Chapter 14, Problem 14.45E

Determine the number of total degrees of freedom and the number of vibrational degrees of freedom for the following molecules. (a) Hydrogen fluoride, $\text{HF}$ (b) Hydrogen telluride, $\text{H}{\text{2}}_{}\text{Te}$ (c) Buckminsterfullerene, ${\text{C}}_{60}$ (d) Phenylalanine, ${\text{C}}_{\text{6}}{\text{H}}_{\text{5}}{\text{CH}}_{\text{2}}{\text{CHNH}}_{\text{2}}\text{COOH}$ (e) Naphthalene, ${\text{C}}_{\text{10}}{\text{H}}_{\text{8}}$ (f) The linear isomer of the ${\text{C}}_{\text{4}}$ radical (g) The bent isomer of ${\text{C}}_{\text{4}}$ radical.

Interpretation Introduction

(a)

Interpretation:

For the molecule hydrogen fluoride, $\text{HF}$, the number of total degrees of freedom and the number of vibrational degrees of freedom are to be determined.

Concept introduction:

To describe the positions of each of the atoms in a molecule having $N$ atoms, it is necessary to use $x,y,$ and $z$ coordinates and thus, a molecule requires a total of $3N$ coordinates to describe its position in space. These $3N$ coordinates are called total degrees of freedom that contain rotational degrees of freedom, translational degrees of freedom and vibrational degrees of freedom. For a linear molecule, the vibrational degrees of freedom is $3N-5$ and for a non-linear molecule, the vibrational degrees of freedom is $3N-6$.

Answer to Problem 14.45E

For the molecule Hydrogen fluoride, $\text{HF}$, the number of total degrees of freedom is $6$ and the number of vibrational degrees of freedom is $1$.

Explanation of Solution

Hydrogen fluoride is a linear molecule. The total number of atoms present in hydrogen fluoride is $2$.

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

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

Where,

• $N$ is the number of atoms.

The value of $N$ for hydrogen fluoride is $2$.

Substitute the value of $N$ in equation (1).

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

Since, $\text{HF}$ is a linear molecule. Therefore, the vibrational degrees of freedom is calculated as shown below.

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

Therefore, the number of total degrees of freedom is $6$ and the number of vibrational degrees of freedom is $1$.

Conclusion

For the molecule Hydrogen fluoride, $\text{HF}$, the number of total degrees of freedom is $6$ and the number of vibrational degrees of freedom is $1$.

Interpretation Introduction

(b)

Interpretation:

For the molecule hydrogen telluride, $\text{H}{\text{2}}_{}\text{Te}$, the number of total degrees of freedom and the number of vibrational degrees of freedom are to be determined.

Concept introduction:

To describe the positions of each of the atoms in a molecule having $N$ atoms, it is necessary to use $x,y,$ and $z$ coordinates and thus, a molecule requires a total of $3N$ coordinates to describe its position in space. These $3N$ coordinates are called total degrees of freedom that contain rotational degrees of freedom, translational degrees of freedom and vibrational degrees of freedom. For a linear molecule, the vibrational degrees of freedom is $3N-5$ and for a non-linear molecule, the vibrational degrees of freedom is $3N-6$.

Answer to Problem 14.45E

For the molecule hydrogen telluride, $\text{H}{\text{2}}_{}\text{Te}$, the number of total degrees of freedom is $9$ and the number of vibrational degrees of freedom is $3$.

Explanation of Solution

Hydrogen telluride is a non-linear molecule. The total number of atoms present in hydrogen telluride is $3$.

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

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

Where,

• $N$ is the number of atoms.

The value of $N$ for hydrogen fluoride is $3$.

Substitute the value of $N$ in equation (1).

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

Since, $\text{H}{\text{2}}_{}\text{Te}$ is a non-linear molecule. Therefore, the vibrational degree of freedom is calculated as shown below.

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

Therefore, the number of total degrees of freedom is $9$ and the number of vibrational degrees of freedom is $3$.

Conclusion

For the molecule hydrogen telluride, $\text{H}{\text{2}}_{}\text{Te}$, the number of total degrees of freedom is $9$ and the number of vibrational degrees of freedom is $3$.

Interpretation Introduction

(c)

Interpretation:

For the molecule Buckminsterfullerene, ${\text{C}}_{60}$, the number of total degrees of freedom and the number of vibrational degrees of freedom are to be determined.

Concept introduction:

To describe the positions of each of the atoms in a molecule having $N$ atoms, it is necessary to use $x,y,$ and $z$ coordinates and thus, a molecule requires a total of $3N$ coordinates to describe its position in space. These $3N$ coordinates are called total degrees of freedom that contain rotational degrees of freedom, translational degrees of freedom and vibrational degrees of freedom. For a linear molecule, the vibrational degrees of freedom is $3N-5$ and for a non-linear molecule, the vibrational degrees of freedom is $3N-6$.

Answer to Problem 14.45E

For the molecule Buckminsterfullerene, ${\text{C}}_{60}$, the number of total degrees of freedom is $180$ and the number of vibrational degrees of freedom is $174$.

Explanation of Solution

Buckminsterfullerene is a non-linear molecule. The total number of atoms present in Buckminsterfullerene is $60$.

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

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

Where,

• $N$ is the number of atoms.

The value of $N$ for hydrogen fluoride is $60$.

Substitute the value of $N$ in equation (1).

$\begin{array}{rll}\text{Total degrees of freedom}& =& 3N\\ & =& \text{3}\times 60\\ & =& 180\end{array}$

Since, ${\text{C}}_{60}$ is a non-linear molecule. Therefore, the vibrational degrees of freedom is calculated as shown below.

$\begin{array}{rll}\text{Vibrational degrees of freedom}& =& 3N-6\\ & =& 3\times 60-6\\ & =& 174\end{array}$

Therefore, the number of total degrees of freedom is $180$ and the number of vibrational degrees of freedom is 174.

Conclusion

For the molecule Buckminsterfullerene, ${\text{C}}_{60}$, the number of total degrees of freedom is $180$ and the number of vibrational degrees of freedom is $174$.

Interpretation Introduction

(d)

Interpretation:

For the molecule phenylalanine, ${\text{C}}_{\text{6}}{\text{H}}_{\text{5}}{\text{CH}}_{\text{2}}{\text{CHNH}}_{\text{2}}\text{COOH}$, the number of total degrees of freedom and the number of vibrational degrees of freedom are to be determined.

Concept introduction:

To describe the positions of each of the atoms in a molecule having $N$ atoms, it is necessary to use $x,y,$ and $z$ coordinates and thus, a molecule requires a total of $3N$ coordinates to describe its position in space. These $3N$ coordinates are called total degrees of freedom that contain rotational degrees of freedom, translational degrees of freedom and vibrational degrees of freedom. For a linear molecule, the vibrational degrees of freedom is $3N-5$ and for a non-linear molecule, the vibrational degrees of freedom is $3N-6$.

Answer to Problem 14.45E

For the molecule Phenylalanine, ${\text{C}}_{\text{6}}{\text{H}}_{\text{5}}{\text{CH}}_{\text{2}}{\text{CHNH}}_{\text{2}}\text{COOH}$, the number of total degrees of freedom is $69$ and the number of vibrational degrees of freedom is $63$.

Explanation of Solution

Phenylalanine is a non-linear molecule. The total number of atoms present in phenylalanine is $23$.

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

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

Where,

• $N$ is the number of atoms.

The value of $N$ for hydrogen fluoride is $23$.

Substitute the value of $N$ in equation (1).

$\begin{array}{rll}\text{Total degrees of freedom}& =& 3N\\ & =& \text{3}\times \text{23}\\ & =& \text{69}\end{array}$

Since, ${\text{C}}_{\text{6}}{\text{H}}_{\text{5}}{\text{CH}}_{\text{2}}{\text{CHNH}}_{\text{2}}\text{COOH}$ is a non-linear molecule. Therefore, the vibrational degrees of freedom is calculated as shown below.

$\begin{array}{rll}\text{Vibrational degrees of freedom}& =& 3N-6\\ & =& 3\times 23-6\\ & =& 63\end{array}$

Therefore, the number of total degrees of freedom is $69$ and the number of vibrational degrees of freedom is $63$.

Conclusion

For the molecule phenylalanine, ${\text{C}}_{\text{6}}{\text{H}}_{\text{5}}{\text{CH}}_{\text{2}}{\text{CHNH}}_{\text{2}}\text{COOH}$, the number of total degrees of freedom is $69$ and the number of vibrational degrees of freedom is $63$.

Interpretation Introduction

(e)

Interpretation:

For the molecule naphthalene, ${\text{C}}_{\text{10}}{\text{H}}_{\text{8}}$, the number of total degrees of freedom and the number of vibrational degrees of freedom are to be determined.

Concept introduction:

To describe the positions of each of the atoms in a molecule having $N$ atoms, it is necessary to use $x,y,$ and $z$ coordinates and thus, a molecule requires a total of $3N$ coordinates to describe its position in space. These $3N$ coordinates are called total degrees of freedom that contain rotational degrees of freedom, translational degrees of freedom and vibrational degrees of freedom. For a linear molecule, the vibrational degrees of freedom is $3N-5$ and for a non-linear molecule, the vibrational degrees of freedom is $3N-6$.

Answer to Problem 14.45E

For the molecule Naphthalene, ${\text{C}}_{\text{10}}{\text{H}}_{\text{8}}$, the number of total degrees of freedom is $54$ and the number of vibrational degrees of freedom is $48$.

Explanation of Solution

Naphthalene is a non-linear molecule. The total number of atoms present in naphthalene is $18$.

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

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

Where,

• $N$ is the number of atoms.

The value of $N$ for hydrogen fluoride is $18$.

Substitute the value of $N$ in equation (1).

$\begin{array}{rll}\text{Total degrees of freedom}& =& 3N\\ & =& \text{3}\times 18\\ & =& 54\end{array}$

Since, ${\text{C}}_{\text{10}}{\text{H}}_{\text{8}}$ is a non-linear molecule. Therefore, the vibrational degrees of freedom is calculated as shown below.

$\begin{array}{rll}\text{Vibrational degrees of freedom}& =& 3N-6\\ & =& 3\times 18-6\\ & =& 48\end{array}$

Therefore, the number of total degrees of freedom is $54$ and the number of vibrational degrees of freedom is $48$.

Conclusion

For the molecule Naphthalene, ${\text{C}}_{\text{10}}{\text{H}}_{\text{8}}$, the number of total degrees of freedom is $54$ and the number of vibrational degrees of freedom is $48$.

Interpretation Introduction

(f)

Interpretation:

For the molecule linear isomer of the ${\text{C}}_{\text{4}}$ radical the number of total degrees of freedom and the number of vibrational degrees of freedom are to be determined.

Concept introduction:

To describe the positions of each of the atoms in a molecule having $N$ atoms, it is necessary to use $x,y,$ and $z$ coordinates and thus, a molecule requires a total of $3N$ coordinates to describe its position in space. These $3N$ coordinates are called total degrees of freedom that contain rotational degrees of freedom, translational degrees of freedom and vibrational degrees of freedom. For a linear molecule, the vibrational degrees of freedom is $3N-5$ and for a non-linear molecule, the vibrational degrees of freedom is $3N-6$.

Answer to Problem 14.45E

For the molecule the linear isomer of the ${\text{C}}_{\text{4}}$ radical, the number of total degrees of freedom is $12$ and the number of vibrational degrees of freedom is $7$.

Explanation of Solution

For the linear isomer of the ${\text{C}}_{\text{4}}$ radical the total number of atoms are 4.

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

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

Where,

• $N$ is the number of atoms.

The value of $N$ for hydrogen fluoride is $4$.

Substitute the value of $N$ in equation (1).

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

Since, ${\text{C}}_{\text{4}}$ is a linear molecule. Therefore, the vibrational degrees of freedom is calculated as shown below.

$\begin{array}{rll}\text{Vibrational degrees of freedom}& =& 3N-5\\ & =& 3\times 4-5\\ & =& 7\end{array}$

Therefore, the number of total degrees of freedom is $12$ and the number of vibrational degrees of freedom is $7$.

Conclusion

For the molecule the linear isomer of the ${\text{C}}_{\text{4}}$ radical, the number of total degrees of freedom is $12$ and the number of vibrational degrees of freedom is $7$.

Interpretation Introduction

(g)

Interpretation:

For the molecule the bent isomer of ${\text{C}}_{\text{4}}$ radical the number of total degrees of freedom and the number of vibrational degrees of freedom are to be determined.

Concept introduction:

To describe the positions of each of the atoms in a molecule having $N$ atoms, it is necessary to use $x,y,$ and $z$ coordinates and thus, a molecule requires a total of $3N$ coordinates to describe its position in space. These $3N$ coordinates are called total degrees of freedom that contain rotational degrees of freedom, translational degrees of freedom and vibrational degrees of freedom. For a linear molecule, the vibrational degrees of freedom is $3N-5$ and for a non-linear molecule, the vibrational degrees of freedom is $3N-6$.

Answer to Problem 14.45E

For the molecule the bent isomer of ${\text{C}}_{\text{4}}$ radical, the number of total degrees of freedom is $12$ and the number of vibrational degrees of freedom is $6$.

Explanation of Solution

For the bent isomer of the ${\text{C}}_{\text{4}}$ radical is a non-linear molecule. The total number of atoms present in bent isomer of the ${\text{C}}_{\text{4}}$ radical is 4.

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

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

Where,

• $N$ is the number of atoms.

The value of $N$ for hydrogen fluoride is $4$.

Substitute the value of $N$ in equation (1).

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

Since, ${\text{C}}_{\text{4}}$ is a non-linear molecule. Therefore, the vibrational degrees of freedom is calculated as shown below.

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

Therefore, the number of total degrees of freedom is $12$ and the number of vibrational degrees of freedom is $6$.

Conclusion

For the molecule the bent isomer of ${\text{C}}_{\text{4}}$ radical, the number of total degrees of freedom is $12$ and the number of vibrational degrees of freedom is $6$.