Spectroscopy

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: vibrational modes
• Linear Molecules: vibrational modes

where is the number of atoms in the molecule.

1. Determine the Molecular Structure and Number of Atoms:
   • (A) CO : Linear molecule with 3 atoms (1 C + 2 O)
   • (B) CH : Nonlinear molecule with 5 atoms (1 C + 4 H)
   • (C) H : Linear molecule with 2 atoms (2 H)
   • (D) C H : Nonlinear molecule with 8 atoms (2 C + 6 H)
2. Calculate the Number of Vibrational Degrees of Freedom:
   • (C) H :
     Linear molecule: vibrational mode
   • (A) CO :
     Linear molecule: vibrational modes
   • (B) CH :
     Nonlinear molecule: vibrational modes
   • (D) C H :
     Nonlinear molecule: vibrational modes
3. Arrange in Increasing Order of Vibrational Degrees of Freedom:
   
4. Conclusion:
   Therefore, the increasing order is:

To determine the reduced mass ( ) of the HCl molecule, we use the formula for the reduced mass of two particles: where:

1. Step 1: Plug in the Values
   
2. Step 2: Calculate the Numerator and Denominator Separately
   
3. Step 3: Compute the Reduced Mass
   
4. Step 4: Round to Appropriate Significant Figures
   Considering the given atomic masses have three significant figures:

Conclusion:
The reduced mass of HCl is approximately . Therefore, the correct option is: Note:
It appears there was a typographical error in the original options provided. Option (3) listed , which is three orders of magnitude larger than the correct value. The accurate reduced mass aligns with option (1) or (4), both indicating .