Which of the following are predicted by the molecular orbital model to be stable diatomic species? a. \(\mathrm{H}_{2}^{+}, \mathrm{H}_{2}, \mathrm{H}_{2}^{-}, \mathrm{H}_{2}^{2-}\) b. \(\mathrm{He}_{2}^{2+}, \mathrm{He}_{2}^{+}, \mathrm{He}_{2}\)

Molecular orbital theory explains how atomic orbitals mix to form molecular orbitals in molecules. These molecular orbitals determine where electrons reside in the molecule, and the overall stability of the molecule. The bond order is calculated as: \( Bond \: Order = \frac{(No. \, of \, electrons \, in \, bonding \, MOs) - (No. \, of \, electrons \, in \, antibonding \, MOs)}{2} \) A positive bond order signifies a stable molecule.

We need to determine the bond orders for the hydrogen species: \(\mathrm{H}_{2}^{+}, \mathrm{H}_{2}, \mathrm{H}_{2}^{-}, \mathrm{H}_{2}^{2-}\) Analyzing these species one by one: 1. \(\mathrm{H}_{2}^{+}\): Total of 1 electron which fills the lower energy bonding sigma orbital Bond order: \(\frac{(1-0)}{2} = \frac{1}{2}\) 2. \(\mathrm{H}_{2}\): Total of 2 electrons, filling the lower energy bonding sigma orbital Bond order: \(\frac{(2-0)}{2} = 1\) 3. \(\mathrm{H}_{2}^{-}\): Total of 3 electrons, filling the bonding sigma orbital and one in the antibonding sigma orbital Bond order: \(\frac{(2-1)}{2} = \frac{1}{2}\) 4: \(\mathrm{H}_{2}^{2-}\): Total of 4 electrons, filling both bonding sigma and antibonding sigma orbitals Bond order: \(\frac{(2-2)}{2} = 0\) All of the hydrogen species except \(\mathrm{H}_{2}^{2-}\) have positive bond orders, making them stable diatomic species.

We need to determine the bond orders for the helium species: \(\mathrm{He}_{2}^{2+}, \mathrm{He}_{2}^{+}, \mathrm{He}_{2}\) Analyzing these species one by one: 1. \(\mathrm{He}_{2}^{2+}\): Total of 2 electrons, filling the lower energy bonding sigma orbital Bond order: \(\frac{(2-0)}{2} = 1\) 2. \(\mathrm{He}_{2}^{+}\): Total of 3 electrons, filling the bonding sigma orbital and one in the antibonding sigma orbital Bond order: \(\frac{(2-1)}{2} = \frac{1}{2}\) 3. \(\mathrm{He}_{2}\): Total of 4 electrons, filling both bonding sigma and antibonding sigma orbitals Bond order: \(\frac{(2-2)}{2} = 0\) In the case of helium species, only \(\mathrm{He}_{2}^{2+}\) and \(\mathrm{He}_{2}^{+}\) have positive bond orders, making them stable diatomic species. In conclusion, stable diatomic species according to the molecular orbital model are: \[ \mathrm{H}_{2}^{+}, \mathrm{H}_{2}, \mathrm{H}_{2}^{-}, \mathrm{He}_{2}^{2+}, \mathrm{and} \, \mathrm{He}_{2}^{+} \]