3502_14_octo12_LG

3502_14_octo12_LG - Chem 3502/5502 Physical Chemistry II...

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Chem 3502/5502 Physical Chemistry II (Quantum Mechanics) 3 Credits Fall Semester 2009 Laura Gagliardi Lecture 14, October 12, 2009 Solved Homework We are given that the lowest microwave absorption of carbon monoxide (CO, with 12 C and 16 O specified) is 115,271,000,000 s 1 (115.271 GHz). We are asked first to compute the moment of inertia of CO. Remember that the first absorption corresponds to the 0 1 rotational transition, and the energy separation between these two levels is Δ E = h ν = 2 B where B = h 2 2 I Rearranging, we have I = h 2 2 B = h 2 h ! = 1.0545 " 10 # 34 kg m 2 s -1 ( ) 2 6.6256 " 10 # 34 kg m 2 s -1 ( ) 115,271,000,000 s -1 ( ) = 1.456 " 10 # 46 kg m 2 We may compute the reduced mass for this system from the atomic masses for 12 C and 16 O μ = m 12 C m 16 O m 12 C + m 16 O = 12 ! 1.66 ! 10 " 27 kg ( ) 15.9949 ! 1.66 ! 10 " 27 kg ( ) 12 ! 1.66 ! 10 " 27 kg ( ) + 15.9949 ! 1.66 ! 10 " 27 kg ( ) = 1.138 ! 10 " 26 kg Recalling that
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14-2 I = μ R 2 where R is the bond length, we have R = I μ = 1.456 ! 10 " 46 kg m 2 1.138 ! 10 " 26 kg = 1.131 ! 10 " 10 m = 1.131 Å The standard tabulated value is 1.128 Å. The difference reflects the very small correction that is associated with CO not being a rigid rotator (it vibrates while it rotates). Finally, we are asked to predict the frequency at which the 1 2 rotational transition occurs. Recall that the spacing between adjacent transitions in the frequency spectrum is precisely equal to the frequency of the 0 1 transition, we may estimate a value of twice the ground state absorption, or 230,542,000,000 s 1 . If you refer back to the DIB website we visited that had the rotational spectrum for CO, you will see that this is about right. In cm 1 units, which are gotten simply by dividing ν by the speed of light c (which is 3 x 10 10 cm s 1 ), this is 7.7 cm 1 . The Hydrogen Atom (again) It’s time to return to our old friend, the H atom. Bohr found a quantum relationship useful for rationalizing the spectra of H, but it violated classical physics by having one charged particle orbiting about another with no loss of energy through radiation. de Broglie made the suggestion that the angular momentum quantization invoked by Bohr was suggestive of wave-particle duality for the electron. We are finally in a position to look at these waves more carefully. Consider a proton and an electron bound together in some way. As these are quantum particles, they may not be still (that would violate the uncertainty principle); instead, they must be in motion. Just as for a rigid rotator, we may divide the total motion into a center of mass motion and a rotational motion about the center of mass.
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3502_14_octo12_LG - Chem 3502/5502 Physical Chemistry II...

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