課程名稱︰物理化學二 課程性質︰系必修 課程教師︰陳逸聰 開課學院：理學院 開課系所︰化學系 考試日期（年月日）︰2012/6/19 考試時限（分鐘）：120 (open book) 是否需發放獎勵金：yes （如未明確表示，則不予發放） 試題 : 1. │ΨA(1) ΨA(2)│ For the wavefunction Ψ=│ │ show that (a)the interchange of │ΨB(1) ΨB(2)│ two columns changes the sign of the wavefunction, (b)the interchange of two rows changes the sign of the wavefunction, and(c)the two electrons cannot have the same spin orbital. 2.Show that one of the Huckel molecule orbitals for 1,3-butadiene, Ψ1, in Eq.11.75 on page 423 of the Textbook is normalized and that it is orthognal to Ψ2, Ψ3, andΨ4 3.Using Huckel molecular-orbital theory, determine whether the linear state + (H-H-H ) or the triangular state of (a) H + is the more stable state ╱ ╲ H —— H - Repeat the calculation for (b)H3 and (c)H3 4.For a molecule, you normally see its emission spectra in the UV-visible spectral range. You might also find its infrared emission spectra, but you hardly hear that it has microwave emission. Explain these facts by using the Einstein coefficient A to calculate the lifetimes for the excited states that are responsible for these electronic, vibrational, and rotational transitions. 5. 12 14 The bond length in C N is 117pm and its force constant is 1630N/m. 12 14 Predict the vibrational-rotational spectrum of C N. 6.The following data are obtained for the vibration-rotation spectrum of 79 ~ ~ ~ ～ H Br. Determine B , B , B , and α from these data. 0 1 e e Line Frequency/(1/cm) ──────────────────────── R(0) 2642.60 R(1) 2658.36 P(1) 2609.67 P(2) 2592.51 7. ～ -1 ～ ～ -1 23 19 Given that ν =536.10 cm and χ ν =3.4 cm for N F, calculate the e e e frquencies of the first and second vibrational overtone transitions. 8.Show that the moment of inertia for a symmetrical tetrahedral molecule, such as CH , is 2 where R is the bond length and m is the mass 4 I= 8mR /3 of each of the four atoms arranged in a tetrahedral manner. 9.The rotational wavefunctions for a diatomic molecule can be represented by M spherical harmonics, i.e., │J,M > = Y (θ,φ). Show that the rotational J transition J=0→J=1 is allowed, but J=0→J=2 is forbidden in microwave spectroscopy (in the rigid-rotator approximation) 10.The formaldehyde molecule, H CO, belongs to a C symmetry group and has 2 2v six vibrational normal modes as illustrated below (a)Try to point out the symmetry representation for each of the six vibrations (b)For the six vibrations, which are infrared active and/or Raman active ↑ ↑ ↑ ○ ○ ○ υ1 │ υ2 │ υ3 │ ↑ │ ↑ ○ ○ ○ ／ ＼ ↖ ／↓＼ ↗ ／ ＼ ○ ○ ○ ○ ○ ○ ↙ ↘ ↘ ↙ + ○→ ←○ ○ υ4 │ υ5 │ υ6 │ │ │ │ ○→ ○→ ○- ／ ↖ ↖ ／ ＼ ／ ＼ ○ ○ ○ ○ ○ ○ ↙ ↙ + + C2v E C2 σv(xz) σv'(yz) ───────────────────────────────────── 2 2 2 A1 1 1 1 1 z,z ,x ,y A2 1 1 -1 -1 xy B1 1 -1 1 -1 x,xz B2 1 -1 -1 1 y,yz 11.The vibrational term of a diatomic molecule is given by ~ 2～ ~ G(v)=(v+0.5) v - (v+0.5) χ v where ν is the vibrational quantum e e e number. (a)Show that the spacing between the adjacent levels ΔG is given by ~ ～ ΔG(v)=G(v+1)-G(v)= v [1-2χ (v+1)] (1) e e The diatomic molecule dissociates in the limit that ΔG(v)→0 (b)Show that the maximum vibrational quantum number, v , is given by ～ max v =(1/2χ )-1 max e (c)Use this result to show that the dissociation energy D of the diatomic e ~ ～2 ～ ~ ～ molecule can be written as D = v (1-χ )/4χ ～ v /4χ (2) e e e e e e ~ ～ (d)Refering to equation 1, explain how the constant v and χ can be e e evaluated from a plot of ΔG versus v+1. This type of plot is called ~ ～ a Birge-Sponer plot. Once the values of v and χ are known. e e Equation 2 can be used to determine the dissociation energy of the molecule (e)Use the following experimental data for H to calculate the dissociation ~ 2 energy D . e v G(v) (1/cm) v G(v) (1/cm) ─────────────────────── 0 4161.12 7 26830.97 1 8087.11 8 29123.93 2 11782.35 9 31150.19 3 15250.36 10 32886.85 4 18497.92 11 34301.83 5 21505.65 12 35351.01 6 24287.83 13 35972.97 (f)Explain why your Birge-Sponer plot is not linear for high values of v. ~ How does the value of D obtained from the Birge-Sponer analysis e -1 compare with the experimental value of 38269.48 cm ? --

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