The mass of the deuterium molecule (D2) is twice that of the hydrogen molecule (H2) . If the vib

step 1 So we're talking about a vibrational frequency here between two hydrogen atoms, right, that are

The mass of the deuterium molecule (D2) is twice that of the...

The mass of the deuterium molecule $\left(\mathrm{D}_{2}\right)$ is twice that of the hydrogen molecule $\left(\mathrm{H}_{2}\right) .$ If the vibrational frequency of $\mathrm{H}_{2}$ is $1.30 \times 10^{14} \mathrm{Hz}$ , what is the vibrational frequency of $\mathrm{D}_{2} ?$ Assume the "spring constant" of attracting forces is the same for the two molecules.

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Key Concepts

Isotopic Substitution Effect

Isotopic substitution involves replacing one atom in a molecule with another isotope, which differs primarily in mass. Since the vibrational frequency in the harmonic oscillator model is inversely proportional to the square root of the reduced mass, increasing the mass (through isotopic substitution) leads to a decrease in the vibrational frequency. This principle is key in understanding spectral shifts in isotopically substituted molecules.

Harmonic Oscillator Model

This model treats the bond between two atoms as a spring connecting two masses. In vibrational spectroscopy and molecular physics, the vibrations of a diatomic molecule can be approximated by a simple harmonic oscillator, where the vibrational frequency is determined by the spring constant (a measure of bond strength) and the effective mass of the system.

Reduced Mass in Molecular Vibrations

In systems involving two interacting particles, such as the atoms in a diatomic molecule, the concept of reduced mass is crucial. It is defined as ? = (m1*m2)/(m1+m2) and allows the two-body vibrational problem to be simplified into an effective one-body problem. For molecules where the two atoms are identical, the reduced mass becomes m/2.

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