The greenhouse-gas carbon dioxide molecule CO2 strongly absorbs infrared radiation when its vibrational normal modes are excited by light at the normal-mode frequencies. CO₂ is a linear triatomic molecule, as shown in (Figure 1), with oxygen atoms of mass mo bonded to a central carbon atom of mass mc. You bonded to care know from chemistry that the atomic masses of carbon pective 12 and 16 and oxygen are, respectively, 12 and 16. Assume that the bond is an ideal spring with spring constant k There are two normal modes of this system for which oscillations take place along the axis. (You can ignore additional bending modes.) In this problem, you will find the normal modes and then use experimental data to determine the bond spring constant. Figure O mo C 2 mc. < 1 of 1 0 3 mo www. 1 X3 > Part A Let ₁, 2, and 3 be the atoms' positions measured from their equilibrium positions. First, use Hooke's law to write the net force on each atom. Pay close attention to signs! For each oxygen, the net force equals mod²x/dt². Carbon has a different mass, so it net force is mcd²x/dt². Define a² = k/mo and 32 = k/mc. Find an equation for the second derivative of ₁ coordinate. Express your answer in terms of some, all, or none of the variables ₁, 2, 3, and the constants a, B. 195] ΑΣΦ dt² Submit ▾ Part B Request Answer d²x₂ dt2 Find an equation for the second derivative of 2 coordinate. Express your answer in terms of some, all, or none of the variables ₁, 2, 3, and the constants a, B. ? VE] ΑΣΦ ?

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In vibrational spectroscopy, the fundamental band refers to a transition from the n = 0 state to n = 1, and the first overtone would be from the n = 0 state to n = 2. For carbon monoxide (CO), the fundamental occurs at 2143.4 cm-¹ and the first overtone at 4259.6 cm-¹. (a) For an anharmonic oscillator, the energy levels can be written (in wavenumbers) as 2 En ) ²³ = ve (n + ¹) - xeve (n +. + Find the values of ve and xeve from the fundamental and overtone transitions in CO. Report your results in units of cm-¹. (b) Use the values found for ve and xeve in part (a) to estimate the number of bound energy levels in CO. (Hint: For an anharmonic oscillator this number is determined by the point at which the spacing between consecutive levels goes to zero.) (c) If CO could be modeled as a Morse oscillator, and the values of ve and xeve provide energy levels consistent with the Morse potential what would be the value of the parameter D in units of cm-¹? How does this compare with the observed…

How do I solve for Problem 12.1? This problem is in a chapter called "Molecules," and it is part of a portion called Systems with two or more Atoms.