Problem 63 - Chemistry An Atoms First Approach

Question

Which of the following are predicted by the molecular orbital model to be stable diatomic species? a. \(\mathrm{H}_{2}^{+}, \mathrm{H}_{2}, \mathrm{H}_{2}^{-}, \mathrm{H}_{2}^{2-}\) b. \(\mathrm{He}_{2}^{2+}, \mathrm{He}_{2}^{+}, \mathrm{He}_{2}\)

Step-by-Step Solution

Answer

The stable diatomic species according to the molecular orbital model are: \(\mathrm{H}_{2}^{+}\), \(\mathrm{H}_{2}\), \(\mathrm{H}_{2}^{-}\), \(\mathrm{He}_{2}^{2+}\), and \(\mathrm{He}_{2}^{+}\).

Step 1: Molecular Orbital Theory Background

Molecular orbital theory explains how atomic orbitals mix to form molecular orbitals in molecules. These molecular orbitals determine where electrons reside in the molecule, and the overall stability of the molecule. The bond order is calculated as: \( Bond \: Order = \frac{(No. \, of \, electrons \, in \, bonding \, MOs) - (No. \, of \, electrons \, in \, antibonding \, MOs)}{2} \) A positive bond order signifies a stable molecule.

Step 2: Analyzing Hydrogen Species

We need to determine the bond orders for the hydrogen species: \(\mathrm{H}_{2}^{+}, \mathrm{H}_{2}, \mathrm{H}_{2}^{-}, \mathrm{H}_{2}^{2-}\) Analyzing these species one by one: 1. \(\mathrm{H}_{2}^{+}\): Total of 1 electron which fills the lower energy bonding sigma orbital Bond order: \(\frac{(1-0)}{2} = \frac{1}{2}\) 2. \(\mathrm{H}_{2}\): Total of 2 electrons, filling the lower energy bonding sigma orbital Bond order: \(\frac{(2-0)}{2} = 1\) 3. \(\mathrm{H}_{2}^{-}\): Total of 3 electrons, filling the bonding sigma orbital and one in the antibonding sigma orbital Bond order: \(\frac{(2-1)}{2} = \frac{1}{2}\) 4: \(\mathrm{H}_{2}^{2-}\): Total of 4 electrons, filling both bonding sigma and antibonding sigma orbitals Bond order: \(\frac{(2-2)}{2} = 0\) All of the hydrogen species except \(\mathrm{H}_{2}^{2-}\) have positive bond orders, making them stable diatomic species.

Step 3: Analyzing Helium Species

We need to determine the bond orders for the helium species: \(\mathrm{He}_{2}^{2+}, \mathrm{He}_{2}^{+}, \mathrm{He}_{2}\) Analyzing these species one by one: 1. \(\mathrm{He}_{2}^{2+}\): Total of 2 electrons, filling the lower energy bonding sigma orbital Bond order: \(\frac{(2-0)}{2} = 1\) 2. \(\mathrm{He}_{2}^{+}\): Total of 3 electrons, filling the bonding sigma orbital and one in the antibonding sigma orbital Bond order: \(\frac{(2-1)}{2} = \frac{1}{2}\) 3. \(\mathrm{He}_{2}\): Total of 4 electrons, filling both bonding sigma and antibonding sigma orbitals Bond order: \(\frac{(2-2)}{2} = 0\) In the case of helium species, only \(\mathrm{He}_{2}^{2+}\) and \(\mathrm{He}_{2}^{+}\) have positive bond orders, making them stable diatomic species. In conclusion, stable diatomic species according to the molecular orbital model are: \[ \mathrm{H}_{2}^{+}, \mathrm{H}_{2}, \mathrm{H}_{2}^{-}, \mathrm{He}_{2}^{2+}, \mathrm{and} \, \mathrm{He}_{2}^{+} \]

Key Concepts

Bond Order

Bond order is a critical concept in understanding the stability of molecules using Molecular Orbital Theory. It is calculated by subtracting the number of electrons in antibonding molecular orbitals (MOs) from those in bonding MOs, then dividing by two. Mathematically, the bond order formula is expressed as: \[ \text{Bond Order} = \frac{(\text{Number of electrons in bonding MOs}) - (\text{Number of electrons in antibonding MOs})}{2} \] A positive bond order indicates that there are more electrons in bonding orbitals than in antibonding ones, suggesting a stable molecular structure. Conversely, a bond order of zero or negative implies instability, as it means the destabilizing effect of antibonding electrons cancels out or outweighs the bonding effects.Let's take a simple example with diatomic hydrogen molecules. In \( \mathrm{H}_{2}^{+} \), there is one electron in a bonding orbital and none in antibonding, resulting in a bond order of \( \frac{1}{2} \). For \( \mathrm{H}_{2} \), with two electrons in bonding orbitals and none in antibonding, the bond order is 1. Both cases show stability due to positive bond orders.

Diatomic Molecules

Diatomic molecules are molecules composed of only two atoms. These can either be of the same or different chemical elements. They are a key focus when studying Molecular Orbital Theory because their simplicity makes them ideal for illustrating basic principles of electron configuration and bonding.In the case of the hydrogen species like \( \mathrm{H}_{2}, \mathrm{H}_{2}^{+}, \mathrm{H}_{2}^{-}, \) and \( \mathrm{H}_{2}^{2-} \), we can see how adding or removing electrons affects stability. As electrons are added to the molecular orbitals, the bond order can change, as seen in the mathematical calculations. For example:

• \( \mathrm{H}_{2} \): Two electrons occupy a low-energy bonding orbital, resulting in optimal stability and a bond order of 1.
• \( \mathrm{H}_{2}^{2-} \): Four electrons fill both bonding and antibonding orbitals, leading to a neutral bond order of 0, indicating instability.

This demonstrates how even within simple diatomic molecules, slight changes in electron count can have notable effects on molecular characteristics.

Stability of Molecules

The stability of molecules in Molecular Orbital Theory can be understood through the concept of bond order and electron configuration. Simply put, more electrons in bonding molecular orbitals compared to antibonding molecular orbitals usually means that the molecule is more stable. When looking at diatomic species like \( \mathrm{He}_{2} \) and its ions, stability can vary significantly:

• \( \mathrm{He}_{2}^{2+} \): With a bond order of 1, this ion is stable because it has electrons predominantly in the bonding orbitals.
• \( \mathrm{He}_{2} \): This species has equal numbers of electrons in both bonding and antibonding orbitals, leading to a bond order of 0. This yields an unstable or nonexistent diatomic molecule under normal conditions.

Hence, understanding which orbitals electrons occupy helps predict whether a molecule will be stable or not. This is crucial for predicting the behavior and interactions of molecules in chemical reactions.