CHAPTER 6: Question 1

The symmetry elements lead to the following symmetry operations: E, C₂¹, s_v, s'_v. Note that these are the only unique symmetry operations since C₂², s_v², and s'_v² are all equivalent to E. The point group can be determined as C_2v either from the set of symmetry elements and Table 6.2, or by using the flow chart in Figure 6.11. The appropriate part of the flow chart is reproduced below.

The symmetry elements lead to the following symmetry operations: E, C₂¹, s_v, s'_v. The point group can be determined as C_2v either from the set of symmetry elements and Table 6.2, or by using the flow chart in Figure 6.11. Note that the symmetry elements, symmetry operations, and point group in this case are identical to those of part a), despite the fact that the benzene ring is disubstituted in this case rather than monosubstituted.

c) The molecule contains the following symmetry elements: the identity element (E); a C₂ axis (shown in red in the diagram below); and two planes of symmetry (s_v and s'_v) shown by the grey or blue boxes in the diagrams below. Note that both planes of symmetry are classified as vertical planes since they are both parallel to the principal axis (the C₂ axis). One of the planes of symmetry is the molecular plane, and the other is orthogonal to the molecular plane along the C₂ axis. Manipulation of the rotating 3D structure shown below may help in the visualization of these symmetry elements.

 The symmetry elements lead to the following symmetry operations: E, C₂¹, s_v, s'_v. The point group can be determined as C_2v either from the set of symmetry elements and Table 6.2, or by using the flow chart in Figure 6.11. Note that the symmetry elements, symmetry operations, and point group in this case are identical to those of parts a and b), despite the different substitution pattern of the benzene ring in each of these cases.

The symmetry elements lead to the following symmetry operations: E, C₂¹, C'₂¹, C"₂¹, s_v, s'_v, s_h, i. The point group can be determined as D_2h either from the set of symmetry elements and Table 6.2, or by using the flow chart in Figure 6.11. The appropriate part of the flow chart is reproduced below.

e) The molecule contains the following symmetry elements: the identity element (E); a C₂ axis (shown in red in the diagram below); and two planes of symmetry (s_v and s'_v) shown by the grey or blue boxes in the diagrams below. Note that both planes of symmetry are classified as vertical planes since they are both parallel to the principal axis (the C₂ axis). One of the planes of symmetry is the molecular plane, and the other is orthogonal to the molecular plane along the C₂ axis. Manipulation of the rotating 3D structure shown below may help in the visualization of these symmetry elements.

The symmetry elements lead to the following symmetry operations: E, C₂¹, s_v, s'_v. The point group can be determined as C_2v either from the set of symmetry elements and Table 6.2, or by using the flow chart in Figure 6.11. It is instructive to compare the answers to parts d and e), since the effect of changing one of the
bromine atoms in part d) to fluorine in part e) is to substantially reduce the number of symmetry elements, and leave the molecule with the same symmetry elements found for bromobenzene in part a).

f) The molecule contains the following symmetry elements: the identity element (E); a C₃ axis (shown in green in the first and red in the other diagrams below) and three C₂ axes (C₂, C'₂, and C"₂ shown in red in the first and magenta in the other diagrams below); four planes of symmetry (s_v, s'_v, s"_v and s_h), three of which are shown by the grey boxes in the first diagram below (the fourth is the molecular plane) and  two of which are also shown by the blue boxes in the other diagrams below; and an S₃ axis which is coincident with the C₃ axis. The C₃ axis is the principal axis, and three of the planes of symmetry are parallel to this axis and so are classified as vertical planes, the fourth plane of symmetry being orthogonal to the principal axis is classified as a horizontal plane. Manipulation of the rotating 3D structure shown below may help in the visualization of these symmetry elements.

The symmetry elements lead to the following symmetry operations: E, C₃¹, C₃², C₂¹, C'₂¹, C"₂¹, sv, s'v, s"v, sh, S₃¹, S₃⁵. Note that S₃², S₃³, S₃⁴, and S₃⁶  are not included in this list as they are equivalent to C₃², sh, C₃¹, and E respectively. The point group can be determined as D_3h either from the set of symmetry elements and Table 6.2, or by using the flow chart in Figure 6.11. The appropriate part of the flow chart is reproduced below.

The symmetry elements lead to the following symmetry operations: E, C₂¹, s_v, s'_v. The point group can be determined as C_2v either from the set of symmetry elements and Table 6.2, or by using the flow chart in Figure 6.11. It is instructive to compare this answer with the answers to parts a) and c), since in each case the
molecule belongs to the same point group despite the different substitution patterns of the benzene ring.

h) The only symmetry elements this molecule contains are the identity element (E) and a plane of symmetry (s) which is coincident with the molecular plane as shown below. Manipulation of the rotating 3D structure shown below may help in the visualization of these symmetry elements. Hence, the symmetry operations are E, s and the point group can be determined as C_s either from the set of symmetry elements and Table 6.2, or by using the flow chart in Figure 6.11. The appropriate part of the flow chart is reproduced below. These eight examples have shown that the symmetry elements, and hence symmetry operations and point groups associated with substituted aromatic compounds depend on the number, location and nature of the substituents. 

 

The symmetry elements lead to the following symmetry operations: E, C₃¹, C₃², s_v, s'_v, and s"_v. The point group can be determined as C_3v either from the set of symmetry elements and Table 6.2, or by using the flow chart in Figure 6.11. The appropriate part of the flow chart is reproduced below.

The symmetry elements lead to the following symmetry operations: E, 4C₃¹, 4C₃², C₂¹, C'₂¹, C"₂¹, S₄¹, S'₄¹, S"₄¹, S₄³, S'₄³, S"₄³, 6s. The point group can be determined as T_d either from the set of symmetry elements and Table 6.2, or by using the flow chart in Figure 6.11. The appropriate part of the flow chart is
reproduced below.

The symmetry elements lead to the following symmetry operations: E, C₃¹, C₃², s_v, s'_v, and s"_v. The point group can be determined as C_3v either from the set of symmetry elements and Table 6.2, or by using the flow chart in Figure 6.11. Comparison of this case with part i) will show that the two molecules have the same shape (a distorted tetrahedron) and hance have the same symmetry elements and belong to the same point group.

The symmetry elements lead to the following symmetry operations: E, C₂¹, s_v, s'_v. The point group can be determined as C_2v either from the set of symmetry elements and Table 6.2, or by using the flow chart in Figure 6.11.

The symmetry elements lead to the following symmetry operations: E, C₃¹, C₃², s_v, s'_v, and s"_v. The point group can be determined as C_3v either from the set of symmetry elements and Table 6.2, or by using the flow chart in Figure 6.11. Comparison of this case with parts i) and k) will show that the three molecules have the same shape (a distorted tetrahedron) and hence have the same symmetry elements and belong to the same point group.

The symmetry elements lead to the following symmetry operations: E, 4C₃¹, 4C₃², C₂¹, C'₂¹, C"₂¹, S₄¹, S'₄¹, S"₄¹, S₄³, S'₄³, S"₄³, 6s. The point group can be determined as T_d either from the set of symmetry elements and Table 6.2, or by using the flow chart in Figure 6.11. Comparison of this example with part j) will
show that the two molecules both have the same shape (tetrahedral), so it should not be surprising that they have the same symmetry elements and belong to the same point group.

 

The symmetry elements lead to the following symmetry operations: E, C₃¹, C₃², C₂¹, C'₂¹, C"₂¹, sv, s'v, s"v, sh, S₃¹, S₃⁵. Note that S₃², S₃³, S₃⁴, and S₃⁶  are not included in this list as they are equivalent to C₃², sh, C₃¹, and E respectively. The point group can be determined as D_3h either from the set of symmetry elements and Table 6.2, or by using the flow chart in Figure 6.11. It is instructive to compare this example with part f), since the two molecules have the same symmetry elements and so belong to the same point group despite having very different shapes.

The symmetry elements lead to the following symmetry operations: E, C₄¹, C₄², C₄³, C'₄¹, C'₄², C'₄³, C"₄¹, C"₄², C"₄³, 4C₃¹, 4C₃², 6C₂¹, i, s_h, s'_h, s"_h, 6s_d, 4S₆¹, 4S₆³, 4S₆⁵, S₄¹, S₄³, S'₄¹, S'₄³, S"₄¹, S"₄³. The point group can be determined as O_h either from the set of symmetry elements and Table 6.2, or by using the flow chart in Figure 6.11. The appropriate part of the flow chart is reproduced below.

The symmetry elements lead to the following symmetry operations: E, C₂¹, s_v, s'_v. The point group can be determined as C_2v either from the set of symmetry elements and Table 6.2, or by using the flow chart in Figure 6.11.

 
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