Problem 208 - A Complete Resource Book in Chemistry for JEE Main

Question

Using MO theory predict which of the following species has the shortest bond length? [2009] (a) \(\mathrm{O}_{2}^{+}\) (b) \(\mathrm{O}_{2}^{-}\) (c) \(\mathrm{O}_{2}^{2}{\underline{\phantom{xx}}}^{2-}\) (d) \(\mathrm{O}_{2}^{2+}\)

Step-by-Step Solution

Answer

\(\mathrm{O}_{2}^{2+}\) has the shortest bond length.

Step 1: Understanding Bond Order

Bond order is directly related to bond length; the higher the bond order, the shorter the bond length. In Molecular Orbital (MO) theory, bond order is calculated using the formula: \(\text{Bond Order} = \frac{(\text{number of bonding electrons}) - (\text{number of antibonding electrons})}{2}\).

Step 2: Electron Configurations for Given Species

Assign the total number of electrons for each species based on their charges:\- \(\mathrm{O}_2\) has 16 electrons. \- \(\mathrm{O}_{2}^{+}\) has 15 electrons. \- \(\mathrm{O}_{2}^{-}\) has 17 electrons. \- \(\mathrm{O}_{2}^{2-}\) has 18 electrons. \- \(\mathrm{O}_{2}^{2+}\) has 14 electrons.

Step 3: Determine MO Electron Configuration

Write out the MO electron configurations: \- \(\mathrm{O}_{2}^{+}\) (15 electrons): \([\sigma(1s)^2][\sigma^*(1s)^2][\sigma(2s)^2][\sigma^*(2s)^2] [\sigma(2p_z)^2][\pi(2p_x)^2][\pi(2p_y)^2][\pi^*(2p_x)^2][\pi^*(2p_y)^1]\) \- \(\mathrm{O}_{2}^{-}\) (17 electrons): \([\sigma(1s)^2][\sigma^*(1s)^2][\sigma(2s)^2][\sigma^*(2s)^2] [\sigma(2p_z)^2][\pi(2p_x)^2][\pi(2p_y)^2][\pi^*(2p_x)^2][\pi^*(2p_y)^3]\) \- \(\mathrm{O}_{2}^{2-}\) (18 electrons): \([\sigma(1s)^2][\sigma^*(1s)^2][\sigma(2s)^2][\sigma^*(2s)^2] [\sigma(2p_z)^2][\pi(2p_x)^2][\pi(2p_y)^2][\pi^*(2p_x)^2][\pi^*(2p_y)^4]\) \- \(\mathrm{O}_{2}^{2+}\) (14 electrons): \([\sigma(1s)^2][\sigma^*(1s)^2][\sigma(2s)^2][\sigma^*(2s)^2] [\sigma(2p_z)^2][\pi(2p_x)^2][\pi(2p_y)^2]\) \Fill them into bonding \((\sigma, \pi)\) and antibonding \((\sigma^*, \pi^*)\) orbitals.

Step 4: Calculation of Bond Order

Calculate the bond order for each species: \- \(\mathrm{O}_{2}^{+}\): Bond order = \((8 - 3)/2 = 2.5\) \- \(\mathrm{O}_{2}^{-}\): Bond order = \((8 - 5)/2 = 1.5\) \- \(\mathrm{O}_{2}^{2-}\): Bond order = \((8 - 6)/2 = 1.0\) \- \(\mathrm{O}_{2}^{2+}\): Bond order = \((8 - 2)/2 = 3.0\)

Step 5: Conclusion Based on Bond Order

The species with the highest bond order will have the shortest bond length. Therefore, among the given species, \(\mathrm{O}_{2}^{2+}\) with a bond order of 3.0 has the shortest bond length.

Key Concepts

Bond Order

Bond order is a fundamental concept in Molecular Orbital (MO) Theory. It indicates the stability and strength of a bond between atoms in a molecule.
To calculate bond order, use this formula:
\[ \text{Bond Order} = \frac{(\text{number of bonding electrons}) - (\text{number of antibonding electrons})}{2} \]The bond order tells us how many chemical bonds exist between a pair of atoms.

• A higher bond order suggests a stronger, more stable bond.
• A low bond order can suggest a weaker bond.
• If the bond order equals zero, the bond does not exist.

For example, in the case of \( \mathrm{O}_2^{2+} \), it has a bond order of 3.0, indicating a very strong triple bond.
By contrast, \( \mathrm{O}_2^{2-} \) has a bond order of 1.0, indicating only a single bond.
This connection between bond order and bond strength is essential when predicting molecular characteristics like bond length.

Bond Length

Bond length is the distance between the nuclei of two bonded atoms. It changes with different bond orders.
A higher bond order generally means a shorter bond length.

• Shorter bonds are typically stronger and more stable.
• Longer bonds are weaker and less stable.

To illustrate this, consider the bond length in \( \mathrm{O}_2^{2+} \) and \( \mathrm{O}_2^{2-} \):

• \( \mathrm{O}_2^{2+} \) has a bond order of 3.0, resulting in a very short bond length.
• \( \mathrm{O}_2^{2-} \) has a bond order of 1.0, leading to a longer bond length.

Shorter bond lengths reflect higher bond energy, as the associated atoms are pulled closer due to the additional bonding interactions.
Therefore, these distances are a useful factor in determining the strength and stability of a molecule.

Electron Configuration

In Molecular Orbital Theory, the electron configuration of molecules helps us understand which electrons are present in bonding and antibonding orbitals.
This configuration is crucial for predicting molecular properties, such as bond order and bond length.
For molecules like oxygen species, the sequence of filling orbitals goes as follows:
\([\sigma(1s)^2][\sigma^*(1s)^2][\sigma(2s)^2][\sigma^*(2s)^2][\sigma(2p_z)^2][\pi(2p_x)^2][\pi(2p_y)^2]\)

• Bonding orbitals (\( \sigma, \pi \)) increase molecule stability.
• Antibonding orbitals (\( \sigma^*, \pi^* \)) decrease stability.

Take for example \( \mathrm{O}_2^{-} \). This includes three \( \pi^* \) electrons, leading to a lower bond order of 1.5.
Meanwhile, \( \mathrm{O}_2^{2+} \), with empty \( \pi^* \) orbitals, has a higher bond order of 3.0.
Understanding these configurations aids in visualizing how molecules will behave and interact.

Ionic Species

Ionic species have an unequal amount of protons and electrons, resulting in a net charge. In Molecular Orbital Theory, ionic species can affect bond characteristics.
For example, when analyzing oxygen species, changes in ionic charges alter electron counts and subsequently bond orders.

• \( \mathrm{O}_2^{+} \): One electron removed, leading to 15 electrons.
• \( \mathrm{O}_2^{-} \): One electron added, leading to 17 electrons.

These differences influence electron configurations in molecular orbitals, thus affecting bond strength and stability.
The neutral species \( \mathrm{O}_2 \) possesses 16 electrons, while its ionic counterparts differ.
Studying ionic species is crucial for a comprehensive understanding of molecular behavior as it helps explain variances in chemical and physical properties due to electron arrangement.

Oxygen Species

Oxygen species, especially in their molecular forms like \( \mathrm{O}_2 \) and its ionic counterparts, play a vital role in chemistries like respiration and combustion.
In chemistry, analyzing oxygen species through MO Theory provides insights into their reactivity and stability.

• \( \mathrm{O}_2 \): A neutral molecule often exhibiting a double bond.
• \( \mathrm{O}_2^{+}, \mathrm{O}_2^{-}, \mathrm{O}_2^{2+}, \mathrm{and} \mathrm{O}_2^{2-} \): Variants with different charges and electron configurations.

Each oxygen species presents unique electron configurations and bond orders. \( \mathrm{O}_2 \) itself is dia-magnetic, while \( \mathrm{O}_2^{-} \) becomes paramagnetic.
This impacts their chemical behavior and interactions. Molecular Orbital Theory enables us to predict these peculiarities accurately, aiding in experimental and theoretical chemistry.