classes_Spring08_172ID39_ProblemSet3_solutions

classes_Spring08_172ID39_ProblemSet3_solutions - Chemistry...

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1 Chemistry 172/273 Problem Set #3 Solutions Due on Friday, May 23, in class Spring 2008 Note : Do not use a pencil. Use black ink only. Try to write as neatly as possible. If I don’t understand your handwriting, you will not get any points. 1 . a) Consider the H 3 + ion in which H is at each corner of an equilateral triangle. Using the SALC of the 3 hydrogen 1s atomic orbitals, construct a molecular diagram for H 3 + , label the orbitals with the apropiate Mulliken symbols, and fill the energy levels with the apropiate number of electrons. b) How many fundamental vibrations do you expect for the cyclic H 3 + ion? What are the symmetries of the vibrations in H 3 + and which are infrared and which are Raman active? H 1 H 3 H 2 Φ 1 is the s orbital of H a ; Φ 2 the s orbital of H 2 and Φ 3 the s orbital of H 3 D 3h E 2 C 3 3 C 2 σ h 2 S 3 3 σ v Г s( H) 3 0 1 3 0 1 Г s(H) = A 1 ’+ E D 3h E C 3 (1) C 3 (2) C 2 (1) C 2 (2) C 2 (3) σ h S 3 (1) S 3 (2) σ v (1) σ v (2) σ v (3) A 1 1 1 1 1 1 1 1 1 1 1 1 1 E 1 1 ε ε * ε * ε 0 0 0 0 0 0 1 1 ε ε * ε * ε 0 0 0 0 0 0 Φ 1 Φ 1 Φ 2 Φ 3 Φ 1 Φ 2 Φ 3 Φ 1 Φ 2 Φ 3 Φ 1 Φ 2 Φ 3 3 2 π ε e = ; ε + ε * =-1; ε - ε * =i 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1 3 2 1 1 ' ) ( 4 1 Φ + Φ + Φ Φ + Φ + Φ = = Φ + Φ + Φ + Φ + Φ + Φ + Φ + Φ + Φ + Φ + Φ + Φ Φ A P => ) ( 3 1 ) ( 3 2 1 Φ + Φ + Φ = Ψ A 3 * 2 1 3 2 1 3 * 2 1 3 2 1 3 * 2 1 1 ' 2 2 2 0 0 0 0 0 0 Φ + Φ + Φ = = Φ + Φ + Φ + Φ + Φ + Φ + Φ + Φ + Φ + Φ + Φ + Φ Φ E a P
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2 3 2 * 1 3 2 1 3 2 * 1 3 2 1 3 2 * 1 1 ' 2 2 2 0 0 0 0 0 0 Φ + Φ + Φ = = Φ + Φ + Φ + Φ + Φ + Φ + Φ + Φ + Φ + Φ + Φ + Φ Φ ε E b P 3 2 1 3 2 1 * 3 * 2 1 ' ' 2 ) 2 1 ( 2 ) 2 1 ( 2 2 ) ( 2 ) ( 2 2 ) ' ( Φ Φ Φ = = Φ + Φ + Φ = + Φ + + Φ + Φ = + = Ψ E b E a P P a E => ) 2 ( 6 1 ) ' ( 3 2 1 Φ Φ Φ = Ψ a E 3 2 3 2 * 3 * 2 ' ' ) 2 1 ( 2 ) 2 1 ( 2 ) ( 2 ) ( 2 ) ' ( Φ Φ = Φ + Φ = Φ + Φ = = Ψ i i i P P b E E b E a => ) ( 2 1 ) ' ( 3 2 Φ Φ = Ψ b E H 3 3 H 1 s atomic orbitals e' a 1 ' Ψ( Α 1 ') Ψ ( E ')
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3 b) How many fundamental vibrations do you expect for the cyclic H 3 + ion? What are the symmetries of the vibrations in H 3 + and which are infrared and which are Raman active? Г vib =3N-6=3 Method (a) D 3h E 2 C 3 3 C 2 σ h 2 S 3 3 σ v Г xyz 3 0 -1 1 -20 1 #unmoved atoms 3 0 1 3 0 1 Г 3N 9 0 -1 3 0 1 Г 3N = A 1 ’+ A 2 ’+2 E ’+ A 2 ”+ E Г 3N = Г vib + Г rot + Г trans From the character table Г rot = A 2 ’+ E ” and Г trans = E ’+ A 2 Г vib = Г 3N - Г trans - Г rot = A 1 ’+ A 2 ’+2 E ’+ A 2 ”+ E ”- A 2 ’- E ”- E ’- A 2 ”= A 1 ’+ E
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4 Г vib = A 1 ’+ E three fundamental vibrations. Method (b) H HH r 1 r 2 r 3 r 1 , r 2 and r 3 are the strechings of the bonds D 3h E 2 C 3 3 C 2 σ h 2 S 3 3 σ v Г ν 3 0 1 3 0 1 Г ν = A 1 ’+ E A 1 ’: Raman E ’: IR and Raman 2 . Cyclopropane, C 3 H 6 , is a remarkably stable molecule (m.p. -127.6 °C, b.p. -32.7 °C), despite the extreme ring strain. Indeed, from a traditional VB approach, assuming sp 3 - hybridized carbon atoms, one might wonder why it exists at all. A delocalized MO model
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This note was uploaded on 10/12/2009 for the course CHEM C172 taught by Professor Zink,j during the Spring '08 term at UCLA.

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classes_Spring08_172ID39_ProblemSet3_solutions - Chemistry...

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