Chapter 10, Problem 10A.1P

(a)

Interpretation Introduction

Interpretation:

The symmetry elements and point group of staggered ${\text{CH}}_{\text{3}}{\text{CH}}_{\text{3}}$ have to be stated.  Whether the molecule is polar or not has to be stated.  Whether the molecule is chiral or not has to be stated.

Concept introduction:

A symmetry operation is defined as an action on an object to reproduce an arrangement that is identical to its original spatial arrangement.  The spatial arrangement of the object remains identical after a symmetry operation.  The point of reference through which a symmetry operation takes place is termed as a symmetry element.

Answer to Problem 10A.1P

The symmetry elements of staggered ${\text{CH}}_{\text{3}}{\text{CH}}_{\text{3}}$ are $E$, $C_2$, $C_3$ and ${\sigma }_{\text{d}}$.  The point group of staggered ${\text{CH}}_{\text{3}}{\text{CH}}_{\text{3}}$ is $D_{\text{3d}}$.

The staggered ${\text{CH}}_{\text{3}}{\text{CH}}_{\text{3}}$ molecule is nonpolar and achiral.

Explanation of Solution

The staggered form of ethane $\left({\text{CH}}_{\text{3}}{\text{CH}}_{\text{3}}\right)$ is shown below.

Figure 1

The staggered ${\text{CH}}_{\text{3}}{\text{CH}}_{\text{3}}$ has a principal axis $C_3$, three rotation axis $C_2$ perpendicular to the principal axis, and three ${\sigma }_{\text{d}}$ planes.  The corresponding Figure is shown below.

Figure 2

The symmetry elements of staggered ${\text{CH}}_{\text{3}}{\text{CH}}_{\text{3}}$ are $E$, $C_2$, $C_3$ and ${\sigma }_{\text{d}}$.  Therefore, staggered ${\text{CH}}_{\text{3}}{\text{CH}}_{\text{3}}$ belongs to the point group $D_{\text{3d}}$.

The molecules which have point group $C_{\text{s}}$, $C_{\text{n}}$ and $C_{\text{nv}}$, are said to be as a polar molecule.  The point group of staggered ${\text{CH}}_{\text{3}}{\text{CH}}_{\text{3}}$ is $D_{\text{3d}}$.  Therefore, the staggered ${\text{CH}}_{\text{3}}{\text{CH}}_{\text{3}}$ molecule is nonpolar.

The molecules which do not have any reflective symmetry elements (plane) but may have rotational axis are said to be as a chiral molecule. The staggered ${\text{CH}}_{\text{3}}{\text{CH}}_{\text{3}}$ molecule has three ${\sigma }_{\text{d}}$ planes. Therefore, the staggered ${\text{CH}}_{\text{3}}{\text{CH}}_{\text{3}}$ molecule is achiral.

(b)

Interpretation Introduction

Interpretation:

The symmetry elements and point group of chair and boat cyclohexane have to be stated.  Whether the molecule is polar or not has to be stated.  Whether the molecule is chiral or not has to be stated

Concept introduction:

As mentioned in the concept introduction in part (a).

Answer to Problem 10A.1P

The symmetry elements of the chair form of cyclohexane are $E$, $C_2$, $C_3$, $S_8$, $i$, and ${\sigma }_{\text{d}}$.  The point group of the chair form of cyclohexane is $D_{\text{3d}}$.

The chair form of cyclohexane molecule is nonpolar and achiral.

The symmetry elements of the boat form of cyclohexane are $E$, $C_2$, and ${\sigma }_{\text{v}}$.  The point group of the boat form of cyclohexane is $C_{\text{2v}}$.

The boat form of cyclohexane molecule is polar and achiral.

Explanation of Solution

The structure of the chair form of cyclohexane is shown below in Figure 3.

Figure 3

The chair form of cyclohexane has a principal axis $C_3$, three rotation axis $C_2$ perpendicular to the principal axis, three ${\sigma }_{\text{d}}$ planes, a center of inversion $i$, and an axis $S_8$.  The corresponding Figure is shown below.

Figure 4

The symmetry elements of the chair form of cyclohexane are $E$, $C_2$, $C_3$, $S_8$, $i$, and ${\sigma }_{\text{d}}$.  Therefore, the chair form of cyclohexane belongs to the point group $D_{\text{3d}}$.

The molecules which have point group $C_{\text{s}}$, $C_{\text{n}}$ and $C_{\text{nv}}$, are said to be as a polar molecule.  The point group of chair form of cyclohexane is $D_{\text{3d}}$.  Therefore, chair form of cyclohexane is nonpolar.

The molecules which do not have any reflective symmetry elements (palne) but may have rotational axis are said to be as a chiral molecule.  The chair form of cyclohexane has three ${\sigma }_{\text{d}}$ planes.  Therefore, chair form of cyclohexane is achiral.

The structure of the boat form of cyclohexane is shown below in Figure 5.

Figure 5

The boat form of cyclohexane has a rotation axis $C_2$ and two ${\sigma }_{\text{v}}$ planes.  The corresponding Figure is shown below.

Figure 6

The symmetry elements of the boat form of cyclohexane are $E$, $C_2$, and ${\sigma }_{\text{v}}$.  Therefore, the point group of the boat form of cyclohexane is $C_{\text{2v}}$.

The molecules which have point group $C_{\text{s}}$, $C_{\text{n}}$ and $C_{\text{nv}}$, are said to be as a polar molecule.  The point group of boat form of cyclohexane is $C_{\text{2v}}$.  Therefore, the boat form of cyclohexane is polar.

The molecules which do not have any reflective symmetry elements (plane) but may have rotational axis are said to be as a chiral molecule.  The boat form of cyclohexane has two ${\sigma }_{\text{v}}$ planes.  Therefore, the boat form of cyclohexane is achiral.

(c)

Interpretation Introduction

Interpretation:

The symmetry elements and point group of diborane $\left({\text{B}}_{\text{2}}{\text{H}}_{\text{6}}\right)$ have to be stated.  Whether the molecule is polar or not has to be stated.  Whether the molecule is chiral or not has to be stated.

Concept introduction:

As mentioned in the concept introduction in part (a).

Answer to Problem 10A.1P

The symmetry elements of diborane $\left({\text{B}}_{\text{2}}{\text{H}}_{\text{6}}\right)$ are $E$, $C_2$, ${\sigma }_{\text{v}}$, and ${\sigma }_{\text{h}}$.  The point group of diborane $\left({\text{B}}_{\text{2}}{\text{H}}_{\text{6}}\right)$ is $D_{\text{2h}}$.

The diborane $\left({\text{B}}_{\text{2}}{\text{H}}_{\text{6}}\right)$ molecule is nonpolar and achiral.

Explanation of Solution

The structure of diborane is shown below in Figure 7.

Figure 7

Diborane $\left({\text{B}}_{\text{2}}{\text{H}}_{\text{6}}\right)$ has three rotation axis $C_2$, two vertical planes ${\sigma }_{\text{v}}$, and one horizontal plane ${\sigma }_{\text{h}}$.  The corresponding Figure is shown below.

Figure 8

The symmetry elements of diborane $\left({\text{B}}_{\text{2}}{\text{H}}_{\text{6}}\right)$ are $E$, $C_2$, ${\sigma }_{\text{v}}$, and ${\sigma }_{\text{h}}$.  Therefore the point group of diborane $\left({\text{B}}_{\text{2}}{\text{H}}_{\text{6}}\right)$ is $D_{\text{2h}}$.

The molecules which have point group $C_{\text{s}}$, $C_{\text{n}}$ and $C_{\text{nv}}$, are said to be as a polar molecule.  The point group of diborane $\left({\text{B}}_{\text{2}}{\text{H}}_{\text{6}}\right)$ is $D_{\text{2h}}$.  Therefore, the diborane $\left({\text{B}}_{\text{2}}{\text{H}}_{\text{6}}\right)$ molecule is nonpolar.

The molecules which do not have any reflective symmetry elements (plane) but may have rotational axis are said to be as a chiral molecule.  The diborane $\left({\text{B}}_{\text{2}}{\text{H}}_{\text{6}}\right)$ has two ${\sigma }_{\text{v}}$, one ${\sigma }_{\text{h}}$ planes.  Therefore, the diborane $\left({\text{B}}_{\text{2}}{\text{H}}_{\text{6}}\right)$ is achiral.

(d)

Interpretation Introduction

Interpretation:

The symmetry elements and point group of ${\left[\text{Co}{\left(\text{en}\right)}_{\text{3}}\right]}^{3+}$ have to be stated.  Whether the molecule is polar or not has to be stated.  Whether the molecule is chiral or not has to be stated.

Concept introduction:

As mentioned in the concept introduction in part (a).

Answer to Problem 10A.1P

The symmetry elements of ${\left[\text{Co}{\left(\text{en}\right)}_{\text{3}}\right]}^{3+}$ are $E$, $C_2$ and $C_3$.  The point group of ${\left[\text{Co}{\left(\text{en}\right)}_{\text{3}}\right]}^{3+}$ is $D_3$.

The ${\left[\text{Co}{\left(\text{en}\right)}_{\text{3}}\right]}^{3+}$ molecule is nonpolar but chiral.

Explanation of Solution

The structure of ${\left[\text{Co}{\left(\text{en}\right)}_{\text{3}}\right]}^{3+}$ molecule is shown below in Figure 9.

Figure 9

The ${\left[\text{Co}{\left(\text{en}\right)}_{\text{3}}\right]}^{3+}$ molecule has a principal axis $C_3$ and three rotation axis $C_2$ perpendicular to the principal axis.  There is no plane of symmetry.  The corresponding Figure is shown below.

Figure 10

The symmetry elements of ${\left[\text{Co}{\left(\text{en}\right)}_{\text{3}}\right]}^{3+}$ are $E$, $C_2$ and $C_3$.  Therefore, the point group of ${\left[\text{Co}{\left(\text{en}\right)}_{\text{3}}\right]}^{3+}$ is $D_3$.

The molecules which have point group $C_{\text{s}}$, $C_{\text{n}}$ and $C_{\text{nv}}$, are said to be as a polar molecule.  The point group of ${\left[\text{Co}{\left(\text{en}\right)}_{\text{3}}\right]}^{3+}$ is $D_3$.  Therefore, ${\left[\text{Co}{\left(\text{en}\right)}_{\text{3}}\right]}^{3+}$ is a molecule nonpolar.

The molecules which do not have any reflective symmetry elements (plane) but may have rotational axis are said to be as a chiral molecule.  The ${\left[\text{Co}{\left(\text{en}\right)}_{\text{3}}\right]}^{3+}$ has no reflective symmetry elements.  Therefore, ${\left[\text{Co}{\left(\text{en}\right)}_{\text{3}}\right]}^{3+}$ is a chiral molecule.

(e)

Interpretation Introduction

Interpretation:

The symmetry elements and point group of crown-shaped ${\text{S}}_8$ have to be stated.  Whether the molecule is polar or not has to be stated.  Whether the molecule is chiral or not has to be stated.

Concept introduction:

As mentioned in the concept introduction in part (a).

Answer to Problem 10A.1P

The symmetry elements of crown-shaped ${\text{S}}_8$ are $E$, $C_4$, $C_2$ and ${\sigma }_{\text{d}}$.  The point group of crown-shaped ${\text{S}}_8$ is $D_{\text{4d}}$.

The crown-shaped ${\text{S}}_8$ molecule is nonpolar and achiral.

Explanation of Solution

The structure of the crown-shaped ${\text{S}}_8$ molecule is shown below in Figure 11.

Figure 11

The crown-shaped ${\text{S}}_8$ molecule has a principal axis $C_4$, four rotation axis $C_2$ perpendicular to the principal axis, and four ${\sigma }_{\text{d}}$ planes.

The symmetry elements of crown-shaped ${\text{S}}_8$ are $E$, $C_4$, $C_2$ and ${\sigma }_{\text{d}}$.  Therefore, the point group of crown-shaped ${\text{S}}_8$ is $D_{\text{4d}}$.

The molecules which have point group $C_{\text{s}}$, $C_{\text{n}}$ and $C_{\text{nv}}$, are said to be as a polar molecule.  The point group of crown-shaped ${\text{S}}_8$ is $D_{\text{4d}}$.  Therefore, the crown-shaped ${\text{S}}_8$ is nonpolar.

The molecules which do not have any reflective symmetry elements (plane) but may have rotational axis are said to be as a chiral molecule.  The crown-shaped ${\text{S}}_8$ has three ${\sigma }_{\text{d}}$ planes.  Therefore, crown-shaped ${\text{S}}_8$ is achiral.