Carbon dioxide CO2 is a linear molecule with a carbon atom in the center and two oxygen atoms on either side. It has a total of 4 atoms, which means it has 12 degrees of freedom 3N, where N is the number of atoms . However, three of these degrees of freedom are translational, and three are rotational, leaving six degrees of freedom for vibrations. Since CO2 is a linear molecule, it has 3N-5 vibrational degrees of freedom, which means it has 4-1=3 vibrational modes.These three vibrational modes are:1. Symmetric stretching mode v1 : In this mode, both oxygen atoms move away from or towards the carbon atom in phase. Since there is no change in the dipole moment during this vibration, it is not IR active and does not appear in the IR spectrum.2. Bending mode v2 : In this mode, the two oxygen atoms move in a plane perpendicular to the molecular axis, causing the molecule to bend. This mode is doubly degenerate, meaning it has two different orientations with the same frequency. The bending mode is IR active and appears in the IR spectrum.3. Asymmetric stretching mode v3 : In this mode, one oxygen atom moves away from the carbon atom while the other moves towards it. This mode is IR active and appears in the IR spectrum.To calculate the vibrational frequencies, we can use the following formula: = 1/2 * k/ where is the vibrational frequency, k is the force constant, and is the reduced mass of the vibrating atoms. The reduced mass can be calculated as: = m1 * m2 / m1 + m2 where m1 and m2 are the masses of the two vibrating atoms.For CO2, the reduced mass for the symmetric stretching mode v1 is: = 12 * 16 / 12 + 16 = 192 / 28 = 6.857 amuFor the bending mode v2 and asymmetric stretching mode v3 , the reduced mass is the same: = 12 * 16 / 12 + 16 = 192 / 28 = 6.857 amuUsing experimental data, the force constants for CO2 are approximately:k1 symmetric stretching = 1900 N/mk2 bending = 440 N/mk3 asymmetric stretching = 2400 N/mNow we can calculate the vibrational frequencies:1 = 1/2 * 1900 / 6.857 = 0 not IR active 2 = 1/2 * 440 / 6.857 6.7 x 10^13 Hz IR active 3 = 1/2 * 2400 / 6.857 9.3 x 10^13 Hz IR active The IR spectrum of CO2 will show two peaks corresponding to the bending mode v2 at around 667 cm^-1 and the asymmetric stretching mode v3 at around 2349 cm^-1. The symmetric stretching mode v1 will not be present in the IR spectrum.