Chapter 11, Problem 11B.2E

(a)

Interpretation Introduction

Interpretation:

Moment of inertia of $\left(i\right){}^{\text{1}}{\text{H}}_{\text{2}}\text{, }\left(ii\right){}^{\text{2}}{\text{H}}_{\text{2}}\text{, }{\left(iii\right)}^{\text{12}}{\text{C}}^{\text{16}}{\text{O}}_{\text{2}}\text{, }\left(iv\right){}^{\text{13}}{\text{CO}}_{\text{2}}$ has to be determined.

Concept Introduction:

Moment of inertia of diatomic molecules is given by,

  $\text{Moment of inertia, I = }\mu {\text{R}}^2$

Effective mass of a homonuclear diatomic molecule is given by,

  $\mu \text{ = }\frac{1}{2}m$

Moment of inertia of the triatomic linear rotors $A{B}_2$ is given as,

  $\begin{array}{l}A-B-A\\ \\ Momentof{\text{ inertia, I = 2m}}_A{R}^2\end{array}$

Answer to Problem 11B.2E

1.         i.            Moment of inertia of ${}^1{\text{H}}_2$ is $\text{4}\text{.58}\times {\text{10}}^{-48}\text{ kg}{\text{.m}}^2$.
2.       ii.            Moment of inertia of ${}^2{\text{H}}_2$ is $\text{9}\text{.16}\times {\text{10}}^{-48}\text{ kg}{\text{.m}}^2$.
3.     iii.            Moment of inertia of ${}^{12}{C}^{16}{O}_2$ is $\text{7}\text{.22}\times {10}^{-46}\text{ kg}\text{.}{m}^2$.
4.     iv.            Moment of inertia of ${}^{13}C{O}_2$ is $\text{7}\text{.22}\times {10}^{-46}\text{ kg}\text{.}{m}^2$.

Explanation of Solution

1.         i.            Moment of inertia of ${}^1{\text{H}}_2$

Given data,

  $\begin{array}{c}m\left({}^{1}H\right)=\text{ 1}{\text{.008m}}_{\text{u}}\\ \\ R\text{ = 74}\times {\text{10}}^{-12}m\end{array}$

Effective mass of ${}^1{\text{H}}_2$ is given by,

  $\begin{array}{c}\mu =\frac{1}{2}m\\ \\ =\frac{1}{2}\times \left(\text{1}{\text{.008m}}_{\text{u}}\right)\\ \\ =\text{ 0}{\text{.504 m}}_u\\ \\ =\text{ 0}\text{.504}\times 1.66054\times {10}^{-27}kg\\ \\ =\text{ 8}\text{.369}\times {10}^{-28}\text{ kg}\end{array}$

Moment of inertia is calculated

  $\begin{array}{c}I\text{ = }\mu R^2\\ \\ =\left(\text{8}\text{.369}\times {10}^{-28}\text{ kg}\right){\left(\text{74}\times {\text{10}}^{-12}m\right)}^2\\ \\ =\text{ 4}\text{.58}\times {\text{10}}^{-48}\text{ kg}{\text{.m}}^2\end{array}$

Moment of inertia of ${}^1{\text{H}}_2$ is $\text{4}\text{.58}\times {\text{10}}^{-48}\text{ kg}{\text{.m}}^2$.

1.       ii.            Moment of inertia of ${}^2{\text{H}}_2$

Given data,

  $\begin{array}{c}m\left({}^{2}H\right)=\text{ 2}{\text{.0141m}}_{\text{u}}\\ \\ R\text{ = 74}\times {\text{10}}^{-12}m\end{array}$

Effective mass of ${}^2{\text{H}}_2$ is given by,

  $\begin{array}{c}\mu =\frac{1}{2}m\\ \\ =\frac{1}{2}\times \left(\text{2}{\text{.0141m}}_{\text{u}}\right)\\ \\ =\text{ 1}{\text{.007 m}}_u\\ \\ =\text{ 1}\text{.007}\times 1.66054\times {10}^{-27}kg\\ \\ =\text{ 1}\text{.672}\times {10}^{-27}\text{ kg}\end{array}$

Moment of inertia is calculated

  $\begin{array}{c}I\text{ = }\mu R^2\\ \\ =\left(\text{1}\text{.672}\times {10}^{-27}\text{ kg}\right){\left(\text{74}\times {\text{10}}^{-12}m\right)}^2\\ \\ =\text{ 9}\text{.16}\times {\text{10}}^{-48}\text{ kg}{\text{.m}}^2\end{array}$

Moment of inertia of ${}^2{\text{H}}_2$ is $\text{9}\text{.16}\times {\text{10}}^{-48}\text{ kg}{\text{.m}}^2$.

1.     iii.            Moment of inertia of ${}^{12}{C}^{16}{O}_2$

Given data,

  $\begin{array}{c}m\left({}^{16}O\right)={\text{ 16m}}_{\text{u}}\\ \\ R\text{ = }116.2\times {\text{10}}^{-12}m\end{array}$

Moment of inertia is calculated

$\begin{array}{c}Momentof{\text{ inertia, I = 2m}}_A{R}^2\\ \\ =\text{ 2}\times \left(\text{16}\times \text{1}\text{.66054}\times {10}^{-27}\text{ kg}\right)\times {\left(116.2\times {\text{10}}^{-12}m\right)}^2\\ \\ =\text{ 7}\text{.17}\times {10}^{-46}\text{ kg}\text{.}{m}^2\end{array}$

Moment of inertia of ${}^{12}{C}^{16}{O}_2$ is $\text{7}\text{.17}\times {10}^{-46}\text{ kg}\text{.}{m}^2$.

1.     iv.            Moment of inertia of ${}^{13}C{O}_2$

Given data,

  $\begin{array}{c}m\left({}^{16}O\right)={\text{ 16m}}_{\text{u}}\\ \\ R\text{ = }116.2\times {\text{10}}^{-12}m\end{array}$

Moment of inertia is calculated

$\begin{array}{c}Momentof{\text{ inertia, I = 2m}}_A{R}^2\\ \\ =\text{ 2}\times \left(\text{16}\times \text{1}\text{.66054}\times {10}^{-27}\text{ kg}\right)\times {\left(116.2\times {\text{10}}^{-12}m\right)}^2\\ \\ =\text{ 7}\text{.17}\times {10}^{-46}\text{ kg}\text{.}{m}^2\end{array}$

Moment of inertia of ${}^{13}C{O}_2$ is $\text{7}\text{.17}\times {10}^{-46}\text{ kg}\text{.}{m}^2$.

(b)

Interpretation Introduction

Interpretation:

Rotational constant of $\left(i\right){}^{\text{1}}{\text{H}}_{\text{2}}\text{, }\left(ii\right){}^{\text{2}}{\text{H}}_{\text{2}}\text{, }{\left(iii\right)}^{\text{12}}{\text{C}}^{\text{16}}{\text{O}}_{\text{2}}\text{, }\left(iv\right){}^{\text{13}}{\text{CO}}_{\text{2}}$ in hertz and as a wavenumber in $c{m}^{-1}$ has to be determined.

Concept Introduction:

Rotational constant of a molecule can be determined using the given formula.

  $\begin{array}{c}\text{Rotational constant, B = }\frac{\hslash }{\text{4πI}}\\ \\ \text{where,}\\ \text{I = moment of inertia }\end{array}$

Wavenumber is the total number of wavelengths observed at a unit distance. Equation for wavenumber can be given as,

  $\begin{array}{c}\text{Wavenumber, }\overline{\text{ν}}\text{ =}\frac{\text{1}}{\text{λ}}\text{ = }\frac{\text{ν}}{\text{c}}\\ \\ \text{where, }\\ \text{λ}\text{=}\text{wavelength}\\ \text{ν}=\text{frequency}\\ \text{c = speed of light}\end{array}$

Answer to Problem 11B.2E

1.         i.            Rotational constant of ${}^1{\text{H}}_2$ is $\text{1}\text{.83 THz}$.

  Rotational constant of ${}^1{\text{H}}_2$ in $c{m}^{-1}$ is $\text{61}{\text{.04 cm}}^{-1}$

1.       ii.            Rotational constant of ${}^2{\text{H}}_2$ is $\text{0}\text{.917 THz}$.

  Rotational constant of ${}^2{\text{H}}_2$ in $c{m}^{-1}$ is $\text{30}{\text{.59 cm}}^{-1}$.

1.     iii.            Rotational constant of ${}^{12}{C}^{16}{O}_2$ is $\text{11}\text{.6 GHz}$.

  Rotational constant of ${}^{12}{C}^{16}{O}_2$ in $c{m}^{-1}$ is $\text{0}{\text{.387 cm}}^{-1}$.

1.     iv.            Rotational constant of ${}^{13}C{O}_2$ is $\text{11}\text{.6 GHz}$.

  Rotational constant of ${}^{13}C{O}_2$ in $c{m}^{-1}$ is $\text{0}{\text{.387 cm}}^{-1}$.

Explanation of Solution

1.         i.            Rotational constant of ${}^1{\text{H}}_2$ in hertz and as a wavenumber in $c{m}^{-1}$

Moment of inertia of ${}^1{\text{H}}_2$ is $\text{4}\text{.58}\times {\text{10}}^{-48}\text{ kg}{\text{.m}}^2$.

Rotational constant of ${}^1{\text{H}}_2$ in $Hz$ is determined as given,

  $\begin{array}{c}\text{Rotational constant, B = }\frac{1.05457\times {10}^{-34}Js}{\text{4}\times \text{3}\text{.14}\times \left(\text{4}\text{.58}\times {\text{10}}^{-48}\text{ kg}{\text{.m}}^2\right)}\\ \\ \text{= 1}\text{.83}\times {\text{10}}^{12}{\text{ s}}^{-1}\\ \\ =\text{ 1}\text{.83 THz}\end{array}$

Rotational constant of ${}^1{\text{H}}_2$ is $\text{1}\text{.83 THz}$.

Rotational constant of ${}^1{\text{H}}_2$ in $c{m}^{-1}$ is determined as given,

  $\begin{array}{c}\stackrel{\sim }{B}\text{ = }\frac{B}{c}\\ \\ =\frac{\left(\text{1}\text{.83}\times {\text{10}}^{12}{\text{ s}}^{-1}\right)}{\left(2.998\times {10}^{8}m{s}^{-1}\right)}\\ \\ ={\text{ 6104 m}}^{-1}\\ \\ =\text{ 61}{\text{.04 cm}}^{-1}\end{array}$

Rotational constant of ${}^1{\text{H}}_2$ in $c{m}^{-1}$ is $\text{61}{\text{.04 cm}}^{-1}$

1.       ii.            Rotational constant of ${}^2{\text{H}}_2$ in hertz and as a wavenumber in $c{m}^{-1}$

Moment of inertia of ${}^2{\text{H}}_2$ is $\text{9}\text{.16}\times {\text{10}}^{-48}\text{ kg}{\text{.m}}^2$.

Rotational constant of ${}^2{\text{H}}_2$ in $Hz$ is determined as given,

  $\begin{array}{c}\text{Rotational constant, B = }\frac{1.05457\times {10}^{-34}Js}{\text{4}\times \text{3}\text{.14}\times \left(\text{9}\text{.16}\times {\text{10}}^{-48}\text{ kg}{\text{.m}}^2\right)}\\ \\ \text{= 9}\text{.17}\times {\text{10}}^{11}{\text{ s}}^{-1}\\ \\ =\text{ 0}\text{.917 THz}\end{array}$

Rotational constant of ${}^2{\text{H}}_2$ is $\text{0}\text{.917 THz}$.

Rotational constant of ${}^2{\text{H}}_2$ in $c{m}^{-1}$ is determined as given,

  $\begin{array}{c}\stackrel{\sim }{B}\text{ = }\frac{B}{c}\\ \\ =\frac{\left(\text{9}\text{.17}\times {\text{10}}^{11}{\text{ s}}^{-1}\right)}{\left(2.998\times {10}^{8}m{s}^{-1}\right)}\\ \\ ={\text{ 3059 m}}^{-1}\\ \\ =\text{ 30}{\text{.59 cm}}^{-1}\end{array}$

Rotational constant of ${}^2{\text{H}}_2$ in $c{m}^{-1}$ is $\text{30}{\text{.59 cm}}^{-1}$.

1.     iii.            Rotational constant of ${}^{12}{C}^{16}{O}_2$ in hertz and as a wavenumber in $c{m}^{-1}$

Moment of inertia of ${}^{12}{C}^{16}{O}_2$ is $\text{9}\text{.16}\times {\text{10}}^{-48}\text{ kg}{\text{.m}}^2$.

Rotational constant of ${}^{12}{C}^{16}{O}_2$ in $Hz$ is determined as given,

  $\begin{array}{c}\text{Rotational constant, B = }\frac{1.05457\times {10}^{-34}Js}{\text{4}\times \text{3}\text{.14}\times \left(\text{7}\text{.17}\times {10}^{-46}\text{ kg}\text{.}{m}^2\right)}\\ \\ \text{= 1}\text{.17}\times {\text{10}}^{10}{\text{ s}}^{-1}\\ \\ =\text{ 11}\text{.7 GHz}\end{array}$

Rotational constant of ${}^{12}{C}^{16}{O}_2$ is $\text{11}\text{.7 GHz}$.

Rotational constant of ${}^{12}{C}^{16}{O}_2$ in $c{m}^{-1}$ is determined as given,

  $\begin{array}{c}\stackrel{\sim }{B}\text{ = }\frac{B}{c}\\ \\ =\frac{\left(\text{1}\text{.17}\times {\text{10}}^{10}{\text{ s}}^{-1}\right)}{\left(2.998\times {10}^{8}m{s}^{-1}\right)}\\ \\ =\text{ 39}{\text{.0 m}}^{-1}\\ \\ =\text{ 0}{\text{.39 cm}}^{-1}\end{array}$

Rotational constant of ${}^{12}{C}^{16}{O}_2$ in $c{m}^{-1}$ is $\text{0}{\text{.39 cm}}^{-1}$.

1.     iv.            Rotational constant of ${}^{13}C{O}_2$ in hertz and as a wavenumber in $c{m}^{-1}$

Moment of inertia of ${}^{13}C{O}_2$ is $\text{9}\text{.16}\times {\text{10}}^{-48}\text{ kg}{\text{.m}}^2$.

Rotational constant of ${}^{13}C{O}_2$ in $Hz$ is determined as given,

  $\begin{array}{c}\text{Rotational constant, B = }\frac{1.05457\times {10}^{-34}Js}{\text{4}\times \text{3}\text{.14}\times \left(\text{7}\text{.17}\times {10}^{-46}\text{ kg}\text{.}{m}^2\right)}\\ \\ \text{= 1}\text{.17}\times {\text{10}}^{10}{\text{ s}}^{-1}\\ \\ =\text{ 11}\text{.7 GHz}\end{array}$

Rotational constant of ${}^{13}C{O}_2$ is $\text{11}\text{.7 GHz}$.

Rotational constant of ${}^{13}C{O}_2$ in $c{m}^{-1}$ is determined as given,

  $\begin{array}{c}\stackrel{\sim }{B}\text{ = }\frac{B}{c}\\ \\ =\frac{\left(\text{1}\text{.17}\times {\text{10}}^{10}{\text{ s}}^{-1}\right)}{\left(2.998\times {10}^{8}m{s}^{-1}\right)}\\ \\ =\text{ 39}{\text{.0 m}}^{-1}\\ \\ =\text{ 0}{\text{.39 cm}}^{-1}\end{array}$

Rotational constant of ${}^{13}C{O}_2$ in $c{m}^{-1}$ is $\text{0}{\text{.39 cm}}^{-1}$.