Lecture 17: Bohr Model of the Atom

Lecture 17: Bohr Model of the Atom • Reading: Zumdahl 12.3, 12.4 • Outline – Emission spectrum of atomic hydrogen. – The Bohr model of the hydrogen atom. – Extension to higher atomic number: Atoms

• Problems (5th Ed, Chapter 12) – 2, 30, 31, 32, 33, 34

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Light is emitted from H (Atoms) http://Jersey.uoregon.edu/vlab/elements/Elements.html

IF “Quantized” spectrum

ΔE

ΔE

“Continuous” spectrum (Sunlight)

Any ΔE is possible

Only certain ΔE are allowed

Atoms relax to lower energy states by giving off energy as light.2

Emission spectrum of H

Light Bulb (Sunlight)

Hydrogen Lamp (video)

3 Quantized, not continuous

Emission

Photon Emission • An atom gives up energy to create a photon. The energy is conserved. • Electrons relax from one energy level to another by emitting a photon. • The energy of the photon matches the energy difference of the electron • With

−ΔE Atom = E photon = hν = hc / λ

λ = 440nm ( Blue Light )

6.6 ⋅10−34 ⋅ 3 ⋅108 −19 = 4.5 ⋅ 10 E photon = hν = hc / λ = J −9 440 ⋅10 4 −19 ΔE Atom = −4.5 ⋅10 J

Emission spectrum of H (cont.)

We can use the emission spectrum to determine the energy levels for the hydrogen atom. Warning: Light oscillations not drawn to scale. 5

Balmer Formula for Frequencies • Joseph Balmer (1885) first noticed that the frequency of visible lines in the H atom spectrum could be reproduced by:

1 1 ν∝ 2− 2 2 n

n = 3, 4,5"

• The above equation predicts that as n increases, the frequencies become more closely spaced.

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Rydberg Formula • Johann Rydberg extends the Balmer Formula (because there were more lines discovered at other frequencies as photographic paper to detect UV and IR were invented)

• For Emission

⎛ 1 1 ⎞ ν = Ry ⎜ 2 − 2 ⎟ ⎝ n2 n1 ⎠

n2 = 1, 2, 3, & " n1 = n2 + 1, n2 + 2" ∞ n1 > n2 R y = 3.29 ⋅ 1015 sec-1

• This suggests that the energy levels of the H atom are proportional to 1/n2 7

H atom energy levels • Use Rydberg formula and assume energy conservation (as with photoelectric effect) ΔEHAtom = E2 − E1 = − hν photon

ν photon

⎛ 1 1 ⎞ = Ry ⎜ 2 − 2 ⎟ ⎝ n2 n1 ⎠

⎛ 1 1 ⎞ E2 − E1 = ( − hRy ) ⎜ 2 − 2 ⎟ ⎝ n2 n1 ⎠ Then

Atom emits energy (exothermic), energy comes out as photon.

n1 > n2

⎛ 1 ⎞ ⎛ 1 ⎞ ⎛ 1 ⎞ E2 = ( − hRy ) ⎜ 2 ⎟ and E1 = ( − hRy ) ⎜ 2 ⎟ or En = − ( hRy ) ⎜ 2 ⎟ < 0 ⎝n ⎠ ⎝ n2 ⎠ ⎝ n1 ⎠ N.B.: There could be an offset constant but it is zero; show later.8

The Bohr Model • Niels Bohr uses the emission spectrum of hydrogen to develop a quantum model for H. • Central idea: electron orbit the “nucleus” and only certain orbits (orbitals) are allowed. • Atoms emit light by electrons moving from one state (or orbit) to another. • Bohr used Coulomb’s law of charge attraction between e- and nucleus. • However, classical physics is unable to explain why an H atom doesn’t simply collapse. P. E .

Vcoulombic

2 ′ e Z =−