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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. ,BCarotene [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. BCarotene 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 halfinﬁnite 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 inﬁnitebox energies (i.e. same
parameters except that Vo=oo). 10. Consider a particle in a twodimensional 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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 Fall '08
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