1. Number of Vibrational Modes

HCl is a linear diatomic molecule, and for any diatomic molecule:

Since HCl has 2 atoms (N = 2):

Number of vibrational modes = 1

2. Properties of the Vibrational Mode in HCl

• Vibrational Frequency (ν): 2885 cm⁻¹
• Vibrational Motion: Symmetric stretching along the H–Cl bond
• Reduced Mass (μ): Calculated using:
  μ = (m₁ × m₂) / (m₁ + m₂)

  Where m₁ = 1.0079 u (H) and m₂ = 35.45 u (Cl)

  μ = 0.980 u = 1.627 × 10⁻²⁷ kg

3. Comparison of HCl and H₂

• Reduced Mass of H₂:
  μ = (1.0079 × 1.0079) / (1.0079 + 1.0079) = 0.504 u = 8.37 × 10⁻²⁸ kg
• Vibrational Frequency of H₂: 4160 cm⁻¹
• H₂ has a lower reduced mass than HCl → leads to a higher vibrational frequency
• Lower mass allows bonds to vibrate faster, consistent with the spring model of molecular vibration

4. IR Intensity Comparison

• HCl:
  • Polar molecule
  • Dipole moment changes during vibration
  • IR Active
• H₂:
  • Homonuclear diatomic molecule
  • No permanent dipole
  • No change in dipole moment during stretching
  • Not IR Active