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.

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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.)

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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

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E = E final − E initial

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E = −2.178x10−18J1

n final2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟− (−2.178x10−18J)

1

ninitial2

⎛

⎝ ⎜

⎞

⎠ ⎟

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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

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E = −2.178x10−18J 1−1

16

⎛

⎝ ⎜

⎞

⎠ ⎟= −2.04x10−18J

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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?

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E = −2.178x10−18J1

n final2

−1

ninitial2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

1

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E = −2.178x10−18J 0 −1( ) = 2.178x10−18J

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E = 2.178x10−18J =hc

λ

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=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?

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E = −2.178x10−18J Z 2( )

1

n final2

−1

ninitial2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

2 1 4

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E = −2.178x10−18J 4( ) 1−1

16

⎛

⎝ ⎜

⎞

⎠ ⎟= −8.16x10−18J

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E = 8.16x10−18J =hc

λ

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=2.43x10−8m = 24.3nm

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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.