Chapter 10, Problem 10.108QP

The molecules cis-dichloroethylene and trans-dichloroethylene shown in Section 10.2 can be interconverted by heating or irradiation. (a) Starting with cis-dichloroethylene, show that rotating the C═C bond by 180° will break only the pi bond but will leave the sigma bond intact. Explain the formation of trans-dichloroethylene from this process. (Treat the rotation as two stepwise 90° rotations.) (b) Account for the difference in the bond enthalpies for the pi bond (about 270 kJ/mol) and the sigma bond (about 350 kJ/mol). (c) Calculate the longest wavelength of light needed to bring about this conversion.

Interpretation Introduction

Interpretation:

In 1,2-dichloroethylene,while rotating the $\text{C=C}$ bond by ${\text{180}}^{\text{o}}$, only pi-bond breaks whereas sigma bond will be intact. The formation of trans-dichloroethylene from this process has to be explained.

Concept Introduction:

$\sigma$-bonds are covalent bonds formed by the end-to-end overlapping of two atomic orbitals. In the $\text{σ}$-bonding, the electronic concentration will be only within the internuclear axis and because of that the $\text{σ}$-bond is being a localized bond.

$\pi$-bonds are also covalent bonds formed by the sideways overlapping of two atomic orbitals. In the $\text{π}$-bonding, the electronic concentration will be above and below the molecular plane of the $\text{σ}$-bond and because of that the $\text{π}$-bond is being a delocalized bond.

Rotation of 1,2-dichloroethylene:

1,2-dichloroethylene has two distinct isomers such as cis- and trans- isomers. The double bond between the two carbon atoms will have one sigma bond and one pi- bond. During ${\text{180}}^{\text{o}}$ rotation, the Pi-bond restricts the rotation about the sigma bond. To convert the isomer from one form to other, both sigma and pi bonds have to rotate together. A significant energy is required to carry out this process.

Explanation of Solution

It is known that sigma bond is formed by end-to-end overlap. So, rotation about ${\text{180}}^{\text{o}}$ cannot break sigma bond easily. Since a pi- bond is formed by sideways overlapping, it can be broken easily by the rotation about ${\text{180}}^{\text{o}}$. If the rotation takes place in two steps like ${\text{90}}^{\text{o}}$ rotations, then in the 1^st rotation the pi-bond gets broken and in the 2^nd rotation it gets reformed. Consequently, the cis-form gets transformed into the trans-form. So only the breaking of pi-bond brings the conversion. This is how the trans-dichloroethylene is being formed from the rotation about ${\text{180}}^{\text{o}}$.

Interpretation Introduction

Interpretation: The difference in the bond enthalpies for the pi and the sigma bond has to be accounted.

Concept Introduction:

Bond enthalpy is the amount of energy required to break one mole of a particular type of bond. Hence, bond enthalpy decides the bond strength.

Trends of bond enthalpy:

The larger bond enthalpy value for a type of bond means that the bond requires more energy for breaking it which implies that the particular bond is being a strong bond. Whereas the smaller bond enthalpy value for a type of bond means that the bond requires less energy for breaking it which implies that the particular bond is being a weak bond.

Explanation of Solution

The bond enthalpy for the sigma bond is given as $350\text{kJ/mol}$

The bond enthalpy for the pi bond is given as $270\text{kJ/mol}$

Clearly, the bond enthalpy value of sigma bond is being higher than that of the pi bond. This difference in the bond enthalpy implies the difference in the bond strength. From this information it can be concluded that sigma bond is stronger bond whereas pi-bond is weaker bond. It is known that sigma bond is formed by end-to-end overlap whereas pi-bond is formed by sideways overlap. The extent of the sideways overlap is less than the end-to-end overlap. Hence, pi-bond is weaker than the sigma bond.

Interpretation Introduction

Interpretation:

In the conversion 1,2-dichloroethylene from cis- to trans-form, the longest wavelength of light needed to bring about the conversion, has to be calculated.

Concept Introduction:

In 1,2-dichloroethylene , conversion from cis- to trans- can be achieved by the rotation of the double bond for about ${\text{180}}^{\text{o}}$.

Rotation of 1,2-dichloroethylene:

1,2-dichloroethylene has two distinct isomers such as cis- and trans- isomers. The double bond between the two carbon atoms will have one sigma bond and one pi- bond. During ${\text{180}}^{\text{o}}$ rotation, the Pi-bond restricts the rotation about the sigma bond. To convert the isomer from one form to other, both sigma and pi bonds have to rotate together. A significant energy is required to carry out this process.

The longest wavelength of light needed to bring about the conversion, can be calculated using the formula shown below:

$\begin{array}{c}\text{E}\text{=}\frac{\text{hc}}{\text{λ}}\\ \text{λ}=\frac{\text{hc}}{\text{E}}\\ \text{E}=\text{Energy}\text{required}\text{for}\text{the}\text{conversion}\text{of}\text{cis}\text{to}\text{trans- form}\text{.}\\ \text{λ}=\text{Wavelength}\text{of}\text{the}\text{light}\text{that}\text{can}\text{cause}\text{the}\text{conversion}\text{.}\end{array}$

$\begin{array}{c}\text{h}\text{=}\text{planck's}\text{constant}\\ \text{c}\text{=}\text{velocity of}\text{light}\end{array}$

Answer to Problem 10.108QP

The longest wavelength of light that is needed for the isomeric conversion of 1,2-dichloroethylene is $\text{λ}=443.43\text{nm}$.

Explanation of Solution

For the conversion from cis- to trans-form in 1,2-dichloroethylene, it is known that only the breaking of pi-bond brings the conversion. The bond enthalpy value of pi-bond is the amount of energy required to break the pi-bond. The bond enthalpy value of pi-bond is

$270\text{kJ/mol}$.

Converting the bond enthalpy value from $\text{kJ/mol}$ to $\text{J/mol}$.

$\begin{array}{c}\text{1 kJ}\text{=}1000\text{J}\\ 270\text{kJ/mol}=\frac{1000\text{J}}{\text{1 kJ}}\times 270\text{kJ/mol}\\ \text{= 270000}\text{J/mol}\end{array}$

Converting the bond enthalpy value from $\text{J/mol}$ into $\text{J/molecule}$

$\begin{array}{c}\text{1 mole}\text{=}6.023\times {10}^{23}\text{molecules}\\ \text{270000}\text{J/mol}=\frac{\text{270000}\text{J}}{\text{1 mol}}\\ \Rightarrow \frac{\text{270000}\text{J}}{6.023\times {10}^{23}\text{molecules}}\\ =\text{4}\text{.4828}\times {\text{10}}^{-19}\text{J}/\text{molecule}\end{array}$

This is the energy required to for the conversion of cis-to trans-form in one molecule. The wavelength corresponding to this energy can be calculated using the formula as follows:

$\begin{array}{c}\text{E}\text{=}\frac{\text{hc}}{\text{λ}}\\ \text{λ}=\frac{\text{hc}}{\text{E}}\\ \text{E}=\text{Energy}\text{required}\text{for}\text{the}\text{conversion}\text{of}\text{cis}\text{to}\text{trans- form}\text{.}\\ \text{=}\text{4}\text{.4828}\times {\text{10}}^{-19}\text{J}/\text{molecule}\end{array}$

$\text{λ}=\text{Wavelength}\text{of}\text{the}\text{light}\text{that}\text{can}\text{cause}\text{the}\text{conversion}\text{.}$

$\begin{array}{c}\text{h}\text{=}\text{planck's}\text{constant}\\ \text{= 6}\text{.626}\times {\text{10}}^{-34}\text{J/s}\\ \text{c}\text{=}\text{velocity of}\text{light}\\ \text{=}\text{3}\times {\text{10}}^8\text{m/s}\end{array}$

Substituting all the known values in the formula and evaluating it:

$\begin{array}{l}\text{λ}=\frac{\text{hc}}{\text{E}}\\ =\frac{\text{6}\text{.626}\times {\text{10}}^{-34}\text{J/s}\times \text{3}\times {\text{10}}^8\text{m/s}}{\text{4}\text{.4828}\times {\text{10}}^{-19}\text{J}}\\ =4.4343\times {\text{10}}^{-7}\text{m}\end{array}$

Converting the wavelength from meter into nanometre:

$\begin{array}{c}\text{1}\text{m}\text{=}{\text{10}}^{\text{-9}}\text{nm}\\ 4.4343\times {\text{10}}^{-7}\text{m}=\frac{{\text{10}}^{\text{9}}\text{nm}}{\text{1}\text{m}}\times 4.4343\times {\text{10}}^{-7}\text{m}\\ =443.43\text{nm}\end{array}$

Therefore, $\text{λ}=443.43\text{nm}$ is the longest wavelength of light that is needed for the isomeric conversion of 1,2-dichloroethylene through rotation.

Conclusion

The longest wavelength of light that is needed for the isomeric conversion of 1,2-dichloroethylene, has been calculated.