Problem 61 - Chemistry: The Central Science

Question

Consider the \(\mathrm{H}_{2}{\underline{\phantom{xx}}}^{+}\) ion. (a) Sketch the molecular orbitals of the ion, and draw its energy-level diagram. (b) How many electrons are there in the \(\mathrm{H}_{2}{\underline{\phantom{xx}}}^{+}\) ion? (c) Draw the electron configuration of the ion in terms of its MOs. (d) What is the bond order in \(\mathrm{H}_{2}{\underline{\phantom{xx}}}^{+}\) ? (e) Suppose that the ion is excited by light so that an electron moves from a lower-energy to a higher-energy MO. Would you expect the excitedstate \(\mathrm{H}_{2}{\underline{\phantom{xx}}}^{+}\) ion to be stable or to fall apart? Explain.

Step-by-Step Solution

Answer

In the H2+ ion, there are two molecular orbitals - a lower-energy bonding orbital (σ1s) and a higher-energy antibonding orbital (σ1s*). This ion has only one electron, which occupies the σ1s orbital, giving it an electron configuration of σ1s^1. Its bond order is 1/2, indicating a weak bond. When the electron is excited to the σ1s* orbital, the bond order becomes -1/2, making the excited-state H2+ ion unstable and likely to fall apart.

Step 1: Sketch molecular orbitals of H2+ ion

For H2+ ion, there are two H atoms in which one of the atoms loses an electron, resulting in only one electron present in the system. Each H atom has one 1s orbital, and when these two 1s orbitals combine, they form molecular orbitals - one bonding orbital (σ1s) and one antibonding orbital (σ1s*). The bonding orbital (σ1s) is of lower energy, while the antibonding orbital (σ1s*) is of higher energy.

Step 2: Draw the energy-level diagram of H2+ ion

Draw two horizontal lines to represent the energy levels of the bonding (σ1s) and antibonding (σ1s*). Place the bonding orbital below the antibonding orbital and label them accordingly. Also, draw the atomic orbitals (1s) for each hydrogen atom, and connect them to the bonding and antibonding orbitals (σ1s and σ1s*), showing how they interact to form the molecular orbitals. #b) Number of electrons in H2+ ion#

Step 3: Determine the number of electrons in the H2+ ion

Since H2+ ion is formed by the loss of one electron from two H atoms, there is only one electron present in the H2+ ion. #c) Electron configuration of H2+ ion in terms of its MOs#

Step 4: Electron configuration of H2+ ion in its molecular orbitals

With only one electron in the H2+ ion, the electron configuration will have the electron occupying the lower energy bonding orbital. Thus, the electron configuration for H2+ ion in terms of molecular orbitals will be σ1s^1. #d) Bond order in H2+ ion#

Step 5: Calculate the bond order in H2+ ion

Bond order is calculated as (number of electrons in bonding orbitals - number of electrons in antibonding orbitals) / 2. In the case of H2+ ion, Bond order = (1 - 0) / 2 = 1/2. #e) Stability of excited-state H2+ ion#

Step 6: Stability of excited-state H2+ ion

When the electron in H2+ ion is excited from the lower-energy bonding orbital (σ1s) to the higher-energy antibonding orbital (σ1s*), the bond order will be (0 - 1) / 2 = -1/2. A negative bond order implies that there is a lack of chemical bonding between the H atoms, leading to an unstable molecule. Thus, the excited-state H2+ ion is expected to be unstable and likely to fall apart.

Key Concepts

H2+ ion

The \(\mathrm{H}_{2}{\underline{\phantom{xx}}}^{+}\) ion, also known as protonated hydrogen or the simplest molecular ion, plays a significant role in understanding the behaviour of molecular orbitals. \(\mathrm{H}_{2}{\underline{\phantom{xx}}}^{+}\) consists of two hydrogen nuclei and only one electron. This asymmetry occurs due to the loss of one electron from the hydrogen molecule \(\mathrm{H}_{2}\).
Since only one electron is present, it is key to determining the ionic properties and interactions. \(\mathrm{H}_{2}{\underline{\phantom{xx}}}^{+}\) is an excellent example for illustrating how molecular orbitals are formed from atomic orbitals.
In the \(\mathrm{H}_{2}{\underline{\phantom{xx}}}^{+}\) ion, two \(1s\) orbitals from individual hydrogen atoms blend to create one bonding molecular orbital (denoted as \(\sigma 1s\)) and one antibonding molecular orbital (denoted as \(\sigma 1s^*\)).
This process will be vital in understanding how the single electron in the \(\mathrm{H}_{2}{\underline{\phantom{xx}}}^{+}\) ion configures itself in these molecular orbitals which affects its chemical properties.

bond order

To determine the \(\mathrm{H}_{2}{\underline{\phantom{xx}}}^{+}\) ion's stability, we calculate something called bond order. The bond order provides insights into the strength and stability of the bond between the two atoms.
The formula to calculate bond order is:

• Bond order = (number of electrons in bonding orbitals - number of electrons in antibonding orbitals) / 2

For the \(\mathrm{H}_{2}{\underline{\phantom{xx}}}^{+}\) ion, we observe that there is 1 electron in the bonding orbital, and 0 electrons in the antibonding orbital.
Using the formula:

• Bond order = (1 - 0) / 2 = 1/2

This bond order of 0.5 suggests that the ion has a weaker bond than a full chemical bond (i.e., bond order of 1). However, it still maintains enough interaction to represent a half-bond, which provides a nuanced view of chemical stability in molecular ions.

energy-level diagram

Creating an energy-level diagram is essential for visualizing how electrons fill molecular orbitals. For the \(\mathrm{H}_{2}{\underline{\phantom{xx}}}^{+}\) ion, such diagrams illustrate the placement of molecular orbitals in terms of energy.
To construct this diagram, follow these steps:

• Start with two horizontal lines, representing energy levels, positioned vertically on your page.
• The lower line, representing the bonding molecular orbital (\