Question

Describe the bond order of diatomic carbon, \(\mathrm{C}_{2},\) with Lewis theory and molecular orbital theory, and explain why the results are different.

Short answer

The bond order of diatomic carbon, \(\mathrm{C}_{2}\), is 1 according to Lewis theory and 2 according to molecular orbital theory. The difference is because Lewis theory only recognizes single bonds, while molecular orbital theory accounts for delocalization of electrons in atomic orbitals.

Step-by-step solution

Understanding Theories

Lewis theory is focused mainly on electron pairs and creating stable by sharing pairs of electrons, while the molecular orbital theory considers the wave properties of electrons and creates molecular orbitals that are spread over the entire molecule.

Using Lewis theory

Each carbon in diatomic carbon, \(\mathrm{C}_{2}\), has 4 electrons in its outer shell. According to Lewis theory, a carbon-carbon bond is created when each carbon atom shares one of its electrons to form a covalent bond. Thus, the bond order according to Lewis theory is 1, meaning there is a single bond between the two carbon atoms.

Using Molecular Orbital Theory

In molecular orbital theory, we combine atomic orbitals to form molecular orbitals, which can be bonding, anti-bonding, or non-bonding. For carbon, we fill the molecular orbitals starting with the lower energy orbitals. We have a total of 8 electrons (4 from each carbon). These electrons will fill the \(\sigma_{1s}\), \(\sigma_{1s}*\), \(\sigma_{2s}\),\(\sigma_{2s}*\), and \(\pi_{2p}\) orbitals, each pair cancelling each other out except in the \(\pi_{2p}\) orbital. The bond order is then \(\frac{1}{2} (4- 0)\) = 2.

Comparing Results and Explaining the Difference

The bond order obtained using Lewis theory is 1 while that obtained using Molecular Orbital theory is 2. This discrepancy is due to the count of \(\pi_{2p}\) electrons in molecular orbital theory. While Lewis theory only recognizes single bonds, molecular orbital theory accounts for delocalization of electrons in atomic orbitals, therefore considering \(\pi_{2p}\) electrons which creates a double bond.

Key concepts

Lewis Theory

Lewis theory is a fundamental concept in chemistry that focuses on the sharing of electron pairs to form chemical bonds. It uses Lewis structures to represent molecules, indicating how atoms bond by sharing electrons. In Lewis theory, each bond is formed when two atoms share a pair of electrons. This creates a stable balance where atoms attain a noble gas electron configuration. For the diatomic carbon molecule, \(\mathrm{C}_2\), Lewis theory suggests that each carbon atom shares one electron, forming a single bond \((C-C)\). This gives a bond order of 1, which signifies one shared pair of electrons. However, Lewis theory does not consider any kind of electron delocalization or resonance that might be present in more advanced theories.

Molecular Orbital Theory

Molecular Orbital (MO) theory provides a more in-depth analysis of how electrons operate in molecules by considering the wave nature of electrons. Instead of localized electron pairs, as seen in Lewis theory, MO theory describes electrons in terms of molecular orbitals that span the entire molecule. When applying MO theory to \(\mathrm{C}_2\), its 8 valence electrons fill available molecular orbitals according to their energy levels. These include bonding and antibonding orbitals such as \(\sigma_{1s}, \sigma_{1s}^*, \sigma_{2s}, \sigma_{2s}^*, \pi_{2p}\), among others. In \(\mathrm{C}_2\), after populating these molecular orbitals, the remaining electrons in the \(\pi_{2p}\) bonding orbitals account for a bond order of 2 because:

• The bond order equation is given by \[\frac{1}{2} (N_b - N_a)\], where \(N_b\) is the number of bonding electrons and \(N_a\) is the number of antibonding electrons.
• For \(\mathrm{C}_2\), \(N_b = 4\) and \(N_a = 0\), resulting in a bond order of 2 \[\frac{1}{2} (4-0) = 2\].

This indicates a double bond, which reflects the delocalization of electrons not accounted for in simple Lewis structures.

Electron Configuration

Electron configuration is essential when constructing and interpreting models such as Lewis structures and molecular orbital diagrams. It tells us how electrons are distributed in each atom, thereby enhancing our understanding of chemical bonds and molecular structure. For diatomic carbon \(\mathrm{C}_2\), each carbon atom starts with an electron configuration of \(1s^2 2s^2 2p^2\). In Lewis theory, the focus is on the easiest picture that each atom shares one electron, while in MO theory, electrons fill molecular orbitals starting from the lowest energy level. These configurations lead to more accurate descriptions of how electrons contribute to bonding. In molecular orbital theory, the electron configuration underpins the filling of bonding and antibonding orbitals:

• Electrons first fill bonding orbitals, \(\sigma\) and \(\pi\), contributing to the stability and bond order.
• Any electron in antibonding orbitals \(\sigma^*\), subtracts from the overall bond order.

Thus, electron configuration becomes a crucial step in predicting bond strength and behavior more accurately.