(a) What are the relationships among bond order, bond length, and bond energy? (b) According to molecular orbital theory, would either \(\mathrm{Be}\), or \(\mathrm{Be}_{2}^{+}\) be expected to exist? Explain.

1. Bond Order: Bond order is defined as the number of chemical bonds between a pair of atoms in a molecule. It can take both integer and fractional values. Bond order can be calculated using molecular orbital theory by taking the difference between the number of bonding electrons and the number of antibonding electrons, divided by 2. The bond order is related to the bond length and bond energy as follows: - Higher bond order leads to a shorter bond length and higher bond energy. - Lower bond order leads to a longer bond length and lower bond energy. 2. Bond Length: Bond length, the distance between the nuclei of two bonded atoms, is determined by the balance between attractive forces (due to the shared electrons) and repulsive forces (due to the positively charged nuclei). As bond order increases, the bond length decreases because there is a stronger attraction between the atoms, which pulls the nuclei closer together. 3. Bond Energy: Bond energy is the energy required to break a chemical bond between two atoms. As bond order increases, the bond strength increases due to a larger electron concentration between the two bonded atoms. Consequently, the bond energy increases, requiring more energy to break the bond. In summary, bond order, bond length, and bond energy are interconnected. Higher bond order is associated with shorter bond length and higher bond energy, while lower bond order corresponds to longer bond length and lower bond energy.

1. Consider Be atom: Beryllium (Be) has an atomic number of 4, which means it has 4 electrons. Its ground state electronic configuration is \(1s^{2}2s^{2}\). Since the electronic configuration does not violate any rules of atomic structure, Be atom can indeed exist. 2. Consider Be2+ ion: To determine the existence of the Be2+ ion, we need to draw its molecular orbital diagram and examine its stability. The molecular orbital diagram for the Be2+ ion would include: - Two 1s orbitals (one from each Be atom) overlapping to form a sigma1s bonding orbital and a sigma*1s antibonding orbital. - Two 2s orbitals (one from each Be atom) overlapping to form a sigma2s bonding orbital and a sigma*2s antibonding orbital. The 9 electrons in Be2+ would fill the molecular orbitals in the following order: 1. Two electrons fill the sigma1s bonding orbital. 2. Two electrons fill the sigma*1s antibonding orbital. 3. Two electrons fill the sigma2s bonding orbital. 4. One electron fills the sigma*2s antibonding orbital (leaving one unpaired electron). The bond order for Be2+ ion is: \(\frac{(2 - 2)}{2} = 0\) Since the bond order of Be2+ is zero, this means that there is no net bond between the two Be atoms. Therefore, Be2+ is not expected to exist according to molecular orbital theory as it is not stable due to the presence of the unpaired electron and zero bond order. In conclusion, while Be can exist, Be2+ is not expected to exist according to the molecular orbital theory.