(4)

$O_2^{+} :\sigma 1s^2 {\sigma }^{*}1s^2 \sigma 2s^2 {\sigma }^{*}2s^2 \sigma 2p_z^2 \pi 2p_x^2=\pi 2p_y^2 {\pi }^{*}2p_x^1$

Due to the presence of one unpaired electron, $O_2^{+}$ is paramagnetic in nature.

(1)

${\mathrm{He}}_2$ does not exist as it has zero bond order.
$H{e}_2 :\sigma 1s^2,{\sigma }^{*}1s^2$

$\text{Bond order}=\frac{1}{2}(N_b-N_a)=\frac{1}{2}(2-2)=0$

(4)

$B{e}_2\left(8\right):KK\sigma {\left(2s\right)}^2 {\sigma }^{*}{\left(2s\right)}^2$

$O_2\left(16\right):KK\sigma {\left(2s\right)}^2 {\sigma }^{*}{\left(2s\right)}^2 \sigma {\left(2p_z\right)}^2\pi {\left(2p_x\right)}^2=\pi {\left(2p_y\right)}^2{\pi }^{*}{\left(2p_x\right)}^1={\pi }^{*}{\left(2p_y\right)}^1$

$N_2\left(14\right):KK\sigma {\left(2s\right)}^2 {\sigma }^{*}{\left(2s\right)}^2\pi {\left(2p_x\right)}^2=\pi {\left(2p_y\right)}^2 \sigma {\left(2p_z\right)}^2$

$C_2\left(12\right):KK\sigma {\left(2s\right)}^2 {\sigma }^{*}{\left(2s\right)}^2 \pi {\left(2p_x\right)}^2=\pi {\left(2p_y\right)}^2$

Therefore, $C_2$ contains 2 $\pi$ bonds as it has 4 electrons in two pi-molecular orbitals.

(4)

$N_2\left(14\right):KK\sigma 2s^2 {\sigma }^{*}2s^2 \pi 2p_x^2=\pi 2p_y^2 \sigma 2p_z^2 ;$ Diamagnetic

$H_2\left(2\right):\sigma 1s^2$ ; Diamagnetic

$L{i}_2\left(6\right):\sigma 1s^2 {\sigma }^{*}1s^2 \sigma 2s^2 ;$ Diamagnetic

$O_2\left(16\right):\sigma 1s^2 {\sigma }^{*}1s^2 \sigma 2s^2 {\sigma }^{*}2s^2 \sigma 2p_z^2 \pi 2p_x^2=\pi 2p_y^2 {\pi }^{*}2p_x^1={\pi }^{*}2p_y^1 ;$ Paramagnetic

(4)

$O_2\left(16\right):KK\sigma 2s^2 {\sigma }^{*}2s^2 \sigma 2p_z^2 \pi 2p_x^2=\pi 2p_y^2 {\pi }^{*}2p_x^1={\pi }^{*}2p_y^1$

$\mathrm{Bond} \mathrm{order}=\frac{1}{2}(8-4)=2$

$O_2^{2-}\left(18\right):KK\sigma 2s^2 {\sigma }^{*}2s^2 \sigma 2p_z^2 \pi 2p_x^2=\pi 2p_y^2 {\pi }^{*}2p_x^2={\pi }^{*}2p_y^2$

Bond order =  $\frac{1}{2}(8-6)=1$

$O_2^{-} \left(17\right):KK\sigma 2s^2 {\sigma }^{*}2s^2 \sigma 2p_z^2 \pi 2p_x^2=\pi 2p_y^2 {\pi }^{*}2p_x^2={\pi }^{*}2p_y^1$

Bond order = $\frac{1}{2}(8-5)=1.5$

$O_2^{+} \left(15\right):KK\sigma 2s^2 {\sigma }^{*}2s^2 \sigma 2p_z^2 \pi 2p_x^2=\pi 2p_y^2 {\pi }^{*}2p_x^1$

Bond order = $\frac{1}{2}(8-3)=2.5$

As, bond order $\propto$ stability

The decreasing order of stability is: ${\text{O}}_2^{+}>{\text{O}}_2>{\text{O}}_2^{-}>{\text{O}}_2^{2-}$

(2)

                $O_2^{-}\mathit{<}{O}_2^{+}\mathit{<}{O}_2^{2+}$
$\mathbf{\text{B.O.}}: 1.5 2.5 3.0$