Post on 21-Dec-2015
Lecture 17: 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.
Photon Emission
• Relaxation from one energy level to another by emitting a photon.
• WithE = hc/
• If = 440 nm,
= 4.5 x 10-19 J
Em
issi
on
Emission spectrum of H
“Continuous” spectrum “Quantized” spectrum
Any E ispossible
Only certain E areallowed
E E
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:
€
ν ∝1
22−
1
n2n = 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:
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ν =Ry1
n12
−1
n22
⎛
⎝ ⎜
⎞
⎠ ⎟
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
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.
€
E = −2.178x10−18JZ 2
n2
⎛
⎝ ⎜
⎞
⎠ ⎟
Z = atomic number (1 for H)
n = integer (1, 2, ….)
• Ry x h = -2.178 x 10-18 J (!)
The Bohr Model (cont.)
€
E = −2.178x10−18JZ 2
n2
⎛
⎝ ⎜
⎞
⎠ ⎟
• 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 E is for any two energy levels
€
E = E final − E initial
€
E = −2.178x10−18J1
n final2
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟− (−2.178x10−18J)
1
ninitial2
⎛
⎝ ⎜
⎞
⎠ ⎟
€
E = −2.178x10−18J1
n final2
−1
ninitial2
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
The Bohr Model (cont.)
• Example: At what wavelength will emission from n = 4 to n = 1 for the H atom be observed?
€
E = −2.178x10−18J1
n final2
−1
ninitial2
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
1 4
€
E = −2.178x10−18J 1−1
16
⎛
⎝ ⎜
⎞
⎠ ⎟= −2.04x10−18J
€
E = 2.04x10−18J =hc
λ
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=9.74x10−8m = 97.4nm
The Bohr Model (cont.)
• Example: What is the longest wavelength of light that will result in removal of the e- from H?
€
E = −2.178x10−18J1
n final2
−1
ninitial2
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
1
€
E = −2.178x10−18J 0 −1( ) = 2.178x10−18J
€
E = 2.178x10−18J =hc
λ
€
=9.13x10−8m = 91.3nm
Extension to Higher Z• 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 = −2.178x10−18JZ 2
n2
⎛
⎝ ⎜
⎞
⎠ ⎟
Z = atomic number
n = integer (1, 2, ….)
Extension to Higher Z (cont.)
• Example: At what wavelength will emission from n = 4 to n = 1 for the He+ atom be observed?
€
E = −2.178x10−18J Z 2( )
1
n final2
−1
ninitial2
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
2 1 4
€
E = −2.178x10−18J 4( ) 1−1
16
⎛
⎝ ⎜
⎞
⎠ ⎟= −8.16x10−18J
€
E = 8.16x10−18J =hc
λ
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=2.43x10−8m = 24.3nm
€
H > λHe +
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.