The fundamental vibration frequency of H^35 Cl is 8.967 ×10^13 s^-1 and that of D^35 Cl is 6.428

step 1 Hi everyone, I'm here with Problem 14 from Chapter 7. In this problem, we're given two different

The fundamental vibration frequency of H^35 Cl is 8.967...

The fundamental vibration frequency of $\mathrm{H}^{35} \mathrm{Cl}$ is $8.967 \times$ $10^{13} \mathrm{s}^{-1}$ and that of $\mathrm{D}^{35} \mathrm{Cl}$ is $6.428 \times 10^{13} \mathrm{s}^{-1} .$ What would the separation be between infrared absorption lines of $\mathrm{H}^{35} \mathrm{Cl}$ and $\mathrm{H}^{37} \mathrm{Cl}$ on one hand and those of $\mathrm{D}^{35} \mathrm{Cl}$ and $\mathrm{D}^{37} \mathrm{Cl}$ on the other, if the force constants of the bonds are assumed to be the same in each pair?

The fundamental vibration frequency of $\mathrm{H}^{35} \mathrm{Cl}$ is $8.967 \times$ $10^{13} \mathrm{s}^{-1}$ and that of $\mathrm{D}^{35} \mathrm{Cl}$ is $6.428 \times 10^{13} \mathrm{s}^{-1} .$ What would the
separation be between infrared absorption lines of $\mathrm{H}^{35} \mathrm{Cl}$ and $\mathrm{H}^{37} \mathrm{Cl}$ on one hand and those of $\mathrm{D}^{35} \mathrm{Cl}$ and $\mathrm{D}^{37} \mathrm{Cl}$ on the other, if the force constants of the bonds are assumed to be the same in each pair?

Key Concepts

Infrared Spectroscopy

Infrared (IR) spectroscopy is an analytical technique used to measure the absorption of infrared light by molecules as they undergo vibrational transitions. The positions of absorption lines provide information about the vibrational frequencies, which are influenced by forces like bond strength and reduced mass. This technique is crucial for distinguishing between isotopologues and understanding molecular structure and bonding.

Isotope Effect

The isotope effect refers to the changes in physical and chemical properties, such as vibrational frequencies, that occur when one isotope is substituted for another in a molecule. Since isotopes differ in mass, the reduced mass of the molecule changes, shifting the vibrational frequency as predicted by the harmonic oscillator model. This effect is particularly observable in infrared spectroscopy when analyzing molecules with different isotopic compositions.

Reduced Mass

Reduced mass is a critical concept in diatomic physics that represents an effective inertial mass appearing in the equations of motion for the two-body system. It is determined by the formula ? = (m1*m2)/(m1 + m2), where m1 and m2 are the masses of the two atoms. Variations in reduced mass, such as those introduced by isotopic substitution, directly affect the vibrational frequency, making it essential for analyzing spectral shifts.

Harmonic Oscillator Model

This model treats the chemical bond between two atoms as a spring-like system where the atoms oscillate about their equilibrium positions. The vibrational frequency derived from this model depends on the square root of the force constant divided by the reduced mass of the diatomic molecule. It provides the foundational basis for understanding molecular vibrations and interpreting spectroscopic data, especially in the infrared region.

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