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6.20 a) The three sp³ carbons of cyclopropane form an equilateral triangle with an internuclear angle of 60°. This angle is much smaller than the tetrahedral angle of 109.5° required for the atomic orbitals of the sp³ hybridized carbons, causing a large amount of angle strain in the molecule. In addition to the angle strain, the rigid and flat cyclopropane ring also has a significant amount of torsional strain due to the eclipsed conformation about each C-C bond. b) Planar cyclobutane has angle strain and torsional strain due to its eclipsed conformations about each C-C bond. As the ring distorts from planarity, torsional strain decreases while angle strain increases. At first, the increase in angle strain is less than the decrease in torsional strain, resulting in a net decrease in ring strain. Eventually, angle strain increases faster than torsional strain decreases. As a result, the lowest energy conformer of cyclobutane is slightly nonplanar. c) The internuclear angle of planar cyclopentane ring is 108°, so it has little or no angle strain. However, planar cyclopentane does have a considerable amount of torsional strain due to eclipsed conformations at all C-C bonds. By distorting from planarity the cyclopentane ring can relieve most of its torsional strain without increasing its angle strain significantly. Therefore the low energy conformation of cyclopentane is nonplanar and shaped somewhat like an envelope. d) Planar cyclohexane has angle strain because its internuclear angle is 120°, which is larger than the tetrahedral angle of 109.5°. It also has a substantial amount of torsional strain because all its bonds are eclipsed. When the ring distorts from planarity its angle strain is decreased. There are two nonplanar conformations of cyclohexane, called the chair and the boat, that are free of angle strain. The chair conformer is the lowest energy conformation because it has neither torsional nor angle strain. The boat conformer has torsional and steric strain. e) Cyclodecane is nonplanar. It has some angle strain and some torsional strain. In addition, cyclodecane has transannular strain that results from steric interactions of C-H bonds that point towards the center of the ring. 6.21 Both the -C=N and -C=CH groups have linear geometry. In the axial position both are aligned parallel to the ring axis and their 1,3-diaxial interactions with other groups are not very large. The -CH₃ group has tetrahedral geometry. One hydrogen of the methyl group must be pointed 89