chapter 7 to 8 - Chem140B Physical Chemistry II Lecture...

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Chem140B – Physical Chemistry II Lecture Homework #5 March 3, 2009 due March 10, 2009 Q7.7) Why does the energy of a rotating molecule depend on l , but not on m l ? The energy depends only on the frequency of rotation, and not on the orientation of the rotation axis. The quantum number l determines the speed, and m l determines the orientation of the rotational axis. Q7.10) What makes the z direction special such that and commute, whereas and 2 ˆ , l ˆ , H ˆ z l 2 ˆ , l ˆ , H ˆ x l do not commute? There is nothing special about the z direction. We choose the z direction because the angular momentum operator for this component has a simple form. We can rotate our coordinate system to make any direction that we choose to lie along the z axis. P7.14) 1 H 35 Cl has a force constant k = N m –1 and a moment of inertia of 2.644 × 10 –47 kg m 2 . Calculate the frequency of the light corresponding to the lowest energy pure vibrational and pure rotational transitions. In what regions of the electromagnetic spectrum do the transitions lie? 2 13 1 27 1 1 516 kg s 8.963 10 s 1.008 34.969 1.661 10 kg 22 amu 1.008 34.969 amu This frequency lies in the infrared region of the eletromagnetic spectrum. The lowest energy transition is = 0 = 1. k JJ ν πμ π == = × ×× × + Δ () 2 34 22 4 7 2 00 11 1 1.055 10 J s 2 1 4.20 10 J 2 2.644 10 kg m 6.35 10 s This frequency lies in the microwave region of the eletromagnetic spectrum. rot rot EJ rr E h μμ × =+ × Δ ×
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P7.21) By substituting in the Schrödinger equation for rotation in three dimensions, show that the rotational wave function ( 1/2 2 5 3cos 1 16 θ π ⎛⎞ ) ⎜⎟ ⎝⎠ is an eigenfunction of the total energy operator. Determine the energy eigenvalue. ( ) ( ) () 2 2 22 2 0 2 2 2 0 2 2 0 ,, 11 sin , 2s i n s i n 55 1 1 16 16 sin i n s i n 5 21 6 YY EY r r r θφ θθφ μθ φ θθ ππ μπ ⎡⎤ ∂∂ −+ = ⎢⎥ ⎣⎦ ⎧⎫ ⎪⎪ ∂− ⎨⎬ ⎜⎪ ⎩⎭ =− = = = {} 2 23 2 2 0 2 2 2 2 00 0 15 1 6cos sin 6sin 12cos sin sin 2 16 sin 6 6 1 cos 6 18cos 3cos 1 6 6 6 The energy r rr r μ ⎞⎡ ⎛ ⎞⎡ −= = = == = 5 2 0 3 eigenvalue is ,corresponding to ( 1) with = 2.
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chapter 7 to 8 - Chem140B Physical Chemistry II Lecture...

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