

wavelengths with dark spaces between them.

Such spectra are called line spectra or atomic

spectra because the emitted radiation is

identified by the appearance of bright lines in

the spectra (Fig. 2.10 page 45).

Line emission spectra are of great

interest in the study of electronic structure.

Each element has a unique line emission

spectrum. The characteristic lines in atomic

spectra can be used in chemical analysis to

identify unknown atoms in the same way as

fingerprints are used to identify people. The

exact matching of lines of the emission

spectrum of the atoms of a known element with

the lines from an unknown sample quickly

establishes the identity of the latter, German

chemist, Robert Bunsen (1811-1899) was one

of the first investigators to use line spectra to

identify elements.

Elements like rubidium (Rb), caesium (Cs)

thallium (Tl), indium (In), gallium (Ga) and

scandium (Sc) were discovered when their

minerals were analysed by spectroscopic

methods. The element helium (He) was

discovered in the sun by spectroscopic method.

Line Spectrum of Hydrogen

When an electric discharge is passed through

gaseous hydrogen, the H

molecules dissociate

and the energetically excited hydrogen atoms

produced emit electromagnetic radiation of

discrete frequencies. The hydrogen spectrum

consists of several series of lines named after

their discoverers. Balmer showed in 1885 on

the basis of experimental observations that if

* The restriction of any property to discrete values is called quantization.

2020-21

STRUCTURE OF ATOM

spectral lines are expressed in terms of

wavenumber ( ), then the visible lines of the

hydrogen spectrum obey the following formula:

(2.8)

where n is an integer equal to or greater than

3 (i.e., n = 3,4,5,....)

The series of lines described by this formula

are called the Balmer series. The Balmer series

of lines are the only lines in the hydrogen

spectrum which appear in the visible region

of the electromagnetic spectrum. The Swedish

spectroscopist, Johannes Rydberg, noted that

all series of lines in the hydrogen spectrum

could be described by the following

expression :

(2.9)

where n

=1,2........

= n

+ 1, n

+ 2......

The value 109,677 cm

–1

is called the

Rydberg constant for hydrogen. The first five

series of lines that correspond to n

= 1, 2, 3,

4, 5 are known as Lyman, Balmer, Paschen,

Bracket and Pfund series, respectively,

Table 2.3 shows these series of transitions in

the hydrogen spectrum. Fig 2.11 (page, 46)

shows the Lyman, Balmer and Paschen series

of transitions for hydrogen atom.

Of all the elements, hydrogen atom has the

simplest line spectrum. Line spectrum becomes

(a)

(b)

Fig. 2.10 (a) Atomic emission. The light emitted by a sample of excited hydrogen atoms (or any other

element) can be passed through a prism and separated into certain discrete wavelengths. Thus an

emission spectrum, which is a photographic recording of the separated wavelengths is called as line

spectrum. Any sample of reasonable size contains an enormous number of atoms. Although a single

atom can be in only one excited state at a time, the collection of atoms contains all possible excited

states. The light emitted as these atoms fall to lower energy states is responsible for the spectrum. (b)

Atomic absorption. When white light is passed through unexcited atomic hydrogen and then through

a slit and prism, the transmitted light is lacking in intensity at the same wavelengths as are emitted in

(a) The recorded absorption spectrum is also a line spectrum and the photographic negative of the

emission spectrum.

Table 2.3 The Spectral Lines for Atomic

Hydrogen

2020-21

46 CHEMISTRY

more and more complex for heavier atom. There

are, however, certain features which are

common to all line spectra, i.e., (i) line spectrum

of element is unique and (ii) there is regularity

in the line spectrum of each element. The

questions which arise are: What are the

reasons for these similarities? Is it something

to do with the electronic structure of atoms?

These are the questions need to be answered.

We shall find later that the answers to these

questions provide the key in understanding

electronic structure of these elements.

2.4 BOHR’S MODEL FOR HYDROGEN

ATOM

Neils Bohr (1913) was the first to explain

quantitatively the general features of the

structure of hydrogen atom and its spectrum.

He used Planck’s concept of quantisation of

energy. Though the theory is not the modern

quantum mechanics, it can still be used to

Fig. 2.11 Transitions of the electron in the

hydrogen atom (The diagram shows

the Lyman, Balmer and Paschen series

of transitions)

rationalize many points in the atomic structure

and spectra. Bohr’s model for hydrogen atom

is based on the following postulates:

i) The electron in the hydrogen atom can

move around the nucleus in a circular path

of fixed radius and energy. These paths are

called orbits, stationary states or allowed

energy states. These orbits are arranged

concentrically around the nucleus.

ii) The energy of an electron in the orbit does

not change with time. However, the

electron will move from a lower stationary

state to a higher stationary state when

required amount of energy is absorbed

by the electron or energy is emitted when

electron moves from higher stationary

state to lower stationary state (equation

2.16). The energy change does not take

place in a continuous manner.

Angular Momentum

Just as linear momentum is the product

of mass (m) and linear velocity (v), angular

momentum is the product of moment of

inertia (I) and angular velocity (ω). For an

electron of mass m

, moving in a circular

path of radius r around the nucleus,

angular momentum = I × ω

Since I = m

, and ω = v/r where v is the

linear velocity,

∴angular momentum = m

× v/r = m

iii) The frequency of radiation absorbed or

emitted when transition occurs between

two stationary states that differ in energy

by ∆E, is given by:

= =

−∆E

E E

2 1

(2.10)

Where E

and E

are the energies of the

lower and higher allowed energy states

respectively. This expression is commonly

known as Bohr’s frequency rule.

iv) The angular momentum of an electron is

quantised. In a given stationary state it

can be expressed as in equation (2.11)

m r n

v = .

2π

n = 1,2,3..... (2.11)

2020-21

STRUCTURE OF ATOM

Where m

is the mass of electron, v is the

velocity and r is the radius of the orbit in which

electron is moving.

Thus an electron can move only in those

orbits for which its angular momentum is

integral multiple of h/2π. That means angular

momentum is quantised. Radiation is emitted

or obsorbed only when transition of electron

takes place from one quantised value of angular

momentum to another. Therefore, Maxwell’s

electromagnetic theory does not apply here that

is why only certain fixed orbits are allowed.

The details regarding the derivation of

energies of the stationary states used by Bohr,

are quite complicated and will be discussed in

higher classes. However, according to Bohr’s

theory for hydrogen atom:

a) The stationary states for electron are

numbered n = 1,2,3.......... These integral

numbers (Section 2.6.2) are known as

Principal quantum numbers.

b) The radii of the stationary states are

expressed as:

= n

(2.12)

where a

= 52.9 pm. Thus the radius of

the first stationary state, called the Bohr

orbit, is 52.9 pm. Normally the electron

in the hydrogen atom is found in this orbit

(that is n=1). As n increases the value of r

will increase. In other words the electron

will be present away from the nucleus.

c) The most important property associated

with the electron, is the energy of its

stationary state. It is given by the

expression.

n H

= −













n = 1,2,3.... (2.13)

where R

is called Rydberg constant and its

value is 2.18×10

–18

J. The energy of the lowest

state, also called as the ground state, is

= –2.18×10

–18

(

) = –2.18×10

–18

J. The

energy of the stationary state for n = 2, will

be : E

= –2.18×10

–18

J (

)= –0.545×10

–18

Fig. 2.11 depicts the energies of different

stationary states or energy levels of hydrogen

atom. This representation is called an energy

level diagram.

When the electron is free from the influence

of nucleus, the energy is taken as zero. The

electron in this situation is associated with the

stationary state of Principal Quantum number

= n = ∞ and is called as ionized hydrogen atom.

When the electron is attracted by the nucleus

and is present in orbit n, the energy is emitted

What does the negative electronic

energy (E

) for hydrogen atom mean?

The energy of the electron in a hydrogen

atom has a negative sign for all possible

orbits (eq. 2.13). What does this negative

sign convey? This negative sign means that

the energy of the electron in the atom is

lower than the energy of a free electron at

rest. A free electron at rest is an electron

that is infinitely far away from the nucleus

and is assigned the energy value of zero.

Mathematically, this corresponds to

setting n equal to infinity in the equation

(2.13) so that E

∞

=0. As the electron gets

closer to the nucleus (as n decreases), E

becomes larger in absolute value and more

and more negative. The most negative

energy value is given by n=1 which

corresponds to the most stable orbit. We

call this the ground state.

Niels Bohr

(1885–1962)

Niels Bohr, a Danish

physicist received his Ph.D.

from the University of

Copenhagen in 1911. He

then spent a year with J.J.

Thomson and Ernest Rutherford in England.

In 1913, he returned to Copenhagen where

he remained for the rest of his life. In 1920

he was named Director of the Institute of

theoretical Physics. After first World War,

Bohr worked energetically 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.

2020-21

48 CHEMISTRY

and its energy is lowered. That is the reason

for the presence of negative sign in equation

(2.13) and depicts its stability relative to the

reference state of zero energy and n = ∞.

d) Bohr’s theory can also be applied to the

ions containing only one electron, similar

to that present in hydrogen atom. For

example, He

, Be

and so on. The

energies of the stationary states associated

with these kinds of ions (also known as

hydrogen like species) are given by the

expression.

J= − ×













−

2 18 10

(2.14)

and radii by the expression

r pm

52 9

. ( )n

(2.15)

where Z is the atomic number and has values

2,3 for the helium and lithium atoms

respectively. From the above equations, it is

evident that the value of energy becomes more

negative and that of radius becomes smaller

with increase of Z . This means that electron

will be tightly bound to the nucleus.

e) It is also possible to calculate the velocities

of electrons moving in these orbits.

Although the precise equation is not given

here, qualitatively the magnitude of

velocity of electron increases with increase

of positive charge on the nucleus and

decreases with increase of principal

quantum number.

2.4.1 Explanation of Line Spectrum of

Hydrogen

Line spectrum observed in case of hydrogen

atom, as mentioned in section 2.3.3, can be

explained quantitatively using Bohr’s model.

According to assumption 2, radiation (energy)

is absorbed if the electron moves from the orbit

of smaller Principal quantum number to the

orbit of higher Principal quantum number,

whereas the radiation (energy) is emitted if the

electron moves from higher orbit to lower orbit.

The energy gap between the two orbits is given

by equation (2.16)

∆E = E

– E

(2.16)

Combining equations (2.13) and (2.16)

∆ E

n n

= −













− −













R R

2 2

(where n

and n

stand for initial orbit and final orbits)

∆E

= −













= × −













−

R J

i f i f

1 1

2 18 10

1 1

2 2

n n n n

(2,17)

The frequency (

) associated with the

absorption and emission of the photon can be

evaluated by using equation (2.18)

= = −













∆ E

h h n n

i f

1 1

2 2

−













−

2 18 10

6 626 10

1 1

34 2 2

i f

n n

(2.18)

= × −













3 29 10

1 1

2 2

n n

i f

(2.19)

and in terms of wavenumbers ( )

= = −













h n n

1 1

2 2

i f

(2.20)

3 29 10

3 10

1 1

15 1

8 2 2

. ×

−













−

m s

i f

n n

= 1 09677 10

1 1

2 2

. × −













−

n n

i f

(2.21)

In case of absorption spectrum, n

> n

and

the term in the parenthesis is positive and energy

is absorbed. On the other hand in case of

emission spectrum n

> n

, ∆ E is negative and

energy is released.

The expression (2.17) is similar to that used

by Rydberg (2.9) derived empirically using the

experimental data available at that time. Further,

each spectral line, whether in absorption or

emission spectrum, can be associated to the

particular transition in hydrogen atom. In case

of large number of hydrogen atoms, different

possible transitions can be observed and thus

leading to large number of spectral lines. The

brightness or intensity of spectral lines depends

upon the number of photons of same wavelength

or frequency absorbed or emitted.

2020-21

STRUCTURE OF ATOM

Problem 2.10

What are the frequency and wavelength

of a photon emitted during a transition

from n = 5 state to the n = 2 state in the

hydrogen atom?

Solution

Since n

= 5 and n

= 2, this transition gives

rise to a spectral line in the visible region

of the Balmer series. From equation (2.17)

∆E =

2 18 10

4 58 10

2 2

× −













− ×

−

It is an emission energy

The frequency of the photon (taking

energy in terms of magnitude) is given by

∆E

= 6.91×10

Problem 2.11

Calculate the energy associated with the

first orbit of He

. What is the radius of this

orbit?

Solution

= −

−

( . )2 18 10

18 2

atom

–1

For He

, n = 1, Z = 2

18 2

2 18 10 2

8 72 10= −

= − ×

−

( . )( )

The radius of the orbit is given by equation

(2.15)

( . )0 0529

Since n = 1, and Z = 2

= =

( . )

0 0529 1

0 02645

2.4.2 Limitations of Bohr’s Model

Bohr’s model of the hydrogen atom was no

doubt an improvement over Rutherford’s

nuclear model, as it could account for the

stability and line spectra of hydrogen atom and

hydrogen like ions (for example, He

, Li

, Be

and so on). However, Bohr’s model was too

simple to account for the following points.

i) It fails to account for the finer details

(doublet, that is two closely spaced lines)

of the hydrogen atom spectrum observed

by using sophisticated spectroscopic

techniques. This model is also unable to

explain the spectrum of atoms other than

hydrogen, for example, helium atom which

possesses only two electrons. Further,

Bohr’s theory was also unable to explain

the splitting of spectral lines in the presence

of magnetic field (Zeeman effect) or an

electric field (Stark effect).

ii) It could not explain the ability of atoms to

form molecules by chemical bonds.

In other words, taking into account the

points mentioned above, one needs a better

theory which can explain the salient features

of the structure of complex atoms.

2.5 TOWARDS QUANTUM MECHANICAL

MODEL OF THE ATOM

In view of the shortcoming of the Bohr’s model,

attempts were made to develop a more suitable

and general model for atoms. Two important

developments which contributed significantly

in the formulation of such a model were :

1. Dual behaviour of matter,

2. Heisenberg uncertainty principle.

2.5.1 Dual Behaviour of Matter

The French physicist, de Broglie, in 1924

proposed that matter, like radiation, should

also exhibit dual behaviour i.e., both particle

and wavelike properties. This means that just

as the photon has momentum as well as

wavelength, electrons should also have

momentum as well as wavelength, de Broglie,

from this analogy, gave the following relation

between wavelength (λ

) and momentum (p) of

a material particle.

2020-21

50 CHEMISTRY

Solution

According to de Brogile equation (2.22)

λ = =

−

kg m s

( . )

( . )( )

6 626 10

0 1 10

= 6.626 × 10

–34

m (J = kg m

–2

)

Problem 2.13

The mass of an electron is 9.1×

–31

kg. If

its K.E. is 3.0×10

–25

J, calculate its

wavelength.

Solution

Since K. E. = ½ mv

K E

kg m s

= =

2 3 0 10

9 1 10

1 2

25 2 2

1 2

. .













× ×













− −

−

= 812 m s

–1

λ = =

−

− −

m v

kg m s

6 626 10

9 1 10

812

( . )( )

= 8967 × 10

–10

m = 896.7 nm

Problem 2.14

Calculate the mass of a photon with

wavelength 3.6 Å.

Solution

λ = 3.6 Å = 3.6 × 10

–10

Velocity of photon = velocity of light

= 6.135 × 10

–29

2.5.2 Heisenberg’s Uncertainty Principle

Werner Heisenberg a German physicist in

1927, stated uncertainty principle which is the

consequence of dual behaviour of matter and

radiation. It states that it is impossible to

determine simultaneously, the exact

position and exact momentum (or velocity)

of an electron.

Mathematically, it can be given as in

equation (2.23).

λ = =

(2.22)

where m is the mass of the particle, v its

velocity and p its momentum. de Broglie’s

prediction was confirmed experimentally

when it was found that an electron beam

undergoes diffraction, a phenomenon

characteristic of waves. This fact has been put

to use in making an electron microscope,

which is based on the wavelike behaviour of

electrons just as an ordinary microscope

utilises the wave nature of light. An electron

microscope is a powerful tool in modern

scientific research because it achieves a

magnification of about 15 million times.

It needs to be noted that according to de

Broglie, every object in motion has a wave

character. The wavelengths associated with

ordinary objects are so short (because of their

large masses) that their wave properties cannot

be detected. The wavelengths associated with

electrons and other subatomic particles (with

very small mass) can however be detected

experimentally. Results obtained from the

following problems prove these points

qualitatively.

Problem 2.12

What will be the wavelength of a ball of

mass 0.1 kg moving with a velocity of 10

m s

–1

Louis de Broglie (1892 – 1987)

Louis de Broglie, a French

physicist, studied history as an

undergraduate in the early

1910’s. His interest turned to

science as a result of his

assignment to radio

communications in World War I.

He received his Dr. Sc. from the University of

Paris in 1924. He was professor of theoretical

physics at the University of Paris from 1932 untill

his retirement in 1962. He was awarded the

Nobel Prize in Physics in 1929.

2020-21

STRUCTURE OF ATOM

(2.23)

where ∆x is the uncertainty in position and ∆p

(or ∆v

) is the uncertainty in momentum (or

velocity) of the particle. If the position of the

electron is known with high degree of accuracy

(∆x is small), then the velocity of the electron

will be uncertain [

∆(v

) is large]. On the other

hand, if the velocity of the electron is known

precisely (

∆(v

) is small), then the position of

the electron will be uncertain

(∆x will be large). Thus, if we carry out some

physical measurements on the electron’s

position or velocity, the outcome will always

depict a fuzzy or blur picture.

The uncertainty principle can be best

understood with the help of an example.

Suppose you are asked to measure the

thickness of a sheet of paper with an

unmarked metrestick. Obviously, the results

obtained would be extremely inaccurate and

meaningless, In order to obtain any accuracy,

you should use an instrument graduated in

units smaller than the thickness of a sheet of

the paper. Analogously, in order to determine

the position of an electron, we must use a

meterstick calibrated in units of smaller than

the dimensions of electron (keep in mind that

an electron is considered as a point charge and

is therefore, dimensionless). To observe an

electron, we can illuminate it with “light” or

electromagnetic radiation. The “light” used

must have a wavelength smaller than the

dimensions of an electron. The high

momentum photons of such light













would change the energy of electrons by

collisions. In this process we, no doubt, would

be able to calculate the position of the electron,

but we would know very little about the

velocity of the electron after the collision.

Significance of Uncertainty Principle

One of the important implications of the

Heisenberg Uncertainty Principle is that it

rules out existence of definite paths or

trajectories of electrons and other similar

particles. The trajectory of an object is

determined by its location and velocity at

various moments. If we know where a body is

at a particular instant and if we also know its

velocity and the forces acting on it at that

instant, we can tell where the body would be

sometime later. We, therefore, conclude that the

position of an object and its velocity fix its

trajectory. Since for a sub-atomic object such

as an electron, it is not possible simultaneously

to determine the position and velocity at any

given instant to an arbitrary degree of

precision, it is not possible to talk of the

trajectory of an electron.

The effect of Heisenberg Uncertainty

Principle is significant only for motion of

microscopic objects and is negligible for

that of macroscopic objects. This can be

seen from the following examples.

If uncertainty principle is applied to an

object of mass, say about a milligram (10

–6

kg),

then

Werner Heisenberg (1901 – 1976) Werner Heisenberg (1901 – 1976) received his Ph.D. in

physics from the University of Munich in 1923. He then spent a year working with Max

Born at Gottingen and three years with Niels Bohr in Copenhagen. He was professor of

physics at the University of Leipzig fr

om 1927 to 1941. During World War II, Heisenberg

was in charge of German research on the atomic bomb. After the war he was named

director of Max Planck Institute for physics in Gottingen. He was also accomplished

mountain climber. Heisenberg was awarded the Nobel Prize in Physics in 1932.

2020-21

52 CHEMISTRY

The value of ∆v∆x obtained is extremely

small and is insignificant. Therefore, one may

say that in dealing with milligram-sized or

heavier objects, the associated

uncertainties are hardly of any real

consequence.

In the case of a microscopic object like an

electron on the other hand. ∆v.∆x obtained is

much larger and such uncertainties are of real

consequence. For example, for an electron

whose mass is 9.11×10

–31

kg., according to

Heisenberg uncertainty principle

It, therefore, means that if one tries to find

the exact location of the electron, say to an

uncertainty of only 10

–8

m, then the uncertainty

∆v in velocity would be

4 2 1

4 1

− −

−

≈

m s

which is so large that the classical picture of

electrons moving in Bohr’s orbits (fixed) cannot

hold good. It, therefore, means that the

precise statements of the position and

momentum of electrons have to be

replaced by the statements of probability,

that the electron has at a given position

and momentum. This is what happens in

the quantum mechanical model of atom.

= 0.579×10

m s

–1

(1J = 1 kg m

–2

)

= 5.79×10

m s

–1

Problem 2.16

A golf ball has a mass of 40g, and a speed

of 45 m/s. If the speed can be measured

within accuracy of 2%, calculate the

uncertainty in the position.

Solution

The uncertainty in the speed is 2%, i.e.,

Using the equation (2.22)

= 1.46×10

–33

This is nearly ~ 10

times smaller than

the diameter of a typical atomic nucleus.

As mentioned earlier for large particles, the

uncertainty principle sets no meaningful

limit to the precision of measurements.

Reasons for the Failure of the Bohr Model

One can now understand the reasons for the

failure of the Bohr model. In Bohr model, an

electron is regarded as a charged particle

moving in well defined circular orbits about

the nucleus. The wave character of the electron

is not considered in Bohr model. Further, an

orbit is a clearly defined path and this path

can completely be defined only if both the

position and the velocity of the electron are

known exactly at the same time. This is not

possible according to the Heisenberg

uncertainty principle. Bohr model of the

hydrogen atom, therefore, not only ignores

dual behaviour of matter but also contradicts

Heisenberg uncertainty principle. In view of

Problem 2.15

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

x = or x v =

4 4

h h

p m∆ ∆ ∆ ∆

π π

2020-21

STRUCTURE OF ATOM

these inherent weaknesses in the Bohr model,

there was no point in extending Bohr model

to other atoms. In fact an insight into the

structure of the atom was needed which could

account for wave-particle duality of matter and

be consistent with Heisenberg uncertainty

principle. This came with the advent of

quantum mechanics.

2.6 QUANTUM MECHANICAL MODEL OF

ATOM

Classical mechanics, based on Newton’s laws

of motion, successfully describes the motion

of all macroscopic objects such as a falling

stone, orbiting planets etc., which have

essentially a particle-like behaviour as shown

in the previous section. However it fails when

applied to microscopic objects like electrons,

atoms, molecules etc. This is mainly because

of the fact that classical mechanics ignores the

concept of dual behaviour of matter especially

for sub-atomic particles and the uncertainty

principle. The branch of science that takes into

account this dual behaviour of matter is called

quantum mechanics.

Quantum mechanics is a theoretical

science that deals with the study of the motions

of the microscopic objects that have both

observable wave like and particle like

properties. It specifies the laws of motion that

these objects obey. When quantum mechanics

is applied to macroscopic objects (for which

wave like properties are insignificant) the

results are the same as those from the classical

mechanics.

Quantum mechanics was developed

independently in 1926 by Werner Heisenberg

and Erwin Schrödinger. Here, however, we

shall be discussing the quantum mechanics

which is based on the ideas of wave motion.

The fundamental equation of quantum

mechanics was developed by Schrödinger and

it won him the Nobel Prize in Physics in 1933.

This equation which incorporates wave-

particle duality of matter as proposed by de

Broglie is quite complex and knowledge of

higher mathematics is needed to solve it. You

will learn its solutions for different systems in

higher classes.

For a system (such as an atom or a

molecule whose energy does not change with

time) the Schrödinger equation is written as

where is a mathematical operator

called Hamiltonian. Schrödinger 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. Solution of this equation

gives E and

Hydrogen Atom and the Schrödinger

Equation

When Schrödinger equation is solved for

hydrogen atom, the solution gives the possible

energy levels the electron can occupy and the

corresponding wave function(s) (

) of the

electron associated with each energy level.

These quantized energy states and

corresponding wave functions which are

characterized by a set of three quantum

numbers (principal quantum number n,

azimuthal quantum number l and

magnetic quantum number m

) arise as a

natural consequence in the solution of the

Schrödinger equation. When an electron is in

Erwin Schrödinger

(1887-1961)

Erwin Schrödinger, an

Austrian physicist

received his Ph.D. in

theoretical physics from

the University of Vienna

in 1910. In 1927

Schrödinger succeeded

Max Planck at the

University of Berlin at

Planck’s request. In 1933,

Schrödinger left Berlin

because of his opposition to Hitler and Nazi

policies and returned to Austria in 1936. After

the invasion of Austria by Germany,

Schrödinger was forcibly removed from his

professorship. He then moved to Dublin, Ireland

where he remained for seventeen years.

Schrödinger shared the Nobel Prize for Physics

with P.A.M. Dirac in 1933.

2020-21

54 CHEMISTRY

any energy state, the wave function

corresponding to that energy state contains all

information about the electron. The wave

function is a mathematical function whose

value depends upon the coordinates of the

electron in the atom and does not carry any

physical meaning. Such wave functions of

hydrogen or hydrogen like species with one

electron are called atomic orbitals. Such wave

functions pertaining to one-electron species

are called one-electron systems. The

probability of finding an electron at a point

within an atom is proportional to the |

that point. The quantum mechanical results

of the hydrogen atom successfully predict all

aspects of the hydrogen atom spectrum

including some phenomena that could not be

explained by the Bohr model.

Application of Schrödinger equation to

multi-electron atoms presents a difficulty: the

Schrödinger equation cannot be solved exactly

for a multi-electron atom. This difficulty can

be overcome by using approximate methods.

Such calculations with the aid of modern

computers show that orbitals in atoms other

than hydrogen do not differ in any radical way

from the hydrogen orbitals discussed above.

The principal difference lies in the consequence

of increased nuclear charge. Because of this

all the orbitals are somewhat contracted.

Further, as you shall see later (in subsections

2.6.3 and 2.6.4), unlike orbitals of hydrogen

or hydrogen like species, whose energies

depend only on the quantum number n, the

energies of the orbitals in multi-electron atoms

depend on quantum numbers n and l.

Important Features of the Quantum

Mechanical Model of Atom

Quantum mechanical model of atom is the

picture of the structure of the atom, which

emerges from the application of the

Schrödinger equation to atoms. The

following are the important features of the

quantum-mechanical model of atom:

1. The energy of electrons in atoms is

quantized (i.e., can only have certain

specific values), for example when

electrons are bound to the nucleus in

atoms.

2. The existence of quantized electronic

energy levels is a direct result of the wave

like properties of electrons and are

allowed solutions of Schrödinger wave

equation.

3. Both the exact position and exact velocity

of an electron in an atom cannot be

determined simultaneously (Heisenberg

uncertainty principle). The path of an

electron in an atom therefore, can never

be determined or known accurately.

That is why, as you shall see later on,

one talks of only probability of finding

the electron at different points in

an atom.

4. An atomic orbital is the wave function

ψψ

for an electron in an atom.

Whenever an electron is described by a

wave function, we say that the electron

occupies that orbital. Since many such

wave functions are possible for an

electron, there are many atomic orbitals

in an atom. These “one electron orbital

wave functions” or orbitals form the

basis of the electronic structure of atoms.

In each orbital, the electron has a

definite energy. An orbital cannot

contain more than two electrons. In a

multi-electron atom, the electrons are

filled in various orbitals in the order of

increasing energy. For each electron of

a multi-electron atom, there shall,

therefore, be 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

and quantum mechanics

makes it possible to extract this

information out of

5. The probability of finding an electron at

a point within an atom is proportional

to the square of the orbital wave function

i.e., |

at that point. |

is known

as probability density and is always

positive. From the value of |

ψψ

different points within an atom, it is

possible to predict the region around

the nucleus where electron will most

probably be found.

2.6.1 Orbitals and Quantum Numbers

A large number of orbitals are possible in an

atom. Qualitatively these orbitals can be

2020-21

STRUCTURE OF ATOM

distinguished by their size, shape and

orientation. An orbital of smaller size means

there is more chance of finding the electron near

the nucleus. Similarly shape and orientation

mean that there is more probability of finding

the electron along certain directions than

along others. Atomic orbitals are precisely

distinguished by what are known as quantum

numbers. Each orbital is designated by three

quantum numbers labelled as n, l and m

The principal quantum number ‘n’ is

a positive integer with value of n = 1,2,3.......

The principal quantum number determines the

size and to large extent the energy of the

orbital. For hydrogen atom and hydrogen like

species (He

, Li

, .... etc.) energy and size of

the orbital depends only on ‘n’.

The principal quantum number also

identifies the shell. With the increase in the

value of ‘n’, the number of allowed orbital

increases and are given by ‘n

’ All the

orbitals of a given value of ‘n’ constitute a

single shell of atom and are represented by

the following letters

n = 1 2 3 4 ............

Shell = K L M N ............

Size of an orbital increases with increase of

principal quantum number ‘n’. In other words

the electron will be located away from the

nucleus. Since energy is required in shifting

away the negatively charged electron from the

positively charged nucleus, the energy of the

orbital will increase with increase of n.

Azimuthal quantum number. ‘l’ is also

known as orbital angular momentum or

subsidiary quantum number. It defines the

three-dimensional shape of the orbital. For a

given value of n, l can have n values ranging

from 0 to n – 1, that is, for a given value of n,

the possible value of l are : l = 0, 1, 2, ..........

(n–1)

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 l values are 0, 1

and 2.

Each shell consists of one or more sub-

shells or sub-levels. The number of sub-shells

in a principal shell is equal to the value of n.

For example in the first shell (n = 1), there is

only one sub-shell which corresponds to l = 0.

There are two sub-shells (l = 0, 1) in the second

shell (n = 2), three (l = 0, 1, 2) in third shell (n =

3) and so on. Each sub-shell is assigned an

azimuthal quantum number (l). Sub-shells

corresponding to different values of l are

represented by the following symbols.

Value for l : 0 1 2

3 4 5 ............

notation for s p d f g h ............

sub-shell

Table 2.4 shows the permissible values of

‘l ’ for a given principal quantum number and

the corresponding sub-shell notation.

Table 2.4 Subshell Notations

Magnetic orbital quantum number. ‘m

’

gives information about the spatial

orientation of the orbital with respect to

standard set of co-ordinate axis. For any

sub-shell (defined by ‘l’ value) 2l+1 values

of m

are possible and these values are given

by :

= – l, – (l–1), – (l–2)... 0,1... (l–2), (l–1), l

Thus for l = 0, the only permitted value of

= 0, [2(0)+1 = 1, one s orbital]. For l = 1, m

can be –1, 0 and +1 [2(1)+1 = 3, three p

orbitals]. For l = 2, m

= –2, –1, 0, +1 and +2,

[2(2)+1 = 5, five d orbitals]. It should be noted

2020-21

56 CHEMISTRY

that the values of m

are derived from l and that

the value of l are derived from n.

Each orbital in an atom, therefore, is

defined by a set of values for n, l and m

. An

orbital described by the quantum numbers

n = 2, l = 1, m

= 0 is an orbital in the p sub-shell

of the second shell. The following chart gives

the relation between the subshell and the

number of orbitals associated with it.

electron has, besides charge and mass,

intrinsic spin angular quantum number. Spin

angular momentum of the electron — a vector

quantity, can have two orientations relative to

the chosen axis. These two orientations are

distinguished by the spin quantum numbers

which can take the values of +½ or –½.

These are called the two spin states of the

electron and are normally represented by two

arrows, ↑ (spin up) and ↓ (spin down). Two

electrons that have different m

values (one +½

and the other –½) are said to have opposite

spins. An orbital cannot hold more than two

electrons and these two electrons should have

opposite spins.

To sum up, the four quantum numbers

provide the following information :

i) n defines the shell, determines the size of

the orbital and also to a large extent the

energy of the orbital.

ii) There are n subshells in the n

shell. l

identifies the subshell and determines the

shape of the orbital (see section 2.6.2).

There are (2l+1) orbitals of each type in a

subshell, that is, one s orbital (l = 0), three

p orbitals (l = 1) and five d orbitals (l = 2)

per subshell. To some extent l also

determines the energy of the orbital in a

multi-electron atom.

iii) m

designates the orientation of the orbital.

For a given value of l, m

has (2l+1) values,

Orbit, orbital and its importance

Orbit and orbital are not synonymous. An orbit, as proposed by Bohr, is a circular path

around the nucleus in which an electron moves. A precise description of this path of the

electron is impossible according to Heisenberg uncertainty principle. Bohr orbits, therefore,

have no real meaning and their existence can never be demonstrated experimentally. An

atomic orbital, on the other hand, is a quantum mechanical concept and refers to the one

electron wave function

in an atom. It is characterized by three quantum numbers (n, l and

) and its value depends upon the coordinates of the electron.

has, by itself, no physical

meaning. It is the square of the wave function i.e., |

which has a physical meaning. |

at any point in an atom gives the value of probability density at that point. Probability density

) is the probability per unit volume and the product of |

and a small volume (called a

volume element) yields the probability of finding the electron in that volume (the reason for

specifying a small volume element is that |

varies from one region to another in space but

its value can be assumed to be constant within a small volume element). The total probability

of finding the electron in a given volume can then be calculated by the sum of all the products

of |

and the corresponding volume elements. It is thus possible to get the probable

distribution of an electron in an orbital.

Value of l 0

3 4 5

Subshell notation s p d f g h

number of orbitals 1 3 5 7 9 11

Electron spin ‘s’ : The three quantum

numbers labelling an atomic orbital can be

used equally well to define its energy, shape

and orientation. But all these quantum

numbers are not enough to explain the line

spectra observed in the case of multi-electron

atoms, that is, some of the lines actually occur

in doublets (two lines closely spaced), triplets

(three lines, closely spaced) etc. This suggests

the presence of a few more energy levels than

predicted by the three quantum numbers.

In 1925, George Uhlenbeck and Samuel

Goudsmit proposed the presence of the fourth

quantum number known as the electron

spin quantum number (m

). An electron

spins around its own axis, much in a similar

way as earth spins around its own axis while

revolving around the sun. In other words, an

2020-21

STRUCTURE OF ATOM

the same as the number of orbitals per

subshell. It means that the number of

orbitals is equal to the number of ways in

which they are oriented.

iv) m

refers to orientation of the spin of the

electron.

Problem 2.17

What is the total number of orbitals

associated with the principal quantum

number n = 3 ?

Solution

For n = 3, the possible values of l are 0, 1

and 2. Thus there is one 3s orbital

(n = 3, l = 0 and m

= 0); there are three 3p

orbitals (n = 3, l = 1 and m

= –1, 0, +1);

there are five 3d orbitals (n = 3, l = 2 and

= –2, –1, 0, +1+, +2).

Therefore, the total number of orbitals is

1+3+5 = 9

The same value can also be obtained by

using the relation; number of orbitals

= n

, i.e. 3

= 9.

Problem 2.18

Using s, p, d, f notations, describe the

orbital with the following quantum

numbers

(a) n = 2, l = 1, (b) n = 4, l = 0, (c) n = 5,

l = 3, (d) n = 3, l = 2

Solution

n l orbital

a) 2 1 2p

b) 4 0 4s

c) 5

3 5f

d) 3 2 3d

2.6.2 Shapes of Atomic Orbitals

The orbital wave function or

for an electron

in an atom has no physical meaning. It is

simply a mathematical function of the

coordinates of the electron. However, for

different orbitals the plots of corresponding

wave functions as a function of r (the distance

from the nucleus) are different. Fig. 2.12(a),

gives such plots for 1s (n = 1, l = 0) and 2s (n =

2, l = 0) orbitals.

According to the German physicist, Max

Born, the square of the wave function

(i.e.,

) at a point gives the probability density

of the electron at that point. The variation of

as a function of r for 1s and 2s orbitals is

given in Fig. 2.12(b). Here again, you may note

that the curves for 1s and 2s orbitals are

different.

It may be noted that for 1s orbital the

probability density 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 decreases again and

approaches zero as the value of r increases

further. The region where this probability

density function reduces to zero is called

nodal surfaces or simply nodes. In general,

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 variation can be

visualised in terms of charge cloud diagrams

[Fig. 2.13(a)]. In these diagrams, the density

Fig. 2.12 The plots of (a) the orbital wave

function

(r )

; (b) the variation of

probability density

(r)

as a function

of distance r of the electron from the

nucleus for 1s and 2s orbitals.

2020-21

58 CHEMISTRY

of the dots in a region represents electron

probability density in that region.

Boundary surface diagrams of constant

probability density for different orbitals give a

fairly good representation of the shapes of the

orbitals. In this representation, a boundary

surface or contour surface is drawn in space

for an orbital on which the value of probability

density |

is constant. 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%. The boundary surface diagram for

1s and 2s orbitals are given in Fig. 2.13(b).

One may ask a question : Why do we not draw

a boundary surface diagram, which bounds a

region in which the probability of finding the

electron is, 100 %? The answer to this question

is that the probability density |

has always

some value, howsoever small it may be, at any

finite distance from the nucleus. It is therefore,

not possible to draw a boundary surface

diagram of a rigid size in which the probability

of finding the electron is 100%. Boundary

surface diagram for a s orbital is actually a

sphere centred on the nucleus. In two

dimensions, this sphere looks like a circle. It

encloses a region in which probability of

finding the electron is about 90%.

Thus, we see that 1s and 2s orbitals are

spherical in shape. In reality all the s-orbitals

are spherically symmetric, that is, the probability

of finding the electron at a given distance is equal

in all the directions. It is also observed that the

size of the s orbital increases with increase in n,

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

located further away from the nucleus as the

principal quantum number increases.

Boundary surface diagrams for three 2p

orbitals (l = 1) are shown in Fig. 2.14. In these

diagrams, the nucleus is at the origin. Here,

unlike s-orbitals, the boundary surface

diagrams are not spherical. Instead each

p orbital consists of two sections called lobes

that are on either side of the plane that passes

through the nucleus. The probability density

function is zero on the plane where the two

lobes touch each other. The size, shape and

If probability density |

is constant on a given surface, |

| is also constant over the surface. The boundary

surface for |

and |

| are identical.

Fig. 2.14 Boundary surface diagrams of the

three 2p orbitals.

Fig. 2.13 (a) Probability density plots of 1s and

2s atomic orbitals. The density of the

dots represents the probability

density of finding the electron in that

region. (b) Boundary surface diagram

for 1s and 2s orbitals.

2020-21

STRUCTURE OF ATOM

energy of the three orbitals are identical. They

differ however, in the way the lobes are

oriented. Since the lobes may be considered

to lie along the x, y or z axis, they are given the

designations 2p

, 2p

, and 2p

. It should be

understood, however, that there is no simple

relation between the values of m

(–1, 0 and

+1) and the x, y and z directions. For our

purpose, it is sufficient to remember that,

because there are three possible values of m

there are, therefore, three p orbitals whose axes

are mutually perpendicular. 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 functions for

p-orbital also pass through value zero, besides

at zero and infinite distance, as the distance

from the nucleus increases. The number of

nodes are given by the n –2, that is number of

radial node is 1 for 3p orbital, two for 4p orbital

and so on.

For l = 2, the orbital is known as d

-orbital

and the minimum value of principal quantum

number (n) has to be 3. as the value of l cannot

be greater than n–1. There are five m

values (–

2, –1, 0, +1 and +2) for l = 2 and thus there are

five d orbitals. The boundary surface diagram

of d orbitals are shown in Fig. 2.15.

The five d-orbitals are designated as d

, d

–y

and d

. The shapes of the first four d-

orbitals are similar to each other, where as that

of the fifth one, d

, is different from others, but

all five 3d orbitals are equivalent in energy. The

d orbitals for which n is greater than 3 (4d,

5d...) also have shapes similar to 3d orbital,

but differ in energy and size.

Besides the radial nodes (i.e., probability

density function is zero), the probability

density functions for the np and nd orbitals

are zero at the plane (s), passing through the

nucleus (origin). For example, in case of p

orbital, xy-plane is a nodal plane, in case of d

orbital, there are two nodal planes passing

through the origin and bisecting the xy plane

containing z-axis. These are called angular

nodes and number of angular nodes are given

by ‘l’, i.e., one angular node for p orbitals, two

angular nodes for ‘d’ orbitals and so on. The

total number of nodes are given by (n–1),

i.e., sum of l angular nodes and (n – l – 1)

radial nodes.

2.6.3 Energies of Orbitals

The energy of an electron in a hydrogen atom

is determined solely by the principal quantum

Fig. 2.15 Boundary surface diagrams of the five

3d orbitals.

2020-21

60 CHEMISTRY

number. Thus the energy of the orbitals in

hydrogen atom increases as follows :

1s < 2s = 2p < 3s = 3p = 3d <4s = 4p = 4d

= 4f < (2.23)

and is depicted in Fig. 2.16. Although the

shapes of 2s and 2p orbitals are different, an

electron has the same energy when it is in the

2s orbital as when it is present in 2p orbital.

The orbitals having the same energy are called

degenerate. The 1s orbital in a hydrogen

atom, as said earlier, corresponds to the most

stable condition and is called the ground state

and an electron residing in this orbital is most

strongly held by the nucleus. An electron in

the 2s, 2p or higher orbitals in a hydrogen atom

is in excited state.

The energy of an electron in a multi-

electron atom, unlike that of the hydrogen

atom, depends not only on its principal

quantum number (shell), but also on its

azimuthal quantum number (subshell). That

is, for a given principal quantum number, s,

p, d

, f ... all have different energies. Within a

given principal quantum number, the energy

of orbitals increases in the order s<p<d<f. For

higher energy levels, these differences are

sufficiently pronounced and straggering of

orbital energy may result, e.g., 4s<3d and

6s<5d

;

4f<6p. The main reason for having

different energies of the subshells is the mutual

repulsion among the electrons in multi-

electron atoms. The only electrical interaction

present in hydrogen atom is the attraction

between the negatively charged electron and

the positively charged nucleus. In multi-

electron atoms, besides the presence of

attraction between the electron and nucleus,

there are repulsion terms between every

electron and other electrons present in the

atom. Thus the stability of an electron in a

multi-electron atom is because total attractive

interactions are more than the repulsive

interactions. In general, the repulsive

interaction of the electrons in the outer shell

with the electrons in the inner shell are more

important. On the other hand, the attractive

interactions of an electron increases with

increase of positive charge (Ze) on the nucleus.

Due to the presence of electrons in the inner

shells, the electron in the outer shell will not

experience the full positive charge of the

nucleus (Ze). The effect will be lowered due to

the partial screening of positive charge on the

nucleus by the inner shell electrons. This is

known as the shielding of the outer shell

electrons from the nucleus by the inner

shell electrons,

and the net positive charge

experienced by the outer electrons is known

as effective nuclear charge (Z

eff

e). Despite

the shielding of the outer electrons from the

nucleus by the inner shell electrons, the

attractive force experienced by the outer shell

electrons increases with increase of nuclear

charge. In other words, the energy of

interaction between, the nucleus and electron

Fig. 2.16 Energy level diagrams for the few

electronic shells of (a) hydrogen atom

and (b) multi-electronic atoms. Note that

orbitals for the same value of principal

quantum number, have the same

energies even for different azimuthal

quantum number for hydrogen atom.

In case of multi-electron atoms, orbitals

with same principal quantum number

possess different energies for different

azimuthal quantum numbers.

2020-21

STRUCTURE OF ATOM

(that is orbital energy) decreases (that is

more negative) with the increase of atomic

number (Z).

Both the attractive and repulsive

interactions depend upon the shell and shape

of the orbital in which the electron is present.

For example electrons present in spherical

shaped, s orbital shields the outer electrons

from the nucleus more effectively as compared

to electrons present in p orbital. Similarly

electrons present in p orbitals shield the outer

electrons from the nucleus more than the

electrons present in d orbitals, even though all

these orbitals are present in the same shell.

Further within a shell, due to spherical shape

of s orbital, the s orbital electron spends more

time close to the nucleus in comparison to p

orbital electron which spends more time in the

vicinity of nucleus in comparison to d

orbital

electron. In other words, for a given shell

(principal quantum number), the Z

eff

experienced by the electron decreases with

increase of azimuthal quantum number (l),

that is, the s orbital electron will be more tightly

bound to the nucleus than p orbital electron

which in turn will be better tightly bound than

the d orbital electron. The energy of electrons

in s orbital will be lower (more negative) than

that of p orbital electron which will have less

energy than that of d orbital electron and so

on. Since the extent of shielding from the

nucleus is different for electrons in different

orbitals, it leads to the splitting of energy levels

within the same shell (or same principal

quantum number), that is, energy of electron

in an orbital, as mentioned earlier, depends

upon the values of n and l. Mathematically,

the dependence of energies of the orbitals on n

and l are quite complicated but one simple rule

is that, the lower the value of (n + l) for an

orbital, the lower is its energy. If two

orbitals have the same value of (n + l), the

orbital with lower value of n will have the

lower energy. The Table 2.5 illustrates the (n

+ l ) rule and Fig. 2.16 depicts the energy levels

of multi-electrons atoms. It may be noted that

different subshells of a particular shell have

different energies in case of multi–electrons

atoms. However, in hydrogen atom, these have

the same energy. Lastly it may be mentioned

here that energies of the orbitals in the

same subshell decrease with increase in

the atomic number (Z

eff

). For example, 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, that

is, E

(H) > E

(Li) > E

(Na) > E

(K).

2.6.4 Filling of Orbitals in Atom

The filling of electrons into the orbitals of

different atoms takes place according to the

aufbau principle which is based on the Pauli’s

exclusion principle, the Hund’s rule of

maximum multiplicity and the relative

energies of the orbitals.

Aufbau Principle

The word ‘aufbau’ in German means ‘building

up’. The building up of orbitals means the

Table 2.5 Arrangement of Orbitals with

Increasing Energy on the Basis of

(n+l) Rule

2020-21

62 CHEMISTRY

filling up of orbitals with electrons. The

principle states : In the ground state of the

atoms, the orbitals are filled in order of

their increasing energies. In other words,

electrons first occupy the lowest energy orbital

available to them and enter into higher energy

orbitals only after the lower energy orbitals are

filled. As you have learnt above, energy of a

given orbital depends upon effective nuclear

charge and different type of orbitals are affected

to different extent. Thus, there is no single

ordering of energies of orbitals which will be

universally correct for all atoms.

However, following order of energies of the

orbitals is extremely useful:

1s, 2s, 2p, 3s, 3p, 4

s, 3d, 4p, 5s, 4d, 5p, 4f,

5d, 6p, 7s...

The order may be remembered by using

the method given in Fig. 2.17. Starting from

the top, the direction of the arrows gives the

order of filling of orbitals, that is starting from

right top to bottom left. With respect to

placement of outermost valence electrons, it is

remarkably accurate for all atoms. For

example, valence electron in potassium must

choose between 3d and 4s orbitals and as

predicted by this sequence, it is found in 4s

orbital. The above order should be assumed

to be a rough guide to the filling of energy

levels. In many cases, the orbitals are similar

in energy and small changes in atomic

structure may bring about a change in the

order of filling. Even then, the above series is a

useful guide to the building of the electronic

structure of an atom provided that it is

remembered that exceptions may occur.

Pauli Exclusion Principle

The number of electrons to be filled in various

orbitals is restricted by the exclusion principle,

given by the Austrian scientist Wolfgang Pauli

(1926). According to this principle : No two

electrons in an atom can have the same

set of four quantum numbers. Pauli

exclusion principle can also be stated as : “Only

two electrons may exist in the same orbital

and these electrons must have opposite

spin.” This means that the two electrons can

have the same value of three quantum numbers

n, l and m

, but must have the opposite spin

quantum number. The restriction imposed by

Pauli’s exclusion principle on the number of

electrons in an orbital helps in calculating the

capacity of electrons to be present in any

subshell. For example, subshell 1s comprises

one orbital and thus the maximum number of

electrons present in 1s subshell can be two, in

p and d subshells, the maximum number of

electrons can be 6 and 10 and so on. This can

be summed up as : the maximum number

of electrons in the shell with principal

quantum number n is equal to 2n

Hund’s Rule of Maximum Multiplicity

This rule deals with the filling of electrons into

the orbitals belonging to the same subshell

(that is, orbitals of equal energy, called

Fig.2.17 Order of filling of orbitals

2020-21

STRUCTURE OF ATOM

degenerate orbitals). It states : pairing of

electrons in the orbitals belonging to the

same subshell (p, d or f) does not take place

until each orbital belonging to that

subshell has got one electron each i.e., it

is singly occupied.

Since there are three p, five d and seven f

orbitals, therefore, the pairing of electrons will

start in the p, d and f orbitals with the entry of

4th, 6th and 8th electron, respectively. It has

been observed that half filled and fully filled

degenerate set of orbitals acquire extra stability

due to their symmetry (see Section, 2.6.7).

2.6.5 Electronic Configuration of Atoms

The distribution of electrons into orbitals of an

atom is called its

electronic configuration.

If one keeps in mind the basic rules which

govern the filling of different atomic orbitals,

the electronic configurations of different atoms

can be written very easily.

The electronic configuration of different

atoms can be represented in two ways. For

example :

(i) s

...... notation

(ii) Orbital diagram

s p d

In the first notation, the subshell is

represented by the respective letter symbol and

the number of electrons present in the subshell

is depicted, as the super script, like a, b, c, ...

etc. The similar subshell represented for

different shells is differentiated by writing the

principal quantum number before the

respective subshell. In the second notation

each orbital of the subshell is represented by

a box and the electron is represented by an

arrow (↑) a positive spin or an arrow (↓) a

negative spin. The advantage of second notation

over the first is that it represents all the four

quantum numbers.

The hydrogen atom has only one electron

which goes in the orbital with the lowest

energy, namely 1s. The electronic

configuration of the hydrogen atom is 1s

meaning that it has one electron in the 1s

orbital. The second electron in helium (He) can

also occupy the 1s orbital. Its configuration

is, therefore, 1s

. As mentioned above, the two

electrons differ from each other with opposite

spin, as can be seen from the orbital diagram.

The third electron of lithium (Li) is not

allowed in the 1s orbital because of Pauli

exclusion principle. It, therefore, takes the next

available choice, namely the 2s orbital. The

electronic configuration of Li is 1s

. The 2s

orbital can accommodate one more electron.

The configuration of beryllium (Be) atom is,

therefore, 1s

(see Table 2.6, page 66 for

the electronic configurations of elements).

In the next six elements—boron

(B, 1s

), carbon (C, 1s

), nitrogen

(N, 1s

), oxygen (O, 1s

), fluorine

(F, 1s

) and neon (Ne, 1s

), the 2p

orbitals get progressively filled. This process

is completed with the neon atom. The orbital

picture of these elements can be represented

as follows :

The electronic configuration of the elements

sodium (Na,1s

) to argon

(Ar,1s

), follow exactly the same

pattern as the elements from lithium to neon

with the difference that the 3s and 3p orbitals

are getting filled now. This process can be

simplified if we represent the total number of

electrons in the first two shells by the name of

element neon (Ne). The electronic configuration

2020-21

64 CHEMISTRY

of the elements from sodium to argon can be

written as (Na, [Ne]3s

) to (Ar, [Ne] 3s

). The

electrons in the completely filled shells are

known as core electrons and the electrons that

are added to the electronic shell with the

highest principal quantum number are called

valence electrons. For example, the electrons

in Ne are the core electrons and the electrons

from Na to Ar are the valence electrons. In

potassium (K) and calcium (Ca), the 4s orbital,

being lower in energy than the 3d orbitals, is

occupied by one and two electrons respectively.

A new pattern is followed beginning with

scandium (Sc). The 3d

orbital, being lower in

energy than the 4p orbital, is filled first.

Consequently, in the next ten elements,

scandium (Sc), titanium (Ti), vanadium (V),

chromium (Cr), manganese (Mn), iron (Fe),

cobalt (Co), nickel (Ni), copper (Cu) and zinc

(Zn), the five 3d orbitals are progressively

occupied. We may be puzzled by the fact that

chromium and copper have five and ten

electrons in 3d orbitals rather than four and

nine as their position would have indicated with

two-electrons in the 4s orbital. The reason is

that fully filled orbitals and half-filled orbitals

have extra stability (that is, lower energy). Thus

, p

, d

, f

etc. configurations, which

are either half-filled or fully filled, are more

stable. Chromium and copper therefore adopt

the d

and d

configuration (Section

2.6.7)[caution: exceptions do exist]

With the saturation of the 3d orbitals, the

filling of the 4p orbital starts at gallium (Ga)

and is complete at krypton (Kr). In the next

eighteen elements from rubidium (Rb) to xenon

(Xe), the pattern of filling the 5s, 4d and 5p

orbitals are similar to that of 4s, 3d and 4p

orbitals as discussed above. Then comes the

turn of the 6s orbital. In caesium (Cs) and the

barium (Ba), this orbital contains one and two

electrons, respectively. Then from lanthanum

(La) to mercury (Hg), the filling up of electrons

takes place in 4f and 5d orbitals. After this,

filling of 6p, then 7s and finally 5f and 6d

orbitals takes place. The elements after

uranium (U) are all short-lived and all of them

are produced artificially. The electronic

configurations of the known elements (as

determined by spectroscopic methods) are

tabulated in Table 2.6 (page 66).

One may ask what is the utility of knowing

the electron configuration? The modern

approach to the chemistry, infact, depends

almost entirely on electronic distribution to

understand and explain chemical behaviour.

For example, questions like why two or more

atoms combine to form molecules, why some

elements are metals while others are non-

metals, why elements like helium and argon

are not reactive but elements like the halogens

are reactive, find simple explanation from the

electronic configuration. These questions have

no answer in the Daltonian model of atom. A

detailed understanding of the electronic

structure of atom is, therefore, very essential

for getting an insight into the various aspects

of modern chemical knowledge.

2.6.6 Stability of Completely Filled and

Half Filled Subshells

The ground state electronic configuration of the

atom of an element always corresponds to the

state of the lowest total electronic energy. The

electronic configurations of most of the atoms

follow the basic rules given in Section 2.6.5.

However, in certain elements such as Cu, or

Cr, where the two subshells (4s and 3d) differ

slightly in their energies, an electron shifts from

a subshell of lower energy (4s) to a subshell of

higher energy (3d), provided such a shift

results in all orbitals of the subshell of higher

energy getting either completely filled or half

filled. The valence electronic configurations of

Cr and Cu, therefore, are 3d

and 3d

respectively and not 3d

and 3d

. It has

been found that there is extra stability

associated with these electronic configurations.

2020-21

STRUCTURE OF ATOM

The completely filled and completely

half-filled subshells are stable due to

the following reasons:

1.Symmetrical distribution of

electrons: It is well known that symmetry

leads to stability. The completely filled or

half filled subshells have symmetrical

distribution of electrons in them and are

therefore more stable. Electrons in the

same subshell (here 3d) have equal energy

but different spatial distribution.

Consequently, their shielding of one-

another is relatively small and the

electrons ar

e more strongly attracted by

the nucleus.

2. Exchange Energy : The stabilizing

effect arises whenever two or more

electrons with the same spin are pr

esent

in the degenerate orbitals of a subshell.

These electrons tend to exchange their

positions and the energy released due to

this exchange is called exchange energy.

The number of exchanges that can take

place is maximum when the subshell is

either half filled or completely filled (Fig.

2.18). As a result the exchange energy is

maximum and so is the stability.

You may note that the exchange

energy is at the basis of Hund’s rule that

electrons which enter orbitals of equal

energy have parallel spins as far as

possible. In other words, the extra

stability of half-filled and completely filled

subshell is due to: (i) relatively small

shielding, (ii) smaller coulombic repulsion

energy, and (iii) larger exchange energy.

Details about the exchange energy will be

dealt with in higher classes.

Fig. 2.18 Possible exchange for a d

configuration

Causes of Stability of Completely Filled and Half-filled Subshells

2020-21

66 CHEMISTRY

Table 2.6 Electronic Configurations of the Elements

* Elements with exceptional electronic configurations

2020-21

STRUCTURE OF ATOM

** Elements with atomic number 112 and above have been reported but not yet fully authenticated and named.

2020-21

68 CHEMISTRY

SUMMARY

Atoms are the building blocks of elements. They are the smallest parts of an element

that chemically react. The first atomic theory, proposed by John Dalton in 1808, regarded

atom as the ultimate indivisible particle of matter. Towards the end of the nineteenth

century, it was proved experimentally that atoms are divisible and consist of three

fundamental particles: electrons, protons and neutrons. The discovery of sub-atomic

particles led to the proposal of various atomic models to explain the structure of atom.

Thomson in 1898 proposed that an atom consists of uniform sphere of positive electricity

with electrons embedded into it. This model in which mass of the atom is considered to be

evenly spread over the atom was proved wrong by Rutherford’s famous alpha-particle

scattering experiment in 1909. Rutherford concluded that atom is made of a tiny positively

charged nucleus, at its centre with electrons revolving around it in circular orbits.

Rutherford model, which resembles the solar system, was no doubt an improvement over

Thomson model but it could not account for the stability of the atom i.e., why the electron

does not fall into the nucleus. Further, it was also silent about the electronic structure of

atoms i.e., about the distribution and relative energies of electrons around the nucleus.

The difficulties of the Rutherford model were overcome by Niels Bohr in 1913 in his model

of the hydrogen atom. Bohr postulated that electron moves around the nucleus in circular

orbits. Only certain orbits can exist and each orbit corresponds to a specific energy. Bohr

calculated the energy of electron in various orbits and for each orbit predicted the distance

between the electron and nucleus. Bohr model, though offering a satisfactory model for

explaining the spectra of the hydrogen atom, could not explain the spectra of multi-electron

atoms. The reason for this was soon discovered. In Bohr model, an electron is regarded as

a charged particle moving in a well defined circular orbit about the nucleus. The wave

character of the electron is ignored in Bohr’s theory. An orbit is a clearly defined path and

this path can completely be defined only if both the exact position and the exact velocity of

the electron at the same time are known. This is not possible according to the Heisenberg

uncertainty principle. Bohr model of the hydrogen atom, therefore, not only ignores the

dual behaviour of electron but also contradicts Heisenberg uncertainty principle.

Erwin Schrödinger, in 1926, proposed an equation called Schrödinger equation to describe

the electron distributions in space and the allowed energy levels in atoms. This equation

incorporates de Broglie’s concept of wave-particle duality and is consistent with Heisenberg

uncertainty principle. When Schrödinger equation is solved for the electron in a hydrogen

atom, the solution gives the possible energy states the electron can occupy [and the

corresponding wave function(s) (ψ) (which in fact are the mathematical functions) of the

electron associated with each energy state]. These quantized energy states and corresponding

wave functions which are characterized by a set of three quantum numbers (principal

quantum number n, azimuthal quantum number l and magnetic quantum number m

)

arise as a natural consequence in the solution of the Schrödinger equation. The restrictions

on the values of these three quantum numbers also come naturally from this solution. The

quantum mechanical model of the hydrogen atom successfully predicts all aspects of the

hydrogen atom spectrum including some phenomena that could not be explained by the

Bohr model.

According to the quantum mechanical model of the atom, the electron distribution of an

atom containing a number of electrons is divided into shells. The shells, in turn, are thought

to consist of one or more subshells and subshells are assumed to be composed of one or

more orbitals, which the electrons occupy. While for hydrogen and hydrogen like systems

(such as He

, Li

etc.) all the orbitals within a given shell have same energy, the energy of

the orbitals in a multi-electron atom depends upon the values of n and l: The lower the

value of (n + l ) for an orbital, the lower is its energy. If two orbitals have the same (n + l )

value, the orbital with lower value of n has the lower energy. In an atom many such orbitals

are possible and electrons are filled in those orbitals in order of increasing energy in

2020-21

STRUCTURE OF ATOM

accordance with Pauli exclusion principle (no two electrons in an atom can have the

same set of four quantum numbers) and Hund’s rule of maximum multiplicity (pairing

of electrons in the orbitals belonging to the same subshell does not take place until

each orbital belonging to that subshell has got one electron each, i.e., is singly occupied).

This forms the basis of the electronic structure of atoms.

EXERCISES

2.1 (i) Calculate the number of electrons which will together weigh one gram.

(ii) Calculate the mass and charge of one mole of electrons.

2.2 (i) Calculate the total number of electrons present in one mole of methane.

(ii) Find (a) the total number and (b) the total mass of neutrons in 7 mg of

(Assume that mass of a neutron = 1.675 × 10

–27

kg).

(iii) Find (a) the total number and (b) the total mass of protons in 34 mg of NH

STP.

Will the answer change if the temperature and pressure are changed ?

2.3 How many neutrons and protons are there in the following nuclei ?

C O Mg Fe Sr, , , ,

2.4 Write the complete symbol for the atom with the given atomic number (Z) and

atomic mass (A)

(i) Z = 17 , A = 35.

(ii) Z = 92 , A = 233.

(iii) Z = 4 , A = 9.

2.5 Yellow light emitted from a sodium lamp has a wavelength (λ) of 580 nm. Calculate

the frequency (ν) and wavenumber (

) of the yellow light.

2.6 Find energy of each of the photons which

(i) correspond to light of frequency 3×10

Hz.

(ii) have wavelength of 0.50 Å.

2.7 Calculate the wavelength, frequency and wavenumber of a light wave whose period

is 2.0 × 10

–10

2.8 What is the number of photons of light with a wavelength of 4000 pm that provide

1J of energy?

2.9 A photon of wavelength 4 × 10

–7

m strikes on metal surface, the work function of

the metal being 2.13 eV. Calculate (i) the energy of the photon (eV), (ii) the kinetic

energy of the emission, and (iii) the velocity of the photoelectron

(1 eV= 1.6020 × 10

–19

J).

2.10 Electromagnetic radiation of wavelength 242 nm is just sufficient to ionise the

sodium atom. Calculate the ionisation energy of sodium in kJ mol

–1

2.11 A 25 watt bulb emits monochromatic yellow light of wavelength of 0.57µm.

Calculate the rate of emission of quanta per second.

2.12 Electrons are emitted with zero velocity from a metal surface when it is exposed to

radiation of wavelength 6800 Å. Calculate threshold frequency (

) and work function

) of the metal.

2.13 What is the wavelength of light emitted when the electron in a hydrogen atom

undergoes transition from an energy level with n = 4 to an energy level with n = 2?

2020-21

70 CHEMISTRY

2.14 How much energy is required to ionise a H atom if the electron occupies n = 5

orbit? Compare your answer with the ionization enthalpy of H atom ( energy required

to remove the electron from n =1 orbit).

2.15 What is the maximum number of emission lines when the excited electron of a

H atom in n = 6 drops to the ground state?

2.16 (i) The energy associated with the first orbit in the hydrogen atom is

–2.18 × 10

–18

J atom

–1

. What is the energy associated with the fifth orbit?

(ii) Calculate the radius of Bohr’s fifth orbit for hydrogen atom.

2.17 Calculate the wavenumber for the longest wavelength transition in the Balmer

series of atomic hydrogen.

2.18 What is the energy in joules, required to shift the electron of the hydrogen atom

from the first Bohr orbit to the fifth Bohr orbit and what is the wavelength of the

light emitted when the electron returns to the ground state? The ground state

electron energy is –2.18 × 10

–11

ergs.

2.19 The electron energy in hydrogen atom is given by E

= (–2.18 × 10

–18

)/n

J. Calculate

the energy required to remove an electron completely from the n = 2 orbit. What is

the longest wavelength of light in cm that can be used to cause this transition?

2.20 Calculate the wavelength of an electron moving with a velocity of 2.05 × 10

m s

–1

2.21 The mass of an electron is 9.1 × 10

–31

kg. If its K.E. is 3.0 × 10

–25

J, calculate its

wavelength.

2.22 Which of the following are isoelectronic species i.e., those having the same number

of electrons?

, K

, Mg

, Ca

, S

2–

, Ar.

2.23 (i) Write the electronic configurations of the following ions: (a) H

–

(b) Na

2–

(d) F

–

(ii) What are the atomic numbers of elements whose outermost electrons are

represented by (a) 3s

(b) 2p

and (c) 3p

(iii) Which atoms are indicated by the following configurations ?

(a) [He] 2s

(b) [Ne] 3s

2.24 What is the lowest value of n that allows g orbitals to exist?

2.25 An electron is in one of the 3d orbitals. Give the possible values of n, l and m

for

this electron.

2.26 An atom of an element contains 29 electrons and 35 neutrons. Deduce (i) the

number of protons and (ii) the electronic configuration of the element.

2.27 Give the number of electrons in the species

2.28 (i) An atomic orbital has n = 3. What are the possible values of l and m

(ii) List the quantum numbers (m

and l ) of electrons for 3d orbital.

(iii) Which of the following orbitals are possible?

1p, 2s, 2p and 3f

2.29 Using s, p, d notations, describe the orbital with the following quantum numbers.

(a) n=1, l=0; (b) n = 3; l=1 (c) n = 4; l =2; (d) n=4; l=3.

2.30 Explain, giving reasons, which of the following sets of quantum numbers are not

possible.

(a) n = 0, l = 0, m

= 0, m

= + ½

(b) n = 1, l = 0, m

= 0, m

= – ½

= 0, m

= + ½

(d) n = 2, l = 1, m

= 0, m

= – ½

2020-21

STRUCTURE OF ATOM

(e) n = 3, l = 3, m

= –3, m

= + ½

(f) n = 3, l = 1, m

= 0, m

= + ½

2.31 How many electrons in an atom may have the following quantum numbers?

(a) n = 4, m

= – ½ (b) n = 3, l = 0

2.32 Show that the circumference of the Bohr orbit for the hydrogen atom is an integral

multiple of the de Broglie wavelength associated with the electron revolving around

the orbit.

2.33 What transition in the hydrogen spectrum would have the same wavelength as the

Balmer transition n = 4 to n = 2 of He

spectrum ?

2.34 Calculate the energy required for the process

(g) g He

(g) + e

–

The ionization energy for the H atom in the ground state is 2.18 × 10

–18

J atom

–1

2.35 If the diameter of a carbon atom is 0.15 nm, calculate the number of carbon atoms

which can be placed side by side in a straight line across length of scale of length

20 cm long.

2.36 2 ×10

atoms of carbon are arranged side by side. Calculate the radius of carbon

atom if the length of this arrangement is 2.4 cm.

2.37 The diameter of zinc atom is 2.6 Å.Calculate (a) radius of zinc atom in pm and (b)

number of atoms present in a length of 1.6 cm if the zinc atoms are arranged side

by side lengthwise.

2.38 A certain particle carries 2.5 × 10

–16

C of static electric charge. Calculate the number

of electrons present in it.

2.39 In Milikan’s experiment, static electric charge on the oil drops has been obtained

by shining X-rays. If the static electric charge on the oil drop is –1.282 × 10

–18

calculate the number of electrons present on it.

2.40 In Rutherford’s experiment, generally the thin foil of heavy atoms, like gold, platinum

etc. have been used to be bombarded by the α-particles. If the thin foil of light

atoms like aluminium etc. is used, what difference would be observed from the

above results ?

2.41 Symbols

and

Br can be written, whereas symbols

and

Br are not

acceptable. Answer briefly.

2.42 An element with mass number 81 contains 31.7% more neutrons as compared to

protons. Assign the atomic symbol.

2.43 An ion with mass number 37 possesses one unit of negative charge. If the ion

conatins 11.1% more neutrons than the electrons, find the symbol of the ion.

2.44 An ion with mass number 56 contains 3 units of positive charge and 30.4% more

neutrons than electrons. Assign the symbol to this ion.

2.45 Arrange the following type of radiations in increasing order of frequency: (a) radiation

from microwave oven (b) amber light from traffic signal (c) radiation from FM radio

(d) cosmic rays from outer space and (e) X-rays.

2.46 Nitrogen laser produces a radiation at a wavelength of 337.1 nm. If the number of

photons emitted is 5.6 × 10

, calculate the power of this laser.

2.47 Neon gas is generally used in the sign boards. If it emits strongly at 616 nm,

calculate (a) the frequency of emission, (b) distance traveled by this radiation in

30 s (c) energy of quantum and (d) number of quanta present if it produces 2 J of

energy.

2020-21

72 CHEMISTRY

2.48 In astronomical observations, signals observed from the distant stars are

generally weak. If the photon detector receives a total of 3.15 × 10

–18

J from the

radiations of 600 nm, calculate the number of photons received by the detector.

2.49 Lifetimes of the molecules in the excited states are often measured by using

pulsed radiation source of duration nearly in the nano second range. If the

radiation source has the duration of 2 ns and the number of photons emitted

during the pulse source is 2.5 × 10

, calculate the energy of the source.

2.50 The longest wavelength doublet absorption transition is observed at 589 and

589.6 nm. Calcualte the frequency of each transition and energy difference

between two excited states.

2.51 The work function for caesium atom is 1.9 eV. Calculate (a) the threshold

wavelength and (b) the threshold frequency of the radiation. If the caesium

element is irradiated with a wavelength 500 nm, calculate the kinetic energy

and the velocity of the ejected photoelectron.

2.52 Following results are observed when sodium metal is irradiated with different

wavelengths. Calculate (a) threshold wavelength and, (b) Planck’s constant.

λ (nm) 500 450 400

v × 10

–5

(cm s

–1

) 2.55

4.35 5.35

2.53 The ejection of the photoelectron from the silver metal in the photoelectric effect

experiment can be stopped by applying the voltage of 0.35 V when the radiation

256.7 nm is used. Calculate the work function for silver metal.

2.54 If the photon of the wavelength 150 pm strikes an atom and one of tis inner bound

electrons is ejected out with a velocity of 1.5 × 10

m s

–1

, calculate the energy with

which it is bound to the nucleus.

2.55 Emission transitions in the Paschen series end at orbit n = 3 and start from orbit n

and can be represeted as v = 3.29 × 10

(Hz) [ 1/3

– 1/n

]

Calculate the value of n if the transition is observed at 1285 nm. Find the region of

the spectrum.

2.56 Calculate the wavelength for the emission transition if it starts from the orbit having

radius 1.3225 nm and ends at 211.6 pm. Name the series to which this transition

belongs and the region of the spectrum.

2.57 Dual behaviour of matter proposed by de Broglie led to the discovery of electron

microscope often used for the highly magnified images of biological molecules and

other type of material. If the velocity of the electron in this microscope is 1.6 × 10

–1

, calculate de Broglie wavelength associated with this electron.

2.58 Similar to electron diffraction, neutron diffraction microscope is also used for the

determination of the structure of molecules. If the wavelength used here is 800 pm,

calculate the characteristic velocity associated with the neutron.

2.59 If the velocity of the electron in Bohr’s first orbit is 2.19 × 10

–1

, calculate the

de Broglie wavelength associated with it.

2.60 The velocity associated with a proton moving in a potential difference of 1000 V is

4.37 × 10

–1

. If the hockey ball of mass 0.1 kg is moving with this velocity,

calcualte the wavelength associated with this velocity.

2.61 If the position of the electron is measured within an accuracy of

+ 0.002 nm, calculate

the uncertainty in the momentum of the electron. Suppose the momentum of the

electron is h/4π

× 0.05 nm, is there any problem in defining this value.

2.62 The quantum numbers of six electrons are given below. Arrange them in order of

increasing energies. If any of these combination(s) has/have the same energy lists:

1. n = 4, l = 2, m

= –2 , m

= –1/2

2. n = 3, l = 2, m

= 1 , m

= +1/2

2020-21

STRUCTURE OF ATOM

3. n = 4, l = 1, m

= 0 , m

= +1/2

4. n = 3, l = 2, m

= –2 , m

= –1/2

5. n = 3, l = 1, m

= –1 , m

= +1/2

6. n = 4, l = 1, m

= 0 , m

= +1/2

2.63 The bromine atom possesses 35 electrons. It contains 6 electrons in 2p orbital,

6 electrons in 3p orbital and 5 electron in 4p orbital. Which of these electron

experiences the lowest effective nuclear charge ?

2.64 Among the following pairs of orbitals which orbital will experience the larger effective

nuclear charge? (i) 2s and 3s, (ii) 4d and 4f, (iii) 3d and 3p.

2.65 The unpaired electrons in Al and Si are present in 3p orbital. Which electrons will

experience more effective nuclear charge from the nucleus ?

2.66 Indicate the number of unpaired electrons in : (a) P, (b) Si, (c) Cr, (d) Fe and (e) Kr.

2.67 (a) How many subshells are associated with n = 4 ? (b) How many electrons will

be present in the subshells having m

value of –1/2 for n = 4 ?

2020-21