Matriculation Physics Bohrs Model of Hydrogen Atom.pdf

8/12/2019 Matriculation Physics Bohrs Model of Hydrogen Atom.pdf

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1

PHYSICS CHAPTER 11

CHAPTER 11:CHAPTER 11:

Bohrs model of hydrogen atomBohrs model of hydrogen atom(3 Hors!(3 Hors!


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PHYSICS CHAPTER 11

2

At the end of this chapter, students should be able to:At the end of this chapter, students should be able to:

ExplainExplainBohrs postulates of hydrogen atom.Bohrs postulates of hydrogen atom.

Learning Outcome:

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&(s11#1 Bohrs atom& model (1hor!


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PHYSICS CHAPTER 11

4

"utherfords model of atom"utherfords model of atom

In 1*11+ Ernest Rtherford %erformed a r&t&al e%er&ment that

sho"ed the Thomsons model &s not orret and %ro%osed h&sne" atom& model $no"n as Rtherfords %lanetary model of the

atom as sho"n &n /&gre 11#2a#

Aord&ng to Rtherfords model+ the atom "as %&tred as

eletrons or.&t&ng arond a entral nles "h&h onentrated

of %os&t&-e harge#

The eletrons are aelerat&ng .ease the&r d&ret&ons are

onstantly hang&ng as they &rle the nles#

nles

eletron

!igure 11.#a!igure 11.#a


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PHYSICS CHAPTER 11

5

Based on the "a-e theory+ an aelerat&ng harge em&ts energy#

Hene the eletrons mst em&t the E rad&at&on as they re-ol-e

arond the nles# As a reslt of the ont&nos loss of energy+ the rad&& of the

eletron or.&ts "&ll .e dereased stead&ly#

Th&s "old lead the eletrons s%&ral and falls &nto the nles+

hene the atom "old olla%se as sho"n &n /&gre 11#2.#

!igure 11.#b!igure 11.#b

++Ze ee

0%lo%

energy loss


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PHYSICS CHAPTER 11

6

+e

ev

r

eF

In 1*13+ 4e&ls Bohr %ro%osed a ne" atom& model .ased onhydrogen atom#

Aord&ng to Bohrs odel+ he assmes that each electroneach electron

mo$es in a circular orbit %hich is centred on the nucleusmo$es in a circular orbit %hich is centred on the nucleus+

the neessary centripetal force being pro$ided by thecentripetal force being pro$ided by the

electrostatic force of attraction bet%een the positi$elyelectrostatic force of attraction bet%een the positi$elycharged nucleus and the negati$ely charged electroncharged nucleus and the negati$ely charged electronas

sho"n &n /&gre 11#3#

11.1.# Bohrs model of hydrogen atom

!igure 11.&!igure 11.&


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PHYSICS CHAPTER 11

7

5n th&s .as&s he "as a.le to sho" that the energy of anenergy of an

orbiting electron depends on the radius of its orbitorbiting electron depends on the radius of its orbit#

Th&s model has se-eral featres "h&h are desr&.ed .y the%ostlates (assm%t&ons! stated .elo" :

1# The electronselectronsmo-e only &n erta&n &rlar or.&ts+ alled

'A(O)A"* 'AE''A(O)A"* 'AE'or E)E"+* LEEL'E)E"+* LEEL'# 6hen &t &s

&n one of these or.&ts+ &t does not radiate energydoes not radiate energy#

2# The only permissible orbits arepermissible orbits arethose &n the d&srete setfor "h&h the angular momentum of the electronangular momentum of the electronLL

e-uals an integer timese-uals an integer times h/h/22# athemat&ally+

2

nhL=

2

nhmvr= 11.1/11.1/

and mvrL=

"here

orbittheofradius:r electrontheofmass:m

,...,,n 321numberquantumprincipal: =


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PHYSICS CHAPTER 11

8

3# Emission or absorptionEmission or absorptionof rad&at&on ors only "hen an

electron ma0es a transition from one orbit to anotherelectron ma0es a transition from one orbit to another#

The fre7enyfof the em&tted (a.sor.ed! rad&at&on &sg&-en .y

if EEhfE == 11.#/11.#/

"here

constantsPlanck':hstateenergyfinal:fE

energyofchange:E

stateenergyinitial:iE)ote:)ote:

IfEf>E

i

IfEf


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PHYSICS CHAPTER 11

9

At the end of this chapter, students should be able to:At the end of this chapter, students should be able to: eri$eeri$eBohrs radius and energy le$el in hydrogen atom.Bohrs radius and energy le$el in hydrogen atom. 2se2se

efineefine

ground state energy, excitation energy andground state energy, excitation energy and

ionisation energy.ionisation energy.

Learning Outcome:

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&(s

11#2 Energy le-el of hydrogen atom (1hor!

==

22

2

2

! mkehnanrn

andand

=

2

2 1

2 na

keEn


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PHYSICS CHAPTER 11

10

11.#.1 Bohrs radius in hydrogen atom

Cons&der one eletron of hargeeand mass mmo-es &n a

&rlar or.&t of rad&s rarond a %os&t&-ely harged nles

"&th a -elo&ty vas sho"n &n /&gre 11#3# The electrostatic force bet%een electron and nucleuselectrostatic force bet%een electron and nucleus

contributes the centripetal forcecontributes the centripetal forceas "r&te &n the relat&on .elo":

11.# Energy le$el of hydrogen atom

ce FF = entr&%etal foreeletrostat& fore

r

mv

r

QQ 2

2

21

!

1

=

and eQQ == 21

r

emv

22

! = 11.&/11.&/


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PHYSICS CHAPTER 11

11

/rom the Bohrs seond %ostlate:

By ta$&ng s7are of .oth s&de of the e7at&on+ "e get

By d&-&d&ng the e7s# (11#8! and (11#3!+ ths

2

nh

mvr=

11.3/11.3/2

22222

!

hnrvm =

=

re

hn

mv

rvm

2

2

22

2

222

!

!

2

22

me

hnr= and

k

!

1=

electrostaticelectrostatic

constantconstant


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PHYSICS CHAPTER 11

12

"h&h rn&s radii of the permissible orbitsradii of the permissible orbitsfor the Bohrs atom# E7# (11#9! an also .e "r&tten as

"here a&s alled the Bohrs radiusBohrs radiusof hydrogen atom#

=kme

hnr

!

12

22

11.4/11.4/"""3#2#1$! 22

22 =

= n

mke

hnrn

2anrn=

22

2

! mke

ha

=

11.5/11.5/

and


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PHYSICS CHAPTER 11

13

The Bohrs rad&s &s def&ned as the radius of the most stablethe radius of the most stable

lo%est/ orbit or ground state lo%est/ orbit or ground state nn=1=1//in the hydrogen atomin the hydrogen atom

and &ts -ale &s

n&t on-ers&on:

The rad&& of the or.&ts asso&ated "&th allo"ed or.&ts or states

n % 2#3#&are !a,'a,+ ths the orbits radii areorbits radii are-uanti6ed-uanti6ed#

( )( )( )( )21''312

23!

1("11"'111"'!

1(3"(

=

a

m131") 11 =a 5R ")31 * (angstrom!

1 * % 1" 11 m


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PHYSICS CHAPTER 11

14

&s def&ned as a fixed energy corresponding to the orbits ina fixed energy corresponding to the orbits in%hich its electrons mo$e around the nucleus%hich its electrons mo$e around the nucleus#

The energy le-els of atoms are -uanti6ed-uanti6ed#

The total energy le$eltotal energy le$elEEof the hydrogen atomof the hydrogen atom&s g&-en .y

7otential energy7otential energy UUof the electronof the electron &s g&-en .y

11.#.# Energy le$el in hydrogen atom

KUE += 11.8/11.8/9inetic energy of the electron9inetic energy of the electron

7otential energy of the electron7otential energy of the electron

r

QkQ

U21

= eQeQ == 21 $"here 2

anr=and

2

2

an

keU = 11./11./

nucleusnucleus electronelectron


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PHYSICS CHAPTER 11

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9inetic energy9inetic energyKKof the electronof the electron &s g&-en .y

Therefore the e7# (11#;! an .e "r&tten as

2

2

1mvK=

11.;/11.;/

.t r

emv

22

! =

=

r

eK

2

!2

1

"here k=

!

1

=

2

2

2

1

an

keK

+=

2

2

2

2

21

anke

ankeEn

and 2anr=

=2

2 1

2 na

keEn 11.1


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PHYSICS CHAPTER 11

16

In general+ the total energy le-elEfor the atom&s

s&ng nmer&al -ale of k+ eand a+ ths the e7# (11#1

"r&tten as

= 2

2

2

2 n

Z

a

ke

En 11.11/11.11/

( )( )( )

=

211

21'' 1

131")2

1("11"'

nEn

=

21'

1+ 1e,

1("1

11-"2

n

1#2#3#"""e,$("132

== nn

En 11.1#/11.1#/

)ote:)ote:

E-s. 11.1


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PHYSICS CHAPTER 11

17

The negati$e signnegati$e sign&n the e7# (11#12! &nd&ates that %or0 has to%or0 has to

be done to remo$e the electron from the bound of the atombe done to remo$e the electron from the bound of the atom

to infinityto infinity+ "here &t &s ons&dered to ha-e 6ero energy6ero energy# The energy le-els of the hydrogen atom are "hen

n=1+ the ground stateground state(the state of the lo%est energy le$ello%est energy le$el! >

n=2+ the first excited statefirst excited state>

n=3+ the second excited statesecond excited state>

n=8+ the third excited statethird excited state>

n=+ the energy le-el &s

( )

e,(13e,

1

("1321 .E ==

( )e,!3e,

2

("1322 .E ==

( )

e,("132

=

=E

( )e,)11e,

3

("1323 .E ==

( )e,+)e,

!

("132! .E ==

electron is completelyelectron is completely

remo$ed from the atomremo$ed from the atom#


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PHYSICS CHAPTER 11

19

The eletron &n the hydrogen atom ma$es a trans&t&on from the

energy state of


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PHYSICS CHAPTER 11

20

The lo"est energy state for hydrogen atom &s 13# e@# eterm&ne

the fre7eny of the %hoton re7&red to &on&De the atom#

(?&-en the s%eed of l&ght &n the -am+ c=3#


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PHYSICS CHAPTER 11

21

/or an eletron &n a hydrogen atom harater&Ded .y the %r&n&%al

7antm nm.er n=2+ allatea# the or.&tal rad&s+.# the s%eed+

# the $&net& energy#

(?&-en c=3#


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PHYSICS CHAPTER 11

22

'olution :'olution :

.# By a%%ly&ng the Bohrs 2nd%ostlate+ ths

# The $&net& energy of the or.&t&ng eletron &s g&-en .y

( ) ( )

3!131 1(3"(112"2111" = v

1( sm1"1 =v

2nhmvrn=

2=n

2

22

hmvr =

2

21 mvK=

( )( )2(31 1"1111"2

1=

1!1")1

=K


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PHYSICS CHAPTER 11

23

A hydrogen atom em&ts rad&at&on of "a-elengths 221#9 nm and

2


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PHYSICS CHAPTER 11

24

'olution :'olution :

a# The energy of the %hoton de to trans&t&on from 2nde&ted state

to the grond state &s

.#

Therefore the "a-elength of the em&tted %hoton de to the

trans&t&on from 2nde&ted state to the 1ste&ted state &s

m1!"22m$1)"221 2

1 ==

( )( )

+3!

21!"22

1"31(3"(

=E

1+3" 12=E

4E1 4E2

4E3

?rond state

1st e&ted state

2nd e&ted state

123 EEE =11

3 1+"+1+3" =E

1)"+2

3

=E

3

3

hcE =

( )( )

3

+3!2 1"31(3"(1)"+

=

m13!"2 (3 =


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PHYSICS CHAPTER 11

25

At the end of this chapter, students should be able to:At the end of this chapter, students should be able to:

ExplainExplainthe emission of line spectrum by using energythe emission of line spectrum by using energy

le$el diagram.le$el diagram.

'tate'tatethe line series of hydrogen spectrum.the line series of hydrogen spectrum. 2se2seformula,formula,

Learning Outcome:

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11#3 &ne s%etrm (1hor!

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1


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PHYSICS CHAPTER 11

26

The em&ss&on l&nes orres%ond to the %hotons of d&sreteenerg&es that are em&tted "hen e&ted atom& states &n the gasma$e trans&t&ons .a$ to lo"er energy le-els#

/&gre 11#9 sho"s l&ne s%etra %roded .y em&ss&on &n the-&s&.le range for hydrogen (H!+ merry (Hg! and neon (4e!#

11.& Line spectrum

!igure 11.4!igure 11.4


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PHYSICS CHAPTER 11

27

Em&ss&on %roesses &n hydrogen g&-e r&se to ser&es+ "h&h are

se7enes of l&nes orres%ond&ng to atom& trans&t&ons# The ser&es &n the hydrogen em&ss&on l&ne s%etrm are

Lyman seriesLyman series&n-ol-eselectron transitionselectron transitions thatend at theend at the

ground state of hydrogen atomground state of hydrogen atom# It &s &n the ltra$ioletltra$iolet

2/ range2/ range#

Balmer seriesBalmer series&n-ol-eselectron transitionselectron transitions that end atend atthe 1the 1ststexcited state of hydrogen atomexcited state of hydrogen atom# It &s &n the $isible$isible

light rangelight range#

7aschen series7aschen series&n-ol-eselectron transitionselectron transitions thatend atend at

the #the #ndndexcited state of hydrogen atomexcited state of hydrogen atom# It &s &n the infraredinfrared

("/ range("/ range# Brac0ett seriesBrac0ett series&n-ol-eselectron transitionselectron transitions that end atend at

the &the &rdrdexcited state of hydrogen atomexcited state of hydrogen atom# It &s &n the ("("

rangerange#

7fund series7fund series &n-ol-eselectron transitionselectron transitions that end at theend at the

33thth

excited state of hydrogen atomexcited state of hydrogen atom# It &s &n the (" range(" range#

11.&.1 =ydrogen emission line spectrum


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PHYSICS CHAPTER 11

29

/&gre 11#; sho"s F%erm&ttedG or.&ts of an eletron &n the Bohrmodel of a hydrogen atom#

!igure 11.5: not to scale!igure 11.5: not to scale

P&tre 11#1
http://opt/scribd/conversion/tmp/scratch2594/G:/P&P/Semester1/Latest/p11_01_01_01a.swfhttp://opt/scribd/conversion/tmp/scratch2594/G:/P&P/Semester1/Latest/p11_01_01_01a.swfhttp://balmer_series.jpg/


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PHYSICS CHAPTER 11

30

If an eletron ma$es a trans&t&on from an oter or.&t of le-el nito

an &nner or.&t of le-el nf+ ths the energy &s rad&ated#

The energy radiatedenergy radiated&n form of E> radiation photon/form of E> radiation photon/"here

the "a-elength &s g&-en .y

/rom the Bohrs 3rd%ostlate+ the e7# (11#13! an .e "r&tten as

11.&.# ?a$elength of hydrogen emission line

spectrum

hcE=hcE=

1 11.1&/11.1&/

if11 nn EEhc ="here

= 2

f

2

12f nakeEn

and

=

2

i

2 1

2i na

keEn


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PHYSICS CHAPTER 11

31

=

2

i

2

2

f

2 1

2

1

2

11

na

ke

na

ke

hc

=

2

i

2

f

2 11

2

1

nna

ke

hc

=

2

i

2

f

2 11

2 nnhca

keand HR

hca

ke=

2

2

= 2i2

f

111

nnRH 11.13/11.13/

"here1- m1-"1constants5ydberd': =HR

nn of0aluefinal:f

nn of0alueinitial:i


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PHYSICS CHAPTER 11

32

)ote:)ote:

/or the hydrogen l&ne s%etrm+

LymanLyman ser&es(nnff=1=1!

BalmerBalmer ser&es(nnff=2=2!

7aschen7aschen ser&es(nnff=3=3!

Brac0ettBrac0ett ser&es(nnff=4=4!

7fund7fund ser&es(nnff=5=5!

To allate the shortest %a$elength in any seriesshortest %a$elength in any series+ ta$e nnii== #

= 2i21

1

11

nR

H

=

2

i

2

1

2

11

nRH

=

2

i

21

311

nRH

=

2

i

2

1

!

11

nRH

=

2

i

2

1

)

11

nRH


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PHYSICS CHAPTER 11

33

The Bohrs model of hydrogen atom

%red&ts sessflly the energy le-els of the hydrogen atom.t fails to explain the energy le$els of more complexfails to explain the energy le$els of more complexatomsatoms#

an e%la&n the s%etrm for hydrogen atom .t somedeta&ls of the s%etrm cannot be explained especiallycannot be explained especially%hen the atom is placed in a magnetic field%hen the atom is placed in a magnetic field#

annot e%la&n the eeman effet (/&gre 11#;!# @eeman effect@eeman effect&s def&ned as the splitting of spectralthe splitting of spectral

lines %hen the radiating atoms are placed in alines %hen the radiating atoms are placed in amagnetic field.magnetic field.

11.&.& Limitation of Bohrs model of hydrogen atom

agnet& f&eld

Trans&t&ons

4o magnet& f&eld

11

##Energy e-els

S%etra!igure 11.8!igure 11.8


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PHYSICS CHAPTER 11

34

The Balmer ser&es for the hydrogen atom orres%onds to eletron&

trans&t&ons that term&nate at energy le-el n=2 as sho"n &n the/&gre 11#)#

Callatea# the longest "a-elength+ and

.# the shortest "a-elength of the %hoton em&tted &n th&s ser&es#

(?&-en the s%eed of l&ght &n the -am c=3#


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PHYSICS CHAPTER 11


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PHYSICS CHAPTER 11

37

eterm&ne the "a-elength for a l&ne s%etrm &n yman ser&es

"hen the eletron ma$es a trans&t&on from n=3 le-el#

(?&-en Ryd.ergs onstant +RH= 1#


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PHYSICS CHAPTER 11

38

Eer&se 11#1 :?&-en c=3#


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PHYSICS CHAPTER 11

Next ChapterCHAPTER 12 :

JKrays

Matriculation Physics Bohrs Model of Hydrogen Atom.pdf

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