Problem_Set_2

Problem_Set_2 - Chemistry 341 Physical Chemistry I Fall 201...

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Unformatted text preview: Chemistry 341 Physical Chemistry I Fall 201 1 Problem Set 2 Due: Wednesday September 14 Answer ANY FIVE (5) questions. 1. ,B-Carotene [responsible for the orange color in certain vegetables such as carrots] is a linear polyene in which 10 C—C single and 11 C=C double bonds alternate along a chain of 22 carbon atoms. B-Carotene Use the flee—electron model to consider the 22 7t electrons and thereby predict the wavelength of the lowest Jr—Mr* transition . In estimating the length over which the Jr electrons move use l(C-—C)=154 pm and l(C=C)=l35 pm. The experimentally observed wavelength is 497 nm, in the Visible range of the electromagnetic spectrum. 2. Suppose that you wanted to synthesize a polyene that has its lowest excitation energy at a wavelength of 600 nm. What would the structure of the molecule be, according to the free electron molecular orbital (F EMO) theory? [Ans.CH2 = CH~ (CH: CH)” — CH: CH2] 3. Solve the Schrodinger equation for a particle in a half-infinite box 0 a, (a) Write down the solutions for x < 0, 0 S x S a , and x > a. Consider E < V0 only. (b) Apply boundary conditions at x=0, x=a, and x=00, generating a formula for the energy. You do not need to solve this formula explicitly to determine the energy as a function of the quantum number. (0) Use a calculator or a computer to solve numerically for the four lowest energy levels (in eV) for the case where a=0.l nm, M=l g mol_1, and V0=4 eV. Compare your result with the corresponding infinite-box energies (i.e. same parameters except that Vo=oo). 10. Consider a particle in a two-dimensional box of sides a X b, with a=3 b. Take the potential to be infinite outside the box and zero inside. (a) What are the energy eigenvalues? (b) Calculate the energies and the degeneracies of the first 10 states. (c) Show plots of the first four eigenfunctions. SAB 9.31, p.345. SAB 9.52, p.346. SAB Computer Problem 9A, p.346. SAB Computer Problem 9.B, p.346. SAB Computer Problem 9.E, p.346. For the v=2 state of the harmonic oscillator [le {/12 = N2{4ax2~1}e 2 1 a 1/4 N2 = (8)1/2 - For the v=2 state of the harmonic oscillator: (a) calculate the standard deviation of the X coordinate, A x , (b) calculate the standard deviation of the momentum p, A p , and with h (c) Show that the Heisenberg uncertainty principle A x A p 2 —2— is followed. ...
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Problem_Set_2 - Chemistry 341 Physical Chemistry I Fall 201...

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