Chapter D, Problem D.15YT

Interpretation Introduction

Interpretation:

The atomic orbital overlap and MO energy diagrams for ${{\text{(CH}}_{\text{3}}\text{)}}_{\text{2}}{\text{CHC}}^{+}{\text{HCH}}_{\text{3}}$ are to be constructed. The HOMO and LUMO are to be identified, and whether they agree with the ones shown in Figure $\text{D-12}$ is to be determined.

Concept introduction:

According to Valence Bond Theory, the hybridized atom is the one, which is bonded to two or more atoms. Hybridization is the mixing of atomic orbitals. When the valence atomic orbitals of an atom, on mixing, form a new set of hybrid orbitals of equivalent energy, the atom is said to be hybridized. The mixing of two atomic orbitals, one from each adjacent atom, forms new orbitals. The hybridization of an atom can be determined based on the electron geometry about the atom. The VSEPR chart depicts the electron and molecular geometry on the basis of numbers of electron groups. The electron geometry about an atom describes the orientation of the group of electrons around it.

The molecular orbitals are formed by overlapping of atomic orbitals of adjacent atoms. The numbers of molecular orbitals formed are equal to the number of atomic orbitals overlap. The two atomic orbitals, on mixing along bonding axes, form two molecular orbitals, one is $\text{σ}$ and other is $\text{σ*}$. The $\text{σ}$ is a bonding MO, and $\text{σ*}$ is an antibonding MO. The $\text{σ}$ MOs have lower energy and greater stability than the corresponding $\text{σ*}$ MOs. Each single bond in the Lewis structure corresponds to a filled $\text{σ}$ MO, and hence each single bond refers as $\text{σ}$ bond. The HOMO is the highest energy MO that contains an electron, and the LUMO is the lowest energy MO that is empty.

Answer to Problem D.15YT

The atomic orbital overlap and MO energy diagrams for ${{\text{(CH}}_{\text{3}}\text{)}}_{\text{2}}{\text{CHC}}^{+}{\text{HCH}}_{\text{3}}$ are constructed indicating HOMO and LUMO in the following figure:

The constructed MO energy diagrams for ${{\text{(CH}}_{\text{3}}\text{)}}_{\text{2}}{\text{CHC}}^{+}{\text{HCH}}_{\text{3}}$ is in agreement with Figure $\text{D-12}$, indicating that the HOMO is a bonding $\text{σ}$ orbital, and the LUMO is the empty $\text{p}$ orbital.

Explanation of Solution

In ${{\text{(CH}}_{\text{3}}\text{)}}_{\text{2}}{\text{CHC}}^{+}{\text{HCH}}_{\text{3}}$, the four carbon atoms are ${\text{sp}}^{\text{3}}$ hybridized, and the positively charged carbon is ${\text{sp}}^2$ hybridized. Thus, according to VSEPR theory, the geometry at the ${\text{sp}}^{\text{3}}$ hybridized carbon is tetrahedral, and that at the ${\text{sp}}^2$ hybridized carbon is trigonal planar. Therefore, the Lewis structure is

In forming the $\text{C}-\text{H}$ bond, the ${\text{sp}}^{\text{3}}$ hybrid orbital of carbon overlaps with $\text{1s}$ atomic orbital of h the hydrogen atom, and in forming $\text{C}-\text{C}$ bonds, the two carbon atoms contribute their ${\text{sp}}^{\text{3}}$ hybrid orbitals in end-on overlap. The ${\text{sp}}^2$ hybridized carbon has its three ${\text{sp}}^2$ hybrid orbitals and one unhybridized pure $\text{p}$ orbital. Thus, there are overall $\text{15}$ interactions in ${{\text{(CH}}_{\text{3}}\text{)}}_{\text{2}}{\text{CHC}}^{+}{\text{HCH}}_{\text{3}}$. All these interactions are along the bonding axes, thus forming $\text{15}$ $\text{σ}$ MOs and $\text{15}$ $\text{σ*}$ MOs, and one $\text{p}$ AO remains as the nonbonding orbital. Thus, total of $\text{31}$ AOs mix to form $\text{31}$ MOs. The overlap of atomic orbitals is shown below:

According to Aufbau Principle, the $\text{30}$ electrons fill the lower energy $\text{15}$ $\text{σ}$ MOs, and one $\text{p}$ AO remains, and $\text{15}$ $\text{σ*}$ MOs remains unoccupied. Thus, each single bond in the Lewis structure corresponds to a filled $\text{σ}$ MO. Therefore, the $\text{σ}$ MOs are HOMO (Highest Occupied Molecular Orbitals). The $\text{p}$ AO is unoccupied and lower in energy than the unoccupied $\text{σ*}$ MOs; thus, $\text{p}$ AO is LUMO (Lowest Unoccupied Molecular Orbital).

The energy diagram for the formation of ${{\text{(CH}}_{\text{3}}\text{)}}_{\text{2}}{\text{CHC}}^{+}{\text{HCH}}_{\text{3}}$ ion from its constituent atoms indicating formation of molecular orbitals is shown below:

The Figure $\text{D-12}$ is:

The constructed MO energy diagrams for ${{\text{(CH}}_{\text{3}}\text{)}}_{\text{2}}{\text{CHC}}^{+}{\text{HCH}}_{\text{3}}$ is in agreement with the Figure $\text{D-12}$ as both indicate that the HOMO is the bonding $\text{σ}$ orbital and the LUMO is the empty $\text{p}$ orbital.

Conclusion

The energy diagram for the formation of ${{\text{(CH}}_{\text{3}}\text{)}}_{\text{2}}{\text{CHC}}^{+}{\text{HCH}}_{\text{3}}$ ion from its constituent atoms is drawn by indicating the formation of bonding and anti-bonding molecular orbitals.