Question

The diatomic molecule OH exists in the gas phase. OH plays an important part in combustion reactions and is a reactive oxidizing agent in polluted air. The bond length and bond energy have been measured to be 97.06 \(\mathrm{pm}\) and 424.7 \(\mathrm{kJ} / \mathrm{mol}\) respectively. Assume that the OH molecule is analogous to the HF molecule discussed in the chapter and that the MOs result from the overlap of a \(p_{z}\) orbital from oxygen and the 1\(s\) orbital of hydrogen (the O-H bond lies along the z axis). a. Draw pictures of the sigma bonding and antibonding molecular orbitals in OH. b. Which of the two MOs has the greater hydrogen 1\(s\) character? c. Can the 2\(p_{x}\) orbital of oxygen form MOs with the 1\(s\) orbital of hydrogen? Explain. d. Knowing that only the 2\(p\) orbitals of oxygen interact significantly with the 1\(s\) orbital of hydrogen, complete the MO energy-level diagram for OH. Place the correct number of electrons in the energy levels. e. Estimate the bond order for OH. f. Predict whether the bond order of \(\mathrm{OH}^{+}\) is greater than, less than, or the same as that of OH. Explain.

Short answer

The sigma bonding molecular orbital (σ) in OH is formed by the in-phase overlap of the \(p_{z}\) orbital from oxygen and the 1\(s\) orbital from hydrogen, while the sigma antibonding molecular orbital (σ*) is formed by their out-of-phase overlap. The bonding MO has more hydrogen 1\(s\) character. The 2\(p_{x}\) orbital of oxygen cannot form MOs with the 1\(s\) orbital of hydrogen due to inappropriate symmetry. The MO energy-level diagram consists of a bonding MO (σ) and an antibonding MO (σ*) with one electron each. The bond order for OH is calculated as 0, which is an unexpected result suggesting that other orbitals or interactions are significant in the actual OH molecule. Predicting bond order for \(\mathrm{OH}^{+}\) based on the simplified MO diagram is not reliable; a more detailed analysis is required.

Step-by-step solution

Identify the atomic orbitals involved

The exercise mentions that the MOs result from the overlap of a \(p_{z}\) orbital from oxygen and the 1\(s\) orbital of hydrogen.

Draw the sigma bonding MO

The sigma bonding molecular orbital (σ) is formed by the in-phase, or head-on, overlap of the \(p_{z}\) orbital of oxygen and the 1\(s\) orbital of hydrogen. Draw the σ orbital as the two overlapping orbitals, indicating the regions where there is a positive or negative phase (usually with different shading or colors).

Draw the sigma antibonding MO

The sigma antibonding molecular orbital (σ*) is formed by the out-of-phase, or head-on, overlap of the \(p_{z}\) orbital of oxygen and the 1\(s\) orbital of hydrogen. Draw the σ* orbital as the two orbitals overlapping, but with a node between them, indicating where there is a positive or negative phase (again with different shading or colors). #b. Which of the two MOs has the greater hydrogen 1\(s\) character?#

Compare the characters of the MOs

In general, the bonding molecular orbitals have more character of the atom with the lower energy atomic orbitals, while the antibonding molecular orbitals have more character of the atom with the higher energy atomic orbitals. Since the hydrogen 1\(s\) orbital is lower in energy than the oxygen \(p_{z}\) orbital, the bonding MO (σ) will have more hydrogen 1\(s\) character. #c. Can the 2\(p_{x}\) orbital of oxygen form MOs with the 1\(s\) orbital of hydrogen? Explain.#

Consider the symmetry of orbitals

Orbitals can form molecular orbitals (bonding or antibonding) only if they have suitable symmetry. In the case of the 2\(p_{x}\) orbital from oxygen and the 1\(s\) orbital of hydrogen, they do not have the appropriate symmetry to overlap or interact effectively. Therefore, the 2\(p_{x}\) orbital of oxygen cannot form MOs with the 1\(s\) orbital of hydrogen. #d. Knowing that only the 2\(p\) orbitals of oxygen interact significantly with the 1\(s\) orbital of hydrogen, complete the MO energy-level diagram for OH. Place the correct number of electrons in the energy levels.#

Build the MO energy-level diagram

Knowing that only the 2\(p\) orbitals of oxygen interact significantly with the 1\(s\) orbital of hydrogen, we just need to consider the 2\(p_{z}\) orbital interacting with the 1\(s\) orbital of hydrogen. First, draw the energy-level diagram with the atomic orbitals of oxygen and hydrogen on different sides of the diagram. Next, draw the bonding MO (σ) and antibonding MO (σ*) between the atomic orbitals. After that, place the correct number of electrons in the energy levels (one from hydrogen and 6 from oxygen, in which 2 electrons are filled in the \(p_{z}\) orbital). Put one electron in the bonding MO (σ) and one electron in the antibonding MO (σ*). #e. Estimate the bond order for OH.#

Calculate the bond order

Bond order is calculated using the formula: Bond order = (No. of electrons in bonding MOs - No. of electrons in antibonding MOs) / 2 For OH, we have 1 electron in the bonding MO (σ) and 1 electron in the antibonding MO (σ*). Therefore, Bond order = (1 - 1) / 2 = 0 Thus, the bond order for OH is 0. This result is unexpected, considering that OH exists in reality. It suggests that other orbitals or interactions may be significant in the actual OH molecule. #f. Predict whether the bond order of \(\mathrm{OH}^{+}\) is greater than, less than, or the same as that of OH. Explain.#

Compare the bond order of OH and \(\mathrm{OH}^{+}\)

Since our calculated bond order for OH is 0, which is unexpected, it is not easy to predict the bond order for \(\mathrm{OH}^{+}\) based on our simplified MO diagram. In reality, other orbitals or interactions may be significant to the actual molecule, changing its bond order. A more detailed analysis and calculation would be necessary to make a reliable prediction.

Key concepts

Diatomic Molecules

Diatomic molecules, as the name suggests, are composed of two atoms. These can be of the same element, like O₂, or different elements, such as OH or HF. They are fundamental in chemistry because they form the simplest molecular systems and help in understanding more complex molecules.
In the context of Molecular Orbital (MO) Theory, diatomic molecules are particularly useful because they allow us to study molecular bonding through a simplified model. By analyzing diatomic molecules like OH, we gain insights into the bonding interactions resulting from the overlap of atomic orbitals. This helps in determining properties such as bond length and energy, which are crucial for predicting the molecule's behavior in chemical reactions.

• Diatomic molecules involve just two atoms.
• They serve as fundamental models to study molecular bonding.
• Understanding their interactions helps in predicting chemical behaviors.

Bond Length and Bond Energy

Bond length refers to the average distance between the nuclei of two bonded atoms. In the OH molecule, this length is measured to be 97.06 pm. Bond energy, on the other hand, is the energy required to break one mole of bonds in gaseous molecules. For OH, this is a significant 424.7 kJ/mol.
Both these properties, bond length and energy, are indicators of the strength and stability of the chemical bond. Shorter bond lengths typically correlate with stronger bonds because the atoms are held together more tightly. Conversely, higher bond energies indicate that more energy is needed to break the bond, signifying more robust bonding.
In terms of MO Theory, the overlap of atomic orbitals (in the case of OH, the 1s orbital of hydrogen and the pz orbital of oxygen) determines these properties. The nature of this overlap affects the resulting molecular bond strength and distance.

• Bond length is the distance between two bonded atoms' nuclei.
• Bond energy is the energy needed to break a bond.
• Both indicate bonding strength and stability.
• Overlaps of atomic orbitals determine these properties.

Orbital Symmetry

Orbital symmetry plays a crucial role in forming molecular orbitals. For effective interaction, orbitals must have compatible symmetries. In the case of OH, focus is on the interaction between the hydrogen 1s orbital and the oxygen's pz orbital, aligning along the same axis. This overlap leads to the formation of sigma (σ) bonding and antibonding (σ*) molecular orbitals.
Orbital symmetry dictates that overlapping orbitals must either be symmetrical or anti-symmetrical around the bond axis for interaction to occur. As shown with the 2px orbital of oxygen, due to its lack of proper symmetry with respect to the 1s orbital of hydrogen, it cannot participate effectively in forming molecular orbitals.
Symmetry considerations not only aid in whether orbitals overlap, but also how they interact, thus dictating the energy and shape of the resulting molecular orbitals. This concept is vital for predicting properties of the molecule based on its shape and energy levels.

• Compatible symmetry is needed for orbital interactions.
• In OH, the 1s and pz orbitals form σ and σ* orbitals.
• 2px orbital's symmetry does not align with hydrogen's 1s.
• Helps determine molecular orbitals' energy and shape.