Chapter 9, Problem 9B.4E

(a)

Interpretation Introduction

Interpretation:

The ground-state electron configuration of $H_2^{-}$ molecule has to be given.

Concept Introduction:

Molecular orbital (MO) theory is a method of determining molecular structure in which electrons are not assigned to individual bonds between atoms, but are treated as moving under the influence of the nuclei in the whole molecule.

According to this theory there are two types of orbitals,

• Bonding orbitals
• Antibonding orbitals

Electrons in molecules are filled in accordance with the energy; the anti-bonding orbital has more energy than the bonding orbitals.

The electronic configuration of oxygen molecule $O_2$ can be represented as follows,

${\left({\text{σ}}_{\text{1s}}\right)}^2{\left({\text{σ*}}_{\text{1s}}\right)}^2{\left({\text{σ}}_{\text{2s}}\right)}^2{\left({\text{σ*}}_{\text{2s}}\right)}^2{\left({\text{σ}}_{\text{2p}}\right)}^2{\left( {\pi }_{\text{2p}}\right)}^4{\left( \pi {*}_{\text{2p}}\right)}^2$

The symbol * represent the antibonding orbital

Sigma (σ) bonds are the bonds in which shared hybrid orbital’s electron density are concentrated along the internuclear axis.

Pi (π) bonds are the bonds in which shared un-hybridized orbital’s (p, d, etc.) electron density are concentrated in above and below of the plane of the molecule.

Bond order: It is the measure of number of electron pairs shared between two atoms.

$\text{Bond}\text{order}\text{=}\frac{\text{1}}{\text{2}}\left(\begin{array}{l}\text{Number}\text{of}\text{electrons}\text{in}\text{bondoing}\text{MOs-}\\ \text{Number}\text{of}\text{electrons}\text{in}\text{antibonding}\text{MOs}\end{array}\right)$

Bond length is inversely proportional to the bond order.

Atoms with unpaired electrons are called Paramagnetic. Paramagnetic atoms are attracted to a magnet.

Atoms with paired electrons are called diamagnetic. Diamagnetic atoms are repelled by a magnet

Explanation of Solution

• Given species: $H_2^{-}$

There are $3$ electrons in $H_2^{-}$.

MO diagram for $H_2^{-}$ ion can be drawn as follows,

  $\begin{array}{l}\\ --\boxed{\uparrow }--\to (\text{Anti}\text{bonding}\text{electrons})\\ \left({\sigma }_{1s}\right)*\\ \\ --\boxed{\uparrow }----\boxed{\uparrow \downarrow }--\\ 1s1s\\ \\ --\boxed{\uparrow \downarrow }--\to (\text{Bonding}\text{electrons})\\ {\left({\sigma }_{1s}\right)}^2\end{array}$

According to the MO theory its ground-state electron configuration can be written as,

${\left({\text{σ}}_{\text{1s}}\right)}^2{\left({\text{σ*}}_{\text{1s}}\right)}^1$

(b)

Interpretation Introduction

Interpretation:

The ground-state electron configuration of $L{i}_2$ molecule has to be given.

Concept Introduction:

Molecular orbital (MO) theory is a method of determining molecular structure in which electrons are not assigned to individual bonds between atoms, but are treated as moving under the influence of the nuclei in the whole molecule.

According to this theory there are two types of orbitals,

• Bonding orbitals
• Antibonding orbitals

Electrons in molecules are filled in accordance with the energy; the anti-bonding orbital has more energy than the bonding orbitals.

The electronic configuration of oxygen molecule $O_2$ can be represented as follows,

${\left({\text{σ}}_{\text{1s}}\right)}^2{\left({\text{σ*}}_{\text{1s}}\right)}^2{\left({\text{σ}}_{\text{2s}}\right)}^2{\left({\text{σ*}}_{\text{2s}}\right)}^2{\left({\text{σ}}_{\text{2p}}\right)}^2{\left( {\pi }_{\text{2p}}\right)}^4{\left( \pi {*}_{\text{2p}}\right)}^2$

The symbol * represent the antibonding orbital

Sigma (σ) bonds are the bonds in which shared hybrid orbital’s electron density are concentrated along the internuclear axis.

Pi (π) bonds are the bonds in which shared un-hybridized orbital’s (p, d, etc.) electron density are concentrated in above and below of the plane of the molecule.

Bond order: It is the measure of number of electron pairs shared between two atoms.

$\text{Bond}\text{order}\text{=}\frac{\text{1}}{\text{2}}\left(\begin{array}{l}\text{Number}\text{of}\text{electrons}\text{in}\text{bondoing}\text{MOs-}\\ \text{Number}\text{of}\text{electrons}\text{in}\text{antibonding}\text{MOs}\end{array}\right)$

Bond length is inversely proportional to the bond order.

Atoms with unpaired electrons are called Paramagnetic. Paramagnetic atoms are attracted to a magnet.

Atoms with paired electrons are called diamagnetic. Diamagnetic atoms are repelled by a magnet

Explanation of Solution

• Given species: $L{i}_2$, The electronic configuration of Li element is ${\text{1s}}^{\text{2}}{\text{2s}}^{\text{1}}$

MO diagram for $L{i}_2$ species can be drawn as follows,

  $\begin{array}{l}\\ --\boxed{}--\to (\text{Anti}\text{bonding}\text{electrons})\\ \left({\sigma }_{2s}\right)*\\ \\ --\boxed{\uparrow }----\boxed{\uparrow }--\\ 2s2s\\ \\ --\boxed{\uparrow \downarrow }--\to (\text{Bonding}\text{electrons})\\ {\left({\sigma }_{2s}\right)}^2\\ \\ \\ --\boxed{\uparrow \downarrow }--\to (\text{Anti}\text{bonding}\text{electrons})\\ \left({\sigma }_{1s}\right)*\\ \\ --\boxed{\uparrow \downarrow }----\boxed{\uparrow \downarrow }--\\ 1s1s\\ \\ --\boxed{\uparrow \downarrow }--\to (\text{Bonding}\text{electrons})\\ {\left({\sigma }_{1s}\right)}^2\end{array}$

According to the MO theory its ground-state electron configuration can be written as,

${\left({\text{σ}}_{\text{1s}}\right)}^2{\left({\text{σ*}}_{\text{1s}}\right)}^2 {\left({\text{σ}}_{\text{2s}}\right)}^2$

(c)

Interpretation Introduction

Interpretation:

The ground-state electron configuration of $B{e}_2$ molecule has to be given.

Concept Introduction:

Molecular orbital (MO) theory is a method of determining molecular structure in which electrons are not assigned to individual bonds between atoms, but are treated as moving under the influence of the nuclei in the whole molecule.

According to this theory there are two types of orbitals,

• Bonding orbitals
• Antibonding orbitals

Electrons in molecules are filled in accordance with the energy; the anti-bonding orbital has more energy than the bonding orbitals.

The electronic configuration of oxygen molecule $O_2$ can be represented as follows,

${\left({\text{σ}}_{\text{1s}}\right)}^2{\left({\text{σ*}}_{\text{1s}}\right)}^2{\left({\text{σ}}_{\text{2s}}\right)}^2{\left({\text{σ*}}_{\text{2s}}\right)}^2{\left({\text{σ}}_{\text{2p}}\right)}^2{\left( {\pi }_{\text{2p}}\right)}^4{\left( \pi {*}_{\text{2p}}\right)}^2$

The symbol * represent the antibonding orbital

Sigma (σ) bonds are the bonds in which shared hybrid orbital’s electron density are concentrated along the internuclear axis.

Pi (π) bonds are the bonds in which shared un-hybridized orbital’s (p, d, etc.) electron density are concentrated in above and below of the plane of the molecule.

Bond order: It is the measure of number of electron pairs shared between two atoms.

$\text{Bond}\text{order}\text{=}\frac{\text{1}}{\text{2}}\left(\begin{array}{l}\text{Number}\text{of}\text{electrons}\text{in}\text{bondoing}\text{MOs-}\\ \text{Number}\text{of}\text{electrons}\text{in}\text{antibonding}\text{MOs}\end{array}\right)$

Bond length is inversely proportional to the bond order.

Atoms with unpaired electrons are called Paramagnetic. Paramagnetic atoms are attracted to a magnet.

Atoms with paired electrons are called diamagnetic. Diamagnetic atoms are repelled by a magnet

Explanation of Solution

• Given species: $B{e}_2$ The electronic configuration of Beryllium element is ${\text{1s}}^{\text{2}}{\text{2s}}^2$

MO diagram for $B{e}_2$ species can be drawn as follows,

  $\begin{array}{l}\\ --\boxed{\uparrow \downarrow }--\to (\text{Anti}\text{bonding}\text{electrons})\\ \left({\sigma }_{2s}\right)*\\ \\ --\boxed{\uparrow \downarrow }----\boxed{\uparrow \downarrow }--\\ 2s2s\\ \\ --\boxed{\uparrow \downarrow }--\to (\text{Bonding}\text{electrons})\\ {\left({\sigma }_{2s}\right)}^2\\ \\ \\ --\boxed{\uparrow \downarrow }--\to (\text{Anti}\text{bonding}\text{electrons})\\ \left({\sigma }_{1s}\right)*\\ \\ --\boxed{\uparrow \downarrow }----\boxed{\uparrow \downarrow }--\\ 1s1s\\ \\ --\boxed{\uparrow \downarrow }--\to (\text{Bonding}\text{electrons})\\ {\left({\sigma }_{1s}\right)}^2\end{array}$

According to the MO theory its ground-state electron configuration can be written as,

${\left({\text{σ}}_{\text{1s}}\right)}^2{\left({\text{σ*}}_{\text{1s}}\right)}^2 {\left({\text{σ}}_{\text{2s}}\right)}^2{\left({\text{σ*}}_{\text{2s}}\right)}^2$

(d)

Interpretation Introduction

Interpretation:

The ground-state electron configuration of $C_2$ molecule has to be given.

Concept Introduction:

Molecular orbital (MO) theory is a method of determining molecular structure in which electrons are not assigned to individual bonds between atoms, but are treated as moving under the influence of the nuclei in the whole molecule.

According to this theory there are two types of orbitals,

• Bonding orbitals
• Antibonding orbitals

Electrons in molecules are filled in accordance with the energy; the anti-bonding orbital has more energy than the bonding orbitals.

The electronic configuration of oxygen molecule $O_2$ can be represented as follows,

${\left({\text{σ}}_{\text{1s}}\right)}^2{\left({\text{σ*}}_{\text{1s}}\right)}^2{\left({\text{σ}}_{\text{2s}}\right)}^2{\left({\text{σ*}}_{\text{2s}}\right)}^2{\left({\text{σ}}_{\text{2p}}\right)}^2{\left( {\pi }_{\text{2p}}\right)}^4{\left( \pi {*}_{\text{2p}}\right)}^2$

The symbol * represent the antibonding orbital

Sigma (σ) bonds are the bonds in which shared hybrid orbital’s electron density are concentrated along the internuclear axis.

Pi (π) bonds are the bonds in which shared un-hybridized orbital’s (p, d, etc.) electron density are concentrated in above and below of the plane of the molecule.

Bond order: It is the measure of number of electron pairs shared between two atoms.

$\text{Bond}\text{order}\text{=}\frac{\text{1}}{\text{2}}\left(\begin{array}{l}\text{Number}\text{of}\text{electrons}\text{in}\text{bondoing}\text{MOs-}\\ \text{Number}\text{of}\text{electrons}\text{in}\text{antibonding}\text{MOs}\end{array}\right)$

Bond length is inversely proportional to the bond order.

Atoms with unpaired electrons are called Paramagnetic. Paramagnetic atoms are attracted to a magnet.

Atoms with paired electrons are called diamagnetic. Diamagnetic atoms are repelled by a magnet

Explanation of Solution

• Given species: $C_2$ The electronic configuration of Carbon element is ${\text{1s}}^{\text{2}}{\text{2s}}^2{\text{2p}}^2$

MO diagram for $C_2$ species can be drawn as follows,

  $\begin{array}{l}\left({\text{σ*}}_{{\text{2p}}_z}\right)\\ \boxed{}\to (\text{Anti}\text{bonding}\text{electrons})\\ \\ \boxed{}-\boxed{}\\ \left({\text{π*}}_{\text{2p}}{}_{{}_x}\right)\left({\text{π*}}_{{\text{2p}}_y}\right)\\ \\ --\boxed{\uparrow }-\boxed{\uparrow }-\boxed{}----\boxed{\uparrow }-\boxed{\uparrow }-\boxed{}--\\ {\text{2}}_{\text{px}}{\text{2}}_{\text{py}}{\text{2}}_{\text{pz}}{\text{2}}_{\text{px}}{\text{2}}_{\text{py}}{\text{2}}_{\text{pz}}\\ \\ \\ \boxed{}\to (\text{Bonding}\text{electrons})\\ \left({\text{σ}}_{{\text{2p}}_z}\right)\\ \\ \boxed{\uparrow \downarrow }-\boxed{\uparrow \downarrow }\\ \left({\text{π}}_{\text{2p}}{}_{{}_x}\right)\left({\text{π}}_{{\text{2p}}_y}\right)\\ \\ --\boxed{\uparrow \downarrow }--\to (\text{Anti}\text{bonding}\text{electrons})\\ \left({\sigma }_{2s}\right)*\\ \\ --\boxed{\uparrow \downarrow }----\boxed{\uparrow \downarrow }--\\ 2s2s\\ \\ --\boxed{\uparrow \downarrow }--\to (\text{Bonding}\text{electrons})\\ {\left({\sigma }_{2s}\right)}^2\\ \\ --\boxed{\uparrow \downarrow }--\to (\text{Anti}\text{bonding}\text{electrons})\\ \left({\sigma }_{1s}\right)*\\ \\ --\boxed{\uparrow \downarrow }----\boxed{\uparrow \downarrow }--\\ 1s1s\\ \\ --\boxed{\uparrow \downarrow }--\to (\text{Bonding}\text{electrons})\\ {\left({\sigma }_{1s}\right)}^2\end{array}$

According to the MO theory its ground-state electron configuration can be written as,

${\left({\text{σ}}_{\text{1s}}\right)}^2{\left({\text{σ*}}_{\text{1s}}\right)}^2{\left({\text{σ}}_{\text{2s}}\right)}^2 {\left({\text{σ*}}_{\text{2s}}\right)}^2{\left({\pi }_{{\text{2p}}_x}\right)}^2{\left({\pi }_{{\text{2p}}_y}\right)}^2$

(e)

Interpretation Introduction

Interpretation:

The ground-state electron configuration of $N_2$ molecule has to be given.

Concept Introduction:

Molecular orbital (MO) theory is a method of determining molecular structure in which electrons are not assigned to individual bonds between atoms, but are treated as moving under the influence of the nuclei in the whole molecule.

According to this theory there are two types of orbitals,

• Bonding orbitals
• Antibonding orbitals

Electrons in molecules are filled in accordance with the energy; the anti-bonding orbital has more energy than the bonding orbitals.

The electronic configuration of oxygen molecule $O_2$ can be represented as follows,

${\left({\text{σ}}_{\text{1s}}\right)}^2{\left({\text{σ*}}_{\text{1s}}\right)}^2{\left({\text{σ}}_{\text{2s}}\right)}^2{\left({\text{σ*}}_{\text{2s}}\right)}^2{\left({\text{σ}}_{\text{2p}}\right)}^2{\left( {\pi }_{\text{2p}}\right)}^4{\left( \pi {*}_{\text{2p}}\right)}^2$

The symbol * represent the antibonding orbital

Sigma (σ) bonds are the bonds in which shared hybrid orbital’s electron density are concentrated along the internuclear axis.

Pi (π) bonds are the bonds in which shared un-hybridized orbital’s (p, d, etc.) electron density are concentrated in above and below of the plane of the molecule.

Bond order: It is the measure of number of electron pairs shared between two atoms.

$\text{Bond}\text{order}\text{=}\frac{\text{1}}{\text{2}}\left(\begin{array}{l}\text{Number}\text{of}\text{electrons}\text{in}\text{bondoing}\text{MOs-}\\ \text{Number}\text{of}\text{electrons}\text{in}\text{antibonding}\text{MOs}\end{array}\right)$

Bond length is inversely proportional to the bond order.

Atoms with unpaired electrons are called Paramagnetic. Paramagnetic atoms are attracted to a magnet.

Atoms with paired electrons are called diamagnetic. Diamagnetic atoms are repelled by a magnet

Explanation of Solution

• Given species: $N_2$ The electronic configuration of Nitrogen element is ${\text{1s}}^{\text{2}}{\text{2s}}^2{\text{2p}}^3$

MO diagram for $N_2$ species can be drawn as follows,

  $\begin{array}{l}\left({\text{σ*}}_{{\text{2p}}_z}\right)\\ \boxed{}\to (\text{Anti}\text{bonding}\text{electrons})\\ \\ \boxed{}-\boxed{}\\ \left({\text{π*}}_{\text{2p}}{}_{{}_x}\right)\left({\text{π*}}_{{\text{2p}}_y}\right)\\ \\ --\boxed{\uparrow }-\boxed{\uparrow }-\boxed{\uparrow }----\boxed{\uparrow }-\boxed{\uparrow }-\boxed{\uparrow }--\\ {\text{2}}_{\text{px}}{\text{2}}_{\text{py}}{\text{2}}_{\text{pz}}{\text{2}}_{\text{px}}{\text{2}}_{\text{py}}{\text{2}}_{\text{pz}}\\ \\ \\ \boxed{\uparrow \downarrow }\to (\text{Bonding}\text{electrons})\\ \left({\text{σ}}_{{\text{2p}}_z}\right)\\ \\ \boxed{\uparrow \downarrow }-\boxed{\uparrow \downarrow }\\ \left({\text{π}}_{\text{2p}}{}_{{}_x}\right)\left({\text{π}}_{{\text{2p}}_y}\right)\\ \\ --\boxed{\uparrow \downarrow }--\to (\text{Anti}\text{bonding}\text{electrons})\\ \left({\sigma }_{2s}\right)*\\ \\ --\boxed{\uparrow \downarrow }----\boxed{\uparrow \downarrow }--\\ 2s2s\\ \\ --\boxed{\uparrow \downarrow }--\to (\text{Bonding}\text{electrons})\\ {\left({\sigma }_{2s}\right)}^2\\ \\ --\boxed{\uparrow \downarrow }--\to (\text{Anti}\text{bonding}\text{electrons})\\ \left({\sigma }_{1s}\right)*\\ \\ --\boxed{\uparrow \downarrow }----\boxed{\uparrow \downarrow }--\\ 1s1s\\ \\ --\boxed{\uparrow \downarrow }--\to (\text{Bonding}\text{electrons})\\ {\left({\sigma }_{1s}\right)}^2\end{array}$

According to the MO theory its ground-state electron configuration can be written as,

${\left({\text{σ}}_{\text{1s}}\right)}^2{\left({\text{σ*}}_{\text{1s}}\right)}^2{\left({\text{σ}}_{\text{2s}}\right)}^2 {\left({\text{σ*}}_{\text{2s}}\right)}^2{\left({\pi }_{{\text{2p}}_x}\right)}^2{\left({\pi }_{{\text{2p}}_y}\right)}^2{\left({\text{σ}}_{{\text{2p}}_z}\right)}^2$

(f)

Interpretation Introduction

Interpretation:

The ground-state electron configuration of $O_2$ molecule has to be given.

Concept Introduction:

Molecular orbital (MO) theory is a method of determining molecular structure in which electrons are not assigned to individual bonds between atoms, but are treated as moving under the influence of the nuclei in the whole molecule.

According to this theory there are two types of orbitals,

• Bonding orbitals
• Antibonding orbitals

Electrons in molecules are filled in accordance with the energy; the anti-bonding orbital has more energy than the bonding orbitals.

The electronic configuration of oxygen molecule $O_2$ can be represented as follows,

${\left({\text{σ}}_{\text{1s}}\right)}^2{\left({\text{σ*}}_{\text{1s}}\right)}^2{\left({\text{σ}}_{\text{2s}}\right)}^2{\left({\text{σ*}}_{\text{2s}}\right)}^2{\left({\text{σ}}_{\text{2p}}\right)}^2{\left( {\pi }_{\text{2p}}\right)}^4{\left( \pi {*}_{\text{2p}}\right)}^2$

The symbol * represent the antibonding orbital

Sigma (σ) bonds are the bonds in which shared hybrid orbital’s electron density are concentrated along the internuclear axis.

Pi (π) bonds are the bonds in which shared un-hybridized orbital’s (p, d, etc.) electron density are concentrated in above and below of the plane of the molecule.

Bond order: It is the measure of number of electron pairs shared between two atoms.

$\text{Bond}\text{order}\text{=}\frac{\text{1}}{\text{2}}\left(\begin{array}{l}\text{Number}\text{of}\text{electrons}\text{in}\text{bondoing}\text{MOs-}\\ \text{Number}\text{of}\text{electrons}\text{in}\text{antibonding}\text{MOs}\end{array}\right)$

Bond length is inversely proportional to the bond order.

Atoms with unpaired electrons are called Paramagnetic. Paramagnetic atoms are attracted to a magnet.

Atoms with paired electrons are called diamagnetic. Diamagnetic atoms are repelled by a magnet

Explanation of Solution

• Given species: $O_2$ The electronic configuration of Oxygen element is ${\text{1s}}^{\text{2}}{\text{2s}}^2{\text{2p}}^4$

MO diagram for $O_2$ species can be drawn as follows,

  $\begin{array}{l}\left({\text{σ*}}_{{\text{2p}}_z}\right)\\ \boxed{}\to (\text{Anti}\text{bonding}\text{electrons})\\ \\ \boxed{\uparrow }-\boxed{\uparrow }\\ \left({\text{π*}}_{\text{2p}}{}_{{}_x}\right)\left({\text{π*}}_{{\text{2p}}_y}\right)\\ \\ --\boxed{\uparrow \downarrow }-\boxed{\uparrow }-\boxed{\uparrow }----\boxed{\uparrow \downarrow }-\boxed{\uparrow }-\boxed{\uparrow }--\\ {\text{2}}_{\text{px}}{\text{2}}_{\text{py}}{\text{2}}_{\text{pz}}{\text{2}}_{\text{px}}{\text{2}}_{\text{py}}{\text{2}}_{\text{pz}}\\ \\ \\ \boxed{\uparrow \downarrow }-\boxed{\uparrow \downarrow }\\ \left({\text{π}}_{\text{2p}}{}_{{}_x}\right)\left({\text{π}}_{{\text{2p}}_y}\right)\\ \\ \boxed{\uparrow \downarrow }\to (\text{Bonding}\text{electrons})\\ \left({\text{σ}}_{{\text{2p}}_z}\right)\\ \\ --\boxed{\uparrow \downarrow }--\to (\text{Anti}\text{bonding}\text{electrons})\\ \left({\sigma }_{2s}\right)*\\ \\ --\boxed{\uparrow \downarrow }----\boxed{\uparrow \downarrow }--\\ 2s2s\\ \\ --\boxed{\uparrow \downarrow }--\to (\text{Bonding}\text{electrons})\\ {\left({\sigma }_{2s}\right)}^2\\ \\ --\boxed{\uparrow \downarrow }--\to (\text{Anti}\text{bonding}\text{electrons})\\ \left({\sigma }_{1s}\right)*\\ \\ --\boxed{\uparrow \downarrow }----\boxed{\uparrow \downarrow }--\\ 1s1s\\ \\ --\boxed{\uparrow \downarrow }--\to (\text{Bonding}\text{electrons})\\ {\left({\sigma }_{1s}\right)}^2\end{array}$

According to the MO theory its ground-state electron configuration can be written as,

${\left({\text{σ}}_{\text{1s}}\right)}^2{\left({\text{σ*}}_{\text{1s}}\right)}^2{\left({\text{σ}}_{\text{2s}}\right)}^2 {\left({\text{σ*}}_{\text{2s}}\right)}^2{\left({\pi }_{{\text{2p}}_x}\right)}^2{\left({\pi }_{{\text{2p}}_y}\right)}^2{\left({\text{σ}}_{{\text{2p}}_z}\right)}^2{\left(\pi {\text{*}}_{{\text{2p}}_x}\right)}^1{\left(\pi {\text{*}}_{{\text{*2p}}_y}\right)}^1$