Chapter 14, Problem 14.46E

Determine the number of total degrees of freedom and the number of vibrational degrees of freedom for the following species. (a) Hydrogen sulfide, ${\text{H}}_{\text{2}}\text{S}$ (b) Carbonyl sulfide, $\text{OCS}$ (c) The sulfate ion, ${\text{SO}}_4{}^{2-}$ (d) Phosgene, ${\text{COCl}}_{\text{2}}$ (e) Elemental chlorine, ${\text{Cl}}_{\text{2}}$ (f) A linear molecule having $20$ atoms (g) A nonlinear molecule having $20$ atoms

Interpretation Introduction

(a)

Interpretation:

The number of total degrees of freedom and vibrational degrees of freedom of ${\text{H}}_{\text{2}}\text{S}$ is to be calculated.

Concept introduction:

Spectroscopy method is used to identify the structure of the molecule. It is based on the interactions between matter and electromagnetic radiations. An electronic state of energy has its own vibrational states. The energy between the electronic states is large followed by vibrational states and then rotational states. During an electronic transition, electron from ground state moves straight to the excited state keeping the internuclear distance constant.

Answer to Problem 14.46E

The number of total degrees of freedom and vibrational degrees of freedom of ${\text{H}}_{\text{2}}\text{S}$ is $9$ and $3$, respectively.

Explanation of Solution

The number of total degrees of freedom is calculated by the formula shown below.

$\text{Total}\text{degrees}\text{of}\text{freedom}=3N$…(1)

Where,

• $N$ is the number of atoms in a molecule.

Substitute $N=3$ in equation (1).

$\begin{array}{rll}\text{Total}\text{degrees}\text{of}\text{freedom}& =& 3\times 3\\ & =& 9\end{array}$

The number of vibrational degrees of freedom for nonlinear molecule is calculated by the formula shown below.

$\text{Vibrational}\text{degrees}\text{of}\text{freedom}=3N-6$…(2)

Substitute $N=3$ in equation (2).

$\begin{array}{rll}\text{Vibrational}\text{degrees}\text{of}\text{freedom}& =& 3\times 3-6\\ & =& 3\end{array}$

Conclusion

The number of total degrees of freedom and vibrational degrees of freedom of ${\text{H}}_{\text{2}}\text{S}$ is $9$ and $3$, respectively.

Interpretation Introduction

(b)

Interpretation:

The number of total degrees of freedom and vibrational degrees of freedom of $\text{OCS}$ is to be calculated.

Concept introduction:

Spectroscopy method is used to identify the structure of the molecule. It is based on the interactions between matter and electromagnetic radiations. An electronic state of energy has its own vibrational states. The energy between the electronic states is large followed by vibrational states and then rotational states. During an electronic transition, electron from ground state moves straight to the excited state keeping the internuclear distance constant.

Answer to Problem 14.46E

The number of total degrees of freedom and vibrational degrees of freedom of $\text{OCS}$ is $9$ and $4$, respectively.

Explanation of Solution

The number of total degrees of freedom is calculated by the formula shown below.

$\text{Total}\text{degrees}\text{of}\text{freedom}=3N$…(3)

Where,

• $N$ is the number of atoms in a molecule.

Substitute $N=3$ in equation (3).

$\begin{array}{rll}\text{Total}\text{degrees}\text{of}\text{freedom}& =& 3\times 3\\ & =& 9\end{array}$

The number of vibrational degrees of freedom for linear molecule is calculated by the formula shown below.

$\text{Vibrational}\text{degrees}\text{of}\text{freedom}=3N-5$…(4)

Substitute $N=3$ in equation (4).

$\begin{array}{rll}\text{Vibrational}\text{degrees}\text{of}\text{freedom}& =& 3\times 3-5\\ & =& 4\end{array}$

Conclusion

The number of total degrees of freedom and vibrational degrees of freedom of $\text{OCS}$ is $9$ and $4$, respectively.

Interpretation Introduction

(c)

Interpretation:

The number of total degrees of freedom and vibrational degrees of freedom of ${\text{SO}}_4{}^{2-}$ is to be calculated.

Concept introduction:

Spectroscopy method is used to identify the structure of the molecule. It is based on the interactions between matter and electromagnetic radiations. An electronic state of energy has its own vibrational states. The energy between the electronic states is large followed by vibrational states and then rotational states. During an electronic transition, electron from ground state moves straight to the excited state keeping the internuclear distance constant.

Answer to Problem 14.46E

The number of total degrees of freedom and vibrational degrees of freedom of ${\text{SO}}_4{}^{2-}$ is $15$ and $9$, respectively.

Explanation of Solution

The number of total degrees of freedom is calculated by the formula shown below.

$\text{Total}\text{degrees}\text{of}\text{freedom}=3N$…(5)

Where,

• $N$ is the number of atoms in a molecule.

Substitute $N=5$ in equation (5).

$\begin{array}{rll}\text{Total}\text{degrees}\text{of}\text{freedom}& =& 3\times 5\\ & =& 15\end{array}$

The number of vibrational degrees of freedom for nonlinear molecule is calculated by the formula shown below.

$\text{Vibrational}\text{degrees}\text{of}\text{freedom}=3N-6$…(6)

Substitute $N=5$ in equation (6).

$\begin{array}{rll}\text{Vibrational}\text{degrees}\text{of}\text{freedom}& =& 3\times 5-6\\ & =& 9\end{array}$

Conclusion

The number of total degrees of freedom and vibrational degrees of freedom of ${\text{SO}}_4{}^{2-}$ is $15$ and $9$, respectively.

Interpretation Introduction

(d)

Interpretation:

The number of total degrees of freedom and vibrational degrees of freedom of ${\text{COCl}}_2$ is to be calculated.

Concept introduction:

Spectroscopy method is used to identify the structure of the molecule. It is based on the interactions between matter and electromagnetic radiations. An electronic state of energy has its own vibrational states. The energy between the electronic states is large followed by vibrational states and then rotational states. During an electronic transition, electron from ground state moves straight to the excited state keeping the internuclear distance constant.

Answer to Problem 14.46E

The number of total degrees of freedom and vibrational degrees of freedom of ${\text{COCl}}_2$ is $12$ and $6$, respectively.

Explanation of Solution

The number of total degrees of freedom is calculated by the formula shown below.

$\text{Total}\text{degrees}\text{of}\text{freedom}=3N$…(7)

Where,

• $N$ is the number of atoms in a molecule.

Substitute $N=4$ in equation (7).

$\begin{array}{rll}\text{Total}\text{degrees}\text{of}\text{freedom}& =& 3\times 4\\ & =& 12\end{array}$

The number of vibrational degrees of freedom for nonlinear molecule is calculated by the formula shown below.

$\text{Vibrational}\text{degrees}\text{of}\text{freedom}=3N-6$…(8)

Substitute $N=4$ in equation (8).

$\begin{array}{rll}\text{Vibrational}\text{degrees}\text{of}\text{freedom}& =& 3\times 4-6\\ & =& 6\end{array}$

Conclusion

The number of total degrees of freedom and vibrational degrees of freedom of ${\text{COCl}}_2$ is $12$ and $6$, respectively.

Interpretation Introduction

(e)

Interpretation:

The number of total degrees of freedom and vibrational degrees of freedom of ${\text{Cl}}_2$ is to be calculated.

Concept introduction:

Spectroscopy method is used to identify the structure of the molecule. It is based on the interactions between matter and electromagnetic radiations. An electronic state of energy has its own vibrational states. The energy between the electronic states is large followed by vibrational states and then rotational states. During an electronic transition, electron from ground state moves straight to the excited state keeping the internuclear distance constant.

Answer to Problem 14.46E

The number of total degrees of freedom and vibrational degrees of freedom of ${\text{Cl}}_2$ is $6$ and $1$, respectively.

Explanation of Solution

The number of total degrees of freedom is calculated by the formula shown below.

$\text{Total}\text{degrees}\text{of}\text{freedom}=3N$…(9)

Where,

• $N$ is the number of atoms in a molecule.

Substitute $N=2$ in equation (9).

$\begin{array}{rll}\text{Total}\text{degrees}\text{of}\text{freedom}& =& 3\times 2\\ & =& 6\end{array}$

The number of vibrational degrees of freedom for linear molecule is calculated by the formula shown below.

$\text{Vibrational}\text{degrees}\text{of}\text{freedom}=3N-5$…(10)

Substitute $N=2$ in equation (10).

$\begin{array}{rll}\text{Vibrational}\text{degrees}\text{of}\text{freedom}& =& 3\times 2-5\\ & =& 1\end{array}$

Conclusion

The number of total degrees of freedom and vibrational degrees of freedom of ${\text{Cl}}_2$ is $6$ and $1$, respectively.

Interpretation Introduction

(f)

Interpretation:

The number of total degrees of freedom and vibrational degrees of freedom of a linear molecule having $20$ atoms is to be calculated.

Concept introduction:

Spectroscopy method is used to identify the structure of the molecule. It is based on the interactions between matter and electromagnetic radiations. An electronic state of energy has its own vibrational states. The energy between the electronic states is large followed by vibrational states and then rotational states. During an electronic transition, electron from ground state moves straight to the excited state keeping the internuclear distance constant.

Answer to Problem 14.46E

The number of total degrees of freedom and vibrational degrees of freedom of a linear molecule having $20$ atoms is $60$ and $55$, respectively.

Explanation of Solution

The number of total degrees of freedom is calculated by the formula shown below.

$\text{Total}\text{degrees}\text{of}\text{freedom}=3N$…(11)

Where,

• $N$ is the number of atoms in a molecule.

Substitute $N=20$ in equation (11).

$\begin{array}{rll}\text{Total}\text{degrees}\text{of}\text{freedom}& =& 3\times 20\\ & =& 60\end{array}$

The number of vibrational degrees of freedom for linear molecule is calculated by the formula shown below.

$\text{Vibrational}\text{degrees}\text{of}\text{freedom}=3N-5$…(12)

Substitute $N=20$ in equation (12).

$\begin{array}{rll}\text{Vibrational}\text{degrees}\text{of}\text{freedom}& =& 3\times 20-5\\ & =& 55\end{array}$

Conclusion

The number of total degrees of freedom and vibrational degrees of freedom of a nonlinear molecule having $20$ atoms is $60$ and $55$, respectively.

Interpretation Introduction

(g)

Interpretation:

The number of total degrees of freedom and vibrational degrees of freedom of a linear molecule having $20$ atoms is to be calculated.

Concept introduction:

Spectroscopy method is used to identify the structure of the molecule. It is based on the interactions between matter and electromagnetic radiations. An electronic state of energy has its own vibrational states. The energy between the electronic states is large followed by vibrational states and then rotational states. During an electronic transition, electron from ground state moves straight to the excited state keeping the internuclear distance constant.

Answer to Problem 14.46E

The number of total degrees of freedom and vibrational degrees of freedom of a nonlinear molecule having $20$ atoms is $60$ and $54$, respectively.

Explanation of Solution

The number of total degrees of freedom is calculated by the formula shown below.

$\text{Total}\text{degrees}\text{of}\text{freedom}=3N$…(13)

Where,

• $N$ is the number of atoms in a molecule.

Substitute $N=20$ in equation (13).

$\begin{array}{rll}\text{Total}\text{degrees}\text{of}\text{freedom}& =& 3\times 20\\ & =& 60\end{array}$

The number of vibrational degrees of freedom for linear molecule is calculated by the formula shown below.

$\text{Vibrational}\text{degrees}\text{of}\text{freedom}=3N-6$…(14)

Substitute $N=20$ in equation (14).

$\begin{array}{rll}\text{Vibrational}\text{degrees}\text{of}\text{freedom}& =& 3\times 20-6\\ & =& 54\end{array}$

Conclusion

The number of total degrees of freedom and vibrational degrees of freedom of a nonlinear molecule having $20$ atoms is $60$ and $54$, respectively.