Solutions

76. The 1,3,5-hexatriene molecule is a conjugated molecule with 6 pi electrons. Consider thepi electrons free to move back and forth along the molecule through the delocalized pisystem. Using the particle in a box approximation, treat the carbon chain as a linear onedimensional"box". Allow each energy level in the box to hold 2 pi electrons.Treating 1,3,5-hexatriene as a linear chain, we would have the structureC 1 C 2 C 3 C 4 C 5 C 6The pi electrons are free to move through the conjugated system; hence, the length of the molecule wouldrepresent the "box." We can add up the lengths of the single and double bonds in the molecule to give the widthof the box, L. For 1,3,5-hexatriene, this giveswhere r represents the bond length.L = 3r C =C + 2r C −C ,€The energy of the pi electrons in the molecule would be represented as€E n = n 2 h 2, n = 1, 2, 3, …,28mLwhere L is the box width defined above and m is the electron mass.a.) Calculate the energy of the highest filled level, using 1.54 Å as the carbon-carbonsingle bond length, and 1.35 Å as the carbon-carbon double bond length.Usingr C =C = 1.35 Å andr C −C = 1.54 Å, the width of the box can be calculated from the equation above.€L = 3(1.35 Å) + 2(1.54 Å) = 7.13 Å (or 7.13×10 -10 m).€Using the formula above with L = 7.13×10 –10 m for the energy leads to the energy diagram shown below.n=5n=4n=3n=2n=1