Physical Chemistry Third Edition

21.7 The Free-Electron Molecular Orbital Method 893

EXAMPLE21.12

Using the FEMO method, calculate the wavelength of the light absorbed when 1,3-butadiene

makes the transition from the ground-state to the first excited state.

Solution

This transition is the promotion of one electron fromn2ton3, so

∆E

h^2

8 ma^2

(9−4)

(5)(6. 6261 × 10 −^34 Js)^2

(8)(9. 109 × 10 −^31 kg)(6. 94 × 10 −^10 m)^2

 6. 25 × 10 −^19 J

λ

hc

∆E



(6. 6261 × 10 −^34 J s)(2. 9979 × 108 ms−^1 )

6. 25 × 10 −^19 J

 3. 18 × 10 −^7 m318 nm

This is in fairly poor agreement with the experimental value of 217 nm, but it is remarkable

that the agreement is not worse than this. Better results could be attained by adding more or

less than a full bond length at each end of the carbon–carbon chain.

Exercise 21.13

Modify the length of the “box” to recover the experimental wavelength of 217 nm for

1,3-butadiene. What fraction of a bond added at the ends of the box does this correspond to?

Could this fraction be negative?

PROBLEMS

Section 21.7: The Free-Electron Molecular Orbital
Method

21.36Using the same value ofβas obtained for benzene in
Example 21.10, find the wavelength of the lowest-energy
electronic transition in 1,3-butadiene according to the
Hückel method. Compare with the experimental value of
217 nm.

21.37Using the same bond lengths as with 1,3-butadiene, find
the reciprocal wavelength of the longest-wavelength
electronic transition of 1,3,5-hexatriene according to the
FEMO method. Compare with the experimental value,
24,000 cm−^1.

21.38Using the same bond lengths as with 1,3-butadiene, find
the wavelength of the longest-wavelength electronic

transition of 1,3,5,7-octatetraene according to the FEMO

method.

21.39In the free-electron molecular orbital model, the electrons

actually move in three dimensions. For 1,3-butadiene

represent the electrons as particles in a three-dimensional

box with a length in thexdirection equal to 694 pm,

width in theydirection equal to 268 pm, and height in the

zdirection equal to the width. Find the wavelength of the

photons absorbed in the longest-wavelength absorption

due to changes in the quantum numbersnyandnz.

Explain why the representation as a one-dimensional

box can successfully be used to understand the

near-ultraviolet spectrum.