10 34 Fall 2006 Homework 4 Due Date Monday Oct 2nd 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 Schr dinger equation h 2 d 2 V E 8 2 I d 2 Eq 1 where which runs from 0 to 2 is the dihedral angle between substituents on the two carbon atoms making the single bond V 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 where E E 2 E 2 k BT 2 E exp E k T exp E k T j j j B Eq 2 and E 2 B E exp E k T exp E k T 2 j j j B B which you will see soon in 10 40 and again in the Spring in 10 65 The potential energy function V is typically obtained by computing V at a half dozen values of using quantum chemistry techniques e g V 0 0 3 2 1 2 3 0 5 8 6 4 3 0 4 5 3 1 8 radians x10 20 Joules and then interpolating e g V yn cos n n 0 1 N max Eq 3 Cite as William Green Jr course materials for 10 34 Numerical Methods Applied to Chemical Engineering Fall 2006 MIT OpenCourseWare http ocw mit edu Massachusetts Institute of Technology Downloaded on DD Month YYYY For the case of Nmax 4 a least squares fitting yields the following y0 2 067 y1 2 033 y2 1 056 y3 1 767 y4 0 678 10 20 Joules It is very convenient to convert the Schr dinger equation from a differential equation into a linear algebra equation by searching for solutions where xm exp i m m M M 1 M 1 M Eq 4 a Write the linear algebra equation that corresponds to Equation 1 Hint multiply through by exp ip and integrate over Give an algebraic expression for the m n th element in the matrix assuming V is given exactly by Eq 3 b Write a Matlab function that makes use of your answer in part A to compute the energy values E corresponding to Eqn 1 taking in the moment of inertia I and the number of basis functions M For the case of I 3x10 45 kg m2 and M 50 calculate the zero point energy of the system you do not need to show all of the eigenvalues Make a plot showing V from 0 to 2 with a horizontal line shown on the plot for each eigenvalue of the system c Write a set of Matlab functions which cumulatively compute C T M using expansions Eq 4 for the case I 3x10 45 kg m2 Calculate the value of the heat capacity for M 100 at 300 K in J mol K d Make a plot of C T from T 100 K to T 2000 K for a series of M s to show how the calculation converges as M is increased Use the following values of M 20 50 100 300 and 500 Plot the curves N B The smallest E value you obtain is called the zero point energy Quantum mechanically it is impossible to remove the zero point energy from the torsional degree of freedom so even at T 0 K the atoms in the molecule are not quite stationary The zero point energy depends on the value of I and hence on the masses of the atoms in the molecule This mass dependence leads to small differences between the enthalpies and hence chemistry of different isotopes of the same molecule Also note that for a given M and I the energy values do not change meaning that the eigenvalue problem only needs to solved once to calculate an entire C T curve It may be useful to recall that cos n e i n e i n 2 Cite as William Green Jr course materials for 10 34 Numerical Methods Applied to Chemical Engineering Fall 2006 MIT OpenCourseWare http ocw mit edu Massachusetts Institute of Technology Downloaded on DD Month YYYY
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