Chapter 9, Problem 9B.3E

Interpretation Introduction

Interpretation:

Increasing order of bond length of the given species has to be arranged and also the bond order of each has to be identified.

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,

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 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: $O_2^{+}$

There are $15$ electrons in $O_2^{+}$. According to the MO theory its electron configuration can be written as,

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

MO diagram for $O_2^{+}$ ion can be drawn as follows,

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

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

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

$\begin{array}{l}{O}_2^{+}\\ \text{Bond}\text{order}\text{=}\frac{\text{1}}{\text{2}}\left(\begin{array}{l}\text{Number}\text{of}\text{electrons}\text{in}\text{bondoing}\text{MOs-}\\ \text{Number}\text{of}\text{electrons}\text{in}\text{antibonding}\text{MOs}\end{array}\right)\\ \text{=}\frac{\text{10-5}}{\text{2}}\text{=}2.5\end{array}$

• Given species: $O_2$

There are $16$ electrons in $O_2$. According to the MO theory its electron configuration can be written as,

$O_2:{\left({\text{σ}}_{\text{1s}}\right)}^2{\left({\text{σ*}}_{\text{1s}}\right)}^2{\left({\text{σ}}_{\text{2s}}\right)}^2{\left({\text{σ*}}_{\text{2s}}\right)}^2{\left({\text{σ}}_{\text{2p}}\right)}^2{\left( {\pi }_{\text{2p}}{}_{{}_x}\right)}^2{\left( {\pi }_{\text{2p}}{}_{{}_y}\right)}^2{\left( \pi {*}_{{\text{2p}}_x}\right)}^1 {\left( \pi {*}_{{\text{2p}}_y}\right)}^1$

$\begin{array}{l}{O}_2\\ \text{Bond}\text{order}\text{=}\frac{\text{1}}{\text{2}}\left(\begin{array}{l}\text{Number}\text{of}\text{electrons}\text{in}\text{bondoing}\text{MOs-}\\ \text{Number}\text{of}\text{electrons}\text{in}\text{antibonding}\text{MOs}\end{array}\right)\\ \text{=}\frac{\text{10-6}}{\text{2}}\text{=}2\\ \end{array}$

• Given species: $O_2{}^{-}$

There are $17$ electrons in $O_2{}^{-}$ one electron is added to the anti-bonding orbital. According to the MO theory its electron configuration can be written as,

$O_2{}^{-}:{\left({\text{σ}}_{\text{1s}}\right)}^2{\left({\text{σ*}}_{\text{1s}}\right)}^2{\left({\text{σ}}_{\text{2s}}\right)}^2{\left({\text{σ*}}_{\text{2s}}\right)}^2{\left({\text{σ}}_{\text{2p}}\right)}^2{\left( {\pi }_{\text{2p}}{}_{{}_x}\right)}^2{\left( {\pi }_{\text{2p}}{}_{{}_y}\right)}^2{\left( \pi {*}_{{\text{2p}}_x}\right)}^2 {\left( \pi {*}_{{\text{2p}}_y}\right)}^1$

$\begin{array}{l}{O}_2{}^{-}\\ \text{Bond}\text{order}\text{=}\frac{\text{1}}{\text{2}}\left(\begin{array}{l}\text{Number}\text{of}\text{electrons}\text{in}\text{bondoing}\text{MOs-}\\ \text{Number}\text{of}\text{electrons}\text{in}\text{antibonding}\text{MOs}\end{array}\right)\\ \text{=}\frac{\text{10-7}}{\text{2}}\text{=}1.5\\ \end{array}$

• Given species: $O_2{}^{2-}$

There are $18$ electrons in $O_2{}^{2-}$. According to the MO theory its electron configuration can be written as,

$O_2{}^{2-}:{\left({\text{σ}}_{\text{1s}}\right)}^2{\left({\text{σ*}}_{\text{1s}}\right)}^2{\left({\text{σ}}_{\text{2s}}\right)}^2{\left({\text{σ*}}_{\text{2s}}\right)}^2{\left({\text{σ}}_{\text{2p}}\right)}^2{\left( {\pi }_{\text{2p}}{}_{{}_x}\right)}^2{\left( {\pi }_{\text{2p}}{}_{{}_y}\right)}^2{\left( \pi {*}_{{\text{2p}}_x}\right)}^2 {\left( \pi {*}_{{\text{2p}}_y}\right)}^2$

Bond order of $O_2$ molecule and $O_2^{+}$ ion can be calculated as follows,

$\begin{array}{l}{O}_2{}^{2-}\\ \text{Bond}\text{order}\text{=}\frac{\text{1}}{\text{2}}\left(\begin{array}{l}\text{Number}\text{of}\text{electrons}\text{in}\text{bondoing}\text{MOs-}\\ \text{Number}\text{of}\text{electrons}\text{in}\text{antibonding}\text{MOs}\end{array}\right)\\ \text{=}\frac{\text{10-8}}{\text{2}}\text{=}1\\ \end{array}$

Therefore,

The bond order decreases as follows,

$\begin{array}{l}Bondorder:O_2^{+}>O_2>O_2{}^{-}>O_2{}^{2-}\\ 2.5>2>1.5>1\end{array}$

Bond order is inversely proportional to bond length.

Thus, increasing bond length is,

$Bondlength:O_2^{+}<O_2<O_2{}^{-}<O_2{}^{2-}$