Chapter 9, Problem 46QP

Sketch the shapes of the following molecular orbitals: ${\sigma }_{1s},{\sigma }_{1s}^{*.},{\pi }_{2p},{\pi }_{2p}^{.*}$. How do their energies compare?

Interpretation Introduction

Interpretation:

The shapes of ${\text{σ}}_{\text{1s}}\text{,}{\text{σ}}^{\ast }{}_{\text{1s}}\text{,}{\text{π}}_{\text{2p}}$, and ${\text{π}}^{\ast }{}_{\text{2p}}$ orbitals are to be drawn and their energies are to be compared.

Concept introduction:

Two atomic orbitals usually combine in order to form a bonding as well as an antibonding molecular orbital. Orbitals that lie on internuclear axis combine to form $\sigma$ molecular orbitals, and the orbitals parallel to each other combine to form $\pi$

molecular orbitals.

The molecular orbitals formed by the combination of $\text{1s}$ orbitals are a bonding molecular orbital, designated as ${\sigma }_{1s}$, and an antibonding molecular orbital, designated as ${\sigma }^{\ast }{}_{1s}$. The $\text{2s}$ orbital forms corresponding molecular orbitals.

The molecular orbitals formed by the combination of the ${\text{2p}}_{\text{x}}$ orbital are a bonding molecular orbital, designated as ${\sigma }_{2{\text{p}}_{\text{x}}}$, and an antibonding molecular orbital, designated as ${\sigma }^{\ast }{}_{2{\text{p}}_{\text{x}}}$.

The molecular orbitals formed by combining the ${\text{2p}}_{\text{y}}$ and ${\text{2p}}_{\text{z}}$ orbitals are bonding molecular orbitals, designated as ${\text{π}}_{{\text{2p}}_{\text{y}}}$ and ${\text{π}}_{{\text{2p}}_{\text{z}}}$, and antibonding molecular orbitals, designated as ${\text{π}}^{\ast }{}_{{\text{2p}}_{\text{y}}}$

and ${\text{π}}^{\ast }{}_{{\text{2p}}_{\text{z}}}$.

Answer to Problem 46QP

Solution: The shapes of the molecular orbitals, ${\text{σ}}_{\text{1s}}\text{,}{\text{σ}}^{\ast }{}_{\text{1s}}\text{,}{\text{π}}_{\text{2p}}$, and ${\text{π}}^{\ast }{}_{\text{2p}}$, are shown below:

${\text{σ}}_{\text{1s}}$ is lower in energy than the combining atomic orbitals; ${\text{σ}}^{\ast }{}_{\text{1s}}$ is higher in energy than the combining atomic orbitals. ${\text{π}}_{\text{2p}}$ is higher in energy than ${\text{σ}}^{\ast }{}_{\text{1s}}$; ${\text{π}}^{\ast }{}_{\text{2p}}$ is higher in energy than ${\text{π}}_{\text{2p}}$.

Explanation of Solution

The orbital ${\text{σ}}_{\text{1s}}$ is a bonding molecular orbital formed by constructive overlap of the $\text{1s}$ orbitals of two atoms. It is lower in energy than the combining atomic orbitals.

The shape of ${\text{σ}}_{\text{1s}}$ is shown below:

The orbital ${\text{σ}}^{\ast }{}_{\text{1s}}$ is an antibonding molecular orbital formed by destructive overlap of the $\text{1s}$ orbitals of two atoms. It is higher in energy than the combining atomic orbitals.

The shape of the ${\text{σ}}^{\ast }\text{1s}$ orbital is shown below:

The molecular orbital ${\text{π}}_{\text{2p}}$ is the bonding orbital that is formed by the constructive overlap of the ${\text{2p}}_{\text{y}}$ or ${\text{2p}}_{\text{z}}$ orbitals. They are higher in energy than ${\text{σ}}^{\ast }{}_{\text{1s}}$.

The shape of $\text{π2p}$ is shown below:

The orbital ${\text{π}}^{\ast }{}_{\text{2p}}$ is an antibonding molecular orbital formed by destructive overlap of the ${\text{2p}}_{\text{y}}$ or ${\text{2p}}_{\text{z}}$ orbital. They are higher in energy than the ${\text{π}}_{\text{2p}}$ orbital.

The shape of ${\text{π}}_{\text{2p}}$ is shown below:

Conclusion

The shapes of ${\text{σ}}_{\text{1s}}\text{,}{\text{σ}}^{\ast }{}_{\text{1s}}\text{,}{\text{π}}_{\text{2p}}$, and ${\text{π}}^{\ast }{}_{\text{2p}}$ have been drawn. ${\text{π}}^{\ast }{}_{\text{2p}}$ is higher in energy than ${\text{π}}_{\text{2p}}$, which in turn, is higher in energy than ${\text{σ}}^{\ast }{}_{\text{1s}}$. ${\text{σ}}_{\text{1s}}$ is lower in energy than the combining atomic orbitals; ${\text{σ}}^{\ast }{}_{\text{1s}}$ is higher in energy than the combining atomic orbitals.