Problem 36 - Chemistry: Atoms First

Question

Calculate the bond order for an ion with this configuration: \(\left(\sigma_{2 s}\right)^{2}\left(\sigma_{2 s}^{*}\right)^{2}\left(\sigma_{2 p x}\right)^{2}\left(\pi_{2 p y}, \pi_{2 p z}\right)^{4}\left(\pi_{2 p y}^{*}, \pi_{2 p z}^{*}\right)^{3}\)

Step-by-Step Solution

Answer

The bond order of the ion is \(\frac{3}{2}\).

Step 1: Understanding Molecular Orbital Theory

Bond order can be calculated using Molecular Orbital Theory, which provides a method for determining the molecular structure by considering electrons in orbitals that spread out over the entire molecule. According to this theory, the bond order is given by the formula \(\frac{\text{number of bonding electrons} - \text{number of antibonding electrons}}{2}\). Electrons in bonding orbitals stabilize the molecule, while those in antibonding orbitals destabilize it.

Step 2: Count the Number of Bonding and Antibonding Electrons

The given electron configuration is \(\left(\sigma_{2s}\right)^2\left(\sigma_{2s}^{*}\right)^2\left(\sigma_{2px}\right)^2\left(\pi_{2py}, \pi_{2pz}\right)^4\left(\pi_{2py}^{*}, \pi_{2pz}^{*}\right)^3\). We count 2 electrons in the \(\sigma_{2s}\) bonding orbital, 2 electrons in the \(\sigma_{2s}^{*}\) antibonding orbital, 2 electrons in the \(\sigma_{2px}\) bonding orbital, 4 electrons in the \(\pi_{2py}\) and \(\pi_{2pz}\) bonding orbitals, and 3 electrons in the \(\pi_{2py}^{*}\) and \(\pi_{2pz}^{*}\) antibonding orbitals. Hence, we have a total of 2 + 2 + 4 = 8 bonding electrons and 2 + 3 = 5 antibonding electrons.

Step 3: Calculate the Bond Order

Now we can plug these numbers into the bond order formula: \(\text{Bond order} = \frac{\text{number of bonding electrons} - \text{number of antibonding electrons}}{2}\). Plugging in the numbers, we get \(\text{Bond order} = \frac{8 - 5}{2} = \frac{3}{2}\).

Key Concepts

Molecular Orbital Theory

The Molecular Orbital Theory (MOT) is an essential concept in chemistry that helps us understand the distribution of electrons in a molecule. Unlike atomic orbitals which are confined to a single atom, molecular orbitals extend over the entire molecule. This theory proposes the formation of molecular orbitals when atomic orbitals overlap during the formation of a covalent bond.

In MOT, there are bonding orbitals, which are lower in energy and thereby stabilize the molecule, and antibonding orbitals, which have higher energy and can destabilize the molecule if they contain electrons. A molecular orbital can hold a maximum of two electrons, with opposite spins.

To visualize this, think of a molecule as a building, with molecular orbitals as rooms that can be filled with electron 'occupants'. Bonding orbitals are like the main living spaces that stabilize the structure, while antibonding orbitals are like overloaded attics that could compromise the building's integrity when filled.

Bonding and Antibonding Electrons

When we dive deeper into the Molecular Orbital Theory, we encounter two critical terms: bonding and antibonding electrons. Bonding electrons are housed in bonding orbitals, providing the 'glue' that holds atoms together in a molecule. These electrons lower the energy, increasing stability. Antibonding electrons, on the other hand, are placed in antibonding orbitals and work against the bonding force; these increase the energy and if present in sufficient number, can lead to molecule instability.

Consider a school team project, with bonding electrons being the team members who contribute positively and cooperate, while antibonding electrons resemble disagreeing members who challenge team cohesion. The difference in numbers between these collaborating and conflicting members can determine the project's (or molecule's) success or failure.

Electron Configuration

Electron configuration in molecular orbitals is akin to a seating chart that shows where each electron 'sits' within a molecule. It tells us which orbitals are filled and with how many electrons. This orderly arrangement is crucial as it determines the molecule’s bond order, reactivity, and overall stability. In the electronic configuration notation, symbols like \( \sigma \), \( \pi \), and asterisks (*) to denote antibonding orbitals, provide a map of the electron distribution.

For students, understanding electron configuration can be compared to understanding the rules of a card game. Just as a player needs to know the hierarchy of the cards and their arrangements to play effectively, a student needs to understand the 'rules' of electron configuration to grasp how molecules form and behave.