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

1. Bond Order: Bond order is the number of chemical bonds between a pair of atoms in a molecule. It is calculated as the difference between the number of bonding electrons and the number of antibonding electrons, divided by 2. A higher bond order implies a stronger bond. 2. Bond Length: Bond length is the distance between the nuclei of two bonded atoms in a molecule. As the bond order increases, the bond becomes stronger, and the two nuclei are pulled closer to each other. Therefore, there is an inverse relationship between bond order and bond length - as bond order increases, bond length decreases. 3. Bond Energy: Bond energy is the energy required to break a chemical bond and form neutral atoms. A higher bond order corresponds to a stronger bond, which means more energy will be required to break it. So, bond energy and bond order are directly proportional. In summary, as bond order increases, the bond length decreases, and the bond energy increases.

1. Be₂ Electronic Configuration: Be has an electronic configuration of [He] 2s². When two Be atoms come together to form a Be₂ molecule, the 2s orbitals of both Be atoms will combine to form bonding (σ) and antibonding (σ*) molecular orbitals. Thus, the molecular orbital configuration for Be₂ is (σ₂s)²(σ₂s*)². 2. Be₂ Bond Order Calculation: The bond order for Be₂ can be calculated as (number of electrons in bonding orbitals - number of electrons in antibonding orbitals) / 2 = (2 - 2) / 2 = 0 3. Existence of Be₂: Since the bond order for Be₂ is 0, it means that there's essentially no bond, and hence Be₂ is not stable and not expected to exist. 4. Be₂⁺ Electronic Configuration: For Be₂⁺, one electron is removed from the antibonding orbital. So, the molecular orbital configuration becomes (σ₂s)²(σ₂s*)¹. 5. Be₂⁺ Bond Order Calculation: The bond order for Be₂⁺ is (2 - 1) / 2 = 0.5 6. Existence of Be₂⁺: In this case, the bond order for Be₂⁺ is 0.5, meaning that there's a weak bond. However, this weak bond is not enough to overcome the electrostatic repulsion between the two Be nuclei. Therefore, Be₂⁺ is also not expected to exist. In conclusion, according to molecular orbital theory, neither Be₂ nor Be₂⁺ are expected to exist due to their bond orders.