Chapter 9, Problem 47GQ

(a)

Interpretation Introduction

Interpretation:

The highest energy occupied molecular orbital (HOMO) and the electron assigned in to it should be determined.

Concept Introduction:

Molecular orbital (MO) theory:  is a method for 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,

1. (1) Bonding orbitals
2. (2) 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 * represent the antibonding orbital

HOMO and LUMO: This statement stand for highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO), respectively. So this energy difference between the HOMO and LUMO is termed the HOMO–LUMO gap.

Answer to Problem 47GQ

The $\left({\text{σ}}_{\text{2p}}\right)$ electron has occupied in HOMO shell. So this electron is assigned for highest orbitals.

Explanation of Solution

Molecular orbital diagram of (CN) molecule can be drawn as

$\begin{array}{l}{\text{σ*}}_{\text{2p}}{}_{{}_{\text{z}}}\\ --\boxed{}--\to (\text{Anti}\text{bonding}\text{electrons})\\ \\ --\boxed{}-\boxed{}--\\ {\text{π*}}_{\text{2p}}{\text{π*}}_{\text{2p}}\\ \\ --\boxed{}-\boxed{}-\boxed{}----\boxed{}-\boxed{}-\boxed{}--\\ {\text{2}}_{\text{px}}{\text{2}}_{\text{py}}{\text{2}}_{\text{pz}}{\text{2}}_{\text{px}}{\text{2}}_{\text{py}}{\text{2}}_{\text{pz}}\\ \\ \\ {\text{σ}}_{\text{2pz}}\\ --\boxed{\uparrow }--\\ --\boxed{\uparrow \downarrow }-\boxed{\uparrow \downarrow }--\\ {\text{π}}_{\text{2px}}{\text{π}}_{\text{2py}}\to (\text{Bonding}\text{electrons})\\ \end{array}$

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

HOMO-LUMO Analysis: There is one unpaired electron in ${\left({\text{σ}}_{\text{2p}}\right)}^1$ HOMO orbital.

So this molecule have one sigma bond and two pi-bond, the corresponding electronic methods are given above molecular orbital correlation method.

(b)

Interpretation Introduction

Interpretation:

Bond order of the given molecule should be determined.

Concept Introduction:

Molecular orbital (MO) theory:  is a method for 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,

1. (3) Bonding orbitals
2. (4) 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,

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

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

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

The * represent the antibonding orbital

Answer to Problem 47GQ

Cyanide (CN) molecule has a bond order of $2.5$

Explanation of Solution

Molecular orbital diagram of (CN) molecule can be drawn as

$\begin{array}{l}{\text{σ*}}_{\text{2p}}{}_{{}_{\text{z}}}\\ --\boxed{}--\to (\text{Anti}\text{bonding}\text{electrons})\\ \\ --\boxed{}-\boxed{}--\\ {\text{π*}}_{\text{2p}}{\text{π*}}_{\text{2p}}\\ \\ --\boxed{}-\boxed{}-\boxed{}----\boxed{}-\boxed{}-\boxed{}--\\ {\text{2}}_{\text{px}}{\text{2}}_{\text{py}}{\text{2}}_{\text{pz}}{\text{2}}_{\text{px}}{\text{2}}_{\text{py}}{\text{2}}_{\text{pz}}\\ \\ \\ {\text{σ}}_{\text{2pz}}\\ --\boxed{\uparrow }--\\ --\boxed{\uparrow \downarrow }-\boxed{\uparrow \downarrow }--\\ {\text{π}}_{\text{2px}}{\text{π}}_{\text{2py}}\to (\text{Bonding}\text{electrons})\\ \end{array}$

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

Calculation method of bond order of CN molecule

The (CN) molecule has seven electrons available in bonding molecular orbitals and two electrons are available in antibonding molecular orbitals so this bond order has shown below. 

  $\begin{array}{l}\text{Bond}\text{order}\text{=}\frac{\text{1}}{\text{2}}\left(\begin{array}{l}\text{Number}\text{of}\text{electrons}\text{in}\text{bondoing}\text{MOs-}\\ \text{Number}\text{of}\text{electrons}\text{in}\text{antibonding}\text{MOs}\end{array}\right)\\ \text{Bond}\text{order}\text{=}\frac{(7-2)}{\text{2}}=\frac{5}{\text{2}}=2.5\end{array}$

(c)

Interpretation Introduction

Interpretation:

Number of net $\sigma and\pi$ bonds in the given molecule should be determined.

Concept Introduction:

Molecular orbital (MO) theory:  is a method for 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,

1. (1) Bonding orbitals
2. (2) 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,

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

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

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

The * 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 unhybridized orbital’s (p, d, etc) electron density are concentrated in above and below of the plane of the molecule.

Answer to Problem 47GQ

Cyanide (CN) molecule has a bond order of $2.5$ and so net $\sigma and\pi$ bond values are $0.5and2$ respectively.

Explanation of Solution

Molecular orbital diagram of (CN) molecule can be drawn as

$\begin{array}{l}{\text{σ*}}_{\text{2p}}{}_{{}_{\text{z}}}\\ --\boxed{}--\to (\text{Anti}\text{bonding}\text{electrons})\\ \\ --\boxed{}-\boxed{}--\\ {\text{π*}}_{\text{2p}}{\text{π*}}_{\text{2p}}\\ \\ --\boxed{}-\boxed{}-\boxed{}----\boxed{}-\boxed{}-\boxed{}--\\ {\text{2}}_{\text{px}}{\text{2}}_{\text{py}}{\text{2}}_{\text{pz}}{\text{2}}_{\text{px}}{\text{2}}_{\text{py}}{\text{2}}_{\text{pz}}\\ \\ \\ {\text{σ}}_{\text{2pz}}\\ --\boxed{\uparrow }--\\ --\boxed{\uparrow \downarrow }-\boxed{\uparrow \downarrow }--\\ {\text{π}}_{\text{2px}}{\text{π}}_{\text{2py}}\to (\text{Bonding}\text{electrons})\\ \end{array}$

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

Calculation method of bond order of $CN$ molecule

The (CN) molecule has seven electrons available in bonding molecular orbitals and two electrons are available in antibonding molecular orbitals so this bond order has shown below. 

  $\begin{array}{l}\text{Bond}\text{order}\text{=}\frac{\text{1}}{\text{2}}\left(\begin{array}{l}\text{Number}\text{of}\text{electrons}\text{in}\text{bondoing}\text{MOs-}\\ \text{Number}\text{of}\text{electrons}\text{in}\text{antibonding}\text{MOs}\end{array}\right)\\ \text{Bond}\text{order}\text{=}\frac{(7-2)}{\text{2}}=\frac{5}{\text{2}}=2.5\end{array}$

This value implies that the net $\sigma and\pi$ bond values in $CN$ molecule are $0.5and2$ respectively.

(d)

Interpretation Introduction

Interpretation:

It should be checked that whether the $CN$ molecule is paramagnetic or diamagnetic in nature.

Concept Introduction:

Molecular orbital (MO) theory:  is a method for 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,

1. (1) Bonding orbitals
2. (2) 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,

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

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

Molecular orbital diagram of (CN) molecule can be drawn as

$\begin{array}{l}{\text{σ*}}_{\text{2p}}{}_{{}_{\text{z}}}\\ --\boxed{}--\to (\text{Anti}\text{bonding}\text{electrons})\\ \\ --\boxed{}-\boxed{}--\\ {\text{π*}}_{\text{2p}}{\text{π*}}_{\text{2p}}\\ \\ --\boxed{}-\boxed{}-\boxed{}----\boxed{}-\boxed{}-\boxed{}--\\ {\text{2}}_{\text{px}}{\text{2}}_{\text{py}}{\text{2}}_{\text{pz}}{\text{2}}_{\text{px}}{\text{2}}_{\text{py}}{\text{2}}_{\text{pz}}\\ \\ \\ {\text{σ}}_{\text{2pz}}\\ --\boxed{\uparrow }--\\ --\boxed{\uparrow \downarrow }-\boxed{\uparrow \downarrow }--\\ {\text{π}}_{\text{2px}}{\text{π}}_{\text{2py}}\to (\text{Bonding}\text{electrons})\\ \end{array}$

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

Magnetic property:

The CN molecule has one unpaired electron in ${\left({\text{σ}}_{\text{2p}}\right)}^1$ orbital and so this molecule has paramagnetic nature.