Problem 10 - Chemistry: The Central Science in SI Units, Global Edition

Question

The following is part of a molecular orbital energy-level diagram for MOs constructed from 1 s atomic orbitals. (a) What labels do we use for the two MOs shown? (b) For which of the following molecules or ions could this be the energy-level diagram: $$ \mathrm{H}_{2} \mathrm{He}_{2}, \mathrm{H}_{2}^{+}, \mathrm{He}_{2}^{+}, \mathrm{or} \mathrm{H}_{2}^{-} ? $$ (c) What is the bond order of the molecule or ion? (d) If an electron is added to the system, into which of the MOs will it be added? [Section 9.7\(]\)

Step-by-Step Solution

Answer

(a) The two MOs shown are σ (sigma) and σ* (sigma-star) molecular orbitals. (b) This energy-level diagram is for \(\mathrm{He}_{2}\). (c) The bond order is 0. (d) If an electron is added to the system, it will enter a higher-lying σ* anti-bonding orbital.

Step 1: Identifying the molecular orbitals

We need to identify the two MOs shown in the given energy-level diagram. In this case, the MOs are formed from 1s atomic orbitals. These two MOs are σ (sigma) and σ* (sigma-star) molecular orbitals. The σ orbital is the bonding orbital that is lower in energy and is formed due to the constructive combination of atomic orbitals, while the σ* orbital is the anti-bonding orbital that is higher in energy and is formed due to the destructive combination of atomic orbitals.

Step 2: Selecting the appropriate molecule/ion

We should choose one of the molecules/ions from the list, which is: \(\mathrm{H}_{2}, \mathrm{He}_{2}, \mathrm{H}_{2}^{+}, \mathrm{He}_{2}^{+}, \) or \(\mathrm{H}_{2}^{-}\). For this step, it is essential to remember the electron configurations and occupancy of molecular orbitals. For instance: 1. \(\mathrm{H}_{2}\): Both hydrogen atoms have 1 electron in their 1s orbitals, and the molecule has a total of 2 electrons, filling the σ bonding orbital but not the σ* anti-bonding orbital. 2. \(\mathrm{He}_{2}\): Each helium atom has 2 electrons in its 1s orbital, and the molecule has a total of 4 electrons. This means that both the σ and σ* orbitals will be filled with 2 electrons each, canceling each other in terms of bond formation. 3. \(\mathrm{H}_{2}^{+}\): The molecule has a total of 1 electron which will fill the σ orbital (with the other σ* orbital remaining empty), resulting in bond formation. 4. \(\mathrm{He}_{2}^{+}\): The ion has a total of 3 electrons, resulting in 2 electrons occupying the σ orbital and 1 electron in the σ* anti-bonding orbital. 5. \(\mathrm{H}_{2}^{-}\): The ion has a total of 3 electrons, filling the σ orbital with 2 electrons, while the other electron goes into the σ* anti-bonding orbital. The energy-level diagram depicts a molecule/ion with both the σ and σ* orbitals occupied by an equal number of electrons, which results in no bond formation. Hence, the correct molecule/ion is \(\mathrm{He}_{2}\).

Step 3: Calculating the bond order

To calculate the bond order, use the formula: Bond order = \(0.5 \times (\textit{number of electrons in bonding orbitals} - \textit{number of electrons in anti-bonding orbitals})\) For \(\mathrm{He}_{2}\), both the σ and σ* orbitals contain 2 electrons each: Bond order = \(0.5 \times (2 - 2) = 0\)

Step 4: Identifying the molecular orbital for the added electron

If an electron were to be added to the system, it would enter the molecular orbital with the lowest available energy level. Since both the σ and σ* orbitals are already filled, adding an electron to the \(\mathrm{He}_{2}\) system would result in an increased energy level. More specifically, it will be added to the next available orbital, which in this case, is a higher-lying σ* anti-bonding orbital.

Key Concepts

Bond Order

Bond order is a key concept in molecular orbital theory, giving us an idea of the stability and strength of a bond. It's a measure of the difference between bonding and anti-bonding electrons. To calculate bond order, you use the formula:

• Bond order = \(0.5 \times (\text{number of electrons in bonding orbitals} - \text{number of electrons in anti-bonding orbitals})\)

A higher bond order indicates a stronger bond. For example, in a molecule like \(\text{H}_2\), both electrons are in bonding orbitals, leading to a bond order of 1. This means a stable, single bond has formed. Conversely, molecules like \(\text{He}_2\) have equal numbers of electrons in bonding and anti-bonding orbitals, resulting in a bond order of 0. This means no bond forms between the helium atoms. Understanding bond order helps predict if a molecule or ion will exist and be stable.

Sigma Orbitals

Sigma orbitals are types of molecular orbitals formed by linear overlap of atomic orbitals. There are two types:

• \(\sigma\) (sigma) orbitals: These are bonding orbitals where electrons in-phase overlap, leading to a decreased energy level and increased stability. They are essential for bond formation.
• \(\sigma^*\) (sigma-star) orbitals: These are anti-bonding orbitals where electrons out-of-phase overlap, resulting in increased energy and decreased stability. These do not contribute to bond strength; instead, they weaken it.

In the energy-level diagram, \(\sigma\) orbitals are lower in energy compared to \(\sigma^*\) orbitals. For a stable molecule, the \(\sigma\) orbitals will be occupied first, ensuring the overall stability of the molecule.

Energy-Level Diagram

An energy-level diagram visualizes the arrangement of electrons in molecular orbitals. It helps in understanding which orbitals are filled and provides insight into the molecule's properties. Here's how it works:

• Molecular orbitals are depicted at different energy levels.
• The \(\sigma\) orbital is generally placed lower than the \(\sigma^*\) orbital, reflecting its lower energy and bonding nature.
• Electrons fill the lower energy orbitals first (like \(\sigma\)), following Hund's rule and the Aufbau principle.

By analyzing the energy-level diagram, you can determine the electron configuration of a molecule. This helps in predicting its bond order and whether it will form a stable bond or not. In cases like \(\text{He}_2\), where both \(\sigma\) and \(\sigma^*\) are filled equally, the energy-level diagram shows no net bond is formed, highlighting the importance of visualizing electron arrangement.