Question

Explain why the bond in an \(H_{2}\) molecule is almost twice as strong as the bond in the \(H_{2}^{+}\) ion. Explain why the \(H_{2}\) bond is less than twice as strong as the \(H_{2}^{+}\) bond.

Short answer

The \(H_2\) bond is almost twice as strong as the \(H_2^+\) bond due to having double the electron density, but it's less than twice as strong due to quantum effects and electron repulsion.

Step-by-step solution

Understanding Bond Strength

The bond strength in a molecular system is primarily determined by the electron density between the bonded atoms. In the case of an \(H_2\) molecule, it is composed of two hydrogen atoms each contributing one electron, leading to a full bonding molecular orbital with two electrons.

Analyzing \(H_{2}^+\) Ion

The \(H_{2}^+\) ion, on the other hand, consists of two hydrogen nuclei but only one electron. This electron occupies the bonding molecular orbital, but the overall electron density is lower than it is in \(H_2\), resulting in a weaker bond.

Comparing Electron Density

In \(H_2\), the bonding molecular orbital contains two electrons, providing a higher electron density between the hydrogen nuclei. In contrast, \(H_2^+\) has only one electron to stabilize the bond, making it weaker than the bond in \(H_2\).

Explaining 'Almost Twice as Strong'

In a simplistic model, doubling the number of electrons in the bonding orbital would suggest a doubling of bond strength. Hence, \(H_2\) is almost twice as strong as \(H_2^+\) due to having twice the electron density in the bond region.

Explaining 'Less than Twice as Strong'

However, the bond strength is not exactly twice because of electron-electron repulsion and other quantum effects, which diminish the additional stabilization from having a second electron. These effects lead to \(H_2\)'s bond being slightly less than twice as strong as that in \(H_2^+\).

Key concepts

Molecular Orbital Theory

Molecular Orbital Theory is a fundamental concept that helps us understand the chemical bonding mechanism at a quantum level. According to this theory, atomic orbitals of bonded atoms combine to form molecular orbitals. These orbitals belong to the entire molecule rather than just individual atoms.
In the case of the hydrogen molecule ( H_2 ), when two hydrogen atoms approach each other, their 1s atomic orbitals overlap constructively to form a bonding molecular orbital. This bonding orbital accommodates both electrons contributed by the two hydrogen atoms. In contrast, the H_2^+ ion has only one electron and hence, forms a molecular orbital with lower electron density compared to H_2 .
The distribution of electrons in these molecular orbitals determines the molecule's stability. A greater concentration of electron density in the bonding region means a stronger bond, as seen in H_2 compared to H_2^+ .

Bond Strength

Bond strength refers to the energy required to break a bond between two atoms. It is significantly influenced by the electron density within the bond. More electrons in the bonding molecular orbital suggest a stronger bond due to increased electron density.
In H_2 , the bond is stronger because there are two electrons in the bonding orbital, leading to higher electron density and stronger attractive forces holding the hydrogen atoms together. For the H_2^+ ion, with just one electron, the bond is weaker since less electron density exists to mediate the attraction between the hydrogen nuclei.
This explains why the bond in H_2 is almost twice as strong as that in H_2^+ : two electrons provide substantial additional stability compared to just one.

Electron Density

Electron density plays a crucial role in defining the strength and characteristics of molecular bonds. Higher electron density implies more electrons are shared between nuclei in the bonding orbital, enhancing bond strength.
In H_2 , the presence of two electrons in the bonding molecular orbital increases the electron density significantly compared to H_2^+ , which has only one electron. As a result, in H_2 , the higher electron density leads to a stronger bond because the bonding region is richer in electrons.
The reduced electron density in H_2^+ results in a weaker bond since only one electron stabilizes the interaction between the two hydrogen nuclei, providing less attraction and stability than the double electron density in H_2 .

Hydrogen Molecule Ion

The hydrogen molecule ion, represented as H_2^+ , consists of two protons and a shared single electron. This single electron resides in the bonding molecular orbital. Because there is only one electron in the bonding region, the ion has relatively low electron density.
This reduced electron density means there is less attraction holding the hydrogen nuclei together, making the bond in H_2^+ significantly weaker than that in H_2 . Additionally, with only one electron, there is minimal repulsion to counteract besides the proton-proton repulsion, leading to a simpler but weaker bond than in the fully stable hydrogen molecule.

Quantum Effects in Bonding

Quantum effects in bonding account for the fact that chemical bonds are not merely mechanical connections but also involve quantum phenomena like electron wavefunction overlap and electron-electron interactions. In the case of H_2 , while you're doubling electron count thereby increasing bond strength notably, quantum effects such as electron-electron repulsions slightly limit this increment.
Such effects explain why the bond in H_2 isn't exactly twice as strong as that in H_2^+ . Electron repulsions become noticeable as further electrons are added, reducing the overall stabilizing effects. Moreover, the wave nature of electrons means that they can't all reside in the same region without some restrictive interactions.
These quantum mechanics' subtleties are crucial for understanding why addition of a second electron improves bond stabilization, albeit not perfectly doubling it.