Question

(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.

Short answer

(a) Bond order, bond length, and bond energy are interconnected. Higher bond order corresponds to shorter bond length and higher bond energy, while lower bond order corresponds to longer bond length and lower bond energy. (b) According to molecular orbital theory, Be can exist, but Be2+ is not expected to exist due to its zero bond order and the presence of an unpaired electron, making it unstable.

Step-by-step solution

Part (a): Relationship between bond order, bond length, and bond energy

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.

Part (b): Existence of Be or Be2+ according to molecular orbital theory

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.

Key concepts

Bond Length

Bond length is an important concept in chemistry that describes the distance between the nuclei of two bonded atoms. It is influenced by the type of chemical bond between the atoms. When two atoms form a bond, they achieve a balance between the attractive forces, which are due to the sharing of electrons, and the repulsive forces, which come from the positively charged atomic nuclei.

A higher bond order, which indicates more shared electrons between atoms, typically results in a shorter bond length. This is because the increased electron density pulls the atomic nuclei closer together, strengthening the bond. Conversely, a lower bond order often leads to a longer bond length, as the nuclei are not pulled as tightly together. Understanding bond length is crucial for determining the characteristics of the molecule, such as its size and shape. - **Higher Bond Order**: Shorter bond length - **Lower Bond Order**: Longer bond length These relationships help predict the stability and reactivity of molecules, providing insights into how molecules interact with one another.

Bond Energy

Bond energy is the measure of the energy required to break a chemical bond between two atoms, indicating the strength of that bond. The stronger the bond, the more energy will be needed to break it. Bond energy is a direct reflection of the bond order; a higher bond order means that more electrons are involved in bonding, thus creating a stronger bond that takes more energy to break.

Energy is needed to overcome the attraction between the bonded atoms' shared electrons and their positively charged nuclei. When bond energy is high, it suggests the bond is stable and the molecule is less reactive. Conversely, lower bond energy indicates a weaker bond, often leading to greater reactivity as bonds are more easily broken. The relationship is as follows: - **Higher Bond Order**: Higher bond energy - **Lower Bond Order**: Lower bond energy Understanding bond energy is important not only for predicting reactivity but also for practical applications such as energy release in combustion reactions or stability in drug molecules.

Molecular Orbital Theory

Molecular Orbital Theory (MOT) is a fundamental concept in chemistry that explains how atoms bond by considering the formation of molecular orbitals. Unlike simple atomic orbitals which belong to individual atoms, molecular orbitals are spread over a molecule and belong to the molecule as a whole.

This theory helps to explain the existence or non-existence of molecules like \(Be\) or \(Be_{2}^{+}\). According to MOT, electrons in molecules occupy molecular orbitals that are either bonding (which help to hold the atoms together) or antibonding (which can weaken or break a bond). The bond order is calculated by taking the number of electrons in bonding orbitals minus the number in antibonding orbitals, divided by two.

For a species like \(Be_{2}^{+}\), the molecular orbital diagram shows that the bond order is zero. This means there is no effective bond present, rendering the molecule unstable and hence, it is not expected to exist:- **Bond Order Calculation**: \(\frac{(number\ of\ bonding\ electrons - number\ of\ antibonding\ electrons)}{2}\)- **Be_{2}^{+} Example**: Bond order is 0Molecular Orbital Theory provides a more nuanced and accurate picture of bonding compared to other models (like the valence bond theory), especially for molecules with unusual bonding or those that are ionically charged.