1,3,5-Hexatriene (a kind of ‘linear’ benzene) was converted into benzene itself. On the basis of a free-electron molecular orbital model (in which hexatriene is treated as a linear box and benzene as a ring), would you expect the lowest energy absorption to rise or fall in energy as a result of the conversion?

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Step 1The pi electrons of 1,3,5 - hexatriene can be considered as free electrons in a linear box.Therefore, the expression for the nondegenerate energy levels of pi electrons of 1,3,5 - hexatriene (E_(n)) is shown below.E_(n)=(n^(2)h^(2))/(8m_(e)L^(2))Where,n is the state of the energy levels of a linear box.h is the Planck's constant.m_(e) is the mass of the particle.L is the length of the box.Step 2The molecule, 1,3,5 - hexatriene has 6pi electrons.Therefore, the lowest energy transition will be from n=3rarr n=4.Substitute the value of n=3 in equation (1).{:[E_(3)=((3)^(2)h^(2))/(8m_(e)L^(2))],[=(9h^(2))/(8m_(e)L^(2))]:}Substitute the value of n=4 in equation (1).{:[E_(4)=((4)^(2)h^(2))/(8m_(e)L^(2))],[=(16h^(2))/(8m_(e)L^(2))]:}The formula to calculate the energy of transition of pi electrons of 1,3,5 - hexatriene is given below.DeltaE_("hexatriene ")=E_(4)-E_(3)Substitute the value of E_(4) and E_(3) from equation (3) and (2) in equation (4){:[DeltaE_("hexatriene ")=(16h^(2))/(8m_(e)L^(2))-(9h^(2))/(8m_(c)L^(2))],[=(7h^(2))/(8m_(n)L^(2))]:}The length of the box is 6 times the C-C bond distance, d.Therefore, the value of L is given by the expression shown below.R=6dSubstitute the value of L in equation (5).{:[DeltaE_("benzene ")=(7h^(2))/(8m_(e)(6d)^(2))],[=(7h^(2))/(288m_(e)d^(2))]:}The pi electrons of benzene can be considered as free electron ... See the full answer