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problem_set4 - 10.34 Fall 2006 Homework#4 Due Date Monday...

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10.34 – Fall 2006 Homework #4 Due Date: Monday, Oct. 2 nd , 2006 – 7 PM (Turn in hard copies in class or at the TA help session Monday evening.) Problem 1: Do problem 3.A.3 in Beers’s textbook. Problem 2: The heat capacity of many organic molecules is dominated by the torsions or internal-rotations about C-C single bonds, but unfortunately this is a bit tricky to calculate. Often a good approach is to first find the eigenvalues {E} of this 1-D Schrödinger equation: + V φ Ψ Ψ φ h 2 d 2 Ψ ( ) ( ) φ = E ( ) Eq.(1) 8 π 2 I d φ 2 where φ , which runs from 0 to 2 π , is the dihedral angle between substituents on the two carbon atoms making the single bond, V ( φ ) is the potential energy associated with this torsional motion, and I is its effective reduced moment of inertia. With the { E }, one can then compute the heat capacity using the statistical-mechanics formula C T = k T 2 Eq.(2) ( ) ( B E j exp ( E j k T 2 E j exp ( E j k T B ) where: E = B ) and E 2 = exp ( E k T exp ( j E k T B ) j B ) which you will see soon in 10.40 and again in the Spring in 10.65.
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