Problem 104 - Chemistry The Central Science

Question

Write the electron configuration for the first excited state for \(\mathrm{N}_{2}\) - that is, the state with the highest-energy electron moved to the next available energy level. (a) Is the nitrogen in its first excited state diamagnetic or paramagnetic? (b) Is the \(\mathrm{N}-\mathrm{N}\) bond strength in the first excited state stronger or weaker compared to that in the ground state? Explain.

Step-by-Step Solution

Answer

The first excited state electron configuration for N₂ is \(1\sigma_g^2 1\sigma_u^2 2\sigma_g^2 2\sigma_u^2 3\sigma_g^2 1\pi_u^3 1\pi_g^1\), making it paramagnetic. The N-N bond strength in the first excited state is weaker compared to the ground state, as indicated by the bond orders (2.5 in the first excited state vs. 3 in the ground state).

Step 1: Write the ground state electron configuration of N₂

The nitrogen atom has 7 electrons. Their electron configuration in the ground state can be expressed as follows: \(1s^2 2s^2 2p^3\). In a nitrogen molecule (N₂), there are a total of 14 electrons. The electron configuration of N₂ in the ground state can be written as: \(1\sigma_g^2 1\sigma_u^2 2\sigma_g^2 2\sigma_u^2 3\sigma_g^2 1\pi_u^4\)

Step 2: Write the electron configuration for the first excited state

To determine the first excited state of N₂, we need to move the highest-energy electron to the next available energy level. The highest-energy electron resides in the 1\(\pi_{u}\) molecular orbital. We'll move one electron from the 1\(\pi_{u}\) MO to the next available energy level, which is the 1\(\pi_{g}\) MO. This gives us the following electron configuration: \(1\sigma_g^2 1\sigma_u^2 2\sigma_g^2 2\sigma_u^2 3\sigma_g^2 1\pi_u^3 1\pi_g^1\)

Step 3: Determine if the first excited state is diamagnetic or paramagnetic

A molecule is diamagnetic if all its electrons are paired, and it's paramagnetic if there's at least one unpaired electron. In the first excited state of N₂, the electron configuration shows that there are unpaired electrons in the 1\(\pi_{u}\) and 1\(\pi_{g}\) MOs (\(1\pi_u^3 1\pi_g^1\)). Therefore, the first excited state of N₂ is paramagnetic.

Step 4: Compare the N-N bond strength in the first excited state to the ground state

To determine the bond strength, we look at the bond order. The bond order can be calculated using the formula: \(Bond\:Order=\frac{1}{2}(number\:of\:electrons\:in\:bonding\:orbitals - number\:of\:electrons\:in\:antibonding\:orbitals)\) For the ground state electron configuration (\(1\sigma_g^2 1\sigma_u^2 2\sigma_g^2 2\sigma_u^2 3\sigma_g^2 1\pi_u^4\)): Bond Order = \(\frac{1}{2}(10 - 4) = 3\) For the first excited state electron configuration (\(1\sigma_g^2 1\sigma_u^2 2\sigma_g^2 2\sigma_u^2 3\sigma_g^2 1\pi_u^3 1\pi_g^1\)): Bond Order = \(\frac{1}{2}(10 - 5) = 2.5\) Since the bond order for the ground state is higher than the first excited state (3 vs. 2.5), the bond strength in the ground state is stronger compared to that in the first excited state.

Key Concepts

Molecular Orbitals

Molecular orbitals are the result of atomic orbitals overlapping when atoms bond together to form molecules. These orbitals can hold electrons that belong to the entire molecule rather than just one atom. When we talk about molecular orbitals, we classify them into bonding orbitals and antibonding orbitals.

• Bonding orbitals are lower in energy and stabilize the molecule when they are filled with electrons. They are denoted with the sigma (\(\sigma\)) or pi (\(\pi\)) symbols followed by the subscripts \(g\) for "gerade" (even) and \(u\) for "ungerade" (odd), indicating symmetric or asymmetric properties.
  
• Antibonding orbitals, on the other hand, are higher in energy and tend to destabilize the molecule. Antibonding orbitals are denoted with an asterisk, like \(\sigma^*\) or \(\pi^*\).
  

For the nitrogen molecule (\(\text{N}_2\)), we see the filling pattern as follows in the electronic configuration: bonding orbitals like \(1\sigma_g^2\) and antibonding sequences showing partial filling in its excited state electrons like \(1\pi_g^1\). This movement to higher energy levels creates what is referred to as the excited state, where electron excitations occur within these molecular orbitals.

Paramagnetism

Paramagnetism is a type of magnetism that occurs in materials due to the presence of unpaired electrons. When such a material is placed in a magnetic field, the unpaired electrons' magnetic moments align with the field, causing the material to be attracted to the magnet.

In the context of determining whether a molecule like \(\text{N}_2\) is paramagnetic, we analyze the electron configuration. For the ground state electron configuration, all electrons are paired, making the molecule diamagnetic. However, when the molecule is in its first excited state, as electrons are promoted to the next molecular orbital, you'll find unpaired electrons.

The excited electron configuration of \(\text{N}_2\) results in unpaired electrons in \(1\pi_u^3 1\pi_g^1\), indicating that the molecule will respond to a magnetic field and has paramagnetic properties because it has unpaired electrons. This change in the electron configuration from paired in the ground state to unpaired in the excited state is a classic example of how molecular orbital arrangements influence magnetic properties.

Bond Order

Bond order provides a numerical indication of the strength and stability of a bond between two atoms in a molecule. It is calculated as one-half the difference between the number of electrons in bonding orbitals and antibonding orbitals.

The formula for bond order is:\[ Bond\:Order = \frac{1}{2}(\text{number of electrons in bonding orbitals} - \text{number of electrons in antibonding orbitals}) \]In \(\text{N}_2\)'s ground state with an electronic configuration of \(1\sigma_g^2 1\sigma_u^2 2\sigma_g^2 2\sigma_u^2 3\sigma_g^2 1\pi_u^4\), the bond order is 3. This indicates a robust triple bond between the nitrogen atoms. In its first excited state, however, with the configuration \(1\sigma_g^2 1\sigma_u^2 2\sigma_g^2 2\sigma_u^2 3\sigma_g^2 1\pi_u^3 1\pi_g^1\), the bond order drops to 2.5.

This decrease in bond order from 3 in the ground state to 2.5 in the excited state means that the bond is weaker when the molecule is excited. A lower bond order indicates that fewer electrons contribute to bonding, hence a diminished bond strength. Thus, bond order is a crucial concept in understanding molecular stability and electron configuration influence on chemical bonding.