Infrared spectroscopy is a technique used to study the vibrational frequencies of molecules, which can provide information about the bond lengths and force constants of the bonds within the molecule. For carbon dioxide CO2 , we can determine the vibrational frequencies and bond length using the following steps:1. Identify the vibrational modes: CO2 is a linear molecule with three atoms, so it has 3N-5 = 4 vibrational modes N = number of atoms . However, since it is a symmetric molecule, only two of these modes are IR active: the symmetric stretch 1 and the asymmetric stretch 3 .2. Obtain the vibrational frequencies: The vibrational frequencies for CO2 can be found in the literature or measured experimentally. The symmetric stretch 1 is around 1330 cm, and the asymmetric stretch 3 is around 2349 cm.3. Calculate the force constant: The force constant k can be calculated using the formula:k = 2 where is the vibrational frequency, is the reduced mass of the CO2 molecule, and is a constant approximately 3.14159 . The reduced mass for CO2 can be calculated using the atomic masses of carbon 12.01 amu and oxygen 16.00 amu : = mm / m + m = 12.01 * 16.00 / 12.01 + 16.00 = 6.86 amuConverting amu to kg, we get: = 6.86 amu * 1.66054 x 10 kg/amu = 1.139 x 10 kgNow, we can calculate the force constant for the symmetric stretch 1 :k = 2 * 1330 cm * 1.139 x 10 kg = 1.896 x 10 N/m4. Calculate the bond length: The bond length r can be calculated using the formula:r = h / 2c * k / where h is the Planck's constant 6.626 x 10 Js , c is the speed of light 2.998 x 10 cm/s , and k and are the force constant and reduced mass calculated earlier. Using the force constant for the symmetric stretch k :r = 6.626 x 10 Js / 2 * 3.14159 * 2.998 x 10 cm/s * 1.896 x 10 N/m / 1.139 x 10 kg = 1.162 x 10 cmConverting to ngströms 1 = 10 cm , we get:r = 1.162 So, the vibrational frequencies of CO2 are approximately 1330 cm symmetric stretch and 2349 cm asymmetric stretch , and the bond length is approximately 1.162 .