Physical Chemistry
Physical Chemistry
2nd Edition
ISBN: 9781133958437
Author: Ball, David W. (david Warren), BAER, Tomas
Publisher: Wadsworth Cengage Learning,
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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, HF (b) Hydrogen telluride, H 2 Te (c) Buckminsterfullerene, C 60 (d) Phenylalanine, C 6 H 5 CH 2 CHNH 2 COOH (e) Naphthalene, C 10 H 8 (f) The linear isomer of the C 4 radical (g) The bent isomer of C 4 radical.

Expert Solution
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Interpretation Introduction

(a)

Interpretation:

For the molecule hydrogen fluoride, 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 3N5 and for a non-linear molecule, the vibrational degrees of freedom is 3N6.

Answer to Problem 14.45E

For the molecule Hydrogen fluoride, 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.

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).

Total degrees of freedom=3N=3×2=6

Since, HF is a linear molecule. Therefore, the vibrational degrees of freedom is calculated as shown below.

Vibrational degrees of freedom=3N5=3×25=1

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

Expert Solution
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Interpretation Introduction

(b)

Interpretation:

For the molecule hydrogen telluride, H2Te, 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 3N5 and for a non-linear molecule, the vibrational degrees of freedom is 3N6.

Answer to Problem 14.45E

For the molecule hydrogen telluride, H2Te, 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.

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).

Total degrees of freedom=3N=3×3=9

Since, H2Te is a non-linear molecule. Therefore, the vibrational degree of freedom is calculated as shown below.

Vibrational degrees of freedom=3N6=3×36=3

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, H2Te, the number of total degrees of freedom is 9 and the number of vibrational degrees of freedom is 3.

Expert Solution
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Interpretation Introduction

(c)

Interpretation:

For the molecule Buckminsterfullerene, C60, 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 3N5 and for a non-linear molecule, the vibrational degrees of freedom is 3N6.

Answer to Problem 14.45E

For the molecule Buckminsterfullerene, C60, 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.

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).

Total degrees of freedom=3N=3×60=180

Since, C60 is a non-linear molecule. Therefore, the vibrational degrees of freedom is calculated as shown below.

Vibrational degrees of freedom=3N6=3×606=174

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, C60, the number of total degrees of freedom is 180 and the number of vibrational degrees of freedom is 174.

Expert Solution
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Interpretation Introduction

(d)

Interpretation:

For the molecule phenylalanine, C6H5CH2CHNH2COOH, 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 3N5 and for a non-linear molecule, the vibrational degrees of freedom is 3N6.

Answer to Problem 14.45E

For the molecule Phenylalanine, C6H5CH2CHNH2COOH, 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.

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).

Total degrees of freedom=3N=3×23=69

Since, C6H5CH2CHNH2COOH is a non-linear molecule. Therefore, the vibrational degrees of freedom is calculated as shown below.

Vibrational degrees of freedom=3N6=3×236=63

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, C6H5CH2CHNH2COOH, the number of total degrees of freedom is 69 and the number of vibrational degrees of freedom is 63.

Expert Solution
Check Mark
Interpretation Introduction

(e)

Interpretation:

For the molecule naphthalene, C10H8, 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 3N5 and for a non-linear molecule, the vibrational degrees of freedom is 3N6.

Answer to Problem 14.45E

For the molecule Naphthalene, C10H8, 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.

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).

Total degrees of freedom=3N=3×18=54

Since, C10H8 is a non-linear molecule. Therefore, the vibrational degrees of freedom is calculated as shown below.

Vibrational degrees of freedom=3N6=3×186=48

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, C10H8, the number of total degrees of freedom is 54 and the number of vibrational degrees of freedom is 48.

Expert Solution
Check Mark
Interpretation Introduction

(f)

Interpretation:

For the molecule linear isomer of the C4 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 3N5 and for a non-linear molecule, the vibrational degrees of freedom is 3N6.

Answer to Problem 14.45E

For the molecule the linear isomer of the C4 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 C4 radical the total number of atoms are 4.

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

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).

Total degrees of freedom=3N=3×4=12

Since, C4 is a linear molecule. Therefore, the vibrational degrees of freedom is calculated as shown below.

Vibrational degrees of freedom=3N5=3×45=7

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 C4 radical, the number of total degrees of freedom is 12 and the number of vibrational degrees of freedom is 7.

Expert Solution
Check Mark
Interpretation Introduction

(g)

Interpretation:

For the molecule the bent isomer of C4 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 3N5 and for a non-linear molecule, the vibrational degrees of freedom is 3N6.

Answer to Problem 14.45E

For the molecule the bent isomer of C4 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 C4 radical is a non-linear molecule. The total number of atoms present in bent isomer of the C4 radical is 4.

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

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).

Total degrees of freedom=3N=3×4=12

Since, C4 is a non-linear molecule. Therefore, the vibrational degrees of freedom is calculated as shown below.

Vibrational degrees of freedom=3N6=3×46=6

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 C4 radical, the number of total degrees of freedom is 12 and the number of vibrational degrees of freedom is 6.

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