Molecular vibration.

Compare HCl and DCl to see how isotope mass changes vibrational energy spacing and infrared absorption.

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.

Show energy levels for

More mass, closer levels.

From bond to IR band

The curve is the bond’s potential energy. Each horizontal line is an allowed energy. Even the ground state sits half a quantum above the bottom of the well.

V(x) = \frac{1}{2}kx^2

\mu = \frac{m_1m_2}{m_1+m_2}

\tilde{\nu} = \frac{1}{2\pi c}\sqrt{\frac{k}{\mu}}

E_v = hc\tilde{\nu}\left(v+\frac{1}{2}\right)

At fixed stiffness, wavenumber varies as the inverse square root of reduced mass. A heavier isotope lowers every level spacing while leaving the potential curve unchanged.

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.

HCl and DCl spectroscopy lab · NIST isotope masses