Transcript Lecture 17: Bohr Model of the Atom

Lecture 15: Bohr Model of the
Atom
• Reading: Zumdahl 12.3, 12.4
• Outline
– Emission spectrum of atomic hydrogen.
– The Bohr model.
– Extension to higher atomic number.
Light as Quantized Energy
• Comparison of experiment to the “classical”
prediction:
Classical prediction is
for significantly higher
intensity as smaller
wavelengths than what
is observed.
“The Ultraviolet Catastrophe”
Light as Quantized Energy
• Planck found that in order to model this behavior,
one has to envision that energy (in the form of
light) is lost in integer values according to:
DE = nhn
frequency
Energy Change
n = 1, 2, 3 (integers)
h = Planck’s constant = 6.626 x 10-34 J.s
Light as a ‘Particle’
• As frequency
of incident light is increased,
kinetic energy of emitted eincreases linearly.
0
1
2
men  hn photon  
2
n0
Frequency (n)

 = energy needed to release e-
Interference of Light
• Shine light through a crystal and look at pattern
of scattering.
• Diffraction can only be explained by treating light
as a wave instead of a particle.
Particles as waves
• Electrons shine through a crystal and look at pattern
of scattering.
• Diffraction can only be explained by treating electrons
as a wave instead of a particle.
Emission
Photon Emission
• Relaxation from one
energy level to another
by emitting a photon.
• With DE = hc/l
• If l = 440 nm,
DE= 4.5 x 10-19 J
Emission spectrum of H
“Quantized” spectrum
DE
DE
“Continuous” spectrum
Any DE is
possible
Only certain
DE are
allowed
Emission spectrum of H (cont.)
Light Bulb
Hydrogen Lamp
Quantized, not continuous
Emission spectrum of H (cont.)
We can use the emission spectrum to determine the
energy levels for the hydrogen atom.
Balmer Model
• Joseph Balmer (1885) first noticed that the
frequency of visible lines in the H atom spectrum
could be reproduced by:
1 1
n 2 2
2 n
n = 3, 4, 5, …..
• The above equation predicts that as n increases,
the frequencies become more closely spaced.


Rydberg Model
• Johann Rydberg extends the Balmer model by
finding more emission lines outside the visible
region of the spectrum:
 1
1 
n  Ry  2  2 
n1 n 2 
n1 = 1, 2, 3, …..
n2 = n1+1, n1+2, …
Ry = 3.29 x 1015 1/s
• This suggests that the energy levels of the H atom
are proportional to 1/n2
The Bohr Model
• Niels Bohr uses the emission spectrum of
hydrogen to develop a quantum model for H.
• Central idea: electron circles the “nucleus” in
only certain allowed circular orbitals.
• Bohr postulates that there is Coulombic attraction
between e- and nucleus. However, classical
physics is unable to explain why an H atom
doesn’t simply collapse.
The Bohr Model (cont.)
• Bohr model for the H atom is capable of
reproducing the energy levels given by the
empirical formulas of Balmer and Rydberg.
2 

Z = atomic number (1 for H)
Z
18
E  2.178x10 J 2 
n 
n = integer (1, 2, ….)
• Ry x h = -2.178 x 10-18 J (!)
The Bohr Model (cont.)
2 

Z
18
E  2.178x10 J 2 
n 

• Energy levels get closer together
as n increases
• at n = infinity, E = 0
The Bohr Model (cont.)
• We can use the Bohr model to predict what DE is
for any two energy levels
DE  E final  E initial
 1 
 1 
18
18
DE  2.178x10 J
n 2 
 (2.178x10 J)n 2 
 initial 
 final 

 1

1
DE  2.178x1018 J
n 2  n 2 

 final
initial 
The Bohr Model (cont.)
• Example: At what wavelength will emission from
n = 4 to n = 1 for the H atom be observed?
 1

1
DE  2.178x1018 J
n 2  n 2 

 final
initial 
1
4
 1 
DE  2.178x10 J1  2.04x1018 J
 16 
18

18

DE  2.04 x10 J 
hc
l
l  9.74 x108 m  97.4nm
The Bohr Model (cont.)
• Example: What is the longest wavelength of light
that will result in removal of the e- from H?
 1

1
DE  2.178x1018 J
n 2  n 2 

 final
initial 
1



DE  2.178x1018 J0 1  2.178x1018 J
18
DE  2.178x10 J 
hc
l
l  9.13x108 m  91.3nm
Extension to Higher Z
• The Bohr model can be extended to any single
electron system….must keep track of Z
(atomic number).
2 

Z
18
E  2.178x10 J 2 
n 
Z = atomic number
n = integer (1, 2, ….)
• Examples: He+ (Z = 2), Li+2 (Z = 3), etc.
Extension to Higher Z (cont.)
• Example: At what wavelength will emission from
n = 4 to n = 1 for the He+ atom be observed?
 1

1
DE  2.178x1018 J Z 2 
n 2  n 2 

 final
initial 
2
1
4
 1 
DE  2.178x10 J41  8.16x1018 J
 16 
hc
18
l  2.43x108 m  24.3nm
DE  8.16x10 J 
l
l H  l He
18



Where does this go wrong?
• The Bohr model’s successes are limited:
• Doesn’t work for multi-electron atoms.
• The “electron racetrack” picture is incorrect.
• That said, the Bohr model was a pioneering,
“quantized” picture of atomic energy levels.