Question

Find the order of ${\text{O - O}}$ bond length in ${{\text{O}}_{\text{2}}}{\text{,}}{{\text{O}}_{\text{2}}}\left[ {{\text{As}}{{\text{F}}_{\text{4}}}} \right]$ and ${\text{K}}\left[ {{{\text{O}}_{\text{2}}}} \right]$ :
(A) ${{\text{O}}_{\text{2}}}\left[ {{\text{As}}{{\text{F}}_{\text{4}}}} \right]\langle {{\text{O}}_{\text{2}}}\langle {\text{K}}\left[ {{{\text{O}}_{\text{2}}}} \right]$
(B) ${{\text{O}}_{\text{2}}}\left[ {{\text{As}}{{\text{F}}_{\text{4}}}} \right]\langle {\text{K}}\left[ {{{\text{O}}_{\text{2}}}} \right]\langle {{\text{O}}_{\text{2}}}$
(C) ${{\text{O}}_{\text{2}}}\langle {\text{K}}\left[ {{{\text{O}}_{\text{2}}}} \right]\langle {{\text{O}}_{\text{2}}}\left[ {{\text{As}}{{\text{F}}_{\text{4}}}} \right]$
(D) ${\text{K}}\left[ {{{\text{O}}_{\text{2}}}} \right]\langle {{\text{O}}_{\text{2}}}\langle {{\text{O}}_{\text{2}}}\left[ {{\text{As}}{{\text{F}}_{\text{4}}}} \right]$

Answer 1

The bond length or the length of a bond or the bond distance can be determined by the number of bonding electrons. The bond length is found to be proportional to inverse of the bond order.

Complete step by step solution:
The three given molecules are ${{\text{O}}_{\text{2}}}{\text{,}}{{\text{O}}_{\text{2}}}\left[ {{\text{As}}{{\text{F}}_{\text{4}}}} \right]$ and ${\text{K}}\left[ {{{\text{O}}_{\text{2}}}} \right]$ .
We need to find out the order of the oxygen – oxygen, i.e., ${\text{O - O}}$ bond length of these molecules.
Now, in the molecule ${{\text{O}}_{\text{2}}}\left[ {{\text{As}}{{\text{F}}_{\text{4}}}} \right]$ , oxygen is present as ${{\text{O}}_{\text{2}}}^{\text{ + }}$ and in the molecule ${\text{K}}\left[ {{{\text{O}}_{\text{2}}}} \right]$ , oxygen is present as ${{\text{O}}_{\text{2}}}^ -$ . So we need to find out the bond order of ${{\text{O}}_{\text{2}}}^ -$ , ${{\text{O}}_{\text{2}}}$ and ${{\text{O}}_{\text{2}}}^{\text{ + }}$ .
We know from the molecular orbital theory (MOT), bond order is equal to half of the difference between the electrons in the bonding molecular orbital and the electrons in the antibonding molecular orbital. Or, bond order is equal to half of [number of electrons in the bonding molecular orbitals minus number of electrons in the antibonding molecular orbitals ]. In MOT, the antibonding orbital is designated by the asterisk symbol as a superscript on the right.
Now, from the molecular orbital theory (MOT) diagram of ${{\text{O}}_{\text{2}}}^ -$ , the electronic configuration is ${\left( {{{{\sigma }}_{{\text{1s}}}}} \right)^{\text{2}}}{\left( {{{{\sigma }}_{{\text{1s}}}}^{\text{*}}} \right)^{\text{2}}}{\left( {{{{\sigma }}_{{\text{2s}}}}} \right)^{\text{2}}}{\left( {{{{\sigma }}_{{\text{2s}}}}^{\text{*}}} \right)^{\text{2}}}{\left( {{{{\sigma }}_{{\text{2}}{{\text{p}}_{\text{z}}}}}} \right)^{\text{2}}}{\left( {{{{\pi }}_{{\text{2}}{{\text{p}}_{\text{x}}}}}} \right)^{\text{2}}}{\left( {{{{\pi }}_{{\text{2}}{{\text{p}}_{\text{y}}}}}} \right)^{\text{2}}}{\left( {{{{\pi }}_{{\text{2}}{{\text{p}}_{\text{x}}}}}^{\text{*}}} \right)^2}{\left( {{{{\pi }}_{{\text{2}}{{\text{p}}_{\text{y}}}}}^{\text{*}}} \right)^{\text{1}}}$ and we can see that there are 10 electrons in the bonding molecular orbitals and 7 electrons in the antibonding molecular orbitals. So we will have, bond order of ${{\text{O}}_{\text{2}}}^ -$ $= \dfrac{1}{2}\left[ {10 - 7} \right] = 1.5$ .
Now, from the molecular orbital theory (MOT) diagram of ${{\text{O}}_{\text{2}}}$ , the electronic configuration is ${\left( {{{{\sigma }}_{{\text{1s}}}}} \right)^{\text{2}}}{\left( {{{{\sigma }}_{{\text{1s}}}}^{\text{*}}} \right)^{\text{2}}}{\left( {{{{\sigma }}_{{\text{2s}}}}} \right)^{\text{2}}}{\left( {{{{\sigma }}_{{\text{2s}}}}^{\text{*}}} \right)^{\text{2}}}{\left( {{{{\sigma }}_{{\text{2}}{{\text{p}}_{\text{z}}}}}} \right)^{\text{2}}}{\left( {{{{\pi }}_{{\text{2}}{{\text{p}}_{\text{x}}}}}} \right)^{\text{2}}}{\left( {{{{\pi }}_{{\text{2}}{{\text{p}}_{\text{y}}}}}} \right)^{\text{2}}}{\left( {{{{\pi }}_{{\text{2}}{{\text{p}}_{\text{x}}}}}^{\text{*}}} \right)^{\text{1}}}{\left( {{{{\pi }}_{{\text{2}}{{\text{p}}_{\text{y}}}}}^{\text{*}}} \right)^{\text{1}}}$ and we can see that there are 10 electrons in the bonding molecular orbitals and 6 electrons in the antibonding molecular orbitals. So we will have, bond order of ${{\text{O}}_{\text{2}}}$ $= \dfrac{1}{2}\left[ {10 - 6} \right] = 2$ .
And from the molecular orbital theory (MOT) diagram of ${{\text{O}}_{\text{2}}}^ +$ , the electronic configuration is ${\left( {{{{\sigma }}_{{\text{1s}}}}} \right)^{\text{2}}}{\left( {{{{\sigma }}_{{\text{1s}}}}^{\text{*}}} \right)^{\text{2}}}{\left( {{{{\sigma }}_{{\text{2s}}}}} \right)^{\text{2}}}{\left( {{{{\sigma }}_{{\text{2s}}}}^{\text{*}}} \right)^{\text{2}}}{\left( {{{{\sigma }}_{{\text{2}}{{\text{p}}_{\text{z}}}}}} \right)^{\text{2}}}{\left( {{{{\pi }}_{{\text{2}}{{\text{p}}_{\text{x}}}}}} \right)^{\text{2}}}{\left( {{{{\pi }}_{{\text{2}}{{\text{p}}_{\text{y}}}}}} \right)^{\text{2}}}{\left( {{{{\pi }}_{{\text{2}}{{\text{p}}_{\text{x}}}}}^{\text{*}}} \right)^{\text{1}}}$ and we can see that there are 10 electrons in the bonding molecular orbitals and 5 electrons in the antibonding molecular orbitals. So we will have, bond order of ${{\text{O}}_{\text{2}}}^{\text{ + }}$ $= \dfrac{1}{2}\left[ {10 - 5} \right] = 2.5$ .
So the bond order increases as ${{\text{O}}_{\text{2}}}^{\text{ - }}$> ${{\text{O}}_{\text{2}}}$ > ${{\text{O}}_{\text{2}}}^{\text{ + }}$. This means the bond length of the ${\text{O - O}}$ bond decreases as ${{\text{O}}_{\text{2}}}^{\text{ - }}$< ${{\text{O}}_{\text{2}}}$ < ${{\text{O}}_{\text{2}}}^{\text{ + }}$ or increases as ${{\text{O}}_{\text{2}}}^ +$ > ${{\text{O}}_{\text{2}}}$> ${{\text{O}}_{\text{2}}}^ -$. So the bond length of the ${\text{O - O}}$ bond increases as ${{\text{O}}_{\text{2}}}\left[ {{\text{As}}{{\text{F}}_{\text{4}}}} \right]$> ${{\text{O}}_{\text{2}}}$> ${\text{K}}\left[ {{{\text{O}}_{\text{2}}}} \right]$ .
So the option A is correct.

Note:
The order of bond strength can also be determined from the bond order as it is directly proportional to the bond order. Longer bonds have lesser bond strengths and hence, bond strength is inversely proportional to bond length and directly proportional to bond order. Zero bond order indicates that no bond exists between the two atoms in question.