Bohr’s model of H atom
PHY123
7/24/2016
Lecture XIII
1
Niels Bohr (1885-1962)
7/24/2016
Lecture XIII
2
By Iutta Waloschek
Hydrogen atom
• Positively charged nucleus
inside (+Ze), negatively
charged electron (-e) around
• Electron is attracted to
nucleus
• Electron is trapped in a
potential well created by
nucleus (“a box”)
Ze 2
U  eV  
40 r
1
• Energy levels in atom
7/24/2016
Lecture XIII
3
Standing electron waves in
Hydrogen atom
• Standing waves:
• 2rn=nl
lh/mv
mvrn=nh/2
• Angular momentum L=mvrn
is quantized
L=nh/2
• n – orbital quantum number
7/24/2016
Lecture XIII
4
Disclaimer
• Though Bohr’s model was able to predict many
properties of H atom and correctly calculate
some of its characteristics, this model is
incomplete and it is not advisable to think of an
atom as a miniature “solar system”.
• Orbits make sense only as average radii of
electron position (not the same as electron
slightly smeared around circular orbits!!!)
7/24/2016
Lecture XIII
5
Hydrogen atom
• mvrn=nh/2 v=nh/(2mrn)
• Newton’s 2nd law for circular
motion:
mv 2
F
r
1 Ze 2 mv 2
n2h2


2
40 r
r
4 2 mr 3
n 2 h 2 0 n 2
rn 
 r1
2
mZe
Z
h 2 0
10
r1 

0
.
529

10
m
2
me
7/24/2016
Bohr radius
Lecture XIII
6
Hydrogen atom
• Energy in H atom KE+U
mv
m nh  m Z e
KE 

2
2 4 2 m 2 n 4 h 4 02
2
2
Z 2e 4 m
KE  2 2 2
8n h  0
Ze 2
U  eV  
40 r
1
Z 2e 4 m
U   2 2 2  2 KE
4n h  0
7/24/2016
2
2
2
2 4
Z 2e 4 m
En  KE  U   KE   2 2 2
8n h  0
Z2
En  2 E1
n
e4m
E1   2 2  13.6eV
8h  0
Lecture XIII
7
Hydrogen atom
• What did we learn?
• For high n energy approaches -0,
radius approaches infinity
• Energy proportional to Z2e4
– Ze2 from the potential, Ze2 from 1/r
(smaller orbits around larger charges)
• Radius inversely proportional to mass
(F=mv2/r)  Energy proportional to
electron mass
• Energy does not depend on nucleus
mass – assume infinitely heavy
• In classical case (h0) rn0 – electron
falls on nucleus.
7/24/2016
Lecture XIII
n 2 h 2 0
rn 
mZe2
Z 2e 4 m
En   2 2 2
8n h  0
8
Hydrogen atom
• Energy levels in H
13.6
En   2 eV
n
• Infinite number of states n=1infinity
• All energies are negative –
electron is trapped in atom
• Lowest possible energy level
E1=-13.6 eV – ground state
• Positive energy – free electron
–ionization
• Ionization energy =13.6/n2 for
atom in n state
7/24/2016
Lecture XIII
9
Some parameters of the H atom
• Ground state n=1, Z=1
• Electron’s “orbit” radius
h 2 0
10
r1 

0
.
529

10
m
2
me
• Wavelength l  2r  3.11010 m  0.31nm
1
• Momentum p  h / l  hc / lc  1240eV  nm / 0.31nm / c  4 KeV / c
• v/c
p  mc 2 v / c 2  4 KeV / c
0.5 10 6 eV  v / c  4 103 eV
v / c  8 10 3
7/24/2016
Lecture XIII
10
Some parameters of the H atom
• Ground state n=1, Z=1
4
e
m
• Kinetic Energy KE 
 13.6eV
2 2
8h  0
• Potential Energy
e4m
U   2 2  27.2eV
4h  0
e4m
• Total energy E   2 2  13.6eV
8h  0
• Longest wavelength absorbed (l – max E =hc/l – min)
n=1n=2
E1=-13.6eV
E2=-13.6/4=-3.4eV Eg=13.6-3.4=10.2eV
lhc/Eg1243eVnm/10.2eV=122nm
7/24/2016
Lecture XIII
11
A cute problem
• Electrons are accelerated
by 12.3V potential
difference and pass
through hydrogen gas at
room T (ground state
n=1). Which wavelengths
of light can be observed?
7/24/2016
Lecture XIII
12