4. We can construct three sp2 hybrid orbitals (i.e., φ_un = 1,2,3) as follows:

φ₁=a₁(2s)+b₁(2pz)

φ₂=a₂(2s)+_b2(2pz)+c2(2px)
φ3 =a₃(2s)+b₃(2pz)+c₃(2p_x)
where 2s,2px and 2pz are normalized atomic orbitals and a_n,b_n and c_n are constants. Assume these hybrid orbitals are normalized and orthogonal to each other, then which of the following is incorrect.
(A)a₁ = a₂ = a₃ =
(B)b₁=
(C)b2 =b3 =-
(D)c₂=c₃ =
(E)=1

1. Define the operators, whereare the position and momentum operators. What is the value of the commutator? (hint: = =) (A)0 (B)1 (C) ih (D)h (E) 2h

2. What is the , for a non-ideal gas whose state can be expressed as the virial equation: (V=molar volume)? (A) (B)(C)(D)(E)

3. Which valence bond wavefunction for He is not a proper wavefunction because it violates the Pauli exclusion principle? (Please note: the numbers 1 and 2 represent the two electrons of He, N is the normalization constant, 1sA and 1Se represent the atomic orbitals, and a and B are spin wavefunctions.) (A) φ(1,2) = N[1sA(1)1sB；(2)-1SA(2)1sB(1)][a(1)B(2)-a(2)B(1)] (B)φ(1,2)=N[1sA(1)1sB(2)+1sA(2)1sB(1)][a(1)B(2)-a(2)B(1)] (C) φ(1,2)=N[1sA(1)1sB(2)-1sA(2)1SB(1)][(1)β(2)+a(2)β(1)] (D)φ(1,2)=N[1sA(1)1sB(2)-1sA(2)1SB(1)][a(1)a(2)] (E)φ(1,2)=N[1sA(1)1sB(2)-1SA(2)1SB(1)][β(1)β(2)]

5. According to the selection rules in the atomic spectra, which of the following transition is NOT allowed. (A) 2P3/2→2S1/2 (B) 2D3/2→2P1/2 ,→ 2P1/2 (C) 2D5/2→2P1/2 (D) 2P1/2→2S1/2 (E) 2D5/2→2P3/2

6. Two metal blocks with the same mass at different temperatures (T1 and T2) are contacting each other the reach the same temperature. What is AS assuming no heat loss during heat transfer.(A) (B) (C)(D)(E)

7. The first few spherical harmonics(θ, Φ) responsible for the angular parts of the hydrogen wavefunctions are listed below.Which combination of spherical harmonics (θ,Φ) can give you the py orbital?hint: Euler formula e±inΦ = cos(nΦ) isin(nΦ) (A) (B) (C) (D)(E)

8. An unnormalized wavefunction of the hydrogen atom is given by r2e(3 cos2 θ - 1). What are the three quantum numbers (n, I, m) associated with this orbital? (A) (2,2,0) (B)(2,1,1) (C) (3,2,0) (D) (3,1,1) (E) none of the above. 

9. The general structure of the porphyrin molecule is shown on the right side. This molecule is planar and so we can approximate the π electrons as being confined in a square with the length a = 10 The porphyrin molecule has "22"π electrons. According to the particle in a two-dimensional (2D) box, the energy states are given bywhere nx and ny are quantum numbers. Assume that each energy levet in the box is capable of holding no more than two electrons. What is the electronic transition responsible for the promotion of an electron from the highest filled energy level to the lowest empty energy level? (A) E(3,3)→E(4,2) (B)E(3,2)→E(4,1) (C) E(2,3) → E(2,4) (D) E(3,3)→E(4,3) (E) E(3,4) → E(4,4)

10. Following the question 09, what is the transition energy for an electron from the highest filled level to the lowest empty level? (A)8091 cm-1 (B) 11006 cm-1 (C) 7240 cm-1  (D)9230 cm-1  (E) 6061 cm-1 

11. Following the questions 09 and 10, alternately, we can approximate the 22πelectrons of the porphyrin molecule on a circular ring with radius of 440 pm (i.e, r = 440 pm), as shown on the right side. For the particle on a ring model, the Schrodinger equation isWhat is the predicted lowest transition energy?(A) 5.2 x 10-14 Hz (B) 6.8 x 10-14 Hz (C) 2.4 X 10-14 Hz (D) 9.0 x 10-15 Hz (E) 1.2 x 10-14 Hz 

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