8/12/2019 Matriculation Physics Bohrs Model of Hydrogen Atom.pdf
1/39
1
PHYSICS CHAPTER 11
CHAPTER 11:CHAPTER 11:
Bohrs model of hydrogen atomBohrs model of hydrogen atom(3 Hors!(3 Hors!
8/12/2019 Matriculation Physics Bohrs Model of Hydrogen Atom.pdf
2/39
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:
"""
#$m%
h#m
atr
&$#e
d
#my
'%h
ys
&(s
"""
#$m%
h#m
atr
&$#e
d
#my
'%h
ys
&(s11#1 Bohrs atom& model (1hor!
8/12/2019 Matriculation Physics Bohrs Model of Hydrogen Atom.pdf
3/39
8/12/2019 Matriculation Physics Bohrs Model of Hydrogen Atom.pdf
4/39
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 $no"n as Rtherfords %lanetary model of the
atom as sho"n &n /&gre 11#2a#
Aord&ng to Rtherfords model+ the atom "as %&tred as
eletrons or.&t&ng arond a entral nles "h&h onentrated
of %os&t&-e harge#
The eletrons are aelerat&ng .ease the&r d&ret&ons are
onstantly hang&ng as they &rle the nles#
nles
eletron
!igure 11.#a!igure 11.#a
8/12/2019 Matriculation Physics Bohrs Model of Hydrogen Atom.pdf
5/39
PHYSICS CHAPTER 11
5
Based on the "a-e theory+ an aelerat&ng harge em&ts energy#
Hene the eletrons mst em&t the E rad&at&on as they re-ol-e
arond the nles# As a reslt of the ont&nos loss of energy+ the rad&& of the
eletron or.&ts "&ll .e dereased stead&ly#
Th&s "old lead the eletrons s%&ral and falls &nto the nles+
hene the atom "old olla%se as sho"n &n /&gre 11#2.#
!igure 11.#b!igure 11.#b
++Ze ee
0%lo%
energy loss
8/12/2019 Matriculation Physics Bohrs Model of Hydrogen Atom.pdf
6/39
PHYSICS CHAPTER 11
6
+e
ev
r
eF
In 1*13+ 4e&ls Bohr %ro%osed a ne" atom& model .ased onhydrogen atom#
Aord&ng to Bohrs odel+ he assmes that each electroneach electron
mo$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$ely charged electronas
sho"n &n /&gre 11#3#
11.1.# Bohrs model of hydrogen atom
!igure 11.&!igure 11.&
8/12/2019 Matriculation Physics Bohrs Model of Hydrogen Atom.pdf
7/39
PHYSICS CHAPTER 11
7
5n th&s .as&s he "as a.le to sho" that the energy of anenergy of an
orbiting electron depends on the radius of its orbitorbiting electron depends on the radius of its orbit#
Th&s model has se-eral featres "h&h are desr&.ed .y the%ostlates (assm%t&ons! stated .elo" :
1# The electronselectronsmo-e only &n erta&n &rlar or.&ts+ alled
'A(O)A"* 'AE''A(O)A"* 'AE'or E)E"+* LEEL'E)E"+* LEEL'# 6hen &t &s
&n one of these or.&ts+ &t does not radiate energydoes not radiate energy#
2# The only permissible orbits arepermissible orbits arethose &n the d&srete setfor "h&h the angular momentum of the electronangular momentum of the electronLL
e-uals an integer timese-uals an integer times h/h/22# athemat&ally+
2
nhL=
2
nhmvr= 11.1/11.1/
and mvrL=
"here
orbittheofradius:r electrontheofmass:m
,...,,n 321numberquantumprincipal: =
8/12/2019 Matriculation Physics Bohrs Model of Hydrogen Atom.pdf
8/39
PHYSICS CHAPTER 11
8
3# Emission or absorptionEmission or absorptionof rad&at&on ors only "hen an
electron ma0es a transition from one orbit to anotherelectron ma0es a transition from one orbit to another#
The fre7enyfof the em&tted (a.sor.ed! rad&at&on &sg&-en .y
if EEhfE == 11.#/11.#/
"here
constantsPlanck':hstateenergyfinal:fE
energyofchange:E
stateenergyinitial:iE)ote:)ote:
IfEf>E
i
IfEf
8/12/2019 Matriculation Physics Bohrs Model of Hydrogen Atom.pdf
9/39
PHYSICS CHAPTER 11
9
At the end of this chapter, students should be able to:At the end of this chapter, students should be able to: eri$eeri$eBohrs radius and energy le$el in hydrogen atom.Bohrs radius and energy le$el in hydrogen atom. 2se2se
efineefine
ground state energy, excitation energy andground state energy, excitation energy and
ionisation energy.ionisation energy.
Learning Outcome:
"""
#$m%
h#m
atr
&$#e
d
#my
'%h
ys
&(s
"""
#$m%
h#m
atr
&$#e
d
#my
'%h
ys
&(s
11#2 Energy le-el of hydrogen atom (1hor!
==
22
2
2
! mkehnanrn
andand
=
2
2 1
2 na
keEn
8/12/2019 Matriculation Physics Bohrs Model of Hydrogen Atom.pdf
10/39
PHYSICS CHAPTER 11
10
11.#.1 Bohrs radius in hydrogen atom
Cons&der one eletron of hargeeand mass mmo-es &n a
&rlar or.&t of rad&s rarond a %os&t&-ely harged nles
"&th a -elo&ty vas sho"n &n /&gre 11#3# The electrostatic force bet%een electron and nucleuselectrostatic force bet%een electron and nucleus
contributes the centripetal forcecontributes the centripetal forceas "r&te &n the relat&on .elo":
11.# Energy le$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.&/
8/12/2019 Matriculation Physics Bohrs Model of Hydrogen Atom.pdf
11/39
PHYSICS CHAPTER 11
11
/rom the Bohrs seond %ostlate:
By ta$&ng s7are of .oth s&de of the e7at&on+ "e get
By d&-&d&ng the e7s# (11#8! and (11#3!+ ths
2
nh
mvr=
11.3/11.3/2
22222
!
hnrvm =
=
re
hn
mv
rvm
2
2
22
2
222
!
!
2
22
me
hnr= and
k
!
1=
electrostaticelectrostatic
constantconstant
8/12/2019 Matriculation Physics Bohrs Model of Hydrogen Atom.pdf
12/39
PHYSICS CHAPTER 11
12
"h&h rn&s radii of the permissible orbitsradii of the permissible orbitsfor the Bohrs atom# E7# (11#9! an also .e "r&tten as
"here a&s alled the Bohrs radiusBohrs radiusof hydrogen atom#
=kme
hnr
!
12
22
11.4/11.4/"""3#2#1$! 22
22 =
= n
mke
hnrn
2anrn=
22
2
! mke
ha
=
11.5/11.5/
and
8/12/2019 Matriculation Physics Bohrs Model of Hydrogen Atom.pdf
13/39
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
8/12/2019 Matriculation Physics Bohrs Model of Hydrogen Atom.pdf
14/39
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 $"here 2
anr=and
2
2
an
keU = 11./11./
nucleusnucleus electronelectron
8/12/2019 Matriculation Physics Bohrs Model of Hydrogen Atom.pdf
15/39
PHYSICS CHAPTER 11
15
9inetic energy9inetic energyKKof the electronof the electron &s g&-en .y
Therefore the e7# (11#;! an .e "r&tten as
2
2
1mvK=
11.;/11.;/
.t r
emv
22
! =
=
r
eK
2
!2
1
"here k=
!
1
=
2
2
2
1
an
keK
+=
2
2
2
2
21
anke
ankeEn
and 2anr=
=2
2 1
2 na
keEn 11.1
8/12/2019 Matriculation Physics Bohrs Model of Hydrogen Atom.pdf
16/39
PHYSICS CHAPTER 11
16
In general+ the total energy le-elEfor the atom&s
s&ng nmer&al -ale of k+ eand a+ ths the e7# (11#1
"r&tten as
= 2
2
2
2 n
Z
a
ke
En 11.11/11.11/
( )( )( )
=
211
21'' 1
131")2
1("11"'
nEn
=
21'
1+ 1e,
1("1
11-"2
n
1#2#3#"""e,$("132
== nn
En 11.1#/11.1#/
)ote:)ote:
E-s. 11.1
8/12/2019 Matriculation Physics Bohrs Model of Hydrogen Atom.pdf
17/39
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#
8/12/2019 Matriculation Physics Bohrs Model of Hydrogen Atom.pdf
18/39
8/12/2019 Matriculation Physics Bohrs Model of Hydrogen Atom.pdf
19/39
PHYSICS CHAPTER 11
19
The eletron &n the hydrogen atom ma$es a trans&t&on from the
energy state of
8/12/2019 Matriculation Physics Bohrs Model of Hydrogen Atom.pdf
20/39
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#
8/12/2019 Matriculation Physics Bohrs Model of Hydrogen Atom.pdf
21/39
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#
8/12/2019 Matriculation Physics Bohrs Model of Hydrogen Atom.pdf
22/39
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
8/12/2019 Matriculation Physics Bohrs Model of Hydrogen Atom.pdf
23/39
PHYSICS CHAPTER 11
23
A hydrogen atom em&ts rad&at&on of "a-elengths 221#9 nm and
2
8/12/2019 Matriculation Physics Bohrs Model of Hydrogen Atom.pdf
24/39
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$1)"221 2
1 ==
( )( )
+3!
21!"22
1"31(3"(
=E
1+3" 12=E
4E1 4E2
4E3
?rond state
1st e&ted state
2nd e&ted state
123 EEE =11
3 1+"+1+3" =E
1)"+2
3
=E
3
3
hcE =
( )( )
3
+3!2 1"31(3"(1)"+
=
m13!"2 (3 =
8/12/2019 Matriculation Physics Bohrs Model of Hydrogen Atom.pdf
25/39
PHYSICS CHAPTER 11
25
At the end of this chapter, students should be able to:At the end of this chapter, students should be able to:
ExplainExplainthe emission of line spectrum by using energythe emission of line spectrum by using energy
le$el diagram.le$el diagram.
'tate'tatethe line series of hydrogen spectrum.the line series of hydrogen spectrum. 2se2seformula,formula,
Learning Outcome:
"""
#$m%
h#m
atr
&$#e
d
#my
'%h
ys&(
s
"""
#$m%
h#m
atr
&$#e
d
#my
'%h
ys&(
s
11#3 &ne s%etrm (1hor!
hc
E=
1
8/12/2019 Matriculation Physics Bohrs Model of Hydrogen Atom.pdf
26/39
PHYSICS CHAPTER 11
26
The em&ss&on l&nes orres%ond to the %hotons of d&sreteenerg&es that are em&tted "hen e&ted atom& states &n the gasma$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
8/12/2019 Matriculation Physics Bohrs Model of Hydrogen Atom.pdf
27/39
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
8/12/2019 Matriculation Physics Bohrs Model of Hydrogen Atom.pdf
28/39
8/12/2019 Matriculation Physics Bohrs Model of Hydrogen Atom.pdf
29/39
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/8/12/2019 Matriculation Physics Bohrs Model of Hydrogen Atom.pdf
30/39
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
8/12/2019 Matriculation Physics Bohrs Model of Hydrogen Atom.pdf
31/39
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
8/12/2019 Matriculation Physics Bohrs Model of Hydrogen Atom.pdf
32/39
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$e nnii== #
= 2i21
1
11
nR
H
=
2
i
2
1
2
11
nRH
=
2
i
21
311
nRH
=
2
i
2
1
!
11
nRH
=
2
i
2
1
)
11
nRH
8/12/2019 Matriculation Physics Bohrs Model of Hydrogen Atom.pdf
33/39
PHYSICS CHAPTER 11
33
The Bohrs model of hydrogen atom
%red&ts sessflly the energy le-els of the hydrogen atom.t fails to explain the energy le$els of more complexfails to explain the energy le$els of more complexatomsatoms#
an e%la&n the s%etrm for hydrogen atom .t somedeta&ls of the s%etrm cannot be explained especiallycannot be explained especially%hen the atom is placed in a magnetic field%hen the atom is placed in a magnetic field#
annot e%la&n the eeman effet (/&gre 11#;!# @eeman effect@eeman effect&s def&ned as the splitting of spectralthe splitting of spectral
lines %hen the radiating atoms are placed in alines %hen the radiating atoms are placed in amagnetic field.magnetic field.
11.&.& Limitation of Bohrs model of hydrogen atom
agnet& f&eld
Trans&t&ons
4o magnet& f&eld
11
##Energy e-els
S%etra!igure 11.8!igure 11.8
8/12/2019 Matriculation Physics Bohrs Model of Hydrogen Atom.pdf
34/39
PHYSICS CHAPTER 11
34
The Balmer ser&es for the hydrogen atom orres%onds to eletron&
trans&t&ons that term&nate at energy le-el n=2 as sho"n &n the/&gre 11#)#
Callatea# the longest "a-elength+ and
.# the shortest "a-elength of the %hoton em&tted &n th&s ser&es#
(?&-en the s%eed of l&ght &n the -am c=3#
8/12/2019 Matriculation Physics Bohrs Model of Hydrogen Atom.pdf
35/39
8/12/2019 Matriculation Physics Bohrs Model of Hydrogen Atom.pdf
36/39
PHYSICS CHAPTER 11
8/12/2019 Matriculation Physics Bohrs Model of Hydrogen Atom.pdf
37/39
PHYSICS CHAPTER 11
37
eterm&ne the "a-elength for a l&ne s%etrm &n yman ser&es
"hen the eletron ma$es a trans&t&on from n=3 le-el#
(?&-en Ryd.ergs onstant +RH= 1#
8/12/2019 Matriculation Physics Bohrs Model of Hydrogen Atom.pdf
38/39
PHYSICS CHAPTER 11
38
Eer&se 11#1 :?&-en c=3#
8/12/2019 Matriculation Physics Bohrs Model of Hydrogen Atom.pdf
39/39
39
PHYSICS CHAPTER 11
Next ChapterCHAPTER 12 :
JKrays