Question

The increasing order of number of vibrational degrees of freedom from the following:
(A) \( \text{CO}_2 \)
(B) \( \text{CH}_4 \) 
(C) \( \text{H}_2 \) 
(D) \( \text{C}_2\text{H}_6 \)
follows the order:
Choose the correct answer from the options given below:

• A. (A), (B), (C), (D)
• B. (A), (D), (C), (B)
• C. (B), (A), (C), (D)
• D. (C), (A), (B), (D)

Hint:

Larger molecules with more atoms and bonds typically have more vibrational degrees of freedom.

Answer 1

The Correct Option is D

The number of vibrational degrees of freedom (DOF) in a molecule depends on its structure (linear or nonlinear) and the number of atoms present. For a molecule:

• Nonlinear Molecules: \( 3N - 6 \) vibrational modes
• Linear Molecules: \( 3N - 5 \) vibrational modes

where \( N \) is the number of atoms in the molecule.

1. Determine the Molecular Structure and Number of Atoms:
   • (A) CO\(_2\): Linear molecule with 3 atoms (1 C + 2 O)
   • (B) CH\(_4\): Nonlinear molecule with 5 atoms (1 C + 4 H)
   • (C) H\(_2\): Linear molecule with 2 atoms (2 H)
   • (D) C\(_2\)H\(_6\): Nonlinear molecule with 8 atoms (2 C + 6 H)
2. Calculate the Number of Vibrational Degrees of Freedom:
   • (C) H\(_2\):
     Linear molecule: \( 3N - 5 = 3(2) - 5 = 1 \) vibrational mode
   • (A) CO\(_2\):
     Linear molecule: \( 3N - 5 = 3(3) - 5 = 4 \) vibrational modes
   • (B) CH\(_4\):
     Nonlinear molecule: \( 3N - 6 = 3(5) - 6 = 9 \) vibrational modes
   • (D) C\(_2\)H\(_6\):
     Nonlinear molecule: \( 3N - 6 = 3(8) - 6 = 18 \) vibrational modes
3. Arrange in Increasing Order of Vibrational Degrees of Freedom:
   \[ \text{H}_2 \, (C) : 1 < \text{CO}_2 \, (A) : 4 < \text{CH}_4 \, (B) : 9 < \text{C}_2\text{H}_6 \, (D) : 18 \]
4. Conclusion:
   Therefore, the increasing order is: \[ (C) < (A) < (B) < (D) \]