Class11 Chemistry 1 Unit02 NCERT TextBook English Edition

8/6/2019 Class11 Chemistry 1 Unit02 NCERT TextBook English Edition

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The rich diversity of chemical behaviour of different elementscan 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 beable to

know about the discovery ofelectron, proton and neutron andtheir characteristics;

describe Thomson, Rutherfordand Bohr atomic models;

understand the importantfeatures of the quantummechanical model of atom;

understand nature ofelectromagnetic radiation andPlancks quantum theory;

explain the photoelectric effectand describe features of atomicspectra;

state the de Broglie relation andHeisenberg uncertainty principle;

define an atomic orbital in termsof quantum numbers;

state aufbau principle, Pauliexclusion principle and Hundsrule of maximum multiplicity;

write the electronic configurationsof atoms.

The existence of atoms has been proposed since the timeof early Indian and Greek philosophers (400 B.C.) whowere of the view that atoms are the fundamental building blocks of matter. According to them, the continuedsubdivisions of matter would ultimately yield atoms whichwould not be further divisible. The word atom has beenderived 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 themexperimentally. These ideas remained dormant for a verylong time and were revived again by scientists in thenineteenth century.

The atomic theory of matter was first proposed on afirm scientific basis by John Dalton, a British schoolteacher in 1808. His theory, called Daltons atomictheory, regarded the atom as the ultimate particle ofmatter (Unit 1).

In this unit we start with the experimentalobservations made by scientists towards the end ofnineteenth and beginning of twentieth century. Theseestablished that atoms can be further divided into sub-atomic particles, i.e., electrons, protons and neutronsa concept very different from that of Dalton. The majorproblems before the scientists at that time were:

to account for the stability of atom after the discoveryof sub-atomic particles,

to compare the behaviour of one element from otherin terms of both physical and chemical properties,


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27STRUCTURE OF ATOM

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to explain the formation of different kindsof molecules by the combination ofdifferent atoms and,

to understand the origin and nature of thecharacteristics of electromagneticradiation absorbed or emitted by atoms.

2.1 SUB-ATOMIC PARTICLES

Daltons atomic theory was able to explainthe law of conservation of mass, law ofconstant composition and law of multipleproportion very successfully. However, it failedto explain the results of many experiments,for example, it was known that substanceslike glass or ebonite when rubbed with silk orfur generate electricity. Many different kinds

of sub-atomic particles were discovered in thetwentieth century. However, in this sectionwe will talk about only two particles, namelyelectron and proton.

2.1.1 Discovery of Electron

In 1830, Michael Faraday showed that ifelectricity is passed through a solution of anelectrolyte, chemical reactions occurred at theelectrodes, which resulted in the liberationand deposition of matter at the electrodes. Heformulated certain laws which you will studyin class XII. These results suggested theparticulate nature of electricity.

An insight into the structure of atom wasobtained from the experiments on electricaldischarge through gases. Before we discussthese results we need to keep in mind a basicrule regarding the behaviour of chargedparticles : Like charges repel each other andunlike charges attract each other.

In mid 1850s many scientists mainlyFaraday began to study electrical dischargein partially evacuated tubes, known ascathode ray discharge tubes. It is depicted

in Fig. 2.1. A cathode ray tube is made of glasscontaining two thin pieces of metal, calledelectrodes, sealed in it. The electricaldischarge through the gases could beobserved only at very low pressures and atvery high voltages. The pressure of differentgases could be adjusted by evacuation. Whensufficiently high voltage is applied across theelectrodes, current starts flowing through a

stream of particles moving in the tube fromthe negative electrode (cathode) to the positiveelectrode (anode). These were called cathoderays or cathode ray particles. The flow ofcurrent from cathode to anode was further

checked by making a hole in the anode andcoating the tube behind anode withphosphorescent material zinc sulphide. Whenthese rays, after passing through anode, strikethe zinc sulphide coating, a bright spot onthe coating is developed(same thing happensin a television set) [Fig. 2.1(b)].

Fig. 2.1(a)A cathode ray discharge tube

Fig. 2.1(b) A cathode ray discharge tube withperforated anode

The results of these experiments aresummarised below.

(i) The cathode rays start from cathode andmove towards the anode.

(ii) These rays themselves are not visible but

their behaviour can be observed with thehelp of certain kind of materials(fluorescent or phosphorescent) whichglow when hit by them. Televisionpicture tubes are cathode ray tubes andtelevision pictures result due tofluorescence on the television screencoated with certain fluorescent orphosphorescent materials.


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(iii) In the absence of electrical or magneticfield, these rays travel in straight lines(Fig. 2.2).

(iv) In the presence of electrical or magneticfield, the behaviour of cathode rays aresimilar to that expected from negativelycharged particles, suggesting that thecathode rays consist of negativelycharged particles, called electrons.

(v) The characteristics of cathode rays(electrons) do not depend upon thematerial of electrodes and the nature ofthe 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. Thomsonmeasured the ratio of electrical charge (e) tothe mass of electron (m

e) by using cathode

ray tube and applying electrical and magneticfield perpendicular to each other as well as tothe path of electrons (Fig. 2.2). Thomsonargued that the amount of deviation of theparticles from their path in the presence ofelectrical or magnetic field depends upon:(i) the magnitude of the negative charge on

the particle, greater the magnitude of thecharge on the particle, greater is theinteraction with the electric or magneticfield and thus greater is the deflection.

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

(ii) the mass of the particle lighter theparticle, greater the deflection.

(iii) the strength of the electrical or magnetic

field the deflection of electrons fromits original path increases with theincrease in the voltage across theelectrodes, or the strength of themagnetic field.

When only electric field is applied, theelectrons deviate from their path and hit thecathode ray tube at point A. Similarly whenonly magnetic field is applied, electron strikesthe cathode ray tube at point C. By carefully balancing the electrical and magnetic fieldstrength, it is possible to bring back the

electron to the path followed as in the absenceof electric or magnetic field and they hit thescreen at point B. By carrying out accuratemeasurements on the amount of deflectionsobserved by the electrons on the electric fieldstrength or magnetic field strength, Thomsonwas able to determine the value ofe/me as:

e

e

m= 1.758820 1011 C kg1 (2.1)

Where me is the mass of the electron in kgand eis the magnitude of the charge on the

electron in coulomb (C). Since electronsare negatively charged, the charge on electronis e.


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2.1.3 Charge on the Electron

R.A. Millikan (1868-1953) devised a methodknown as oil drop experiment (1906-14), to

determine the charge on the electrons. Hefound that the charge on the electron to be 1.6 1019 C. The present accepted value ofelectrical charge is 1.6022 1019 C. Themass of the electron (me) was determined bycombining these results with Thomsons valueofe/me ratio.

e

e

= =/

e

m

e m

= 9.10941031 kg (2.2)

2.1.4 Discovery of Protons and Neutrons

Electrical discharge carried out in themodified cathode ray tube led to the discoveryof particles carrying positive charge, alsoknown as canal rays. The characteristics ofthese positively charged particles are listedbelow.(i) unlike cathode rays, the positively

charged particles depend upon thenature of gas present in the cathode raytube. These are simply the positivelycharged gaseous ions.

(ii) The charge to mass ratio of the particles

is found to depend on the gas from whichthese originate.

(iii) Some of the positively charged particlescarry a multiple of the fundamental unitof electrical charge.

(iv) The behaviour of these particles in themagnetic or electrical field is opposite tothat observed for electron or cathoderays.

The smallest and lightest positive ion wasobtained from hydrogen and was calledproton. This positively charged particle was

characterised in 1919. Later, a need was feltfor the presence of electrically neutral particleas one of the constituent of atom. Theseparticles were discovered by Chadwick (1932)by bombarding a thin sheet of beryllium by-particles. When electrically neutral particleshaving a mass slightly greater than that ofthe protons was emitted. He named theseparticles as neutrons. The important

Millikans Oil Drop Method

In this method, oil droplets in the form ofmist, produced by the atomiser, were allowedto enter through a tiny hole in the upper plateof electrical condenser. The downward motionof these droplets was viewed through thetelescope, equipped with a micrometer eyepiece. By measuring the rate of fall of thesedroplets, Millikan was able to measure themass of oil droplets.The air inside thechamber was ionized by passing a beam of

X-rays through it. The electrical charge onthese oil droplets was acquired by collisions

with gaseous ions. The fall of these chargedoil droplets can be retarded, accelerated ormade stationary depending upon the charge

on the droplets and the polarity and strengthof the voltage applied to the plate. By carefullymeasuring the effects of electrical fieldstrength on the motion of oil droplets,Millikan concluded that the magnitude ofelectrical charge, q, on the droplets is alwaysan integral multiple of the electrical charge,e, that is, q= ne, where n = 1, 2, 3... .

Fig. 2.3 The Millikan oil drop apparatus formeasuring 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.

properties of these fundamental particles aregiven in Table 2.1.

2.2 ATOMIC MODELS

Observations obtained from the experimentsmentioned in the previous sections havesuggested that Daltons indivisible atom iscomposed of sub-atomic particles carryingpositive and negative charges. Different


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Table 2.1 Properties of Fundamental Particles

atomic models were proposed to explain thedistributions of these charged particles in anatom. Although some of these models werenot able to explain the stability of atoms, twoof these models, proposed by J. J. Thomson

and Ernest Rutherford are discussed below.2.2.1 Thomson Model of Atom

J. J. Thomson, in 1898, proposed that anatom possesses a spherical shape (radiusapproximately 1010 m) in which the positivecharge is uniformly distributed. The electronsare embedded into it in such a manner as togive the most stable electrostatic arrangement(Fig. 2.4). Many different names are given tothis model, for example, plum pudding,raisin pudding or watermelon. This model

In the later half of the nineteenth centurydifferent kinds of rays were discovered,besides those mentioned earlier. WilhalmRentgen (1845-1923) in 1895 showed

that when electrons strike a material inthe cathode ray tubes, produce rayswhich can cause fluorescence in thefluorescent materials placed outside thecathode ray tubes. Since Rentgen didnot know the nature of the radiation, henamed them X-rays and the name is stillcarried on. It was noticed that X-rays areproduced effectively when electronsstrike the dense metal anode, calledtargets. These are not deflected by theelectric and magnetic fields and have a

very high penetrating power through thematter and that is the reason that theserays are used to study the interior of theobjects. These rays are of very short wavelengths (0.1 nm) and possesselectro-magnetic character (Section2.3.1).

Henri Becqueral (1852-1908)observed that there are certain elementswhich emit radiation on their own andnamed this phenomenon asradioactivity and the elements known

as radioactive elements. This field wasdeveloped by Marie Curie, Piere Curie,Rutherford and Fredrick Soddy. It wasobserved that three kinds of rays i.e., ,- and -rays are emitted. Rutherfordfound that-rays consists of high energyparticles carrying two units of positivecharge and four unit of atomic mass. He

Fig.2.4 Thomson model of atom

can be visualised as a pudding or watermelonof positive charge with plums or seeds(electrons) embedded into it. An important

feature of this model is that the mass of theatom is assumed to be uniformly distributed

over the atom. Although this model was ableto explain the overall neutrality of the atom,but was not consistent with the results of laterexperiments. Thomson was awarded NobelPrize for physics in 1906, for his theoreticaland experimental investigations on theconduction of electricity by gases.


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concluded that- particles are heliumnuclei as when - particles combinedwith two electrons yielded helium gas.

-rays are negatively charged particlessimilar to electrons. The -rays are highenergy radiations like X-rays, are neutralin nature and do not consist of particles.As regards penetrating power, -particlesare the least, followed by -rays (100times that of particles) and -rays(1000 times of that-particles).

2.2.2 Rutherfords Nuclear Model of Atom

Rutherford and his students (Hans Geiger andErnest Marsden) bombarded very thin goldfoil with particles. Rutherfords famous particle scattering experiment is

represented in Fig. 2.5. A stream of highenergyparticles from a radioactive sourcewas directed at a thin foil (thickness 100

nm) of gold metal. The thin gold foil had acircular fluorescent zinc sulphide screenaround it. Wheneverparticles struck thescreen, a tiny flash of light was produced atthat point.

The results of scattering experiment werequite unexpected. According to Thomsonmodel of atom, the mass of each gold atom inthe foil should have been spread evenly overthe entire atom, and particles had enoughenergy to pass directly through such auniform distribution of mass. It was expected

that the particles would slow down andchange directions only by a small angles asthey passed through the foil. It was observedthat :

(i) most of the particles passed throughthe gold foil undeflected.

(ii) a small fraction of the particles wasdeflected by small angles.

(iii) a very few particles (1 in 20,000)bounced back, that is, were deflected bynearly 180.

On the basis of the observations,Rutherford drew the following conclusionsregarding the structure of atom :

(i) Most of the space in the atom is emptyas most of the particles passedthrough the foil undeflected.

(ii) A few positively charged particles weredeflected. The deflection must be due toenormous repulsive force showing thatthe positive charge of the atom is notspread throughout the atom as Thomsonhad presumed. The positive charge has

to be concentrated in a very small volumethat repelled and deflected the positivelycharged particles.

(iii) Calculations by Rutherford showed thatthe volume occupied by the nucleus isnegligibly small as compared to the totalvolume of the atom. The radius of theatom is about 1010 m, while that ofnucleus is 1015 m. One can appreciate

Fig.2.5 Schematic view of Rutherfords scatteringexperiment. When a beam of alpha ()

particles is shot at a thin gold foil, most

of them pass through without much effect.Some, however, are deflected.

A. Rutherfords scattering experiment

B. Schematic molecular view of the gold foil


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this difference in size by realising that ifa cricket ball represents a nucleus, thenthe radius of atom would be about 5 km.

On the basis of above observations andconclusions, Rutherford proposed thenuclear model of atom (after the discovery ofprotons). According to this model :

(i) The positive charge and most of the massof the atom was densely concentratedin extremely small region. This very smallportion of the atom was called nucleusby Rutherford.

(ii) The nucleus is surrounded by electronsthat move around the nucleus with a

very high speed in circular paths calledorbits. Thus, Rutherfords model of atomresembles the solar system in which thenucleus plays the role of sun and theelectrons that of revolving planets.

(iii) Electrons and the nucleus are heldtogether by electrostatic forces ofattraction.

2.2.3 Atomic Number and Mass Number

The presence of positive charge on thenucleus is due to the protons in the nucleus.

As established earlier, the charge on theproton is equal but opposite to that ofelectron. The number of protons present inthe nucleus is equal to atomic number (Z).For example, the number of protons in thehydrogen nucleus is 1, in sodium atom it is11, therefore their atomic numbers are 1 and11 respectively. In order to keep the electricalneutrality, the number of electrons in anatom is equal to the number of protons(atomic number, Z). For example, number ofelectrons 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 electronsin a nuetral atom (2.3)

While the positive charge of the nucleusis due to protons, the mass of the nucleus,due to protons and neutrons. As discussed

earlier protons and neutrons present in thenucleus 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 ofneutrons (n) (2.4)

2.2.4 Isobars and Isotopes

The composition of any atom can berepresented by using the normal elementsymbol (X) with super-script on the left handside as the atomic mass number (A) andsubscript (Z) on the left hand side as theatomic number (i.e., AZ X).

Isobars are the atoms with same massnumber but different atomic number forexample, 6

14C and 7

14N. On the other hand,

atoms with identical atomic number butdifferent atomic mass number are known asIsotopes. In other words (according toequation 2.4), it is evident that differencebetween the isotopes is due to the presenceof different number of neutrons present inthe nucleus. For example, considering ofhydrogen atom again, 99.985% of hydrogenatoms contain only one proton. This isotope

is called protium( 11

H). Rest of the percentageof hydrogen atom contains two other isotopes,the one containing 1 proton and 1 neutronis called deuterium (

1

2D, 0.015%) and the

other one possessing 1 proton and 2 neutronsis called tritium(

1

3T ). The latter isotope is

found in trace amounts on the earth. Otherexamples of commonly occuring isotopes are:carbon atoms containing 6, 7 and 8 neutronsbesides 6 protons (12 13 14

6 6 6C, C, C ); chlorine

atoms containing 18 and 20 neutrons besides17 protons ( 35 37

17 17Cl, Cl ).

Lastly an important point to mentionregarding isotopes is thatchemical propertiesof atoms are controlled by the number of

electrons, which are determined by the

number of protons in the nucleus. Number ofneutrons present in the nucleus have verylittle effect on the chemical properties of anelement. Therefore, all the isotopes of a givenelement show same chemical behaviour.


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Problem 2.1

Calculate the number of protons,neutrons and electrons in 80

35

Br.SolutionIn this case, 80

35Br, Z = 35, A = 80, species

is neutralNumber of protons = number of electrons= Z = 35

Number of neutrons = 80 35 = 45,(equation 2.4)

Problem 2.2

The number of electrons, protons andneutrons in a species are equal to 18,16 and 16 respectively. Assign the propersymbol to the species.Solution

The atomic number is equal tonumber of protons = 16. The element issulphur (S). Atomic mass number = number ofprotons + number of neutrons

= 16 + 16 = 32Species is not neutral as the number ofprotons is not equal to electrons. It isanion (negatively charged) with charge

equal to excess electrons = 18 16 = 2.Symbol is 32 2 16

S .

Note : Before using the notation AZX, find

out whether the species is a neutralatom, a cation or an anion. If it is aneutral 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 ofprotons are larger (cation, positive ion)or smaller (anion, negative ion) than thenumber of electrons. Number of neutrons

is always given by AZ, whether thespecies is neutral or ion.

2.2.5 Drawbacks of Rutherford Model

Rutherford nuclear model of an atom islike a small scale solar system with the

* Classical mechanics is a theoretical science based on Newtons laws of motion. It specifies the laws of motion of macroscopicobjects.

nucleus playing the role of the massive sunand the electrons being similar to the lighterplanets. Further, the coulomb force (kq1q2/r

2

where q1 and q2 are the charges, r is thedistance of separation of the charges and k isthe proportionality constant) between electronand the nucleus is mathematically similar to

the gravitational force1 2

2G.

m m

r

where m1

and m2 are the masses, r is the distance ofseparation of the masses and G is thegravitational constant. When classicalmechanics* is applied to the solar system, itshows that the planets describe well-definedorbits around the sun. The theory can also

calculate precisely the planetary orbits andthese are in agreement with the experimentalmeasurements. The similarity between thesolar system and nuclear model suggests thatelectrons should move around the nucleusin well defined orbits. However, when a bodyis moving in an orbit, it undergoesacceleration (even if the body is moving witha constant speed in an orbit, it mustaccelerate because of changing direction). Soan electron in the nuclear model describingplanet like orbits is under acceleration.According to the electromagnetic theory of

Maxwell, charged particles when acceleratedshould emit electromagnetic radiation (Thisfeature does not exist for planets since theyare uncharged). Therefore, an electron in anorbit will emit radiation, the energy carriedby radiation comes from electronic motion. The orbit will thus continue to shrink.Calculations show that it should take anelectron only 108 s to spiral into the nucleus.But this does not happen. Thus, theRutherford model cannot explain the stabilityof an atom. If the motion of an electron is

described on the basis of the classicalmechanics and electromagnetic theory, youmay ask that since the motion of electrons inorbits is leading to the instability of the atom,then why not consider electrons as stationaryaround the nucleus. If the electrons werestationary, electrostatic attraction between


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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 twomutually perpendicular planes.

the dense nucleus and the electrons wouldpull the electrons toward the nucleus to forma miniature version of Thomsons model of

atom. Another serious drawback of the

Rutherford model is that it says nothingabout the electronic structure of atoms i.e.,how the electrons are distributed around thenucleus and what are the energies of theseelectrons.

2.3 DEVELOPMENTS LEADING TO THEBOHRS MODEL OF ATOM

Historically, results observed from the studiesof interactions of radiations with matter haveprovided immense information regarding thestructure of atoms and molecules. Neils Bohrutilised these results to improve upon themodel proposed by Rutherford. Twodevelopments played a major role in theformulation of Bohrs model of atom. Thesewere:

(i) Dual character of the electromagneticradiation which means that radiationspossess both wave like and particle likeproperties, and

(ii) Experimental results regarding atomicspectra which can be explained only byassuming quantized (Section 2.4)electronic energy levels in atoms.

2.3.1 Wave Nature of ElectromagneticRadiation

James Maxwell (1870) was the first to give acomprehensive explanation about theinteraction between the charged bodies andthe behaviour of electrical and magnetic fieldson macroscopic level. He suggested that whenelectrically charged particle moves underaccelaration, alternating electrical andmagnetic fields are produced andtransmitted. These fields are transmitted inthe forms of waves called electromagnetic

waves orelectromagnetic radiation.Light is the form of radiation known from

early days and speculation about its naturedates back to remote ancient times. In earlierdays (Newton) light was supposed to be madeof particles (corpuscules). It was only in the

19th century when wave nature of light wasestablished.

Maxwell was again the first to reveal that

light waves are associated with oscillatingelectric and magnetic character (Fig. 2.6).Although electromagnetic wave motion iscomplex in nature, we will consider here onlya few simple properties.(i) The oscillating electric and magnetic

fields produced by oscillating chargedparticles are perpendicular to each otherand both are perpendicular to thedirection of propagation of the wave.Simplified picture of electromagneticwave is shown in Fig. 2.6.

(ii) Unlike sound waves or water waves,electromagnetic waves do not requiremedium and can move in vacuum.

(iii) It is now well established that there aremany types of electromagneticradiations, which differ from one anotherin wavelength (or frequency). These

constitute what is calledelectromagnetic spectrum (Fig. 2.7).Different regions of the spectrum areidentified by different names. Someexamples are: radio frequency regionaround 106 Hz, used for broadcasting;microwave region around 1010 Hz usedfor radar; infrared region around 1013 Hzused for heating; ultraviolet region


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around 1016Hz a component of sunsradiation. The small portion around 1015

Hz, is what is ordinarily called visible

light. It is only this part which our eyescan see (or detect). Special instrumentsare required to detect non-visibleradiation.

(iv) Different kinds of units are used torepresent electromagnetic radiation.

These radiations are characterised by theproperties, namely, frequency ( ) andwavelength ().

The SI unit for frequency ( ) is hertz(Hz, s1), after Heinrich Hertz. It is defined asthe 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 oflength is meter (m). Since electromagneticradiation consists of different kinds of wavesof much smaller wavelengths, smaller unitsare used. Fig.2.7 shows various types ofelectro-magnetic radiations which differ fromone another in wavelengths and frequencies.

In vaccum all types of electromagneticradiations, regardless of wavelength, travel

Fig. 2.7 (a) The spectrum of electromagnetic radiation. (b) Visible spectrum. The visible region is onlya small part of the entire spectrum .

at the same speed, i.e., 3.0 108 m s1

(2.997925 108 m s1, to be precise). This iscalled speed of light and is given the symbol

c. The frequency (), wavelength () and velocityof light (c) are related by the equation (2.5).

c = (2.5)

The other commonly used quantityspecially in spectroscopy, is thewavenumber

( ). It is defined as the number of wavelengths per unit length. Its units are reciprocal ofwavelength unit, i.e., m1. However commonlyused unit is cm1(not SI unit).

Problem 2.3The Vividh Bharati station of All IndiaRadio, Delhi, broadcasts on a frequencyof 1,368 kHz (kilo hertz). Calculate the wavelength of the electromagneticradiation emitted by transmitter. Whichpart of the electromagnetic spectrumdoes it belong to?

SolutionThe wavelength, , is equal to c/, wherec is the speed of electromagneticradiation in vacuum and is the

(a)

(b)


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frequency. Substituting the given values,we have

c

= 8

8

3

3.00 10=

1368 kH

3.00 10=

1368 10

= 219.3 m

This is a characteristic radiowavewavelength.

Problem 2.4

The wavelength range of the visiblespectrum extends from violet (400 nm)to red (750 nm). Express these wavelengths in frequencies (Hz).(1nm = 109 m)

SolutionUsing equation 2.5, frequency of violetlight

3.00c= =

400

= 7.50 1014 HzFrequency of red light

3.00 1c= =

750

= 4.00 1014 Hz

The range of visible spectrum is from4.0 1014 to 7.5 1014 Hz in terms offrequency units.

Problem 2.5Calculate (a) wavenumber and (b)frequency of yellow radiation havingwavelength 5800 .

Solution

(a) Calculation of wavenumber ( )

=5800=58

= 5

* 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.

1 1= =

5800

=1.724

=1.724

(b) Calculation of the frequency ()

3 10c= =

5800

2.3.2 Particle Nature of ElectromagneticRadiation: Plancks Quantum

Theory

Some of the experimental phenomenon suchas diffraction* and interference** can beexplained by the wave nature of theelectromagnetic radiation. However, followingare some of the observations which could not be explained with the help of even theelectromagentic theory of 19th centuryphysics (known as classical physics):

(i) the nature of emission of radiation fromhot bodies (black -body radiation)

(ii) ejection of electrons from metal surfacewhen radiation strikes it (photoelectric

effect)(iii) variation of heat capacity of solids as a

function of temperature

(iv) l ine spectra of atoms with specialreference to hydrogen.

It is noteworthy that the first concreteexplanation for the phenomenon of the blackbody radiation was given by Max Planck in1900. This phenomenon is given below:

When solids are heated they emitradiation over a wide range of wavelengths.For example, when an iron rod is heated in a

furnace, it first turns to dull red and thenprogressively becomes more and more red asthe temperature increases. As this is heatedfurther, the radiation emitted becomes white and then becomes blue as thetemperature becomes very high. In terms of


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frequency, it means that the radiation emittedgoes from a lower frequency to a higherfrequency as the temperature increases. The

red colour lies in the lower frequency region while blue colour belongs to the higherfrequency region of the electromagneticspectrum. The ideal body, which emits andabsorbs all frequencies, is called a black body

and the radiation emitted by such a body is

called black body radiation. The exactfrequency distribution of the emittedradiation (i.e., intensity versus frequencycurve of the radiation) from a black bodydepends only on its temperature. At a giventemperature, intensity of radiation emittedincreases with decrease of wavelength,reaches a maximum value at a givenwavelength and then starts decreasing withfurther decrease of wavelength, as shown inFig. 2.8.

to its frequency ( ) and is expressed byequation (2.6).

E= h (2.6)

The proportionality constant, h is knownas Plancks constant and has the value6.6261034J s.

With this theory, Planck was able toexplain the distribution of intensity in theradiation from black body as a function offrequency or wavelength at differenttemperatures.

Photoelectric Effect

In 1887, H. Hertz performed a very interestingexperiment in which electrons (or electric

current) were ejected when certain metals (forexample potassium, rubidium, caesium etc.)were exposed to a beam of light as shown inFig.2.9. The phenomenon is called

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

Fig. 2.8 Wavelength-intensity relationship

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.

The above experimental results cannot beexplained satisfactorily on the basis of the

wave theory of light. Planck suggested thatatoms and molecules could emit (or absorb)energy only in discrete quantities and not ina continuous manner, a belief popular at thattime. Planck gave the name quantum to thesmallest quantity of energy that can beemitted or absorbed in the form ofelectromagnetic radiation. The energy (E) ofa quantum of radiation is proportional

Fig.2.9 Equipment for studying the photoelectriceffect. Light of a particular frequency strikesa clean metal surface inside a vacuumchamber. Electrons are ejected from themetal and are counted by a detector thatmeasures their kinetic energy.


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Photoelectric effect. The results observedin this experiment were:

(i) The electrons are ejected from the metal

surface as soon as the beam of lightstrikes the surface, i.e., there is no timelag between the striking of light beamand the ejection of electrons from themetal surface.

(ii) The number of electrons ejected isproportional to the intensity orbrightness of light.

(iii) For each metal, there is a characteristicminimum frequency,

0(also known as

threshold frequency) below whichphotoelectric effect is not observed. At a

frequency > 0, the ejected electronscome out with certain kinetic energy.The kinetic energies of these electronsincrease with the increase of frequencyof the light used.

All the above results could not beexplained on the basis of laws of classicalphysics. According to latter, the energycontent of the beam of light depends uponthe brightness of the light. In other words,number of electrons ejected and kineticenergy associated with them should depend

on the brightness of light. It has beenobserved that though the number of electronsejected does depend upon the brightness oflight, the kinetic energy of the ejectedelectrons does not. For example, red light [= (4.3 to 4.6) 1014 Hz] of any brightness

(intensity) may shine on a piece of potassiummetal for hours but no photoelectrons areejected. But, as soon as even a very weak

yellow light (= 5.15.2 1014

Hz) shines onthe potassium metal, the photoelectric effectis observed. The threshold frequency (0) forpotassium metal is 5.01014 Hz.

Einstein (1905) was able to explain thephotoelectric effect using Plancks quantumtheory of electromagnetic radiation as astarting point,

Shining a beam of light on to a metalsurface can, therefore, be viewed as shootinga beam of particles, the photons. When aphoton of sufficient energy strikes an electron

in the atom of the metal, it transfers its energyinstantaneously to the electron during thecollision and the electron is ejected withoutany time lag or delay. Greater the energypossessed by the photon, greater will betransfer of energy to the electron and greaterthe kinetic energy of the ejected electron. Inother words, kinetic energy of the ejectedelectron is proportional to the frequency ofthe electromagnetic radiation. Since thestriking photon has energy equal to h andthe minimum energy required to eject theelectron is h0 (also called work function, W0; Table 2.2), then the difference in energy(hh

0) is transferred as the kinetic energy

of the photoelectron. Following theconservation of energy principle, the kineticenergy of the ejected electron is given by theequation 2.7.

1

2h h 0= + (2.7)

where meis the mass of the electron and v is

the velocity associated with the ejectedelectron. Lastly, a more intense beam of lightconsists of larger number of photons,consequently the number of electrons ejectedis also larger as compared to that in anexperiment in which a beam of weakerintensity of light is employed.

Dual Behaviour of ElectromagneticRadiation

The particle nature of light posed adilemma for scientists. On the one hand, it

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.


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39STRUCTURE OF ATOM

39

could explain the black body radiation andphotoelectric effect satisfactorily but on theother hand, it was not consistent with theknown wave behaviour of light which couldaccount for the phenomena of interferenceand diffraction. The only way to resolve thedilemma was to accept the idea that lightpossesses both particle and wave-likeproperties, i.e., light has dual behaviour.Depending on the experiment, we find thatlight behaves either as a wave or as a streamof particles. Whenever radiation interacts withmatter, it displays particle like properties incontrast to the wavelike properties(interference and diffraction), which itexhibits when it propagates. This concept wastotally alien to the way the scientists thoughtabout matter and radiation and it took thema long time to become convinced of its validity.It turns out, as you shall see later, that somemicroscopic particles like electrons alsoexhibit this wave-particle duality.

Problem 2.6Calculate energy of one mole of photonsof radiation whose frequency is 5 1014

Hz.Solution

Energy (E) of one photon is given by theexpression

E= h

h= 6.626 1034 J s

= 51014 s1 (given)

E= (6.626 1034 J s) (5 1014 s1)= 3.313 1019 J

Energy of one mole of photons

= (3.313 1019 J) (6.022 1023 mol1)= 199.51 kJ mol1

Problem 2.7

A 100 watt bulb emits monochromaticlight of wavelength 400 nm. Calculate

the number of photons emitted persecond by the bulb.Solution

Power of the bulb = 100 watt

= 100 J s1

Energy of one photon E= h= hc/

36.626 10

=4

1= 4.969 10Number of photons emitted

1

19

100 J s

4.969 10 J

Problem 2.8

When electromagnetic radiation ofwavelength 300 nm falls on the surfaceof sodium, electrons are emitted with akinetic energy of 1.68 105J mol1. Whatis the minimum energy needed to remove

an electron from sodium? What is themaximum wavelength that will cause aphotoelectron to be emitted ?

Solution The energy (E) of a 300 nm photon isgiven by

= c /

6.626 1=

h h

= 6.626 10-19J

The energy of one mole of photons

= 6.626 1019 J 6.022 1023 mol1

= 3.99 105 J mol1

The minimum energy needed to removea mole of electrons from sodium

= (3.99 1.68) 105 J mol1

= 2.31 105 J mol1

The minimum energy for one electron

Metal Li Na K Mg Cu Ag

W0 /ev 2.42 2.3 2.25 3.7 4.8 4.3

Table 2.2 Values of Work Function (W0) for a Few Metals


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40 CHEMISTRY

40

23

19

2.31=

6.022 10

= 3.84 10

This corresponds to the wavelength

c=

6.626 1=

h

E

=517 nm

(This corresponds to green light)Problem 2.9

The threshold frequency0 for a metalis 7.0 1014 s1. Calculate the kineticenergy of an electron emitted whenradiation of frequency =1.0 1015 s1

hits the metal.Solution

According to Einsteins equation

Kinetic energy = mev2=h(0 )

= (6.626 1034 J s) (1.0 1015 s1 7.01014 s1)

= (6.626 1034 J s) (10.0 1014 s1 7.01014 s1)

= (6.626 1034 J s) (3.0 1014 s1)

= 1.988 1019 J

2.3.3 Evidence for the quantized*Electronic Energy Levels: Atomicspectra

The speed of light depends upon the natureof the medium through which it passes. As aresult, the beam of light is deviated orrefracted from its original path as it passesfrom one medium to another. It is observed

that when a ray of white light is passedthrough a prism, the wave with shorterwavelength bends more than the one with alonger wavelength. Since ordinary white lightconsists of waves with all the wavelengths inthe visible range, a ray of white light is spreadout into a series of coloured bands calledspectrum. The light of red colour which has

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

longest wavelength is deviated the least whilethe violet light, which has shortest wavelengthis deviated the most. The spectrum of white

light, that we can see, ranges from violet at7.50 1014 Hz to red at 41014 Hz. Such aspectrum is called continuous spectrum.Continuous because violet merges into blue,blue into green and so on. A similar spectrumis produced when a rainbow forms in the sky.Remember that visible light is just a smallportion of the electromagnetic radiation(Fig.2.7). When electromagnetic radiationinteracts with matter, atoms and moleculesmay absorb energy and reach to a higherenergy state. With higher energy, these are inan unstable state. For returning to theirnormal (more stable, lower energy states)energy state, the atoms and molecules emitradiations in various regions of theelectromagnetic spectrum.

Emission and Absorption Spectra

The spectrum of radiation emitted by asubstance that has absorbed energy is calledan emission spectrum. Atoms, molecules orions that have absorbed radiation are said tobe excited. To produce an emissionspectrum, energy is supplied to a sample byheating it or irradiating it and the wavelength(or frequency) of the radiation emitted, as thesample gives up the absorbed energy, isrecorded.

An absorption spectrum is like thephotographic negative of an emissionspectrum. A continuum of radiation is passedthrough a sample which absorbs radiation ofcertain wavelengths. The missing wavelengthwhich corresponds to the radiation absorbedby the matter, leave dark spaces in the brightcontinuous spectrum.

The study of emission or absorptionspectra is referred to as spectroscopy. Thespectrum of the visible light, as discussedabove, was continuous as all wavelengths (redto violet) of the visible light are representedin the spectra. The emission spectra of atomsin the gas phase, on the other hand, do notshow a continuous spread of wavelength from


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41STRUCTURE OF ATOM

41

red to violet, rather they emit light only atspecific wavelengths with dark spacesbetween them. Such spectra are called line

spectra or atomic spectra because theemitted radiation is identified by theappearance of bright lines in the spectra(Fig, 2.10)

Line emission spectra are of greatinterest in the study of electronic structure.Each element has a unique line emissionspectrum. The characteristic lines in atomicspectra can be used in chemical analysis toidentify unknown atoms in the same way asfinger prints are used to identify people. Theexact matching of lines of the emission

spectrum of the atoms of a known element with the lines from an unknown samplequickly establishes the identity of the latter,German chemist, Robert Bunsen (1811-1899)was one of the first investigators to use linespectra to identify elements.

Elements like rubidium (Rb), caesium (Cs)thallium (Tl), indium (In), gallium (Ga) andscandium (Sc) were discovered when their

(a)

(b)

minerals were analysed by spectroscopicmethods. The element helium (He) wasdiscovered in the sun by spectroscopic

method.Line Spectrum of Hydrogen

When an electric discharge is passed throughgaseous hydrogen, the H2 moleculesdissociate and the energetically excitedhydrogen atoms produced emitelectromagnetic radiation of discrete

frequencies. The hydrogen spectrum consistsof several series of lines named after theirdiscoverers. Balmer showed in 1885 on thebasis of experimental observations that ifspectral lines are expressed in terms of

wavenumber ( ), then the visible lines of thehydrogen spectrum obey the followingformula :

109,677= (2.8)

where nis an integer equal to or greater than3 (i.e., n= 3,4,5,....)

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


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42 CHEMISTRY

42

The series of lines described by thisformula are called the Balmer series. TheBalmer series of lines are the only lines in

the hydrogen spectrum which appear in the visible region of the electromagneticspectrum. The Swedish spectroscopist, Johannes Rydberg, noted that all series oflines in the hydrogen spectrum could bedescribed by the following expression :

109,677

=

(2.9)

where n1=1,2........n

2= n

1+ 1, n

1+ 2......

The value 109,677 cm1 is called theRydberg constant for hydrogen. The first fiveseries of lines that correspond to n1 = 1, 2, 3,4, 5 are known as Lyman, Balmer, Paschen,Bracket and Pfund series, respectively,Table 2.3 shows these series of transitions inthe hydrogen spectrum. Fig 2.11 shows theLyman, Balmer and Paschen series oftransitions for hydrogen atom.

Of all the elements, hydrogen atom hasthe simplest line spectrum. Line spectrumbecomes more and more complex for heavieratom. There are however certain featureswhich are common to all line spectra, i.e.,

(i) line spectrum of element is unique and(ii) there is regularity in the line spectrum ofeach element. The questions which arise are: What are the reasons for these similarities?Is it something to do with the electronicstructure of atoms? These are the questionsneed to be answered. We shall find later thatthe answers to these questions provide thekey in understanding electronic structure ofthese elements.

2.4 BOHRS MODEL FOR HYDROGENATOM

Neils Bohr (1913) was the first to explainquantitatively the general features ofhydrogen atom structure and its spectrum. Though the theory is not the modernquantum mechanics, it can still be used torationalize many points in the atomicstructure and spectra. Bohrs model forhydrogen atom is based on the followingpostulates:

i) The electron in the hydrogen atom canmove around the nucleus in a circularpath of fixed radius and energy. These

paths are called orbits, stationary statesor allowed energy states. These orbits arearranged concentrically around thenucleus.

ii) The energy of an electron in the orbit doesnot change with time. However, the

Table 2.3 The Spectral Lines for AtomicHydrogen

Fig. 2.11 Transitions of the electron in thehydrogen atom (The diagram shows

the Lyman, Balmer and Paschen seriesof transitions)


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43STRUCTURE OF ATOM

43

electron will move from a lower stationarystate to a higher stationary state whenrequired amount of energy is absorbed

by the electron or energy is emitted whenelectron moves from higher stationarystate to lower stationary state (equation2.16). The energy change does not takeplace in a continuous manner.

Angular Momentum

Just as linear momentum is the productof mass (m) and linear velocity (v), angularmomentum is the product of moment ofinertia (I) and angular velocity (). For anelectron of mass me, moving in a circularpath of radius raround the nucleus,

angular momentum = ISince I= mer

2 , and = v/rwhere v is thelinear velocity,

angular momentum = mer2 v/r= mevr

iii) The frequency of radiation absorbed oremitted when transition occurs betweentwo stationary states that differ in energybyE, is given by :

2EE

h h

= = (2.10)

Where E1 and E2 are the energies of thelower and higher allowed energy statesrespectively. This expression is

commonly known as Bohrs frequencyrule.

iv) The angular momentum of an electron

in a given stationary state can beexpressed as in equation (2.11)

v .2

=e

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

Thus an electron can move only in thoseorbits for which its angular momentum isintegral multiple of h/2 that is why onlycertain fixed orbits are allowed.

The details regarding the derivation ofenergies of the stationary states used by Bohr,are quite complicated and will be discussedin higher classes. However, according to

Bohrs theory for hydrogen atom:a) The stationary states for electron are

numbered n= 1,2,3.......... These integralnumbers (Section 2.6.2) are known asPrincipal quantum numbers.

b) The radii of the stationary states areexpressed as :r

n= n2 a

0(2.12)

where a0 = 52,9 pm. Thus the radius ofthe first stationary state, called the Bohrradius, is 52.9 pm. Normally the electronin the hydrogen atom is found in this

orbit (that is n=1). As n increases thevalue ofrwill increase. In other wordsthe electron will be present away fromthe nucleus.

c) The most important property associatedwith the electron, is the energy of itsstationary state. It is given by theexpression.

2

1R

=

n HE

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

where RH is called Rydberg constant and itsvalue is 2.181018 J. The energy of the loweststate, also called as the ground state, is

E1 = 2.181018 ( 2

1

1) = 2.181018 J. The

energy of the stationary state for n = 2, will

be : E2 = 2.181018J ( 2

1

2)= 0.5451018J.

Fig. 2.11 depicts the energies of different

Niels Bohr(18851962)

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.


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44 CHEMISTRY

44

stationary states or energy levels of hydrogenatom. This representation is called an energylevel diagram.

where Zis the atomic number and has values2, 3 for the helium and lithium atomsrespectively. From the above equations, it is

evident that the value of energy becomes morenegative and that of radius becomes smallerwith increase ofZ. This means that electronwill be tightly bound to the nucleus.

e) It is also possible to calculate the velocities of electrons moving in theseorbits. Although the precise equation isnot given here, qualitatively themagnitude of velocity of electronincreases with increase of positive chargeon the nucleus and decreases withincrease of principal quantum number.

2.4.1 Explanation of Line Spectrum ofHydrogen

Line spectrum observed in case of hydrogenatom, as mentioned in section 2.3.3, can beexplained quantitatively using Bohrs model.According to assumption 2, radiation (energy)is absorbed if the electron moves from theorbit of smaller Principal quantum numberto the orbit of higher Principal quantumnumber, whereas the radiation (energy) isemitted if the electron moves from higher orbitto lower orbit. The energy gap between the

two orbits is given by equation (2.16)E= Ef Ei (2.16)Combining equations (2.13) and (2.16)

H

2

f

RE

n

=

(where ni and nf

stand for initial orbit and final orbits)

EH 2 2

i f

1 1R

n n

=

(2,17)

The frequency ( ) associated with theabsorption and emission of the photon canbe evaluated by using equation (2.18)

HRE

h h

= =

18

34

2.18 10

6.626 10

=

(2.18)

What does the negative electronicenergy (E

n) for hydrogen atom mean?

The energy of the electron in a hydrogenatom has a negative sign for all possibleorbits (eq. 2.13). What does this negativesign convey? This negative sign means thatthe energy of the electron in the atom islower than the energy of a free electron atrest. A free electron at rest is an electronthat is infinitely far away from the nucleusand is assigned the energy value of zero.Mathematically, this corresponds to

setting n equal to infinity in the equation(2.13) so thatE=0. As the electron getscloser to the nucleus (as ndecreases), E

n

becomes larger in absolute value and moreand more negative. The most negativeenergy value is given by n=1 whichcorresponds to the most stable orbit. Wecall this the ground state.

When the electron is free from theinfluence of nucleus, the energy is taken aszero. The electron in this situation isassociated with the stationary state of

Principal Quantum number = n= and iscalled as ionized hydrogen atom. When theelectron is attracted by the nucleus and ispresent in orbit n, the energy is emitted andits energy is lowered. That is the reason forthe presence of negative sign in equation(2.13) and depicts its stability relative to thereference state of zero energy and n= .d) Bohrs theory can also be applied to the

ions containing only one electron, similarto that present in hydrogen atom. Forexample, He+ Li2+, Be3+ and so on. Theenergies of the stationary statesassociated with these kinds of ions (alsoknown as hydrogen like species) are givenby the expression.

n2.18 1= E (2.14)

and radii by the expression2

n

52.9 ( )r

n

Z= (2.15)


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45STRUCTURE OF ATOM

45

15

3.29 10

=

(2.19)

and in terms of wavenumbers ( )

HR

c ch

= =

(2.20)

15

8

3.29 10=

3 10 ms

= 1.09677 1 (2.21)

In case of absorption spectrum, nf> ni andthe term in the parenthesis is positive andenergy is absorbed. On the other hand in caseof emission spectrum ni > nf, Eis negativeand energy is released.

The expression (2.17) is similar to thatused by Rydberg (2.9) derived empiricallyusing the experimental data available at thattime. Further, each spectral line, whether inabsorption or emission spectrum, can beassociated to the particular transition inhydrogen atom. In case of large number ofhydrogen atoms, different possible transitionscan be observed and thus leading to largenumber of spectral lines. The brightness orintensity of spectral lines depends upon thenumber of photons of same wavelength orfrequency absorbed or emitted.

Problem 2.10

What are the frequency and wavelengthof a photon emitted during a transitionfrom n= 5 state to the n= 2 state in thehydrogen atom?

SolutionSince ni = 5 and nf= 2, this transitiongives rise to a spectral line in the visibleregion of the Balmer series. Fromequation (2.17)

= 2.18 1

= 4.58 1

E

It is an emission energy The frequency of the photon (takingenergy in terms of magnitude) is given

by

=

E

h

1

3

4.58 10=6.626 10

= 6.911014 Hz

3.0c= =

6.91

Problem 2.11Calculate the energy associated with thefirst orbit of He+ . What is the radius ofthis orbit?

Solution

n

(2.18E

= atom1

For He+, n= 1, Z = 2

1

(2.18E

=

The radius of the orbit is given byequation (2.15)

(0.0529 nrn

Z=

Since n= 1, and Z= 2

(0.0529 nr

2n=

2.4.2 Limitations of Bohrs Model

Bohrs model of the hydrogen atom was no

doubt an improvement over Rutherfordsnuclear model, as it could account for thestability and line spectra of hydrogen atomand hydrogen like ions (for example, He+, Li2+,Be3+, and so on). However, Bohrs model wastoo 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


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46 CHEMISTRY

46

Louis de Broglie (1892-1987)

Louis de Broglie, a French

physicist, studied history as anundergraduate in the early

1910s. 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.

by using sophisticated spectroscopictechniques. This model is also unable toexplain the spectrum of atoms other than

hydrogen, for example, helium atom whichpossesses only two electrons. Further,Bohrs theory was also unable to explainthe splitting of spectral lines in thepresence of magnetic field (Zeeman effect)or an electric field (Stark effect).

ii) It could not explain the ability of atoms toform molecules by chemical bonds.

In other words, taking into account thepoints mentioned above, one needs a bettertheory which can explain the salient featuresof the structure of complex atoms.

2.5 TOWARDS QUANTUM MECHANICALMODEL OF THE ATOM

In view of the shortcoming of the Bohrs model,attempts were made to develop a moresuitable and general model for atoms. Twoimportant developments which contributedsignificantly in the formulation of such amodel were :

1. Dual behaviour of matter,

2. Heisenberg uncertainty principle.

2.5.1 Dual Behaviour of Matter The French physicist, de Broglie in 1924proposed that matter, like radiation, shouldalso exhibit dual behaviour i.e., both particleand wavelike properties. This means that justas the photon has momentum as well as wavelength, electrons should also havemomentum as well as wavelength, de Broglie,from this analogy, gave the following relationbetween wavelength () and momentum (p) ofa material particle.

v

h h

m p = = (2.22)

where m is the mass of the particle, v its velocity andp its momentum. de Brogliesprediction was confirmed experimentallywhen it was found that an electron beamundergoes diffraction, a phenomenoncharacteristic of waves. This fact has been putto use in making an electron microscope,

which is based on the wavelike behaviour ofelectrons just as an ordinary microscopeutilises the wave nature of light. An electronmicroscope is a powerful tool in modernscientific research because it achieves amagnification of about 15 million times.

It needs to be noted that according to deBroglie, every object in motion has a wavecharacter. The wavelengths associated withordinary objects are so short (because of theirlarge masses) that their wave propertiescannot be detected. The wavelengthsassociated with electrons and other subatomicparticles (with very small mass) can howeverbe detected experimentally. Results obtainedfrom the following problems prove thesepoints qualitatively.

Problem 2.12

What will be the wavelength of a ball ofmass 0.1 kg moving with a velocity of 10m s1 ?SolutionAccording to de Brogile equation (2.22)

(6.2

v (0.1

h

m = =

= 6.6261034 m (J = kg m2 s2)

Problem 2.13

The mass of an electron is 9.11031 kg.If its K.E. is 3.01025 J, calculate itswavelength.


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47STRUCTURE OF ATOM

47

Solution

Since K. E. = mv2

1/ 22K.E.v =

m

= 812 m s1

v (9.1

h

m = =

= 8967 1010 m = 896.7 nm

Problem 2.14Calculate the mass of a photon withwavelength 3.6 .

Solution

3.6 3. = =

Velocity of photon = velocity of light

(3.6

hm = =

= 6.135 1029 kg

2.5.2 Heisenbergs Uncertainty Principle

Werner Heisenberg a German physicist in1927, stated uncertainty principle which is

the consequence of dual behaviour of matterand radiation. It states that it is impossibleto determine simultaneously, the exactposition and exact momentum (or velocity)of an electron.

Mathematically, it can be given as inequation (2.23).

xx

4

hp

(2.23)

or x x ( v )m

or x v4

x

h

where xis the uncertainty in position andp

x( orvx) is the uncertainty in momentum

(or velocity) of the particle. If the position ofthe electron is known with high degree ofaccuracy (x is small), then the velocity of theelectron will be uncertain [(vx) is large]. On

the other hand, if the velocity of the electronis known precisely ((v

x) is small), then the

position of the electron will be uncertain

(x will be large). Thus, if we carry out somephysical measurements on the electronsposition or velocity, the outcome will alwaysdepict a fuzzy or blur picture.

The uncertainty principle can be bestunderstood with the help of an example.Suppose you are asked to measure thethickness of a sheet of paper with anunmarked metrestick. Obviously, the resultsobtained would be extremely inaccurate andmeaningless, In order to obtain any accuracy,you should use an instrument graduated in

units smaller than the thickness of a sheet ofthe paper. Analogously, in order to determinethe position of an electron, we must use ameterstick calibrated in units of smaller thanthe dimensions of electron (keep in mind thatan electron is considered as a point chargeand is therefore, dimensionless). To observean electron, we can illuminate it with lightor electromagnetic radiation. The light usedmust have a wavelength smaller than thedimensions of an electron. The high

momentum photons of such light =h

p

would change the energy of electrons bycollisions. In this process we, no doubt, would be able to calculate the position of theelectron, but we would know very little aboutthe velocity of the electron after the collision.

Significance of Uncertainty Principle

One of the important implications of theHeisenberg Uncertainty Principle is that itrules out existence of definite paths ortrajectories of electrons and other similarparticles. The trajectory of an object is

determined by its location and velocity atvarious moments. If we know where a body isat a particular instant and if we also know its velocity and the forces acting on it at thatinstant, we can tell where the body would besometime later. We, therefore, conclude thatthe position of an object and its velocity fixits trajectory. Since for a sub-atomic objectsuch as an electron, it is not possible


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48 CHEMISTRY

48

simultaneously to determine the position andvelocity at any given instant to an arbitrarydegree 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 ofmicroscopic objects and is negligible forthat of macroscopic objects. This can beseen from the following examples.

If uncertainty principle is applied to anobject of mass, say about a milligram (106

kg), then

v. x =4 .

6.6=

4 3.1

h

m

The value ofvx obtained is extremelysmall and is insignificant. Therefore, one maysay thatin dealing with milligram-sized orheavier objects, the associateduncertainties are hardly of any realconsequence.

In the case of a microscopic object like anelectron on the other hand. v.x obtained ismuch larger and such uncertainties are of

real consequence. For example, for an electronwhose mass is 9.111031 kg., according toHeisenberg uncertainty principle

v. x =4

h

m

4 2 1

6.626=

4 3.1416

= 10 m s

It, therefore, means that if one tries to findthe exact location of the electron, say to an

uncertainty of only 108 m, then theuncertaintyv in velocity would be

4 2 1

8

10 m s

110 m

which is so large that the classical picture ofelectrons moving in Bohrs orbits (fixed)cannot hold good. It, therefore, means thatthe precise statements of the position andmomentum of electrons have to bereplaced by the statements of probability,that the electron has at a given positionand momentum. This is what happens inthe quantum mechanical model of atom.

Problem 2.15

A microscope using suitable photons isemployed to locate an electron in anatom within a distance of 0.1 . What isthe uncertainty involved in themeasurement of its velocity?

Solution

x = or 4

hp

v =4 x

h

m

v4 3.14 0

=

= 0.579107 m s1 (1J = 1 kg m2 s2

= 5.79106 m s1

Problem 2.16A golf ball has a mass of 40g, and a speedof 45 m/s. If the speed can be measured within accuracy of 2%, calculate theuncertainty in the position.

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 from 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.


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Erwin Schrdinger(1887-1961)

Solution

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

245 = 0.100

.

Using the equation (2.22)

x =4 v

6=

4 3.14 40

h

m

= 1.461033 m

This is nearly ~ 1018 times smaller thanthe diameter of a typical atomic nucleus.As mentioned earlier for large particles,the uncertainty principle sets nomeaningful limit to the precision ofmeasurements.

Reasons for the Failure of the Bohr Model

One can now understand the reasons for thefailure of the Bohr model. In Bohr model, anelectron is regarded as a charged particlemoving in well defined circular orbits aboutthe nucleus. The wave character of theelectron is not considered in Bohr model.

Further, an orbit is a clearly defined path andthis path can completely be defined only if both the position and the velocity of theelectron are known exactly at the same time. This is not possible according to theHeisenberg uncertainty principle. Bohr modelof the hydrogen atom, therefore, not only

ignores dual behaviour of matter but also

contradicts Heisenberg uncertainty principle.In view of these inherent weaknesses in theBohr model, there was no point in extendingBohr model to other atoms. In fact an insightinto the structure of the atom was needed

which could account for wave-particle dualityof matter and be consistent with Heisenberguncertainty principle. This came with theadvent of quantum mechanics.

2.6 QUANTUM MECHANICAL MODEL OFATOM

Classical mechanics, based on Newtons lawsof motion, successfully describes the motion

of all macroscopic objects such as a fallingstone, orbiting planets etc., which haveessentially a particle-like behaviour as shownin the previous section. However it fails whenapplied to microscopic objects like electrons,atoms, molecules etc. This is mainly becauseof the fact that classical mechanics ignores

the concept of dual behaviour of matterespecially for sub-atomic particles and theuncertainty principle. The branch of sciencethat takes into account this dual behaviourof matter is called quantum mechanics.

Quantum mechanics is a theoreticalscience that deals with the study of themotions of the microscopic objects that haveboth observable wave like and particle likeproperties. It specifies the laws of motion thatthese objects obey. When quantummechanics is applied to macroscopic objects

(for which wave like properties areinsignificant) the results are the same asthose from the classical mechanics.

Quantum mechanics was developedindependently in 1926 by Werner Heisenbergand Erwin Schrdinger. Here, however, weshall be discussing the quantum mechanicswhich is based on the ideas of wave motion. The fundamental equation of quantum

Erwin Schrdinger, an

Austrian physicist

received his Ph.D. in

theoretical physics from

the University of Vienna

in 1910. In 1927

Schrdinger succeeded

Max Planck at the

University of Berlin at

Plancks request. In 1933,

Schrdinger left Berlin

because of his opposition to Hitler and Nazi

policies and returned to Austria in 1936. After

the invasion of Austria by Germany,

Schrdinger was forcibly removed from his

professorship. He then moved to Dublin, Ireland

where he remained for seventeen years.

Schrdinger shared the Nobel Prize for Physics

with P.A.M. Dirac in 1933.


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mechanics was developed by Schrdingerand it won him the Nobel Prize in Physics in1933. This equation which incorporates

wave-particle duality of matter as proposedby de Broglie is quite complex and knowledgeof higher mathematics is needed to solve it. You will learn its solutions for differentsystems in higher classes.

For a system (such as an atom or amolecule whose energy does not change withtime) the Schrdinger equation is written as

H = E where H is a mathematicaloperator called Hamiltonian. Schrdingergave a recipe of constructing this operatorfrom the expression for the total energy ofthe system. The total energy of the systemtakes into account the kinetic energies of allthe sub-atomic particles (electrons, nuclei),attractive potential between the electrons andnuclei and repulsive potential among theelectrons and nuclei individually. Solution ofthis equation gives Eand .

Hydrogen Atom and the SchrdingerEquation

When Schrdinger equation is solved forhydrogen atom, the solution gives thepossible energy levels the electron can occupyand the corresponding wave function(s) ()of the electron associated with each energylevel. These quantized energy states andcorresponding wave functions which arecharacterized by a set of three quantumnumbers (principal quantum number n,azimuthal quantum number l andmagnetic quantum number ml

) arise as a

natural consequence in the solution of theSchrdinger equation. When an electron isin any energy state, the wave functioncorresponding to that energy state contains

all information about the electron. The wavefunction is a mathematical function whosevalue depends upon the coordinates of theelectron in the atom and does not carry anyphysical meaning. Such wave functions ofhydrogen or hydrogen like species with oneelectron are called atomic orbitals. Such wave functions pertaining to one-electronspecies are called one-electron systems. The

probability of finding an electron at a pointwithin an atom is proportional to the ||2 atthat point. The quantum mechanical results

of the hydrogen atom successfully predict allaspects of the hydrogen atom spectrumincluding some phenomena that could notbe explained by the Bohr model.

Application of Schrdinger equation tomulti-electron atoms presents a difficulty: theSchrdinger equation cannot be solvedexactly for a multi-electron atom. Thisdifficulty can be overcome by usingapproximate methods. Such calculationswith the aid of modern computers show thatorbitals in atoms other than hydrogen do not

differ in any radical way from the hydrogenorbitals discussed above. The principaldifference lies in the consequence ofincreased nuclear charge. Because of this allthe orbitals are somewhat contracted.Further, as you shall see later (in subsections2.6.3 and 2.6.4), unlike orbitals of hydrogenor hydrogen like species, whose energiesdepend only on the quantum numbern, theenergies of the orbitals in multi-electronatoms depend on quantum numbers nand l.

Important Features of the Quantum

Mechanical Model of AtomQuantum mechanical model of atom is thepicture of the structure of the atom, whichemerges from the application of theSchrdinger equation to atoms. The followingare the important features of the quantum-mechanical model of atom:

1. The energy of electrons in atoms isquantized (i.e., can only have certainspecific values), for example whenelectrons are bound to the nucleus inatoms.

2. The existence of quantized electronicenergy levels is a direct result of the wavelike properties of electrons and are allowedsolutions of Schrdinger wave equation.

3. Both the exact position and exact velocityof an electron in an atom cannot bedetermined simultaneously (Heisenberguncertainty principle). The path of anelectron in an atom therefore, can never

be determined or known accurately. That


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is why, as you shall see later on, one talksof only probability of finding the electronat different points in an atom.

4.An atomic orbital is the wave functionfor an electron in an atom. Wheneveran electron is described by a wavefunction, we say that the electron occupiesthat orbital. Since many such wavefunctions are possible for an electron,there are many atomic orbitals in an atom.

These one electron orbital wave functionsor orbitals form the basis of the electronicstructure of atoms. In each orbital, theelectron has a definite energy. An orbitalcannot contain more than two electrons.In a multi-electron atom, the electrons arefilled in various orbitals in the order of

increasing energy. For each electron of amulti-electron atom, there shall, therefore,

be an orbital wave function characteristicof the orbital it occupies. All theinformation about the electron in an atomis stored in its orbital wave function andquantum mechanics makes it possible toextract this information out of.

5. The probability of finding an electron at apoint within an atom is proportional to thesquare of the orbital wave function i.e.,||2 at that point. ||2 is known asprobability density and is always

positive. From the value of ||2

atdifferent points within an atom, it ispossible to predict the region aroundthe nucleus where electron will most

probably be found.

2.6.1 Orbitals and Quantum Numbers

A large number of orbitals are possible in anatom. Qualitatively these orbitals can bedistinguished by their size, shape andorientation. An orbital of smaller size meansthere is more chance of finding the electronnear the nucleus. Similarly shape and

orientation mean that there is moreprobability of finding the electron alongcertain directions than along others. Atomicorbitals are precisely distinguished by whatare known as quantum numbers. Eachorbital is designated by three quantumnumbers labelled as n, land ml.

The principal quantum number n is apositive integer with value ofn= 1,2,3....... .

The principal quantum number determinesthe size and to large extent the energy of theorbital. For hydrogen atom and hydrogen like

species (He+

, Li2+

, .... etc.) energy and size ofthe 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 orbitalincreases and are given by n2 All theorbitals of a given value of n constitute asingle shell of atom and are represented bythe 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 otherwords the electron will be located away fromthe nucleus. Since energy is required inshifting away the negatively charged electronfrom the positively charged nucleus, theenergy of the orbital will increase withincrease ofn.

Azimuthal quantum number. l is alsoknown as orbital angular momentum orsubsidiary quantum number. It defines thethree dimensional shape of the orbital. For agiven value ofn, l can have nvalues rangingfrom 0 to n 1, that is, for a given value ofn,the possible value oflare : l= 0, 1, 2, ..........(n1)

For example, when n= 1, value oflis only0. Forn= 2, the possible value oflcan be 0and 1. Forn= 3, the possible lvalues 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 valueofn. For example in the first shell (n= 1),there is only one sub-shell which correspondsto l= 0. There are two sub-shells (l= 0, 1) in

the second shell (n= 2), three (l= 0, 1, 2) inthird shell (n= 3) and so on. Each sub-shellis assigned an azimuthal quantum number(l). Sub-shells corresponding to differentvalues oflare represented by the followingsymbols.Value forl: 0 1 2 3 4 5 ............notation for s p d f g h ............sub-shell


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Table 2.4 Subshell Notations

Value ofl 0 1 2 3 4 5

Subshell notation s p d f g h

number of orbitals 1 3 5 7 9 11

Table 2.4 shows the permissible valuesof l for a given principal quantum numberand the corresponding sub-shell notation.

Orbit, orbital and its importance

Orbit and orbital are not synonymous. An orbit, as proposed by Bohr, is a circular path aroundthe nucleus in which an electron moves. A precise description of this path of the electron isimpossible according to Heisenberg uncertainty principle. Bohr orbits, therefore, have no realmeaning and their existence can never be demonstrated experimentally. An atomic orbital, on theother hand, is a quantum mechanical concept and refers to the one electron wave function inan atom. It is characterized by three quantum numbers (n, land m

l) and its value depends upon

the coordinates of the electron. has, by itself, no physical meaning. It is the square of the wavefunction i.e., ||2 which has a physical meaning. ||2 at any point in an atom gives the value ofprobability density at that point. Probability density (||2) is the probability per unit volume andthe product of ||2 and a small volume (called a volume element) yields the probability of findingthe electron in that volume (the reason for specifying a small volume element is that ||2 variesfrom 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 becalculated by the sum of all the products of ||2 and the corresponding volume elements. It isthus possible to get the probable distribution of an electron in an orbital.

Thus forl= 0, the only permitted value ofm

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

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

orbitals]. Forl= 2, ml = 2, 1, 0, +1 and +2,[2(2)+1 = 5, five dorbitals]. It should be notedthat the values ofm

lare derived from land

that the value oflare derived from n.

Each orbital in an atom, therefore, isdefined by a set of values forn, land m

l. An

orbital described by the quantum numbersn= 2, l= 1, m

l= 0 is an orbital in thepsub-

shell of the second shell. The following chartgives the relation between the sub-shell andthe number of orbitals associated with it.

Electron spin s : The three quantumnumbers labelling an atomic orbital can beused equally well to define its energy, shapeand orientation. But all these quantumnumbers are not enough to explain the linespectra observed in the case of multi-electronatoms, that is, some of the lines actually occurin doublets (two lines closely spaced), triplets(three lines, closely spaced) etc. This suggeststhe presence of a few more energy levels thanpredicted by the three quantum numbers.

In 1925, George Uhlenbeck and SamuelGoudsmit proposed the presence of the fourth

Magnetic orbital quantum number. ml

gives information about the spatialorientation of the orbital with respect tostandard set of co-ordinate axis. For any

sub-shell (defined by l value) 2l+1 valuesof mlare possible and these values are given

by :

ml = l, (l1), (l2)... 0,1... (l2), (l1), l


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quantum number known as the electronspin quantum number (m

s). An electron

spins around its own axis, much in a similar

way as earth spins around its own axis whilerevolving around the sun. In other words, anelectron has, besides charge and mass,intrinsic spin angular quantum number. Spinangular momentum of the electron a vectorquantity, can have two orientations relativeto the chosen axis. These two orientations aredistinguished by the spin quantum numbersms which can take the values of + or .These are called the two spin states of theelectron and are normallyrepresented by twoarrows, (spin up) and (spin down). Two

electrons that have differentms values (one+ and the other ) are said to haveopposite spins. An orbital cannot hold morethan two electrons and these two electronsshould have opposite spins.

To sum up, the four quantum number

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