= 42.51 CP Consider the CO₂ molecule shown in Fig. 42.10c. The oxygen molecules have mass Mo and the carbon atom has mas

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42 51 Cp Consider The Co Molecule Shown In Fig 42 10c The Oxygen Molecules Have Mass Mo And The Carbon Atom Has Mas 1 (91.34 KiB) Viewed 19 times

= 42.51 CP Consider the CO₂ molecule shown in Fig. 42.10c. The oxygen molecules have mass Mo and the carbon atom has mass Mc. Parameterize the positions of the left oxygen atom, the carbon atom, and the right oxygen atom using x₁, x₂, and x3 as the respective rightward de- viations from equilibrium. Treat the bonds as Hooke's-law springs with common spring constant k'. (a) Use Newton's second law to obtain ex- pressions for M₁x; = M₁ d²x/di² in each case i = 1, 2, 3, where M1.2.3 (Mo, Mc, Mo). (Note: We represent time derivatives using dots.) Assume Xc = 0 at t = 0. (b) To ascertain the motion of the asym- metric stretching mode, set x₁ = x3 = Xo and set x₂ = Xc. Write the two independent equations that remain from your previous result. (c) Eliminate the sum Xo + Xc from your equations. Use what remains to ascertain X in terms of Xo. (d) Substitute your expression for Xc into your equation for Xo to derive a harmonic oscillator equation Meff Xo = -kXo. What is Meff? (e) This equation has the solution Xo(t) = A cos (wt). What is the angu- lar frequency w? (f) Using the experimentally determined spring constant k' = 1860 N/m and the atomic masses Mc = 12 u and Mo = 16 u, where u = 1.6605 x 10-27 kg, to determine the oscillation frequency f = w/2π.
Asymmetric stretching mode When carbon atom moves right ... C oxygen atoms move left, and vice versa.