Chapter 15, Problem 15.1P

Interpretation Introduction

(a)

Interpretation:

The $6$ molecular orbital (MOs) of conjugated $\left(E\right)-\text{1, 3, 5}-\text{hexatriene}$ with relative energies $\pm 1.80\beta$, $\pm 1.25\beta$ and $\pm 0.44\beta$ are to be sketched and classified as bonding and anti-bonding orbitals.

Concept introduction:

Dienes are the compounds that contain carbon-carbon double bonds. The term conjugated dienes is used when the double bonds are present alternatively in a hydrocarbon chain. The mathematical function that describes the wave like behavior of the electron in a molecule is expressed by molecular orbital. The physical and chemical properties are found with the help of molecular orbital.

Answer to Problem 15.1P

The bonding molecular orbitals are ${\pi }_1$, ${\pi }_2$ and ${\pi }_3$.

The anti-bonding molecular orbitals are ${\pi }_4^{*}$, ${\pi }_5^{*}$ and ${\pi }_6^{*}$.

The molecular orbitals of conjugated $\left(E\right)-\text{1, 3, 5}-\text{hexatriene}$ are given below.

Explanation of Solution

In $\left(E\right)-\text{1, 3, 5}-\text{hexatriene}$, the combination of $6$ $2p$ orbitals provides the information of $6$ molecular orbitals. Among the six molecular orbitals (MOs)\$3$ are bonding molecular orbitals (BMO) and the other three are anti-bonding (ABMO) molecular orbitals.

The three bonding molecular orbitals are ${\pi }_1$, ${\pi }_2$ and ${\pi }_3$.

The three anti bonding molecular orbitals are ${\pi }_4^{*}$, ${\pi }_5^{*}$ and ${\pi }_6^{*}$.

The sketch showing the molecular orbital of conjugated $\left(E\right)-\text{1, 3, 5}-\text{hexatriene}$ is given below.

Figure 1

Conclusion

The three bonding molecular orbitals are ${\pi }_1$, ${\pi }_2$ and ${\pi }_3$.

The three anti bonding molecular orbitals are ${\pi }_4^{*}$, ${\pi }_5^{*}$ and ${\pi }_6^{*}$.

The sketch showing the molecular orbital of conjugated $\left(E\right)-\text{1, 3, 5}-\text{hexatriene}$ is shown above in Figure 1.

Interpretation Introduction

(b)

Interpretation:

The total number of nodes in the $6$ molecular orbital (MOs) of conjugated $\left(E\right)-\text{1, 3, 5}-\text{hexatriene}$ is to be stated.

Concept introduction:

The term conjugated dienes is used when the double bonds are present alternatively in a hydrocarbon chain. The mathematical function that describes the wave like behavior of an electron in a molecule is expressed by molecular orbital. The molecular orbital contains nodes and lobes. Nodes are the regions in the molecular orbital of conjugated system in which the adjoining orbital lobes undergoes a phase change.

Answer to Problem 15.1P

The total number of nodes that the molecular orbital (MOs) of conjugated $\left(E\right)-\text{1, 3, 5}-\text{hexatriene}$ contains is such that ${\pi }_1$ has zero, ${\pi }_2$ has one, ${\pi }_3$ has two, ${\pi }_4^{*}$ has three, ${\pi }_5^{*}$ has four and ${\pi }_6^{*}$ has five nodes.

Explanation of Solution

There is no change of phase occurs in lobes in the ${\pi }_1$ molecular orbitals and therefore it contains zero nodes.

There is one change of phase occurs in lobes in the ${\pi }_2$ molecular orbitals and therefore it contains 1 nodes.

There is two change of phase occurs in lobes in the ${\pi }_3$ molecular orbitals and therefore it contains 2 nodes.

There are four change of phase occurs in lobes in the ${\pi }_4^{*}$ molecular orbitals and therefore it contains 3 nodes.

There are five change of phase occurs in lobes in the ${\pi }_5^{*}$ molecular orbitals and therefore it contains 4 nodes.

There are six change of phase occurs in lobes in the ${\pi }_6^{*}$ molecular orbitals and therefore it contains 5 nodes.

The number of nodes that molecular orbital (MOs) of conjugated $\left(E\right)-\text{1, 3, 5}-\text{hexatriene}$ contains is shown in the tabular form below.

Molecular Orbital	Number of Nodes
${\pi }_1$	Zero
${\pi }_2$	One
${\pi }_3$	Two
${\pi }_4^{*}$	Three
${\pi }_5^{*}$	Four
${\pi }_6^{*}$	Five

Conclusion

The number of nodes that molecular orbital (MOs) of conjugated $\left(E\right)-\text{1, 3, 5}-\text{hexatriene}$ contains is shown above in the tabular form.

Interpretation Introduction

(c)

Interpretation:

The position of the nodes in ${\pi }_1$, ${\pi }_2$ and ${\pi }_6^{*}$ of the molecular orbital(MOs) of conjugated $\left(E\right)-\text{1, 3, 5}-\text{hexatriene}$ is to be shown.

Concept introduction:

The term conjugated dienes is used when the double bonds are present alternatively in a hydrocarbon chain. The mathematical function that describes the wave like behavior of the electron in a molecule is expressed by molecular orbital. The molecular orbital contains nodes and lobes. Nodes are the regions in the molecular orbital of conjugated system in which the adjoining orbital lobes undergoes a phase change.

Answer to Problem 15.1P

The position of the nodes in ${\pi }_1$, ${\pi }_2$ and ${\pi }_6^{*}$ of the molecular orbital (MOs) of conjugated $\left(E\right)-\text{1, 3, 5}-\text{hexatriene}$ is shown below.

Explanation of Solution

There is no change of phase occurs in lobes in the ${\pi }_1$ molecular orbitals and therefore it contains zero nodes. Also the electron density is spread over the whole $\left(E\right)-\text{1, 3, 5}-\text{hexatriene}$. Therefore, the ${\pi }_1$ MO is delocalized.

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The sketch of ${\pi }_1$ MO is shown below.

Figure 2

There is one change of phase occurs in lobes in the ${\pi }_2$ molecular orbitals and therefore it contains zero nodes. Therefore, ${\pi }_2$ molecular orbital contain one node and the electron density is spread over two lobes separated by one node.

The sketch of ${\pi }_2$ molecular orbital is shown below.

Figure 3

There are 6 change of phase occurs in lobes in the ${\pi }_6^{*}$ molecular orbitals and therefore it contains 5 nodes.

The ${\pi }_6^{*}$ molecular orbital contain zero nodes and in this molecular orbital the electron density is spread in $6$ small lobes separated by $5$ nodes.

The sketch of ${\pi }_6^{*}$ molecular orbital is shown below.

Figure 4

Conclusion

The position of the nodes in ${\pi }_1$, ${\pi }_2$ and ${\pi }_6^{*}$ of the molecular orbital (MOs) of conjugated $\left(E\right)-\text{1, 3, 5}-\text{hexatriene}$ is shown in Figure 2, 3 and 4.