Using MO theory, predict which of the following species has the shortest bond length? $\newline$ ( $1$ ) $\text{O}_2^-$ $\newline$ ( $2$ ) $\text{O}_2^{2-}$ $\newline$ ( $3$ ) $\text{O}_2^{2+}$ $\newline$ ( $4$ ) $\text{O}_2^+$

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Q. Using MO theory, predict which of the following species has the shortest bond length?

\newline

(

1

)

\text{O}_2^-

\newline

(

2

)

\text{O}_2^{2-}

\newline

(

3

)

\text{O}_2^{2+}

\newline

(

4

)

\text{O}_2^+

Define Bond Length Calculation: Molecular Orbital (MO) theory suggests that bond length is inversely related to bond order, where bond order is defined as $(\text{number of bonding electrons} - \text{number of antibonding electrons})/2$ . To determine the bond length, we need to calculate the bond order for each species.
Calculate Bond Order for O_$2$: For O_$2$, the electronic configuration in MO terms is $(\sigma_{2s})^2(\sigma^*_{2s})^2(\sigma_{2p})^2(\pi_{2p})^4(\pi^*_{2p})^2$ , which gives a bond order of $(8 \text{ bonding electrons} - 4 \text{ antibonding electrons})/2 = 2$ .
Calculate Bond Order for $O_2^-$ : For $O_2^-$ , an extra electron is added to the antibonding $\pi^*$ orbital, so the bond order becomes $(8 \text{ bonding electrons} - 5 \text{ antibonding electrons})/2 = 1.5$ .
Calculate Bond Order for $\mathrm{O}_2^{2-}$ : For $\mathrm{O}_2^{2-}$ , another extra electron is added to the antibonding $\pi^*$ orbital, so the bond order becomes $(8 \text{ bonding electrons} - 6 \text{ antibonding electrons})/2 = 1$ .
Calculate Bond Order for $O_2^{2+}$ : For $O_2^{2+}$ , two electrons are removed. According to Hund's rule, these electrons are removed from the antibonding $\pi^*$ orbitals first, so the bond order becomes $(8 \text{ bonding electrons} - 2 \text{ antibonding electrons})/2 = 3$ .
Calculate Bond Order for O_$2$⁽⁺⁾: For O_$2$⁽⁺⁾, one electron is removed from the antibonding $\pi^*$ orbital, so the bond order becomes $(8$ bonding electrons $- 3$ antibonding electrons $)/2 = 2.5$ .
Compare Bond Orders: Comparing the bond orders, we find that $\text{O}_2^{2+}$ has the highest bond order of $3$ , which means it has the shortest bond length among the given species.