Problem 72 - Chemistry The Central Science In Si Units

Question

(a) Sketch the molecular orbitals of the \(\mathrm{H}_{2}^{-}\) ion and draw its energy-level diagram. (b) Write the electron configuration of the ion in terms of its MOs. (c) Calculate the bond order in \(\mathrm{H}_{2}^{-}\). (d) Suppose that the ion is excited by light, so that an electron moves from a lower- energy to a higher-energy molecular orbital. Would you expect the excited- state \(\mathrm{H}_{2}^{-}\) ion to be stable? (e) Which of the following statements about part (d) is correct: (i) The light excites an electron from a bonding orbital to an antibonding orbital, (ii) The light excites an electron from an antibonding orbital to a bonding orbital, or (iii) In the excited state there are more bonding electrons than antibonding electrons?

Step-by-Step Solution

Answer

a. Sketch involves \( \sigma_1s \) and \( \sigma_1s^* \). b. Configuration: \( \sigma_1s^2\sigma_1s^{*1} \). c. Bond order: 0.5. d. Excited state is unstable. e. Statement (i) is correct.

Step 1: Understanding Molecular Orbitals

For diatomic molecules like \( \mathrm{H}_2^- \), we consider the combination of atomic orbitals from two hydrogen atoms to form molecular orbitals. The key orbitals to consider are the bonding \( \sigma_1s \) and antibonding \( \sigma_1s^* \) orbitals.

Step 2: Sketching the Molecular Orbitals

The \( \sigma_1s \) orbital is lower in energy and represents the in-phase combination of the two \( 1s \) atomic orbitals, leading to electron density between the nuclei, which stabilizes the molecule. The \( \sigma_1s^* \) orbital is higher in energy and represents the out-of-phase combination, with a node between the nuclei, making it less stable.

Step 3: Energy Level Diagram

Draw an energy-level diagram with two atomic \( 1s \) orbitals at the bottom, the \( \sigma_1s \) orbital in the middle, and the \( \sigma_1s^* \) orbital above it. Arrange these vertically to show increasing energy.

Step 4: Electron Configuration of \( \mathrm{H}_2^- \)

\( \mathrm{H}_2^- \) has three electrons. Fill the \( \sigma_1s \) orbital with the first two electrons (\( \sigma_1s^2 \)) and the third electron in the \( \sigma_1s^* \) orbital: configuration \( \sigma_1s^2\sigma_1s^{*1} \).

Step 5: Calculating the Bond Order

Bond order is calculated as \( (N_b - N_a)/2 \), where \( N_b \) is the number of electrons in bonding orbitals and \( N_a \) is the number of electrons in antibonding orbitals. Here, \( N_b = 2 \) and \( N_a = 1 \). Bond order = \( (2 - 1)/2 = 0.5 \).

Step 6: Stability of the Excited-State \( \mathrm{H}_2^- \)

In an excited state, an electron goes from \( \sigma_1s \) to \( \sigma_1s^* \), changing the configuration to \( \sigma_1s^1\sigma_1s^{*2} \). The bond order becomes \( (1 - 2)/2 = -0.5 \), indicating the molecule would not be stable.

Step 7: Evaluating the Statements

(i) is correct. Light excites an electron from a bonding orbital (\( \sigma_1s \)) to an antibonding orbital (\( \sigma_1s^* \)). This decreases bonding electrons and increases antibonding electrons, reducing stability.

Key Concepts

Electron Configuration

Electron configuration is a method of describing the distribution of electrons in an atom or molecule's orbitals. In the case of the molecular ion \( \mathrm{H}_2^- \), it consists of three electrons shared between two hydrogen atoms. When describing electron configurations in terms of molecular orbitals (MO), we primarily consider the occupancy in bonding and antibonding orbitals.
For \( \mathrm{H}_2^- \), the configuration in molecular orbitals begins with filling the lower-energy bonding orbital, denoted as \( \sigma_{1s} \). This bonding orbital can hold up to two electrons, which stabilize the molecule. Thus, the first two electrons fill the \( \sigma_{1s} \) orbital: \( \sigma_{1s}^2 \). The remaining third electron then occupies the next available higher energy orbital, which is the antibonding orbital \( \sigma_{1s}^* \): \( \sigma_{1s}^1 \).The complete electron configuration of the \( \mathrm{H}_2^- \) ion in MO terms is \( \sigma_{1s}^2 \sigma_{1s}^{*1} \). This configuration indicates that we have two electrons in the bonding orbital, responsible for all the bonding interactions, and one in the antibonding orbital, which serves to slightly destabilize the bond.By understanding this configuration, one can predict both the bond order and possible stability upon excitation.

Bond Order

Bond order provides insight into the strength and stability of a bond. It is calculated from the difference in the number of electrons in bonding and antibonding molecular orbitals. For any diatomic species like \( \mathrm{H}_2^- \), the bond order can tell you how strong the bond is between the two atoms involved.
Here is how the bond order is calculated:

• Count the number of electrons in bonding orbitals (\( N_b \)). For \( \mathrm{H}_2^- \), this is 2 from the \( \sigma_{1s}^2 \) configuration.
• Count the electrons in antibonding orbitals (\( N_a \)). In this case, 1 from \( \sigma_{1s}^{*1} \).
• Apply the formula for bond order: \[ \text{Bond Order} = \frac{N_b - N_a}{2} \]
• Substitute the values: \[ \text{Bond Order} = \frac{2 - 1}{2} = 0.5 \]

A bond order of 0.5 indicates a relatively weak bond between the two hydrogen atoms in the ion. This lower bond order means \( \mathrm{H}_2^- \) would be more reactive and less stable compared to a bond order of 1 or higher, hence it is partially bonded but still connected.

Energy-Level Diagram

An energy-level diagram illustrates the relative energies of molecular orbitals in a molecule. For molecules like \( \mathrm{H}_2^- \), it helps visualize which orbitals electrons will fill first, due to their respective energy levels.
The diagram for \( \mathrm{H}_2^- \) can be visualized as follows:

• At the lowest energy, you start with the individual atomic 1s orbitals from each hydrogen atom.
• The \( \sigma_{1s} \) orbital, formed by constructive interference, or in-phase overlap of the atomic 1s orbitals, comes next. This orbital has lower energy than the separated atoms and is a bonding orbital.
• Above the \( \sigma_{1s} \), there's the \( \sigma_{1s}^* \) antibonding orbital. It's higher in energy due to its formation from destructive interference, or out-of-phase overlap of the atomic orbitals.

When drawing this diagram: reflect that the energy increases as you go upward. The electrons will occupy the lowest available energy levels first, which are the \( \sigma_{1s} \) before moving to the \( \sigma_{1s}^* \).
Such diagrams are crucial for understanding not just \( \mathrm{H}_2^- \), but also any molecule where electron sharing between atoms occurs. They clarify why certain configurations confer molecular stability, and therefore help predict molecular behavior when exposed to energies such as light, which can cause electronic excitations.