Molecular vibration.
A sample contains HCl and DCl. Use the gap between vibrational levels to predict which molecule absorbs infrared light at the lower wavenumber.
Vibrational energy levels
| Isotope | IR band (cm⁻¹) | Reduced mass (u) |
|---|
Change the isotope
Deuterium adds a neutron to hydrogen. The bonding electrons experience essentially the same electric forces, so the model uses one spring constant (k = 480 N/m) for both bonds. The isotope swap changes the mass that vibrates in that potential well.
From bond to IR band
The curve is the bond’s potential energy, V(x) = ½kx². Each horizontal line is an allowed energy. Even the ground state sits half a quantum above the bottom of the well.
μ = m₁m₂ / (m₁ + m₂)
ν̃ = (1 / 2πc) √(k / μ)
Eᵥ = hcν̃ (v + ½)
The fundamental absorption takes the molecule from v = 0 to 1. On the chart’s energy scale, the gap between those lines reads directly as the predicted IR wavenumber. Use μ in kilograms and c in cm/s with k in N/m.
This harmonic model treats the bond as an ideal spring. Real HCl spectra also show anharmonic shifts and rotational structure.