Problem 58 - General Chemistry: Principles and Modern Applications

Question

Show that both the valence-bond method and molecular orbital theory provide an explanation for the existence of the covalent molecule \(\mathrm{Na}_{2}\) in the gaseous state. Would you predict \(\mathrm{Na}_{2}\) by the Lewis theory?

Step-by-Step Solution

Answer

The valence-bond theory and the molecular orbital theory both support the existence of \( \mathrm{Na}_{2} \). The valence-bond theory suggests that overlapping of atomic orbitals forms this molecule. The molecular orbital theory indicates that a bond in \( \mathrm{Na}_{2} \) arises when the two 3s atomic orbitals of the Sodium atoms combine to form a bonding molecular orbital filled by the one valence electron of each Sodium atom. According to the Lewis theory, the existence of \( \mathrm{Na}_{2} \) is not favoured as the formation of \( \mathrm{Na}_{2} \) does not satisfy the octet rule.

Step 1: Valence-Bond Method

The valence-bond theory suggests that covalent bonds are formed when atomic orbitals on adjacent atoms overlap. For \( \mathrm{Na}_{2} \), each atom of Sodium (Na) has a 3s1 valence-shell electronic configuration. When two sodium atoms collide, the atomic orbitals (3s) of each overlap, leading to the forming of a bond through the sharing of the single electron in each of their 3s orbitals. Consequently, a covalent molecule is formed.

Step 2: Molecular Orbital Theory

The Molecular Orbital (MO) theory discusses the combination of atomic orbitals to lead to the formation of molecular orbitals. The molecular orbital for \( \mathrm{Na}_{2} \) can be constructed by combination of the 3s orbitals of each Sodium atom. The combination leads to two molecular orbitals, a bonding orbital (lower in energy) and an anti-bonding orbital (higher in energy). Both Sodium atoms each contribute one electron (from the 3s orbital) to fill up the bonding molecular orbital, thus leading to the existence of a bond in \( \mathrm{Na}_{2} \), and its existence as a covalent molecule in the gaseous state.

Step 3: Lewis Theory Perspective

The Lewis theory focuses primarily on valence electrons for illustrating chemical bonding. According to Lewis structures, each atom aims for an octet arrangement, i.e., having eight electrons in its outermost shell (except Hydrogen which aims for two). However, in the case of Sodium (Na), the Lewis theory might not predict the formation of \( \mathrm{Na}_{2} \). Each Sodium atom has one valence electron, and sharing of this single electron by each Sodium does lead to a bond. But this doesn't satisfy the octet rule, as each Sodium would then have only 2 electrons in their outermost shell. Thus, it can be predicted that \( \mathrm{Na}_{2} \) might not exist.

Key Concepts

Valence-Bond Theory

The Valence-Bond Theory is a crucial concept in understanding covalent bonding. This theory suggests that covalent bonds form when atomic orbitals on neighboring atoms overlap. In the context of a Sodium molecule (\( \mathrm{Na}_{2} \)), each sodium atom has an electronic configuration with a single electron in the 3s orbital. When two sodium atoms approach each other, their 3s orbitals overlap, allowing them to share these electrons. This overlap of orbitals results in the formation of a covalent bond. The simplicity of this theory lies in its ability to explain how single electrons are used to hold atoms together, allowing them to share electrons effectively. By depicting the overlap of their 3s orbitals, we see the foundation of their bond and can visualize the resulting covalent molecule.

Molecular Orbital Theory

Molecular Orbital Theory provides another perspective on covalent bonding. Unlike Valence-Bond Theory's focus on orbital overlap, Molecular Orbital Theory combines atomic orbitals to create molecular orbitals that encompass the entire molecule. In the case of \( \mathrm{Na}_{2} \), the 3s orbitals from each sodium atom merge to form two new molecular orbitals: a bonding orbital and an anti-bonding orbital. The bonding molecular orbital is lower in energy and is occupied by the electrons from both sodium atoms. This filling stabilizes the molecule by lowering its overall energy. This theory elegantly shows how all electrons in the molecule can be delocalized rather than confined to single atoms. It emphasizes the concept of energy levels and why the bonding molecular orbital results in a stable \( \mathrm{Na}_{2} \) molecule in its gaseous state. Molecular Orbital Theory helps us predict not just existence but also the relative energy of possible molecular configurations.

Lewis Theory

Lewis Theory simplifies covalent bonding by utilizing dot structures that represent valence electrons. The primary aim of this theory is to achieve an octet configuration, where atoms strive to have eight electrons in their valence shell. This satisfies their need for stability, except for hydrogen, which aims for two. In the case of \( \mathrm{Na}_{2} \), each sodium atom only has one valence electron. When sharing occurs, they form a bond by sharing these electrons. However, Lewis Theory might not predict the formation of \( \mathrm{Na}_{2} \) as it doesn't satisfy the octet rule; each sodium atom would end up with only two electrons. This limitation of the Lewis approach illustrates that while it's a useful tool for visualizing electron pairs, it may not always accurately predict molecular formation, especially for simpler molecules like \( \mathrm{Na}_{2} \). Despite its limitations, Lewis Theory remains fundamental in the general teaching and understanding of chemical bonding.