Chapter 11, Problem 11B.4BE

Interpretation Introduction

Interpretation:

The $\text{CO}$ and $\text{CS}$ bond length in $\text{OCS}$ molecule have to be stated.

Concept introduction:

A rotational spectrum is a spectrum in which only the rotation of molecule changes.  For a molecule to exhibit pure rotational spectrum, the molecule must possess a permanent electric dipole moment.  The molecules possessing dipole moment is considered as a handle that stirs the electromagnetic field into oscillation.  The energy for a rotational spectrum relates the quantum number and rotational constant to give the expression as shown below.

  $E=\tilde{B}J(J+1)$

The rotational constant further gives a relation with the moment of inertia of the molecule.

Answer to Problem 11B.4BE

The bond length of $\text{CO}$ in $\text{OCS}$ molecule is $\underset{\_}{1.67\times 10^{-10}m}$ and the bond length of $\text{CS}$ in $\text{OCS}$ molecule is $\underset{\_}{1.2\times 10^{-10}m}$.

Explanation of Solution

The inertia of a triatomic linear molecule $\left(\text{ABC}\right)$ is given by the equation as follows.

  $I=m_{\text{A}}{R}_{\text{AB}}^2+m_{\text{C}}{R}_{\text{BC}}^2-\frac{{\left(m_{\text{A}}{R}_{\text{AB}}-m_{\text{C}}{R}_{\text{BC}}\right)}^2}{m}$        (1)

Where,

• $m_{\text{A}}$ is the mass of $\text{A}$.
• $m_{\text{C}}$ is the mass of $\text{C}$.
• $R_{\text{AB}}$ is the bond distance between $\text{AB}$.
• $R_{\text{BC}}$ is the bonf distance between $\text{BC}$.
• $m$ is the sum of mass of $\text{A,B}$ and $\text{C}$.

From the above stated relation, bond distance can be calculated if isotopic substitution of any one atom is considered and the values of inertia of with and without isotopic substitution are considered.

On isotopic substitution and comparing the values of inertia the bond length is calculated as shown below.

  $R_{\text{AB}}=\frac{Im{m}_{\text{C}}\text{'}-I\text{'}{m}_{C}m\text{'}}{m_{\text{A}}{m}_{\text{B}}\left(m_{\text{C}}\text{'}-m_{\text{C}}\right)}$        (2)

Where,

• $m_{\text{C}}\text{'}$ is the mass of isotope.
• $m\text{'}$ is the mass of molecule after isotopic substitution.

Consider $\text{ABC}$ molecule as $\text{OCS}$ molecule, therefore $\text{A}$ is oxygen atom, $\text{B}$ is carbon atom and $\text{C}$ is ${}^{\text{32}}\text{S}$.

For bond length, ${}^{34}\text{S}$ is substituted in place of ${}^{\text{32}}\text{S}$ and the values of inertia are compared to give the formula of bond length of $\text{CO}$ as shown below.

  $R_{\text{CO}}=\frac{I{m}_{{\text{OC}}^{\text{34}}\text{S}}{m}_{{}^{34}\text{S}}\text{'}-I\text{'}{m}_{{\text{OC}}^{\text{34}}\text{S}}{m}_{{}^{32}\text{S}}}{m_{\text{O}}{m}_{\text{C}}\left(m_{{}^{34}\text{S}}\text{'}-m_{{}^{32}\text{S}}\right)}$        (3)

Where,

• $I$ is the inertia of ${\text{OC}}^{\text{32}}\text{S}$.
• $I\text{'}$ is the inertia of ${\text{OC}}^{\text{34}}\text{S}$.
• $m$ is the total mass of ${\text{OC}}^{\text{32}}\text{S}$.
• $m_{\text{O}}$ is the mass of oxygen atom.
• $m_{\text{C}}$ is the mass of carbon atom.
• $m_{{}^{34}\text{S}}\text{'}$ is the mass of ${}^{34}\text{S}$
• $m_{{}^{32}\text{S}}$ is the mass of ${}^{32}\text{S}$

For the values of inertia, the rotational constants $\left(\tilde{B}\right)$ of ${\text{OC}}^{\text{32}}\text{S}$ and ${\text{OC}}^{\text{34}}\text{S}$ are considered.

The value of rotational constant is given by the equation as shown below.

  $\tilde{B}=\frac{\hslash }{4\text{π}I}$        (4)

Convert the value of $\tilde{B}$ into ${\text{s}}^{-1}$ as shown below.

  $\begin{array}{c}1\text{MHz}={10}^6{\text{s}}^{-1}\\ 6081.5\text{}\text{MHz}=6081.5\times {10}^6{\text{s}}^{-1}\end{array}$

For ${\text{OC}}^{\text{32}}\text{S}$, substitute $\tilde{B}$ as $6081.5\times {10}^6{\text{s}}^{-1}$ and $\hslash$ as $1.05457\times {10}^{-34}\text{}\text{J}\cdot \text{s}$ in equation (4).

  $\begin{array}{c}6081.5\times {10}^6{\text{s}}^{-1}=\frac{1.05457\times {10}^{-34}\text{J}\cdot \text{s}}{4\times 3.14\times I}\\ I=\frac{1.05457\times {10}^{-34}\text{kg}{\text{m}}^2\cdot {\text{s}}^{-2}\cdot \text{s}}{4\times 3.14\times 6081.5\times {10}^6{\text{s}}^{-1}}\\ =\frac{1.05457\times {10}^{-34}}{7.63\times {10}^{10}}\text{kg}{\text{m}}^2\\ =1.38\times {10}^{-45}\text{kg}{\text{m}}^2\end{array}$

Convert the value of inertia into $\text{g}{\text{mol}}^{-\text{1}}{\text{m}}^{\text{2}}$ as shown below.

  $\begin{array}{c}\text{1}\text{kg}{\text{m}}^{\text{2}}=6.022\times {10}^{26}\text{g}{\text{mol}}^{-1}{\text{m}}^{\text{2}}\\ 1.38\times {10}^{-45}\text{kg}{\text{m}}^2=8.31\times {10}^{-19}\text{g}{\text{mol}}^{-1}{\text{m}}^{\text{2}}\end{array}$

Convert the value of $\tilde{B}$ into ${\text{s}}^{-1}$ as shown below.

  $\begin{array}{c}1\text{MHz}={10}^6{\text{s}}^{-1}\\ 5932.8\text{}\text{MHz}=5932.8\times {10}^6{\text{s}}^{-1}\end{array}$

For ${\text{OC}}^{\text{34}}\text{S}$, substitute $\tilde{B}$ as $5932.8\times {10}^6{\text{s}}^{-1}$ and $\hslash$ as $1.05457\times {10}^{-34}\text{}\text{J}\cdot \text{s}$ in equation (4).

  $\begin{array}{c}5932.8\times {10}^6{\text{s}}^{-1}=\frac{1.05457\times {10}^{-34}\text{J}\cdot \text{s}}{4\times 3.14\times I}\\ I\text{'}=\frac{1.05457\times {10}^{-34}\text{J}\cdot \text{s}}{4\times 3.14\times 5932.8\times {10}^6{\text{s}}^{-1}}\\ =\frac{1.05457\times {10}^{-34}}{7.45\times {10}^{10}}\text{kg}{\text{m}}^2\\ =1.41\times {10}^{-45}\text{kg}{\text{m}}^2\end{array}$

Convert the value of inertia into $\text{g}{\text{mol}}^{-\text{1}}{\text{m}}^{\text{2}}$ as shown below.

  $\begin{array}{c}\text{1}\text{kg}{\text{m}}^{\text{2}}=6.022\times {10}^{26}\text{g}{\text{mol}}^{-1}{\text{m}}^{\text{2}}\\ 1.41\times {10}^{-45}\text{kg}{\text{m}}^2=8.49\times {10}^{-19}\text{g}{\text{mol}}^{-1}{\text{m}}^{\text{2}}\end{array}$

The mass of ${\text{OC}}^{\text{32}}\text{S}$ molecule is calculated as shown below.

  $m=m_{\text{O}}+m_{\text{C}}+m_{{}^{32}\text{S}}$        (5)

Substitute $m_{\text{O}}$ as $16\text{g/mol}$, $m_{\text{S}}$ as $32\text{g/mol}$ and $m_{\text{C}}$ as $12\text{g/mol}$ in equation (5).

  $\begin{array}{c}m=\left(16+12+32\right)\text{g/mol}\\ =60\text{g/mol}\end{array}$

The mass of ${\text{OC}}^{\text{34}}\text{S}$ molecule is calculated as shown below.

  $m^{\text{'}}=m_{\text{O}}+m_{\text{C}}+m_{{}^{34}\text{S}}$        (5)

Substitute $m_{\text{O}}$ as $16\text{g/mol}$, $m_{{}^{34}\text{S}}$ as $34\text{g/mol}$ and $m_{\text{C}}$ as $12\text{g/mol}$ in equation (5).

  $\begin{array}{c}m=\left(16+12+34\right)\text{g/mol}\\ =62\text{g/mol}\end{array}$

Substitute $I$ as $8.31\times {10}^{-19}\text{g}{\text{mol}}^{-1}{\text{m}}^{\text{2}}$, $I\text{'}$ as $8.49\times {10}^{-19}\text{g}{\text{mol}}^{-1}{\text{m}}^{\text{2}}$, $m$ as $60\text{g/mol}$, $m_{\text{O}}$ as $16\text{g/mol}$, $m_{\text{S}}$ as $32\text{g/mol}$ , $m_S\text{'}$ as $34\text{g/mol}$ and $m_{\text{C}}$ as $12\text{g/mol}$ in equation (3).

  $\begin{array}{c}{R}_{\text{CO}}{}^2=\frac{\left(\begin{array}{l}8.31\times {10}^{-19}\text{g}{\text{mol}}^{-1}{\text{m}}^2\times 60\text{g/mol}\times 34\text{g/mol}-\\ 8.49\times {10}^{-19}\text{g}{\text{mol}}^{-1}{\text{m}}^2\times 62\text{g/mol}\times 32\text{g/mol}\end{array}\right)}{16\text{g/mol}\times \text{12}\text{g/mol}\left(34\text{g/mol}-32\text{g/mol}\right)}\\ R_{\text{CO}}=\sqrt{\frac{1.69\times {10}^{-15}-1.68\times {10}^{-15}}{192\times 2}{\text{m}}^2}\\ =\underset{\_}{1.67\times 10^{-10}m}\end{array}$

Therefore, the bond length of $\text{CO}$ in $\text{OCS}$ molecule is $\underset{\_}{1.67\times 10^{-10}m}$.

For bond length, ${}^{34}\text{S}$ is substituted in place of ${}^{\text{32}}\text{S}$ and the values of inertia are compared to give the formula of bond length of $\text{CS}$ as shown below.

  $R_{\text{CS}}=\frac{I{m}_{{\text{OC}}^{\text{34}}\text{S}}{m}_{{}^{34}\text{S}}\text{'}-I\text{'}{m}_{{\text{OC}}^{\text{34}}\text{S}}{m}_{{}^{32}\text{S}}}{m_{\text{S}}{m}_{\text{C}}\left(m_{{}^{34}\text{S}}\text{'}-m_{{}^{32}\text{S}}\right)}$        (6)

Where,

• $I$ is the inertia of ${\text{OC}}^{\text{32}}\text{S}$.
• $I\text{'}$ is the inertia of ${\text{OC}}^{\text{34}}\text{S}$.
• $m$ is the total mass of ${\text{OC}}^{\text{32}}\text{S}$.
• $m_{\text{S}}$ is the mass of sulphur atom.
• $m_{\text{C}}$ is the mass of carbon atom.
• $m_{{}^{34}\text{S}}\text{'}$ is the mass of ${}^{34}\text{S}$
• $m_{{}^{32}\text{S}}$ is the mass of ${}^{32}\text{S}$

Substitute $I$ as $8.31\times {10}^{-19}\text{g}{\text{mol}}^{-1}{\text{m}}^{\text{2}}$, $I\text{'}$ as $8.49\times {10}^{-19}\text{g}{\text{mol}}^{-1}{\text{m}}^{\text{2}}$, $m$ as $60\text{g/mol}$, $m_{\text{S}}$ as $32\text{g/mol}$ , $m_S\text{'}$ as $34\text{g/mol}$ and $m_{\text{C}}$ as $12\text{g/mol}$ in equation (3).

$\begin{array}{c}{R}_{\text{CS}}{}^2=\frac{\left(\begin{array}{l}8.31\times {10}^{-19}\text{g}{\text{mol}}^{-1}{\text{m}}^2\times 60\text{g/mol}\times 34\text{g/mol}-\\ 8.49\times {10}^{-19}\text{g}{\text{mol}}^{-1}{\text{m}}^2\times 62\text{g/mol}\times 32\text{g/mol}\end{array}\right)}{32\text{g/mol}\times \text{12}\text{g/mol}\left(34\text{g/mol}-32\text{g/mol}\right)}\\ R_{\text{CO}}=\sqrt{\frac{1.69\times {10}^{-15}-1.68\times {10}^{-15}}{384\times 2}{\text{m}}^2}\\ =\underset{\_}{1.2\times 10^{-10}m}\end{array}$

Therefore, the bond length of $\text{CS}$ in $\text{OCS}$ molecule is $\underset{\_}{1.2\times 10^{-10}m}$.