Question

Use molecular orbital theory to predict the arrangement of electrons in MOs, the bond order, and the number of unpaired electrons in (a) \(\mathrm{BN}\) (b) cyanide ion, \(\mathrm{CN}^{-}\)

Short answer

BN has a bond order of 2 with 0 unpaired electrons; CN⁻ has a bond order of 3 with 0 unpaired electrons.

Step-by-step solution

Determine Electron Configuration of Atoms

Boron ( 2s^2 2p^1 ) and nitrogen ( 2s^2 2p^3 ) contribute their valence electrons to form BN. This gives a total of 8 valence electrons to fill the molecular orbitals.

Arrange Electrons in Molecular Orbitals for BN

For diatomic molecules like BN, the molecular orbitals are filled in this order: σ(2s), σ*(2s), π(2p), σ(2p), π*(2p), σ*(2p). With 8 electrons, fill: σ(2s)^2, σ*(2s)^2, π(2p)^4.

Calculate Bond Order for BN

Bond order is calculated as: \[ \text{Bond Order} = \frac{(\text{number of electrons in bonding MOs}) - (\text{number of electrons in antibonding MOs})}{2} \] For BN, bond order = \((6 - 2)/2 = 2\).

Determine Unpaired Electrons for BN

With all 8 electrons paired in the molecular orbitals, BN has 0 unpaired electrons.

Electron Configuration for CN⁻ Ion

Carbon ( 2s^2 2p^2 ) and nitrogen ( 2s^2 2p^3 ) give 9 valence electrons, add 1 more electron for the CN^− ion, making 10 electrons to arrange.

Arrange Electrons in Molecular Orbitals for CN⁻

Using the same order of filling: σ(2s), σ*(2s), π(2p), σ(2p), π*(2p), σ*(2p). Place 10 electrons: σ(2s)^2, σ*(2s)^2, π(2p)^4, σ(2p)^2.

Calculate Bond Order for CN⁻

For CN^−, bond order = \((8 - 2)/2 = 3\).

Determine Unpaired Electrons for CN⁻

All electrons in CN^− are paired, so there are 0 unpaired electrons.

Key concepts

Bond Order

The concept of bond order is essential in understanding the stability and strength of a chemical bond. Bond order is calculated using Molecular Orbital (MO) theory, which considers both bonding and antibonding electrons. Bond order is determined by this formula:
\[ \text{Bond Order} = \frac{(\text{number of electrons in bonding MOs}) - (\text{number of electrons in antibonding MOs})}{2} \]
This formula essentially tells you how strongly two atoms are bonded. A higher bond order indicates a stronger, more stable bond. For example:

• If the bond order is zero, this implies that the molecule is unstable and does not exist under normal conditions.
• A bond order of one means there is a single bond, as seen in a standard diatomic hydrogen molecule.
• A bond order of two corresponds to a double bond, such as in oxygen.
• A bond order of three suggests a triple bond, similar to nitrogen gas.

The bond order also helps predict the bond length: a higher bond order often means a shorter bond length, contributing to bond strength.

Electron Configuration

Electron configuration in molecular orbital theory involves distributing the molecule's electrons into the available molecular orbitals. This distribution follows the same basic principles as atomic electron configuration, such as the Aufbau principle, Hund's rule, and the Pauli exclusion principle. Let's break these down:

• Aufbau Principle: Electrons enter orbitals of lowest energy first. When filling molecular orbitals, one must fill bonding orbitals before antibonding ones.
• Hund's Rule: When filling degenerate orbitals (orbitals with the same energy), electrons fill them singly before pairing. This minimizes electron-electron repulsion.
• Pauli Exclusion Principle: An orbital can hold a maximum of two electrons, which must have opposite spins.

For example, in BN, the electron configuration is \( \sigma(2s)^2, \sigma^*(2s)^2, \pi(2p)^4 \), reflecting 8 valence electrons filled into the corresponding molecular orbital energy levels. Understanding this configuration allows us to determine other properties such as bond order and magnetic behavior.

Unpaired Electrons

Unpaired electrons play a key role in a molecule's magnetic properties and reactivity. Within molecular orbital theory, unpaired electrons are those which do not have a complementary electron with opposite spin in the same orbital. Here’s why this matters:

• Molecules with one or more unpaired electrons are typically paramagnetic, meaning they are attracted to magnetic fields. This occurs because the unpaired electrons' magnetic fields are not canceled out.
• If all the electrons in a molecule are paired, it is diamagnetic, repelling from magnetic fields due to the balance of the magnetic moments.

In the molecular cases of BN and CN^-, all electrons are paired, meaning both are diamagnetic with no unpaired electrons. Determining the presence of unpaired electrons gives insights into the magnetic characteristics of a compound, which can be crucial in fields like materials science and chemistry.