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 \(\mathrm{Be}_{2}^{+}\)be expected to exist? Explain.

Short answer

(a) Bond order, bond length, and bond energy are interconnected. As bond order increases, bond length decreases and bond energy increases. This is because an increased bond order results in stronger and shorter bonds due to the greater attraction between electrons and nuclei. (b) Using molecular orbital theory, Be2 has a bond order of 0, indicating it does not form a stable molecule. Be2+ has a bond order of 0.5, suggesting the possibility of a weak bond. However, because its bond order is significantly lower than 1, the existence of Be2+ under normal conditions is doubtful.

Step-by-step solution

(a) Relationship among bond order, bond length, and bond energy

Bond order, bond length, and bond energy are related properties that describe the stability and nature of chemical bonds in molecules. Bond order represents the number of chemical bonds between two atoms in a molecule. It can be calculated as the difference between the number of bonding electrons and antibonding electrons, divided by 2: \[Bond\,order = \frac{(Number\,of\,bonding\,electrons - umber\,of\,antibonding\,electrons)}{2}\] Bond length is the distance between the nuclei of two bonded atoms. Bond energy is the energy required to break a chemical bond and form separate atoms. These properties are interconnected, as follows: 1. As bond order increases, bond length decreases. This happens because as more electrons are participating in forming the bond, they are more attracted to the positively charged nuclei, leading to a stronger and shorter bond. 2. As bond order increases, bond energy also increases. Greater bond order means that electrons are more strongly attracted to the nuclei, resulting in a more stable bond and requiring more energy to break it.

(b) Molecular orbital theory for Be2 and Be2+

Molecular orbital (MO) theory is a powerful tool to predict the stability and electronic configuration of molecules. To determine whether Be2 or Be2+ exist, we will use MO diagrams to evaluate their bond orders. For Be2: 1. Identify the atomic orbitals: Both beryllium atoms have an electronic configuration of 1s² 2s². 2. Combine the atomic orbitals to form molecular orbitals: For a diatomic molecule, the atomic orbitals of the two atoms combine to form bonding σ and antibonding σ* molecular orbitals. Note that for Be2, 2s electrons are involved. 3. Fill the molecular orbitals with electrons: Be2 has a total of 4 electrons in the valence shell (2 electrons from each Be atom). In the MO diagram, these electrons fill both the bonding σ and antibonding σ* orbitals. 4. Calculate bond order: The bond order is 0 since the number of bonding and antibonding electrons is equal. For Be2+: 1. Identify the atomic orbitals: One of the Be2 electrons is removed, resulting in an electronic configuration of 2s² for one Be and 2s¹ for the other. 2. Combine atomic orbitals to form molecular orbitals: As previously, diatomic molecule has σ and σ* molecular orbitals. 3. Fill the molecular orbitals with electrons: Be2+ has a total of 3 valence electrons. These electrons will fill both the bonding σ and antibonding σ* orbitals (σ: 2 electrons, σ*: 1 electron) 4. Calculate bond order: Bond order = (2 - 1) / 2 = 0.5 Conclusion: According to molecular orbital theory, Be2 has a bond order of 0, meaning that it does not form a stable molecule, whereas Be2+ has a bond order of 0.5, suggesting that it may exist as a relatively weak bond. However, given that the bond order is significantly lower than 1, the existence of Be2+ would be in doubt under normal conditions.

Key concepts

Bond Order

Bond order is a fundamental concept in molecular orbital theory. It gives us insight into the number of chemical bonds between a pair of atoms. Essentially, bond order indicates how many bonding interactions are present over the antibonding interactions in a molecule.
It is calculated using the formula:

• \[Bond\,order = \frac{(Number\,of\,bonding\,electrons - Number\,of\,antibonding\,electrons)}{2}\]

A higher bond order implies a stronger and more stable bond. For instance, a bond order of 1 suggests a single bond, while a bond order of 2 suggests a double bond. If the bond order is zero, it’s unlikely for a stable bond to exist between the atoms. Understanding bond order helps predict the stability and existence of molecules, such as whether the beryllium molecule \(\text{Be}_2\) can form.

Bond Length

Bond length refers to the distance between the nuclei of two bonded atoms. It is a critical parameter in understanding molecular structure.

Typically, a shorter bond length denotes a stronger bond due to greater attractive forces between the atoms' nuclei and the electrons participating in the bond. As bond order increases, bond length generally decreases. This is because additional bonding interactions pull the atoms closer together. Therefore, evaluating bond length can inform us about the bond's strength and stability. For example, a double bond is shorter than a single bond because of its increased bond order.

Bond Energy

Bond energy is the energy required to break a bond between two atoms. It is a direct reflection of bond strength and is usually expressed in kilojoules per mole.
The relationship between bond energy and bond order is vital. As bond order increases, bond energy generally increases as well. More bonding interactions make a bond harder and more energy-intensive to break. Understanding this concept helps clarify why molecules with higher bond order are often more stable. In practical applications, knowledge about bond energy is essential for understanding reactions and stability under different conditions.

Be2 Molecule

The \(\text{Be}_2\) molecule provides an intriguing example in molecular orbital theory. According to molecular orbital theory, for \(\text{Be}_2\), the 1s and 2s orbitals of the two beryllium atoms combine to form molecular orbitals.
When filled with electrons, these orbitals result in equal numbers of bonding and antibonding electrons, leading to a bond order of zero. This calculation implies that \(\text{Be}_2\) is not stable and therefore unlikely to exist under normal conditions. The absence of a net bonding interaction due to zero bond order can conceptually explain the instability and lack of existence for the \(\text{Be}_2\) molecule.

Chemical Bond Stability

Chemical bond stability refers to the likelihood of a bond remaining intact under various conditions. The stability of a bond is influenced by several factors:

• Bond order: Higher bond orders generally suggest more stable bonds.
• Bond energy: Greater bond energy implies a more stable bond, as more energy is required to break it.
• Electronegativity: Significant differences in electronegativity between bonded atoms result in stronger bonds.

Stability is crucial in determining whether certain molecules can exist. For instance, while \(\text{Be}_2\) shows instability due to its bond order of zero, \(\text{Be}_2^{+}\) might exist with a bond order of 0.5, albeit with weak bond stability. Chemical bond stability is critical for understanding molecular behavior and predicting reactions in chemistry.