# Quantum Chemistry/Example 18

Write an example question showing the determination of the force constant of CO using IR spectroscopy.

## Question edit

Using IR spectroscopy, calculate the force constant of carbon monoxide (CO) and report the value in kN/m.

## Solution edit

To solve this question, the IR spectrum of CO is required. The website, webbook.nist.gov/chemistry/ is a database that is available to the public for access to IR spectra. If you are unfamiliar on acquiring an IR spectrum, follow these steps: with the nist website open, go to Search Option -> Name -> Enter: Carbon Monoxide -> Select: IR Spectrum -> Change Transmittance to Absorbance -> Finally, take a screen shot.

The CO IR spectrum earlier acquired is a low resolution rotational-vibrational spectrum. In IR spectrum's, the fundamental frequency can be determined by utilizing the P-branch, Q-branch, and R-branch. Remember that the Q-branch is pure vibrational, which is forbidden, therefore the peak does not exist and is located between the P-branch and R-branch.

Using the website, apps.automeris.io/wpd/ will allow for an accurate collection of data points from the image of the CO IR spectrum taken earlier. For low resolution spectrums, locate the R-branch and P-branch, use the peak maxima as the data point. If you are unfamiliar on acquiring the data points from a image, follow these steps: Open the website -> Load Image -> Choose File -> Open -> 2D(X-Y) Plot -> Align Axes -> Plot Known -> Complete -> Insert Known Values -> Ok -> Add Points -> View Data.

Equation 1: Using the data points from the image, the fundamental frequency can be determined by the following relationship.

Where,

is the fundamental frequency

is the highest wavenumber

is the lowest wavenumber

Equation 2: The following equation is used to relate the fundamental frequency to the force constant.

Where,

is the fundamental frequency

is the speed of light

is the force constant

is the reduced mass

Equation 3: The reduced mass equation is required for relating the fundamental frequency to the force constant. Remember to use exact masses, and since the isotopes were not specified, that means we assume the most abundant version is to be used.

For oxygen:

For carbon:

Where,

is the reduced mass

is the exact mass 1

is the exact mass 2

Equation 4: Rearrange equation 2 to solve for the force constant.

The fundamental frequency of CO from IR spectra is determined using equation 1:

The reduced mass of CO is calculated using equation 3:

The force constant of CO is calculated using equation 4: