Chapter 27, Problem 27.4P

(a) How many $\text{π}$ molecular orbitals are present in deca $-1,3,5,7,9-$ pentaene

$\left({\text{CH}}_{\text{2}}=\text{CH}-\text{CH}=\text{CH}-\text{CH}=\text{CH}-\text{CH}={\text{CH}}_{\text{2}}-\text{CH}={\text{CH}}_{\text{2}}\right)$? (b) How many are bonding MOs and how many are antibonding MOs? (c) How many nodes are present in ${\text{ψ}}_{\text{1}}$? (d) How many nodes are present in ${\text{ψ}}_{\text{10}}{}^{\ast }$?

Interpretation Introduction

(a)

Interpretation: The number of $\text{π}$ molecular orbitals present in deca $-1,3,5,7,9-$ pentaene is to be stated.

Concept introduction: The electronic structures of the compounds are described by drawing the molecular orbitals. Molecular orbitals are formed from the combination of atomic orbitals. The interactions of similar and opposite phases of $p$ orbitals present in starting material result in the formation of bonding and antibonding molecular orbitals respectively.

The occurrence of more number of bonding interactions than number of nodes indicate that bonding molecular orbital is present, whereas the fewer bonding interactions than number of nodes indicate that antibonding molecular orbital is present.

Answer to Problem 27.4P

Ten $\text{π}$ molecular orbitals are present in deca $-1,3,5,7,9-$ pentaene due to the presence of five $\text{π}$ bonds.

Explanation of Solution

The molecular formula of deca $-1,3,5,7,9-$ pentaene is ${\text{CH}}_{\text{2}}=\text{CH}-\text{CH}=\text{CH}-\text{CH}=\text{CH}-\text{CH}={\text{CH}}_{\text{2}}-\text{CH}={\text{CH}}_{\text{2}}$. The formula indicates that five $\text{π}$ bonds are present in the compound. These bonds are formed by the overlap of ten $p$ orbitals, which are present on ten adjacent carbon atoms. Ten $p$ orbitals interacted in different ways to form ten molecular orbitals represented as ${\text{ψ}}_{\text{1}}-{\text{ψ}}_{\text{10}}$.

Therefore, ten $\text{π}$ molecular orbitals are present in deca $-1,3,5,7,9-$ pentaene.

Conclusion

Ten $\text{π}$ molecular orbitals are present in deca $-1,3,5,7,9-$ pentaene due to the presence of five $\text{π}$ bonds.

Interpretation Introduction

(b)

Interpretation: The numbers of bonding MOs and antibonding MOs are to be stated.

Concept introduction: The electronic structures of the compounds are described by drawing the molecular orbitals. Molecular orbitals are formed from the combination of atomic orbitals. The interactions of similar and opposite phases of $p$ orbitals present in starting material result in the formation of bonding and antibonding molecular orbitals respectively.

The occurrence of more number of bonding interactions than number of nodes indicate that bonding molecular orbital is present, whereas the fewer bonding interactions than number of nodes indicate that antibonding molecular orbital is present.

Answer to Problem 27.4P

In the given compound, five bonding molecular orbitals $\left({\text{ψ}}_{\text{1}}\text{,}{\text{ψ}}_2\text{,}{\text{ψ}}_{\text{3}}\text{,}{\text{ψ}}_4{\text{,ψ}}_5\right)$ and five antibody molecular orbitals $\left({\text{ψ}}_6\ast \text{,}{\text{ψ}}_7\ast \text{,}{\text{ψ}}_8\ast \text{,}{\text{ψ}}_9\ast {\text{,ψ}}_{10}\ast \right)$ are present.

Explanation of Solution

In the given compound, ten $p$ orbitals interacted in different ways to form ten molecular orbitals represented as ${\text{ψ}}_{\text{1}}-{\text{ψ}}_{\text{10}}$. Bonding and antibonding molecular orbitals are generated from additive and subtractive interactions respectively. Out of ten molecular orbitals, five are bonding molecular orbitals $\left({\text{ψ}}_{\text{1}}\text{,}{\text{ψ}}_2\text{,}{\text{ψ}}_{\text{3}}\text{,}{\text{ψ}}_4{\text{,ψ}}_5\right)$ and five are antibody molecular orbitals $\left({\text{ψ}}_6\ast \text{,}{\text{ψ}}_7\ast \text{,}{\text{ψ}}_8\ast \text{,}{\text{ψ}}_9\ast {\text{,ψ}}_{10}\ast \right)$.

Conclusion

In the given compound, five bonding molecular orbitals $\left({\text{ψ}}_{\text{1}}\text{,}{\text{ψ}}_2\text{,}{\text{ψ}}_{\text{3}}\text{,}{\text{ψ}}_4{\text{,ψ}}_5\right)$ and five antibody molecular orbitals $\left({\text{ψ}}_6\ast \text{,}{\text{ψ}}_7\ast \text{,}{\text{ψ}}_8\ast \text{,}{\text{ψ}}_9\ast {\text{,ψ}}_{10}\ast \right)$ are present.

Interpretation Introduction

(c)

Interpretation: The number of nodes present in ${\text{ψ}}_{\text{1}}$ is to be stated.

Concept introduction: The electronic structures of the compounds are described by drawing the molecular orbitals. Molecular orbitals are formed from the combination of atomic orbitals. The interactions of similar and opposite phases of $p$ orbitals present in starting material result in the formation of bonding and antibonding molecular orbitals respectively.

The occurrence of more number of bonding interactions than number of nodes indicate that bonding molecular orbital is present, whereas the fewer bonding interactions than number of nodes indicate that antibonding molecular orbital is present.

Answer to Problem 27.4P

The number of nodes present in ${\text{ψ}}_{\text{1}}$ molecular orbital is zero.

Explanation of Solution

The possibility of the presence of electrons is zero in nodes. The increase of molecular orbital energy results in decrease of bonding interactions, which leads to the formation of more number of nodes. The given molecular orbital ${\text{ψ}}_{\text{1}}$ is the first bonding molecular orbital. Therefore, the number of nodes present in ${\text{ψ}}_{\text{1}}$ molecular orbital is zero.

Conclusion

The number of nodes present in ${\text{ψ}}_{\text{1}}$ molecular orbital is zero.

Interpretation Introduction

(d)

Interpretation: The number of nodes present in ${\text{ψ}}_{\text{10}}{}^{\ast }$ is to be stated.

Concept introduction: The electronic structures of the compounds are described by drawing the molecular orbitals. Molecular orbitals are formed from the combination of atomic orbitals. The interactions of similar and opposite phases of $p$ orbitals present in starting material result in the formation of bonding and antibonding molecular orbitals respectively.

The occurrence of more number of bonding interactions than number of nodes indicate that bonding molecular orbital is present, whereas the fewer bonding interactions than number of nodes indicate that antibonding molecular orbital is present.

Answer to Problem 27.4P

The number of nodes present in ${\text{ψ}}_{\text{10}}{}^{\ast }$ molecular orbital is nine.

Explanation of Solution

The possibility of the presence of electrons is zero in nodes. The increase of molecular orbital energy results in decrease of bonding interactions, which leads to the formation of more number of nodes. The given molecular orbital ${\text{ψ}}_{\text{10}}{}^{\ast }$ is the tenth antibonding molecular orbital, which has highest energy in the molecular orbital diagram. Therefore, the number of nodes present in ${\text{ψ}}_{\text{10}}{}^{\ast }$ molecular orbital is nine.

Conclusion

The number of nodes present in ${\text{ψ}}_{\text{10}}{}^{\ast }$ molecular orbital is nine.