ps4_ans_1982

ps4_ans_1982 - MIT OpenCourseWare http/ocw.mit.edu 5.80...

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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 5.80 Small-Molecule Spectroscopy and Dynamics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . MASSACHUSETTS INSTITUTE OF TECHNOLOGY Chemistry 5.76 Spring 1982 Problem Set #4 ANSWERS 1. The corners of a cube are numbered 1, 2, 3, 4 clockwise around the top face of the cube, and 5, 6, 7, 8 clockwise around the bottom face, corner 5 lying under corner 1, corner 6 under corner 2, and so on. A face center is denoted by the two numbers of the corners between which a face diagonal can be drawn which passes through that face center (for example, either 13 or 24 would denote the center of the top face). (a) The structures of several kinds of AB 4 molecules are described as follows with the above numbering system. The A –atom is placed at the center of the cube, and the A — B bond distances are given by the cube dimensions. However, the B –atoms are not necessarily all equivalent, the actual equivalence being determined by the symmetry elements remaining in the AB 4 structures. (i) AB 4 (1 , 2 , 3 , 4) (ii) AB 4 (1 , 3 , 5 , 7) (iii) AB 4 (1 , 3 , 6 , 8) (iv) AB 4 (1 , 5 , 16 , 18) (v) AB 4 (13 , 36 , 68 , 18) Give the point-group symbol for each AB 4 structure (for example, C 3 v , D 6 h , and so on). Answer: B B B B B B T d (i) A C 4v (ii) A D 2h (iii) A B B B B B B B B B A C 2v (v) A (iv) D 4h B B B B B (b) Classify the five molecules of Part (a) as to type of molecular rotator (linear, symmetrical top, and so on). Answer: (i) Prolate Symmetric Top (ii) Asymmetric Top (iii) Spherical Top (iv) Asymmetric Top (v) Oblate Symmetric Top 5.76 Problem Set #4 ANSWERS Spring, 1982 page 2 (c) Which of the five molecules of Part (a) will give a pure-rotational spectrum in the far-infrared or mi- crowave region? Which will give a pure-rotational Raman e ff ect? Answer: Pure Rotation in Far IR or Microwave: (i) and (iv) Pure Rotational Raman: (i), (ii), (iv), and (v) 2. The harmonic oscillator (mass m ) in two dimensions has a potential energy V expressed in polar coordinates r , θ , of the general form 2 V = k 1 r 2 cos 2 θ + k 2 r 2 sin 2 θ , where k 1 and k 2 are force constants. For the special case k 1 = k 2 = k , the oscillator has a single frequency v = ( k / m ) 1 / 2 / 2 π , and its Schr¨odinger equation has solutions of the form 2 parenleftBigg − α r parenrightBigg ψ v ,ℓ = N v ,ℓ exp exp( i ℓθ ) P ( r ) 2 where v , ℓ are quantum numbers ( v = , 1 , 2 , . . . , ∞ ; ± ℓ = , 2 , 4 , . . . , v for v even, ± ℓ = 1 , 3 , 5 , . . . v for v odd); N v ,ℓ is a normalization constant; α = 4 π 2 mv / h ; P ( r ), a polynomial in r only, depends for its form on the values of v and ℓ and is an even function for even v , odd for odd v ....
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This note was uploaded on 11/28/2011 for the course CHEM 5.74 taught by Professor Robertfield during the Spring '04 term at MIT.

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ps4_ans_1982 - MIT OpenCourseWare http/ocw.mit.edu 5.80...

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