Question

(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 ${\mathrm{Be}}_2{}^7$ be expected to exist? Explain.

Short answer

(a) Bond order is directly proportional to bond energy and inversely proportional to bond length. Higher bond order corresponds to stronger bonds, shorter bond lengths, and higher bond energies. (b) Using molecular orbital theory, ${\mathrm{Be}}_2$ has a bond order of 0, and ${\mathrm{Be}}_2{}^7$ has a bond order of -1.5. Since both bond orders are less than or equal to 0, neither ${\mathrm{Be}}_2$ nor ${\mathrm{Be}}_2{}^7$ is expected to exist.

Step-by-step solution

(Part a: Understand the relationships among bond order, bond length, and bond energy)

: 1. Bond order: It is defined as the number of chemical bonds between a pair of atoms. A higher bond order corresponds to a stronger bond between atoms. 2. Bond length: It is the distance between the two atomic nuclei in a molecule. It typically decreases with increasing bond order. 3. Bond energy: It is the energy required to break a chemical bond between two atoms. Higher bond energy corresponds to a more stable bond. Bond energy usually increases with increasing bond order. Thus, bond order is directly proportional to bond energy, whereas bond length is inversely proportional to bond order.

(Part b: Assess the existence of $B{e}_2$ and $B{e}_2^7$ using molecular orbital theory)

: 1. Construct the molecular orbital diagram for ${\mathrm{Be}}_2$ and ${\mathrm{Be}}_2{}^7$ For this purpose, we consider the atomic orbitals of the Be atom first. Be's electron configuration is $1s^{2}2s^2$. Therefore, the valence electrons are in the 2s orbital. In molecular orbital theory, we combine the valence orbitals of each Be atom to form bonding (lower energy) and antibonding (higher energy) molecular orbitals. 2. Fill the molecular orbitals with electrons For ${\mathrm{Be}}_2$, there are 4 valence electrons (2 from each Be). The bonding molecular orbital (2s bonding) will be filled first, followed by the antibonding molecular orbital (2s antibonding). As a result, ${\mathrm{Be}}_2$ will have 2 electrons in the bonding and 2 electrons in the antibonding molecular orbitals. For ${\mathrm{Be}}_2{}^7$, there are 7 valence electrons. The bonding molecular orbital (2s bonding) will be filled first, followed by the antibonding molecular orbital (2s antibonding). As a result, ${\mathrm{Be}}_2{}^7$ will have 2 electrons in the bonding and 5 electrons in the antibonding molecular orbitals. 3. Calculate bond order for each species Bond order is calculated as: $(\text{number of electrons in bonding orbitals}-\text{number of electrons in antibonding orbitals})/2$ For ${\mathrm{Be}}_2$: Bond order = $(2-2)/2=0$ For ${\mathrm{Be}}_2{}^7$: Bond order = $(2-5)/2=-1.5$ 4. Assess the stability and existence of ${\mathrm{Be}}_2$ and ${\mathrm{Be}}_2{}^7$ A species with bond order greater than 0 is more stable, and thus more likely to exist. Since ${\mathrm{Be}}_2$ and ${\mathrm{Be}}_2{}^7$ both have bond orders less than or equal to 0, they are not expected to exist based on molecular orbital theory.

Key concepts

Bond Order

Bond order is a key concept in molecular orbital theory and deeply influences the bond's stability. It is defined as the number of covalent bonds shared between a pair of atoms in a molecule. This can be easily understood with a simple formula: bond order equals the difference between the number of electrons in bonding orbitals and antibonding orbitals, divided by two.

• A higher bond order indicates a greater number of shared electrons, leading to stronger attractions between the atoms.
• A bond order of zero suggests no net bonding interactions, implying instability and the molecule’s nonexistence.

In the exercise regarding $ext{Be}_2$ and $ext{Be}_2^7$, calculations showed bond orders of 0 and -1.5 respectively, indicating that these molecules are not stable as there are more electrons in antibonding than in bonding orbitals.

Bond Length

Bond length refers to the average distance between the nuclei of two bonded atoms. It is a critical determinant of the molecule’s overall structure. Bond length inversely correlates to bond order, meaning that as bond order increases, bond length usually decreases. This happens because more shared electrons pull the bonded atoms closer together.

• This shorter distance due to higher bond orders leads to a more stable bond as the atoms are closer and hold more tension together.
• Conversely, a lower bond order results in a longer bond length, as there are fewer electrons to attract the two atoms together.

Understanding this relationship helps in predicting molecular behavior and bonding characteristics in chemistry.

Bond Energy

Bond energy is the measure of bond strength in a chemical bond. Specifically, it represents the energy required to break a bond between two atoms, thereby separating them into individual atoms.

• Higher bond orders contribute to higher bond energies because more shared electrons result in a stronger attraction between the atoms, requiring more energy to break them apart.
• Conversely, bonds with lower energy will have lesser bond orders, reflecting weaker bonds that are easier to break.

In essence, bond energy provides valuable insight into the stability of molecules; the more energy needed to break a bond, the more stable that bond is considered. This is why molecules with higher bond orders and subsequently shorter bond lengths tend to have higher bond energies.