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Zumdahl

Scanning tunneling micrograph showing a ring of 48 Fe atoms on a copper surface.

QUANTUM MECHANICS

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Quantum mechanics was developed because classicalmechanics did not work in some situations including foratoms and molecules.

Why Quantum Mechanics?

Difficulties already discussed:

Dual nature of light and matter.

Quantization of energy levels for oscillators.

• Ultraviolet catastrophe

• Photoelectric effect

 Another difficulty:

• Emission and absorption spectra in atoms and molecules

Classically: all energies allowed.

Observed: only certain energies allowed (discrete or quantized energies)  FE2

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EMISSION SPECTRA

 ABSORPTION SPECTRUM

Continuous

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Rydberg equation:

and na and nb are whole positive numbers with na < nb.

EXPERIMENTAL OBSERVATIONS FOR H ATOM

where R h = Rydberg constant = 1.097 x 107 m-1.

(Empirical formula)

Light of this wavelength is green.

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EARLY THEORY TO EXPLAIN RYDBERG EQUATION FOR H ATOM

Bohr’s Model of the Atom:

1. The electron moves in circular orbits about the protonunder the influence of the Coulomb force of attraction.

2. Only certain orbits are stable. These stable orbits are

ones in which the electron does not radiate.

3. Radiation is emitted by the atom when the electron"jumps" from a high energy state (E i ) to a lower energystate (E 

f ).

photon energy = hf = |E f  – E i |

4. The size of the allowed electron orbits is determined by an additional

quantum condition imposed on the electron's orbital angular momentum.

mvr = nh/(2 π  ) where n = 1, 2, 3, . . .

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EARLY THEORY TO EXPLAIN THE H ATOM SPECTRUM

Results of Bohr’s Model for the hydrogen atom:

2

18

2

0.0529 0.529 52.9

2.178x10 joule n 1, 2, 3, . . .

o

n o o

n

r a n a Bohr radius nm pm

 E n

−

= = = = Α =

−= =

where r n is the radius and E n is the energy for the nth level for an electron in

the hydrogen atom.

Note, n = 1 is the ground state and n = 2 or more are excited states for theelectron. n is called the principle quantum number.

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18

2

2.178x10 joule n 1, 2, 3, . . .

n E n

−−

= =

Example: (a) Calculate the energy released when an electrondrops from n = 4 excited state to n = 2 excited state. (b) What isthe frequency of the photon. (c) What is its wavelength?

( )

18 18

192 4 4 2 2 22.178 10 2.178 10 4.08 104 2

 photon  x x E E E E E E x J 

− −

−

− −= −∆ = − − = − = − =

18 1819

4 2 2 4 2 2

2.178 10 2.178 104.08 10

4 2

 photon

 x x E E E or E E x J 

− −

−− −= − − = − =

1914 1

34

4.08 10; 6.16 10

6.63 10

 photon

 photon

 E    x J hf E f x s

h x J s

−

−

−= = = =

⋅

( )8

7 7 9

14 1

3.00 10 /; 4.87 10 487 10 10 487

6.16 10

c x m sc f x m x nm nm

 f x sλ λ 

  − −

−= = = = = =

Note, this is the same wavelength asfound using the Rydberg equation.

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The Bohr model works for single electron atoms, e.g. He+, Li2+.The only modification includes the effect of the increased nuclear

charge (Z). Thus,

 Z 

noanr 

2=

18 2

2

2.178x10 joule n 1, 2, 3, . . .

n

 Z  E 

n

−

= − =

o  Artificially, has stable orbitals.

o Correctly explains the spectra for the H atom and other 1electron ions (He+, Li2+, etc.)

o Fails miserable for molecules and multielectron atoms.

o Violates the Heisenberg uncertainty principle.

o Model has limited usefulness.

o Uses classical physics and the quantization of angular momentumfor an electron (mvr).

Summary of Bohr Model