# Quantum Chemistry/Example 18

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

## Question

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

## Solution

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 ${\displaystyle X_{1},X_{2}.Y_{1},Y_{2}}$  -> 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.

${\displaystyle v_{0}=\left({\frac {num_{H}-num_{L}}{2}}\right)+num_{L}\equiv num_{H}-\left({\frac {num_{H}-num_{L}}{2}}\right)}$

Where,

${\displaystyle v_{0}}$  is the fundamental frequency ${\displaystyle ({\text{cm}}^{-1})}$

${\displaystyle num_{H}}$  is the highest wavenumber ${\displaystyle ({\text{cm}}^{-1})}$

${\displaystyle num_{L}}$  is the lowest wavenumber ${\displaystyle ({\text{cm}}^{-1})}$

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

${\displaystyle v_{0}={\frac {1}{2\pi c}}\left({\frac {k}{\mu }}\right)^{1/2}}$

Where,

${\displaystyle v_{0}}$  is the fundamental frequency ${\displaystyle ({\text{m}}^{-1})}$

${\displaystyle c}$  is the speed of light ${\displaystyle ({\text{m/s}})}$

${\displaystyle k}$  is the force constant ${\displaystyle ({\text{N/m}})}$

${\displaystyle \mu }$  is the reduced mass ${\displaystyle ({\text{kg}})}$

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: ${\displaystyle ^{16}{\text{O}}:15.994915{\text{ amu}}}$

For carbon: ${\displaystyle ^{12}{\text{C}}:12.000000{\text{ amu}}}$

${\displaystyle \mu ={\frac {m_{1}\cdot m_{2}}{m_{1}+m_{2}}}}$

Where,

${\displaystyle \mu }$  is the reduced mass ${\displaystyle ({\text{kg}})}$

${\displaystyle m_{1}}$  is the exact mass 1 ${\displaystyle ({\text{amu}})}$

${\displaystyle m_{2}}$  is the exact mass 2 ${\displaystyle ({\text{amu}})}$

${\displaystyle {\text{amu}}=1.66054{\text{ x}}10^{-27}{\text{kg}}}$

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

${\displaystyle k=4\pi ^{2}v_{0}^{2}c^{2}\mu }$

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

${\displaystyle v_{0}=\left({\frac {2176.206896{\text{ cm}}^{-1}-2122.41379{\text{ cm}}^{-1}}{2}}\right)+2122.41379{\text{ cm}}^{-1}=2149.310343{\text{ cm}}^{-1}}$

The reduced mass of CO is calculated using equation 3:

${\displaystyle \mu ={\frac {15.994915{\text{ amu}}\cdot 12.000000{\text{ amu}}}{15.994915{\text{ amu}}+12.000000{\text{ amu}}}}=6.856208708{\text{ amu}}\cdot (1.66054{\text{ x}}10^{-27}{\text{kg/amu}})=1.13850088{\text{ x}}10^{-26}{\text{ kg}}}$

The force constant of CO is calculated using equation 4:

${\displaystyle k=4\pi ^{2}\left(2149.310343{\text{ cm}}^{-1}\cdot \left({\frac {100{\text{ cm}}}{1{\text{ m}}}}\right)\right)^{2}(2.998{\text{ x}}10^{8}{\text{ m/s}})^{2}(1.13850088{\text{ x}}10^{-26}{\text{ kg}})=1886.184688{\text{ N/m}}}$

${\displaystyle k=1.886184688{\text{ kN/m}}}$

Answer: Using IR spectroscopy, the force constant of carbon monoxide was determined to be 1.87 kN/m.