Place the species \(\mathrm{B}_{2}^{+}, \mathrm{B}_{2},\) and \(\mathrm{B}_{2}^{-}\) in order of increasing bond length and increasing bond energy.

Count the valence electrons of each species to determine their electron configurations. - \(\mathrm{B}_{2}^{+}\): Each boron atom contributes 3 valence electrons, and there is a charge of +1, so we have a total of 3 + 3 - 1 = 5 valence electrons. - \(\mathrm{B}_{2}\): Each boron atom contributes 3 valence electrons, so we have a total of 3 + 3 = 6 valence electrons. - \(\mathrm{B}_{2}^{-}\): Each boron atom contributes 3 valence electrons, and there is a charge of -1, so we have a total of 3 + 3 + 1 = 7 valence electrons.

Utilize the formula for bond order which is: \[Bond \, Order = \frac{1}{2} \times (number \, of \, bonding \, electrons - number \, of \, antibonding \, electrons)\] - For \(\mathrm{B}_2^+\), there are 5 valence electrons. According to molecular orbital theory, the electron configuration is \(\sigma_{1s}^2\sigma^*_{1s}^2\sigma_{2s}^1\). Therefore, the bond order is \(\frac{1}{2}(3-2)=0.5\). - For \(\mathrm{B}_2\), there are 6 valence electrons. The electron configuration is \(\sigma_{1s}^2\sigma^*_{1s}^2\sigma_{2s}^2\). Therefore, the bond order is \(\frac{1}{2}(4-2)=1\). - For \(\mathrm{B}_2^-\), there are 7 valence electrons. The electron configuration is \(\sigma_{1s}^2\sigma^*_{1s}^2\sigma_{2s}^2\sigma_{2p_x}^1\). Therefore, the bond order is \(\frac{1}{2}(5-2)=1.5\).

As mentioned, higher bond order corresponds to shorter bond length. Arrange the species based on their bond orders found in step 2: - \(\mathrm{B}_{2}^{+}\): Bond order = 0.5 (Longest bond length) - \(\mathrm{B}_{2}\): Bond order = 1 (Medium bond length) - \(\mathrm{B}_{2}^{-}\): Bond order = 1.5 (Shortest bond length) So the order of increasing bond length is: \(\mathrm{B}_{2}^{+} < \mathrm{B}_{2} < \mathrm{B}_{2}^{-}\).

Higher bond order also corresponds to greater bond energy. Using the bond orders calculated in step 2, arrange the species based on their bond energies: - \(\mathrm{B}_{2}^{+}\): Bond order = 0.5 (Lowest bond energy) - \(\mathrm{B}_{2}\): Bond order = 1 (Medium bond energy) - \(\mathrm{B}_{2}^{-}\): Bond order = 1.5 (Highest bond energy) So the order of increasing bond energy is: \(\mathrm{B}_{2}^{+} < \mathrm{B}_{2} < \mathrm{B}_{2}^{-}\).