Methane CH4 is a tetrahedral molecule with four C-H bonds. In the harmonic oscillator approximation, we can calculate the vibrational frequencies of the molecule by considering the stretching and bending modes of the bonds.There are 3N-6 vibrational modes for a non-linear molecule, where N is the number of atoms. Methane has 5 atoms 1 carbon and 4 hydrogens , so it has 3 5 -6 = 9 vibrational modes. These modes can be classified as follows:1. 3 stretching modes symmetric, asymmetric, and doubly degenerate 2. 6 bending modes 3 doubly degenerate pairs The vibrational frequencies of these modes can be calculated using force constants and reduced masses. However, the exact values of these constants are not provided in the question, so we cannot calculate the exact frequencies.In general, the stretching modes have higher frequencies than the bending modes. The symmetric stretching mode is not IR-active, meaning it does not show up in the infrared spectrum. The asymmetric and doubly degenerate stretching modes are IR-active and show up in the infrared spectrum. The bending modes are also IR-active and show up in the infrared spectrum.The infrared spectrum of methane typically shows absorption bands in the range of 1300-1500 cm^-1 for the stretching modes and 600-1300 cm^-1 for the bending modes. These values are approximate and can vary depending on the specific force constants and reduced masses of the molecule.In summary, the vibrational frequencies and infrared spectrum of methane at 298 K using the harmonic oscillator approximation cannot be calculated exactly without the force constants and reduced masses. However, the general features of the spectrum can be described, with stretching modes appearing in the range of 1300-1500 cm^-1 and bending modes in the range of 600-1300 cm^-1.