structure of atom.pdf

7/27/2019 structure of atom.pdf

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STRUCTURE OF ATOM

CLASS XI


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Introduction

The evidence of the electrical nature of matter preceded the first theoriesabout the structure of matter. In 1832, Michael Faraday discovered

electrolysis. He gave the first important clue relating electricity to matter andabout the electric nature of atoms.

However, the first definite modern theory on the structure of matter wasJohn Dalton's Atomic theory in 1890. It surmised that all matter wascomposed of indivisible and extremely small structureless, hard, sphericalparticles called atoms. However, later discoveries at the turn of the20thcentury showed that atoms had still a further complex structure andconsisted of smaller sub-atomic particles called electrons, protons andneutrons. These were considered to be the fundamental particles of allatoms.

Discovery of Atomic Particles

Electron - Discharge Tube experiments

J.J. Thomson studied the passage of electricity through gases at extremelylow pressure in a cylindrical glass tube in detail. He was discovered that atextremely low pressure, gases become conductors of electricity and emitstreaks of light, which flows in the form of cathode rays.

Experiment and observations

In its simplest form, the electron-discharge tube consists of a cylindricalglass tube about 50 cm long, closed at both ends. It is also called thedischarge tube or Crookes tube. The tube is fitted with two metallicelectrodes, which are connected to a source of high voltage. It is filled withgas and is connected to a side tube, through which the gas can be evacuatedto any desired pressure with the help of a vacuum pump.


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Discharge Tube Experiment

The following observations were discovered:

At normal temperatures, all gases were non-conducting, in spite of highvoltage (5000-10,000 volts) being applied between the electrodes in thedischarge tube.

The residual gas in the tube became 'conducting'. When the pressure of thegas was reduced to 10-2 atm. the gas emitted a flow of light evacuating some

of the gas with the help of vacuum pump, through the side tube.

When the pressure in the discharge tube was further reduced, the residualgas continued to conduct electricity but its light emission glow becameweaker and finally stopped glowing at about 10-4 atm pressure. At thispressure, the glass tube only showed a greenish fluorescence at the anodeend.

The glow in the tube at low gas pressures of 10-2 to 10-4 atm was due to thebombardment at the glass by certain rays, which were emitted from thecathode as streaks of light and moved towards the anode end. The colour of

the light depended on the nature of the gas used.

These rays were named as cathode rays because their point of origin was thecathode.

Nature of cathode rays

Further experiments established the following properties of cathode rays.


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The rays travel in straight lines from the cathode to the anode. This is provedbecause when an object is placed in the path of a cathode ray in thedischarge tube, the light gets blocked in the area of the object andfluorescence is seen only in regions outside the shadow. Further it casts ashadow away from the cathode, on the opposite side).

The rays consist of material particles and can produce a mechanical effect.For example, when a small paddle wheel is placed between the electrodes ofthe discharge tube, it rotated.

These material particles are charged with a negative charge. This is provedon exposing cathode rays to electrical or magnetic fields. They deflecttowards the positively charged plate.

They have a heating effect i.e., when they strike a thin metal foil it getsheated up.

They produce fluorescence or glow on striking glass or certain othermaterials.

They produce X-rays on striking hard metals like copper, tungsten etc.

They penetrate through thin sheets of aluminium or other metal.

Cathode rays affect photographic plates.

The ratio of charge to mass (charge/mass) is the same for all the cathoderays irrespective of the type of gas used in the tube.

The above observations confirmed the existence of negatively chargedparticles called electrons as particles that made up the cathode rays.

Charge and mass of electron

Ratio of the charge of electrons to its mass (e/m):

The charge to mass ratio is found by measuring the deflection of a ray underthe simultaneous influence of electrical and magnetic fields, appliedperpendicularly to each other as well as to the direction of the flow of light.This is illustrated in the figure below:


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Determining e/m ratio

A high voltage charge accelerates cathode ray electrons between cathodeand anode. After the anode, a circular disc selects a straight beam and

directs it past the electric and magnetic fields, which are perpendicular toeach other as well as to the direction of the motion of the light beam. Thebeam is deflected according to the relative strengths of the electric andmagnetic fields and the ratio of e/m controls the deflection. By measuring thedeflection and the field strengths of the two fields the e/m ratio can becalculated.

Determination of the charge of electrons

The charge of electrons is determined by Millikan's experiment in 1909. Smalldrops of oil formed by a sprayer are allowed to fall in between a positivelycharged upper metal plate and a negatively charged lower metal plate. Thespace between the two plates is irradiated with X-rays. This displaces someelectrons of the air molecules, which consequently get attached to the oildroplet. The fall of the oil drop is observed through a microscope. Thecharged plates create an electrical field in the upper direction, whichcounteracts the gravity influence on the drop of oil. By adjusting the electricfield strength to a level equal to the downward gravitational force, the dropremains stationary in mid-air. The charge on the droplet is then determinedby the amount of charge on the plates and the mass of the droplet. The massof the droplet is determined earlier, from the rate of fall of droplet throughthe air when the metal plates were uncharged.

From the experiments of Millikan, the charge on the electron is found to be1.602 x 10-19 coulombs.


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Millikan's oil drop experiment

Calculation of the mass of the electron

Themass of an electron (m) is determined by dividing the value of 'e' bye/m.

Charge/mass (e/m) = 1.76 x 108 coulomb/g

Charge (e) = 1.602 x 10-19 coulombs.

= 9.1 x 10-28g

Electron

An electron is defined as a subatomic particle which carries one unit ofelectrical charge (1.602 x 10-19 C) and has a mass of 9.1 x10-28g.

The mass of an electron is almost negligible, being 1/1837th the mass of anatom of hydrogen. The charge of an electron is the smallest known electricalcharge and is referred to as unit negative charge.

The discharge tube experiments showed thatirrespective of the gas used orthe nature of the material of the cathode, all electrons were found to havethe same mass and same charge and therefore the same e/m ratios. Thuselectronsof all cathode rays are the same and only electrons (no gaseousatoms) make up the fundamental common particles of the rays.

Later on it was found that all electrons emitted from all sources and by all


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methods have the same mass and same charge. Thus, the electron in theatom is the universal constituent of all matter.

Anode rays and the discovery of protons

Since atoms on the whole are neutral, the presence of negatively chargedelectrons suggested the presence of positively charged particles.

Experiment and characteristics

E.Goldstien modified the discharge tube experiments by using a perforatedcathode. After evacuating the tube he applied high voltage across theelectrodes. Apart from the usual cathode rays emerging from theperforations, he found a new set of rays emerging and travelling in theopposite direction. In similar experiments to those of Thomson's involvingelectric and magnetic fields he observed, that like the cathode rays, theserays also deflect but in the opposite direction. They were attracted towardsnegative plates establishing their positive nature. He called these rays asanode rays.

Anode rays Goldstein's experiment

Characteristics of anode rays

These travel in straight lines and cast shadow of the object placed in theirpath.

They are deflected by magnetic and electric fields in the opposite direction to

that of cathode rays.

The anode rays produce mechanical and heating effects also.

The charge to mass ratio is smaller than that of the electrons, showing thatthese particles are heavier than the cathode ray particles.

The charge to mass ratio depends upon the nature of the gas.


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The charge and mass of the positive particles

The e/m ratio of the anode rays obtained from hydrogen gas was found to be

highest and equivalent to 9.58 x 10

4

C g

-1

. These particles also carried acharge of 1.602 x 10-19 C. Thus, the mass of the positive particle fromhydrogen gas is,

As the mass of the electron is 9.1 x 1028 g, the ratio of the mass of positiveparticle obtained from hydrogen to the mass of an electron is,

The positive particle from hydrogen is 1837 times heavier than the electron.This positively charged particle was called proton. The proton is produced bythe loss of an electron from a neutral hydrogen atom and is thus a hydrogenion H+. The mass of H is found to be 1837 times that of an electron and sothe mass of the proton is nearly the same as that of a hydrogen atom.

Proton

The proton has a mass equal to that of hydrogen atom, which is equal to1.67 x 10-24 g or 1.0073 amu while it has an unit positive charge of +1.602 x10-19 C.

Discovery of NeutronsThe whole mass of an atom has found to be in the nucleus, which means thatthe nucleus must contain protons equal to the mass of the atom. As thenumber of protons is equal to the atomic number, the atomic mass should beequal to the atomic number. But for all atoms except hydrogen the atomicmass was found to be more than the atomic number. To account for the

remaining mass Rutherford predicted the presence of neutral particles havingmass equal to that of protons.

In 1932, James Chadwick bombarded a thin sheet of beryllium element withparticles and observed the generation of highly penetrating rays consisting ofneutral particles. These particles had a mass nearly the same as that ofhydrogen atom and did not have a charge. Since these particles wereelectrically neutral, they were named as neutrons.


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Neutron

A neutron is a particle having a mass equal to 1.67 x 10-24 g same as that ofa hydrogen atom and no electrical charge.

Some Important Terms Relating to Atomic Structure

Atomic number

This is the number of protons present in the nucleus or the number ofelectrons present outside the nucleus. It is denoted by the letter 'Z'.

Atomic number (Z) = Nuclear charge or number of protons (p)

= Number of electrons (e)

Mass number

The sum of the number of protons and neutrons in the nucleus of an atom iscalled the mass number. It is represented by the letter A.

Mass number (A)= Number of protons (p) + number of neutrons (n)

From the knowledge of atomic number and mass number the number ofelectrons, protons and neutrons in an atom can be easily predicted.

For example, Lithium has an atomic number = 3 and mass number = 7

Number of electrons = Atomic number = 3

Number of protons = Atomic number = 3

Number of neutrons = A - Z = 7 - 3 = 4

An atom is represented by its symbol for the element (X) with the atomicnumber written on the lower side of the symbol and the mass numberwritten on the upper side.

Problem

1. How many protons and neutrons are there in the following nucleus?


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Solution

Atomic number Z = 8, Mass number A = 17

Number of protons = Z = 8

Number of neutrons + number of protons = A

Number of neutrons + 8 = 17

Number of neutrons = 17- 8 = 9

Atomic number Z = 12, Mass number A = 25

Number of protons = Z = 12

Number of neutrons = A - number of protons

= 25 - 12 = 13

2. Find (i) the total number of neutrons and (ii) total mass of neutrons in 7mg of14C. Assume: mass of neutron = mass of hydrogen atom.

Solution

Mass number of14C is 14.

So, 14 g of14C contain 6.023 x 1023 atoms of14C

0.007g of14e contains

= 3.012 x 1020 atoms

Number of neutrons in 1 atom of14C = 14 - 6 = 8

Number of neutrons in 7 mg of14C = (8 x 3.012 x 1020)


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= 24.1 x1020

Mass of the neutron = mass of hydrogen atom

= 1.675 x 10

-27

kgHence, mass of the neutron in 7 mg of14C

= 1.675 x 10-27 kg x 24.1 x1020

= 40.36 x 10-7 kg

Isotopes

Sometimes, atoms of the same element have the same atomic number butdifferent mass numbers. These are called isotopes.

The nuclei of these atoms have same number of protons but differentnumber of neutrons.

The properties of isotopes differ depending upon their mass

Different isotopes of an element exhibit similar chemical properties.

Hydrogen has three isotopes namely, hydrogen-H (protium), deuterium-Dand tritium-T with mass numbers 1, 2 and 3 respectively.

Three isotopes of hydrogen

Isotopes of some common elements


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Isobars

The atoms of different elements, which have the same mass number butdifferent atomic number are called isobars. These have different number ofprotons but equal sum of the number of protons and neutrons.

Some typical isobars

Isotones

The atoms of different elements, having the same number of neutrons butdifferent atomic number are called isotones.

Some typical isotones


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Thomson's Model and its Limitation

J.J.Thomson proposed the first simple and most primitive model of thestructure of an atom in 1898, when the neutron was not yet discovered. He

tried to explain the arrangement of electrons and protons within an atom byconsidering an atom to consist of a diffused sphere of uniform positivecharge, of radius 10-8 cm, into which the negatively charged electrons wereembedded just like raisins dotted evenly in a plum pudding. The model wasthus called as 'raisin pudding' model of the atom.

Limitations of Thomson's model

There was no distinct place of existence for the electrons or the protons i.e.,they were not concentrated in any particular part of the atom.

It would be difficult to separate the positive and negative charges by thismodel as evidenced by the discharge tube experiments.

The mass of the atom was spread evenly throughout it. Since the modelcould not satisfy experimental facts, scientists later discarded it.

The subsequent detection of radioactive rays by Henry Becquerel anddiscovery of radioactivity by Marie and Pierre Curie led to more discoverieson the structure of the atom. Rutherford found out the nature of theradioactive , and rays and devised an alternate theory of atomic structure.

Properties of, and rays


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Rutherford's model

The discovery of radioactive particles let Rutherford to perform anexperiment where he bombarded a thin sheet of gold with '' particlesobtained from a radioactive substance. On striking the gold foil some of theparticles scattered and produced flashes on a zinc sulphide (ZnS) screenplaced at the back of the gold foil. These tiny flashes were observed by amovable microscope. The observations made in this scattering experimentwere as follows:

Most of the '' particles pass through the metal and are undeflected.

Some of the '' particles are deflected through small angles.

Only a very few of them are deflected through as much as 90o or even largerangles.

As Thomson's atomic model was not in conformity with the results of thescattering experiment.

Rutherford concluded that:

Most of the space inside the atom was empty or hollow since most of the ''particles passed undeflected.

Some of the particles that deflect with large angles indicated that someheavy positively charged body was present inside the body of the atom,which repelled the like charge of the '' particle. This heavy, positivelycharged body was named as nucleus.

As the number of heavy positively charged particles that undergo deflectionwas very small, the volume occupied by the nucleus must also be very small


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compared to the volume of the atom.

As heavy particles like the '' particles deflected, the nucleus of the atommust also have an appreciable mass.

Rutherford's nuclear model

On the basis of the scattering experiment, Rutherford described the structureof the atom as:

An atom consists of a positively charged nucleus surrounded by electronsthat move around it. The positive charge of the nucleus is due to the protons.

Electrons and protons are held together by coulombic force of attraction.

The effective volume of the nucleus is extremely small as compared to theeffective volume of the atom. The volume occupied by the nucleus is smallerby about 10-12 times the volume of the atom.

The entire mass of the atom is concentrated at the nucleus.

Since each atom is electrically neutral, the number of positive charges in thenucleus of an atom is equal to the number of electrons in it.

Rutherford's model of an atom

Limitations of Rutherford's model

Rutherford's model suffered some drawback. It could not explain the stabilityof the atom in-spite of the electrons revolving around the nucleus. Moving

electrons should in principle emit radiations and lose energy. This shouldcause them to slow down, gradually get pulled towards the nucleus byfollowing a spiral path and then ultimately fall into the nucleus. This shouldmake the atom collapse and hence unstable. But this is not so in reality.

Planck's quantum theory

At around the same time of the discovery of radio activity, Max Planck, in


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1900, presented the results of his famous black body radiation experiments,which showed that light has a dual character, behaving like a particle as wellas a wave. He gave the Quantum Theory of Radiation explainingelectromagnetic radiation and energy.

Its main features were:Radiant energy is emitted or absorbed discontinuously in the form of smallpackets of energy called 'quanta' (and not continuously as thought earlier).Each such quantum is associated with a definite amount of energy. In thecase of light, the quantum of energy is called 'photon'.

The amount of energy (e) associated with a quantum of radiation is

where 'h' is proportionality constant, universally referred to as Planck'sconstant. It has a fixed value of:

h = 6.63 x 1034 joule/sec or h = 3.99 x 10-13 kJ sec mol-1

The total amount of energy emitted or absorbed by a body will be somewhole number multiple of the quantum, i.e. E = nh , where n = 1, 2, 3, 4, ....

In other words, a body can emit or absorb energy equal to 1h , 2h , 3h etc.,and not as 1.6h , 2.4h , 3.2h etc.

From electromagnetic wave nature of light

this can be written as

where '' is the frequency, '' is the wavelength and 'c' is the velocity of thelight wave.

We know that

Thus, the energy associated with a quantum of radiation depends inverselyon its wavelength (or conversely with its frequency) i.e., higher thewavelength of radiation, lesser the energy associated with its quantum. Forexample, a photon of violet light will have more energy than that of a red


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light because the former has a lower wavelength. The concept of energypackets of light supports the corpuscular character.

Quantum theory of radiation

Albert Einstein used the Planck's quantum theory in 1905 to explain thephotoelectric effect. According to Einstein when a photon struck a metalsurface, a part of the total energy of the incident, called binding energy (theenergy that binds the electron to the nucleus), was utilized in ejectingelectrons from the metal atom. The remaining part of the total energy isgiven to the ejected electron in the form of kinetic energy.

As light is propagated in the form of energy packets called photons, it can betreated as particles of light. On the other hand as light also exhibits thephenomena of interference and diffraction, which indicates that light, alsohas wave like character. These facts suggest that light has dual characteri.e., particle as well as wave character.

Problem

5. Calculate the energy of a photon of light having frequency of 3.0 x 1015 s-1(Planck's constant h = 6.63 x 10-34 Js)

Solution

The energy of a photon is given by E = h

where the frequency of light = 3.0 x 1015 s-1

Planck's constant, h = 6.63 x 10-34 J s

E = (6.63 x 10-34 J s) x (3.0 x 101 s-1)

= 6.63 x 3 x 1019 J

Bohr's Atomic Model

In 1913, Neils Bohr proposed a model of an atom based on the Planck'squantum theory of radiation. The basic postulates of Bohr's theory are:

An atom consists of a small, positively charged nucleus heavily around whichelectrons revolve in definite circular paths called orbits.


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These orbits are associated with definite energies called energy shells/energylevels. They are designated as K, L, M, N, .... etc. shells or numbered as 1, 2, 3,4, ......etc. from the nucleus.

As long as the electron remains in a particular orbit /energy shell its energy

remains constant. This accounts for the stability of an atom.Only those orbits are permitted in which the angular momentum of the electronis a whole number multiple of h/ 2 where h is Planck's constant.

Any moving body taking a circular orbit has an angular momentum equal to theproduct of its mass (m), velocity of movement (v) and radius of orbit (r). Inother words the angular momentum of an electron (m x v x r) is a wholenumber multiple of h/ 2 .

Thus,

mvr = n * h/ 2 ; where n = 1,2,3,.i.e. for (n=1), h/ 2 is the electrons angular momentum

for (n=2), h/ is the electrons angular momentumfor (n=3), 3h/ 2 is the electrons angular momentum and so on.

This postulate introduces the concept of quantisation of angular momentum.

Electrons can jump from one energy level to another and when they do so, theyeither lose or absorb energy abruptly. For instance when an electron moves fromthe 'normal or ground state - E2' of an atom i.e., the state of lowest energy as

required by its 'n' and 'l' values, to a higher level, it causes the atom to be inwhat is called 'excited state-E2'. This is where electrons in an atom occupyenergy levels higher than those permitted by its 'n' and 'l' values. The reverse isalso true and the change in energy is E,

E = E2 - E1 = h

Energy changes in an electron jump


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Bohr's atomic model explained successfully:

The stability of an atom. Bohr postulated that as long an electron remains in aparticular orbit it does not emit radiation i.e. lose energy. Hence it does notbecome unstable.

Atomic spectra

A spectrum is an assembly of energy levels in the form of radiations emitted byan atom in its excited state. Every atom gives discontinuous line spectra. Eachline in the spectra corresponds to a specific wavelength and it is unique to agiven element. The atoms of two elements give same pattern of lines in theirspectra.

Atomic emission spectra

When a substance is heated to a high temperature, the atoms in the vapours getenergized. These energized atoms then return to the ground state by emittingelectromagnetic light radiations of certain definite wavelength. These radiationsappear as a series of bright lines separated from each other by dark spaceswhen made strike a photographic plate. This representation of emitted radiationis called atomic emission spectra.

Mechanism of emission spectra

Atomic absorption spectra

When the atomic vapours from a sample are placed in the path of white light

from an arc lamp, it absorbs the light of certain characteristic wavelengths andthe light of other wavelengths get transmitted. In such conditions a series ofdark lines on a white background are formed on the photographic plate. This iscalled an absorption spectrum.

The dark lines in the absorption spectrum and the bright lines in the emissionspectrum of a given element appear at the same wavelength.


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Comparison of Absorption and emission spectra of sodium vapour

Since each element gives a definite pattern of lines at certain definitefrequencies or wavelengths, the atomic spectra is used in chemical analysis toidentify and estimate the elements present in any sample.

Atomic spectra of hydrogen atom

Hydrogen is the simplest element with its atom having only one electron. Hence,the atomic spectrum of hydrogen has played a significant role in the

development of atomic structure. In the emission spectrum of hydrogen, whenan electric discharge is passed through hydrogen gas, the molecules of hydrogenbreak into atoms. The hydrogen atoms get energized and go into an excitedstate. The excited atoms then return to the ground state by emitting light.Hydrogen atoms emit a bluish light. On passing this light through a prism, adiscontinuous line spectrum consisting of several sharp lines is obtained. This isthe line spectrum of hydrogen.

Four sharp coloured lines were observed in the visible region of this spectrum byBalmer, in the ultra violet region by Lyman, and in the infrared region byPaschen, Brackett and Pfund. These series of lines are named after these

scientists who discovered them. Balmer expressed these lines in terms ofinverse of their wavelength ( ) by a mathematical relation, which was latermodified by Rydberg.

where 'RH' is the Rydberg's constant and 'n1', 'n2' are integers with values equalto or greater than 3 and ' ' is the wavelength.


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Line spectrum of hydrogen atom

The atomic spectrum of hydrogen: This was explained due to the concept ofdefinite energy levels. The one electron of hydrogen being closest to the nucleusis in its lowest energy shell (n =1) or normal ground state can absorb a definiteamount of energy and jump to a higher energy state. This excited state beingunstable, the electron comes back to a lower energy level. When the emitted

energy during transition, strikes a photographic plate, it gives its impression inthe form of a line. This difference is also the energy of photon expressed as E2 -E1 = h .

The frequency of the emitted radiation is:

Since E2 and E1 have only definite values and are characteristic of energy levelsof atoms, the values of '' will also be definite and characteristic of the atoms.

Thus each transition will produce a light of definite wavelength, which isobserved as a line in the spectrum.

For example, if the electron jumps down from the third to the first energy levelhaving energies E3 and E1 respectively, then the wavelength of the spectral linewould be

Similarly, when the electron jumps down from the fourth to the first energy levelhaving energies E4 and E1 respectively or from the fifth to the second i.e., E5 andE2, then we have

These will give different lines in the spectrum of the atom corresponding todifferent transitions having definite wavelengths.

Different spectral lines: Any sample of hydrogen gas contains a large number ofatoms and when energy is supplied, the electrons in different hydrogen atoms


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absorb different amounts of energies. These are raised to different energystates. For example, the electrons in some atoms may jump to second energylevel (L), while in others it may be to the third (M), fourth (N) or fifth (O) and soon. These electrons come back from the higher energy levels to the ground statein one or more jumps emitting different amount of energies.

Different routes to the ground state from n = 4

The different spectral lines depending upon the difference in energies of thelevels concerned can be summarized in the form of series named after thescientists who have discovered them.

Lyman series from n = 2, 3, 4, 5....to n = 1

Balmer series from n = 3, 4, 5, 6.....to n = 2

Paschen series from n = 4, 5, 6, 7.....to n = 3

Brackett series from n = 5, 6, 7, 8.....to n = 4

Pfund series from n = 6, 7, 8, 9.....to n = 5.

The energy of the electron in a particular orbit of hydrogen atom. This could becalculated by Bohr's theory. The energy of the electron in the 'nth' orbit hasbeen found to be

where 'm' is the mass and 'e' is the charge of the electron. The energyexpression for hydrogen like ions such as He, Li can be written as:

where 'Z' is the nuclear charge, which is equal to atomic number.


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Note :

The electronic energy is negative because when unlike charges are broughttogether energy is released by the system due to the force of attraction. Thuswhen electron moves closer to the nucleus its energy decreases and it becomes

less than zero i.e., negative.Although Bohr's model successfully explained the stability and the line spectrumof hydrogen, it still had its limitations. These were:

Limitations and problems

It could not explain the line spectrum of multi electron atoms.

This model failed to explain the effect of magnetic field on the spectra of atoms(Zeeman effect).

The effect of an electric field on the spectra could also not be explained byBohr's model (Stark effect).

The shapes of molecules arising out of directional bonding could not beexplained.

The dual nature of electrons (both as wave and particle) was not explained,further the path of motion of electrons in well defined orbits were not correct.

Problems

7. If the energy difference between the electronic states of hydrogen atom is214.68 kJ mol-1, what will be the frequency of light emitted when the electron

jumps from the higher to the lower energy state? (Planck's constant = 39.79 x10-14 kJ mol-1)

Solution

The frequency () of emitted light is related to the energy difference of twolevels ( E) as

E = 214.68 kJ mol-1, h = 39.79 x 10-14 kJ mol-1


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= 5.39 x 1014 s-1

8. The wavelength of first spectral line in the Balmer series is 6561 units.Calculate the wavelength of the second spectral line in Balmer series.

Solution

According to Rydberg equation:

For the first line in Balmer series, n1 = 2, n2 = 3

For the second line in Balmer series, n1 = 2, n2 = 4

Dividing equations (i) by (ii)

Quantum Mechanical Model of the Atom

Two new concepts came to substantially modify Bohr's Atomic model. These are:

1. Wave nature of materials objects

2. Heisenberg's uncertainty principle

Wave nature of material objects

In 1924, de Broglie's suggested that all material objects including an electronhave a dual character; they behave as particles as well as waves. The


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wavelength associated with a particle of mass 'm', moving with velocity 'v' isgiven by de Broglie's relation as:

Note :

This discovery of the wave like character of the electron helped in the making ofthe modern electron microscope.

Heisenberg's uncertainity principle

Heisenberg, in 1927 pointed out that it is not possible to measuresimultaneously the momentum (or velocity) as well as the position of amicroscopic particle with absolute accuracy. Mathematically, this may beexpressed as:

where, x = uncertainity in position

p = uncertainity in momentum

The constant on the right side of the equation (the product of the twouncertainties) tells us that the two uncertainties are inversely related. If themomentum of the particle is measured with more accuracy there is a largeruncertainity in its position and vice versa.

Uncertainity is not due to the lack of refined techniques available formeasurement but observes on microscopic bodies cannot be made withoutdisturbing them. Observations made as result of the impact of light suffer achange in the position or velocity of these microscopic objects. This does nothold good for large objects of daily life, as the changes that occur are negligible.

Probability picture of an electron

According to Heisenberg's uncertainty principle, it is impossible to describe the

exact position of an electron at a given moment in terms of position. We canspeak of most probable regions where the probability of finding an electron inthe space around the nucleus of an atom is high. The electron does not alwaysremain at a fixed distance from the nucleus. It keeps moving in the whole spacearound the nucleus but tends to remain most of the time within a small volumearound the nucleus, where the probability of locating the electron is maximum.Hence a new atomic model, was needed to explain the


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Wave nature (dual character) of atoms

The idea of uncertainity in the position of electrons in an atom

The concept of fixed energy states

In view of these facts, Schrodinger put forward the wave model or quantummechanical model of atomic structure. In this model, the behavior of an electronis defined by the mathematical equation:

where, = (psi) is a wave function of space coordinates 'x', 'y', 'z' and representsthe amplitude of the electron wave.

m = mass of the electron

E = the total permissible energy level, which the electron can have.

V = potential energy of the electron given by ze2/r.

h = Planck's constant having the value 6.626 x 10-34 J s. = (delta) stands forinfinitesimal change.

The wave length function (psi) describes a number of possible states of anelectron in an atom. Since a large number of solutions are possible, only

meaningful permissible values of energy and location of the electron withrespect to the nucleus are considered. Four important solutions in the name of"quantum numbers describe the position of the electron accurately.

Atomic orbitals

Electrons cannot exist at a particular point or in a well-defined orbit (path),according to the above new approach of wave mechanics. This led to theconcept of 'most probable regions' in space around the nucleus. Theseregions are called atomic orbitals, where the probability (chances) of findingthe electron is maximum (90%-95%)'.

An atomic orbital differs from an orbit (Bohr's orbit) in the following ways.


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Quantum Numbers

Orbitals of electrons in atoms differ in size, shape and orientation. Definiteenergies and angular movements characterize atomic orbitals. The state ofan electron in any atom is defined by certain permissible values of energyand angular momentum, which describe its location with respect to itsnucleus and its energy level. These permissible states are called orbitals andare expressed by a set of four numbers 'n', 'l', 'm' and 's' called quantumnumbers. These numbers serve as the signature of the electrons, uniquelydescribing its position in the atom. The 'n', 'l' and 'm' indicate the spatialdistribution while 's' indicates the spin orientation of the electrons.

Principal quantum number

The principal quantum number gives the average distance of the electronfrom the nucleus and the energy associated with it, i.e., it determines themain energy shell or energy level in which the electron is present.

It is denoted by the letter 'n' that can take whole number values startingfrom 1, 2, 3, 4,..... The shell with n = 1 is called first shell or 'K' shell. The


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shell with n = 2 is the 'L' shell and so on. The first shell is closest to thenucleus. As the value of 'n' increases, the distance from the nucleus as wellas the energy of the electrons increases.

Azimuthal quantum number or angular quantum number

The Azimuthal quantum number determines the angular momentum of theelectron, denoted by the letter 'l'. The value of 'l' gives the sub level or subshell in a given principal energy shell to which the electron belongs. It canhave only positive integral values from zero to (n-1) where 'n' is the principalquantum number. The various sub shell values of 'l' are also designated bythe letters s, p, d, f,... For any main energy level, the energies of the sub-shell follow the order s > p > d > f.

The different sub-shells are represented by, first writing the value of 'n' andthen the letter designated for the value of 'l'.

To illustrate,

n = 1 l = 0 one sub shell 1s

n = 2 l = 0,1 two sub shells 2s, 2p

n = 3 l = 0,1,2 three sub shells 3s, 3p, 3d

n = 4 l = 0,1,2,3 four sub shells 4s, 4p, 4d, 4f

Thus for each value of 'n' there are 'n' values of 'l'.

The value of azimuthal quantum number gives the shape of the sub shell ororbital. So it is also called as orbital quantum number.

Magnetic quantum number

The magnetic quantum number denoted by the letter 'm' describes thebehaviour of electron in a magnetic field. In the absence of an externalmagnetic field, electrons orbitals having same values of 'n' and 'l' but

different values of 'm' have the same energies. They are called degenerateorbitals. However, in the presence of an external magnetic field the orbitalsvary in their energies slightly. This happens because the preferred orientationof the orbital in space is a result of interaction of its own magnetic field withthat of the external magnetic field.

The magnetic quantum number 'm', depend on 'l' for its values. Thisquantum number can have all integral values from '-l' to '+l' including 0.


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Thus for a given 'l' value there are (2l + 1) values of 'm'. Two orbitals in thesame shell can have identical 'n' and 'l' values but they must have differentfixed values of 'm'.

The number of orbitals in each sub shell are given below:

s sub shell l = 0 m = 0 only one orientation one orbital

p sub shell l = 1 m = +1,0, -1 three orientations three orbitals

d sub shell l = 2 m = +2,+1,0,-1,-2 five orientations five orbitals

Spin quantum numbers

The orientation of spin of an electron is designated by its spin quantumnumber 's'. The spin orientation is an intrinsic characteristic of the electronconnected more with its magnetic behaviour rather than rotation of anelectron about its own axis. This number can have only two valuescorresponding to clockwise and anticlockwise spins i.e., + and -. Theclockwise spin is represented by an arrow pointing upwards (h). The anticlockwise spin is represented by an arrow pointing downwards (i). Eachorbital can accommodate a maximum of two electrons provided they haveopposite spins.

Number of sub shells and orbitals in the K, L and M shellsProblem

9.(a) What are the permissible values for 'l' and 'm' when 'n' = 3?

(b) Which orbital is specified by 'l' = 2 and 'n' = 3?


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Solution

(a) For n = 3, the permissible values for 'l' and 'm' are: 'l' = 0, 1, 2

For 'l' = 0 m = 0 (s-orbital)

For 'l' = 1 m = +1, 0, -1 (p-orbital)

For 'l' = 2 m = +2, +1, 0, -1, -2 (d-orbital)

(b) For 'n' = 3 and 'l' = 2:

'l' = 2 means 'd' orbitals

The given orbital is '3d'.

Shapes of Atomic Orbitals

An atomic orbital is the space around the nucleus in which the probability offinding the electron is maximum. These most probable regions can bediagrammatically represented by cloud density or dot diagrams. The densityof dots (or lack of them) in any region of the cloud diagram, indicates thedegree of probability of finding the electron in that region. It is not alwaysconvenient to draw dot diagrams of orbitals, since the probability of findingan electron decreases with distance (but does not become zero), thus not

giving it any definite shape. Drawing boundary surfaces, which enclose 95-99% of the probability of locating an electron, is the method generally usedto show the shape of an orbital.

As definite energies and angular momentums characterize atomic orbitals,the permissible values of these parameters are expressed in terms ofquantum numbers. The azimuthul quantum 'l', is related to orbital angularmomentum of the electron in terms of the quantity given below:

Thus different values of 'l' correspond to different orbital angularmomentums. As a result the azimuthul quantum 'l' determines the shape oforbitals.

Shape of 's' orbitals

The orbital angular momentum of a 's' orbital is zero, as 'l' = 0. This means


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that the probability of finding an electron at a particular distance from thenucleus is the same in all directions or at all angles. Since the distribution ofelectron density is symmetrical, the shape representing the 's' orbital is asphere.

It is to be noted that firstly the total number of concentric spheres at anygiven main energy level in an 's' orbital equals the principle quantum numberof that level. Thus for example '1s' orbital consists of only one sphere while a'3s' orbital consists of three concentric spheres. Secondly, as the value of theprincipal quantum number 'n' increases, the 's' orbital becomes larger andthe energy of the 's' orbital increases, while retaining the sphericalsymmetry. The energies of the various 's' orbitals follow the order 1s < 2s

structure of atom.pdf

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