Bohr theory

2. Outline
◦ Emission spectrum of atomic hydrogen.
◦ The Bohr model.
◦ Extension to higher atomic number.

3.  Relaxation from one
energy level to
another by emitting a
photon.
 With ∆E = hc/λ
 If λ = 440 nm,
∆Ε = 4.5 x 10-19
J
Emission

4. “Continuous” spectrum “Quantized” spectrum
Any ∆E is
possible
Only certain
∆E are
allowed
∆E
∆E

5. Light Bulb
Hydrogen Lamp
Quantized, not continuous

6. We can use the emission spectrum to determine the
energy levels for the hydrogen atom.

7.  Joseph Balmer (1885) first noticed that the
frequency of visible lines in the H atom
spectrum could be reproduced by:
ν∝
1
22
−
1
n2
n = 3, 4, 5, …..
• The above equation predicts that as n increases,
the frequencies become more closely spaced.

8.  Johann Rydberg extends the Balmer model by
finding more emission lines outside the visible
region of the spectrum:
ν=Ry
1
n1
2
−
1
n2
2






n1 = 1, 2, 3, …..
• This suggests that the energy levels of the H atom
are proportional to 1/n2
n2 = n1+1, n1+2, …
Ry = 3.29 x 1015
1/s

9.  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 orbits.
• Bohr postulates that there is Coulomb attraction
between e- and nucleus. However, classical
physics is unable to explain why an H atom
doesn’t simply collapse.

10. • Bohr model for the H atom is capable of
reproducing the energy levels given by the
empirical formulas of Balmer and Rydberg.
E=−2.178x10−18
J
Z2
n2






Z = atomic number (1 for H)
n = integer (1, 2, ….)
• Ry x h = -2.178 x 10-18
J (!)

11. E=−2.178x10−18
J
Z2
n2






• Energy levels get closer together
as n increases
• at n = infinity, E = 0

12. • We can use the Bohr model to predict what ∆E is
for any two energy levels
∆E=Efinal−Einitial
∆E=−2.178x10−18
J
1
nfinal
2





−(−2.178x10−18
J)
1
ninitial
2






∆E=−2.178x10−18
J
1
nfinal
2
−
1
ninitial
2







13. • Example: At what wavelength will emission from
n = 4 to n = 1 for the H atom be observed?
∆E=−2.178x10−18
J
1
nfinal
2
−
1
ninitial
2






1 4
∆E=−2.178x10−18
J1−
1
16





=−2.04x10−18
J
∆E=2.04x10−18
J=
hc
λ
λ=9.74x10−8
m=97.4nm

14. • Example: What is the longest wavelength of light
that will result in removal of the e-
from H?
∆E=−2.178x10−18
J
1
nfinal
2
−
1
ninitial
2






∞ 1
∆E=−2.178x10−18
J0−1( )=2.178x10−18
J
∆E=2.178x10−18
J=
hc
λ
λ=9.13x10−8
m=91.3nm

15. • The Bohr model can be extended to any single
electron system….must keep track of Z
(atomic number).
• Examples: He+
(Z = 2), Li+2
(Z = 3), etc.
E=−2.178x10−18
J
Z2
n2






Z = atomic number
n = integer (1, 2, ….)

16. • Example: At what wavelength will emission from
n = 4 to n = 1 for the He+
atom be observed?
∆E=−2.178x10−18
JZ2
( )
1
nfinal
2
−
1
ninitial
2






2
1 4
∆E=−2.178x10−18
J4()1−
1
16





=−8.16x10−18
J
∆E=8.16x10−18
J=
hc
λ
λ=2.43x10−8
m=24.3nm
λH>λHe+

17.  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.