Question

Place the species \(\mathrm{B}_{2}+, \mathrm{B}_{2}\), and \(\mathrm{B}_{2}^{-}\) in order of increasing bond length and increasing bond energy.

Short answer

The species \(\mathrm{B}_{2}+, \mathrm{B}_{2}\), and \(\mathrm{B}_{2}^{-}\) can be ordered by increasing bond length as \(\mathrm{B_{2}^{+}}\), \(\mathrm{B}_{2}\), \(\mathrm{B}_{2}^{-}\) and by increasing bond energy as \(\mathrm{B}_{2}^{-}\), \(\mathrm{B}_{2}\), \(\mathrm{B_{2}^{+}}\).

Step-by-step solution

1. Determine the Total Number of Electrons in Each Species

First, let's determine the total number of electrons in each species: - \(\mathrm{B_{2}^{+}}\): Boron has 5 electrons, but this species has a positive charge, so it has lost an electron. Therefore, it has (2 × 5) - 1 = 9 electrons total. - \(\mathrm{B}_{2}\): This is a neutral molecule with each boron atom having 5 electrons, so it has 2 × 5 = 10 electrons total. - \(\mathrm{B}_{2}^{-}\): This species has a negative charge, indicating that it has gained an electron. So, it has (2 × 5) + 1 = 11 electrons total.

2. Fill Molecular Orbitals

Now, let's fill the molecular orbitals with electrons using the molecular orbital diagram for B₂. Use the Aufbau principle (fill lower energy orbitals first), the Pauli Exclusion Principle (maximum of 2 electrons per orbital), and Hund's rule (maximize the number of parallel spins). The molecular orbitals for B₂ are in the following order (starting from lowest energy): 1. σ(2s) 2. σ*(2s) 3. [σ(2p) and two π(2p) orbitals degenerate in energy, i.e., at the same energy level] 4. [σ(2p)* and two π*(2p) orbitals degenerate in energy] Energies of molecular orbitals decrease with increasing bond order. Bond order determines bond strength and length. Higher bond order correlates with shorter bond length and greater bond strength. Now, fill the molecular orbitals for each molecular species based on their total number of electrons: - \(\mathrm{B_{2}^{+}}\) (9 electrons): σ(2s)², σ*(2s)², σ(2p)², π(2p)³ - \(\mathrm{B}_{2}\) (10 electrons): σ(2s)², σ*(2s)², σ(2p)², π(2p)⁴ - \(\mathrm{B}_{2}^{-}\) (11 electrons): σ(2s)², σ*(2s)², σ(2p)², π(2p)⁴, σ(2p)*¹

3. Calculate the Bond Order

Next, we'll calculate the bond order for each species by using the formula: Bond order = (number of electrons in bonding orbitals - number of electrons in antibonding orbitals)/2 - \(\mathrm{B_{2}^{+}}\): Bond order = (6-3)/2 = 1.5 - \(\mathrm{B}_{2}\): Bond order = (6-4)/2 = 1 - \(\mathrm{B}_{2}^{-}\): Bond order = (6-5)/2 = 0.5

4. Order Species by Bond Length and Bond Energy

We know that higher bond order means greater bond energy (strength) and shorter bond length. Based on the bond order values we calculated, we can arrange the species as follows: - Bond length (increasing order): \(\mathrm{B_{2}^{+}}\) < \(\mathrm{B}_{2}\) < \(\mathrm{B}_{2}^{-}\) - Bond energy (increasing order): \(\mathrm{B}_{2}^{-}\) < \(\mathrm{B}_{2}\) < \(\mathrm{B_{2}^{+}}\) So, the species \(\mathrm{B}_{2}+, \mathrm{B}_{2}\), and \(\mathrm{B}_{2}^{-}\) can be ordered by increasing bond length as \(\mathrm{B_{2}^{+}}\), \(\mathrm{B}_{2}\), \(\mathrm{B}_{2}^{-}\) and by increasing bond energy as \(\mathrm{B}_{2}^{-}\), \(\mathrm{B}_{2}\), \(\mathrm{B_{2}^{+}}\).

Key concepts

Bond Order

In molecular orbital theory, bond order is a crucial concept that helps us understand the strength and stability of a bond between two atoms. It is calculated using the formula: \( \text{Bond order} = \frac{\text{Number of electrons in bonding orbitals} - \text{Number of electrons in antibonding orbitals}}{2} \). A higher bond order implies a more stable and stronger bond. For instance, in our exercise, the boron molecule species \( \mathrm{B}_{2}^{+} \), \( \mathrm{B}_{2} \), and \( \mathrm{B}_{2}^{-} \) have bond orders of 1.5, 1, and 0.5, respectively. This means that \( \mathrm{B}_{2}^{+} \) is the most stable with the strongest bond, whereas \( \mathrm{B}_{2}^{-} \) is less stable with the weakest bond.

• Higher bond order = stronger bond.
• Lower bond order = weaker bond.

Understanding bond order allows us to predict and compare the properties of different molecular species.

Bond Length

Bond length is the distance between the nuclei of two bonded atoms. It inversely relates to bond order, meaning that a higher bond order results in a shorter bond length. In the molecular species discussed, the bond lengths can be arranged in the following order: \( \mathrm{B}_{2}^{+} < \mathrm{B}_{2} < \mathrm{B}_{2}^{-} \). This order reflects their bond orders; \( \mathrm{B}_{2}^{+} \) has the shortest bond length due to its highest bond order of 1.5, while \( \mathrm{B}_{2}^{-} \) has the longest bond length with its lowest bond order of 0.5.

• When bond order increases, bond length decreases.
• When bond order decreases, bond length increases.

Recognizing how bond order affects bond length helps us understand molecular geometry and reactivity.

Bond Energy

Bond energy is a measure of bond strength in a chemical bond. Higher bond energy means it takes more energy to break the bond, indicating a stronger and more stable bond. From the exercise, the bond energies of our species can be ordered as: \( \mathrm{B}_{2}^{-} < \mathrm{B}_{2} < \mathrm{B}_{2}^{+} \). This order showcases that \( \mathrm{B}_{2}^{+} \) not only has the strongest and shortest bond, but it also requires the most energy to break.

• Higher bond energy correlates with greater bond strength.
• Lower bond energy correlates with less bond strength.

Understanding bond energy is essential for grasping how chemical reactions occur and how much energy might be involved.