Methanal, H2 C=O, exhibits an absorption in its IR spectrum at 2783 cm^-1 assigned to a C-H stre

Methanal, H2 C=O, exhibits an absorption in its IR spectrum...

Methanal, $\mathrm{H}_{2} \mathrm{C}=\mathrm{O},$ exhibits an absorption in its IR spectrum at $2783 \mathrm{cm}^{-1}$ assigned to a $\mathrm{C}-\mathrm{H}$ stretching mode. What will be the shift in vibrational wavenumber for this band on going from $\mathrm{H}_{2} \mathrm{C}=\mathrm{O}$ to $\mathrm{D}_{2} \mathrm{C}=\mathrm{O} ?$ (Assume that the force constants of the $\mathrm{C}-\mathrm{H}$ and $\mathrm{C}-\mathrm{D}$ bonds are the same; for exact masses, see problem $12.2 .$ )

Key Concepts

Isotopic Substitution Effects

Isotopic substitution, such as replacing hydrogen with deuterium, alters the mass of the atom involved in the bond without changing the force constant. This increase in mass results in an increased reduced mass, leading to a decrease in the vibrational frequency according to the harmonic oscillator formula. This shift in vibrational wavenumbers is a key indicator in studies of molecular dynamics and structure.

Reduced Mass

The concept of reduced mass is critical in the analysis of molecular vibrations. It is calculated from the masses of the two atoms in a bond and is used in the harmonic oscillator model. A change in the masses, such as substituting one atom for a heavier isotope, leads to a different reduced mass, which in turn shifts the vibrational frequency measured in an infrared spectrum.

Infrared Spectroscopy

Infrared spectroscopy is a technique used to study the vibrational transitions of molecules. Molecules absorb infrared light at specific frequencies that correspond to the vibrational modes of their bonds, and these absorptions can provide insights into molecular structure and bond characteristics.

Harmonic Oscillator Model

The harmonic oscillator model is a simplified way to describe the vibrational motions of a diatomic bond as if it were a mass on a spring. In this model, the vibrational frequency is proportional to the square root of the force constant divided by the reduced mass, which explains how changes in mass or bond stiffness affect the frequency of vibration.

Transcript

00:02 So this question actually asks about the vibration number difference between the stage stretching bond modes and cd stretching modes.

00:16 And basically, you say they are same, right, bonds, but you have different isotopes.

00:25 And as we learned, right, for isotopes, it doesn't change the chemical reactivity.

00:30 So the bond strength for the ch and the cd, we can regard them as similar for both.

00:47 Which means this two will have the same force constant.

00:57 K is the same.

01:02 And now we can write down the expression of the wave number.

01:07 Let's record it.

01:09 So remember the wave number of the ch is actually equals to 1.

01:17 Over 2 pi speed of light all times square root fourth constant, right? over the effective mass, which is the stage here.

01:38 And for the cd, right, vibration bonds equals to 1, 2 pi c times kcd, the fourth constant over mu the cd and if we use equation 1 over 2 we'll find 1 over 2 right we'll find the wave number c h over the wave number of cd this 2 which is cancel right so equals to vector mass over as we learned previously right the fourth count of these two molecules, two stretching is the same, so they can be cancelled.

03:02 And then this just equals to the effective mass of the cd over the effective mass of the ch.

03:14 So actually, we know this with number of the ch stretching is known.

03:21 So now if we want to know this one, right, we just need to calculate the effective mass.

03:27 Of the ch and c .d.

03:31 So now let's calculate the effective mass.

03:35 So the effective mass of the c .h is equal to the mass of the c times mass of the hydrogen over mass of the carbon plus the mass of the hydrogen, which here equals to 12 times 1 .1 .1 .1...

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