Randall D. Knight — Oscillations: Problem 82

The greenhouse-gas carbon dioxide molecule \(CO_2\) strongly absorbs infrared radiation when its vibrational normal modes are excited by light at the normal-mode frequencies. \(CO_2\) is a linear triatomic molecule, as shown in FIGURE CP15.82, with oxygen atoms of mass \(m_O\) bonded to a central carbon atom of mass \(m_C\). You know from chemistry that the atomic masses of carbon 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. a. Let \(x_1, x_2\), and \(x_3\) 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 \(m_O d^2x/dt^2\). Carbon has a different mass, so its net force is \(m_C d^2x/dt^2\). Define \(\alpha^2 = k/m_O\) and \(\beta^2 = k/m_C\), then write three equations for the second derivatives of the three coordinates. The main task is to find three new variables whose differential equation is the SHM equation, meaning that they oscillate at a single frequency that you can identify. The first is \(w = x_1 + (m_C/m_O) x_2 + x_3\). Notice that the ratio \(m_C/m_O = \alpha^2/\beta^2\). If you calculate \(d^2w/dt^2\), you’ll find that it equals zero. (Check this to verify that your work is correct to this point.) This seems odd, but notice that multiplying \(w\) by \(m_O\) gives an expression for the center of mass. So what you’ve learned is that the center of mass doesn’t accelerate. It could be drifting through space, but for this problem you can let the center of mass be at rest at the origin. So the solution to \(d^2w/dt^2 = 0\) is \(w = 0\). b. A second new variable (call it \(u\)) can be formed from only \(x_1\) and \(x_3\), the oxygen coordinates. You’re looking for a combination such that \(d^2u/dt^2 = -\text{constant} \times u\). Determine \(u\) and find an expression for its angular frequency (call it \(\omega\)) in terms of \(k\) and the masses. The solution to the equation for \(u\) is \(u = 2A \cos(\omega t + \phi_a)\). c. The third variable (call it \(v\)) is \(v = x_1 - cx_2 + x_3\), where \(c\) is a constant you need to determine so that the equation for \(v\) is the SHM equation. Determine \(v\) and find an expression for its angular frequency (call it \(\Omega\)) in terms of \(k\) and the masses. The solution to the equation for \(v\) is \(v = 2B \cos(\Omega t + \phi_b)\). d. Use the definitions of \(u, v\), and \(w\) to solve for \(x_1, x_2\), and \(x_3\) in terms of \(u, v\), and \(w\). Using \(M = 2m_O + m_C\) as the total mass of the molecule will keep the expressions simple. This is a general solution for the positions of the three masses. e. Let \(B = 0\) and \(\phi_a = \phi_b = 0\). This gives a solution that oscillates at frequency \(\omega\). Find expressions for \(x_1, x_2\), and \(x_3\). You should find that this normal mode at frequency \(\omega\) consists of the carbon atom sitting at rest while the oxygen atoms oscillate in and out in opposite directions. This is called the symmetric stretch mode. f. Now let \(A = 0\) and \(\phi_a = \phi_b = 0\). This gives a solution that oscillates at frequency \(\Omega\). Find expressions for \(x_1, x_2\), and \(x_3\). This normal mode at frequency \(\Omega\) consists of the carbon atom moving in one direction while the two oxygen atoms move in the opposite direction with a different amplitude. This is called the antisymmetric stretch mode. Notice that the center of mass remains stationary. g. The symmetric stretch frequency is known to be \(4.00 \times 10^{13} \text{ Hz}\). What is the spring constant of the C-O bond? Use \(1 \text{ u} = 1 \text{ atomic mass unit} = 1.66 \times 10^{-27} \text{ kg}\) to find the atomic masses in SI units. Interestingly, the spring constant is similar to that of springs you might use in the lab. h. Use the frequency of the symmetric stretch to predict the frequency of the antisymmetric stretch. The measured frequency is \(7.05 \times 10^{13} \text{ Hz}\), so your prediction is close but not perfect. The reason is that the bonds are not ideal springs but have a slight amount of anharmonicity. Nonetheless, you’ve learned a great deal about the \(CO_2\) molecule from a simple model of oscillating masses.