Final Exam of Physical Chemistry      2005

(120 minutes)

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1. For hydrogen atom, the energy associated with spin-orbit splitting is

where the constant  is approximately  Hartrees. Compute the change in energy of spin-orbit splitting between  and  states of the hydrogen atom in atomic unit. （4 points）

 

2. In a certain Compton-scattering experiment, a photon of wavelength 1 nm knocks an electron out of the atom. The scattered photon has a wavelength of 2 nm. Calculate the kinetic energy of the ejected electron in electron volts. The initial kinetic energy of the electron can be assumed to be negligible. (4 points)

 

3. For the operator

1)      Is the function  an eigen-function of ? If it is, find the corresponding eigen-value. (5 points)

2)      Normalize  in spherical polar coordinates, so that

You might need the definite integral . (4 points)

4．Azulene C₁₀H₈ is an aromatic （芳烃） hydrocarbon containing delocalized -electrons.

(1) How many -electrons does azulene have? (2 points)

(2) As a model for this -electron system, consider the mobile electrons in a rectangular（矩形的） two-dimensional box of dimensions 5.00A by 4.65A. Identify the quantum numbers of the HOMO and LUMO of the -electron system. (6 points)

(3) Calculate the wavelength (in nm) of the lowest-energy -electron transition. (4 points)

(4) Should azulene be a colored compound? If so, what color? (2 points)

 

5. The only metal that crystallizes in a primitive cubic lattice is polonium, which has a unit cell side of 334.5 pm. What are the perpendicular distances between planes with indices (110), (111), (210)? (5 points)

6. Liner combination for two of the three  hybrid orbitals are given by the functions:

1)      Show that  is normalized; (3 points)

2)      Determine whether  and  are orthogonal; (3 points)

3)      Deter the angle between these two hybrid orbitals. (4 points)

 

7. An electron in a  orbital of a hydrogen atom has the wavefunction

1)      Show that the  is an eigenfunction of the Hamiltonian for the hydrogen atom and find the eigenvalue. (4 points)

2)      In a single measurement of  for this electron, what is the probability of measuring the value ? (3 points)

3）In a single measurement of  for this electron, what is the probability of measuring the value ? (3 points)

8. (1) Determine the term symbols for the electron configuration . (6 points)

 (2) Which term symbol corresponds to the lowest energy state? (2 points)

(3) How many energy states correspond to the lowest energy term symbol? (4 points)

 

9. Using Huckel molecular-orbital theory, determine whether the linear state or the triangular ring of H₃⁺ is more stable state. (6 points)

 

10. Consider a free particle in a spherical box. The potential is zero within the box of radius  and infinite outside the box. Please write the Hamiltonian. (3 points)

 

11. Titanium forms hexagonal close-packet crystals. Given the atomic radius of 146 pm, what are the unit cell dimensions, and what is the density of crystals? ( Atomic mass: Ti-48) (5 points)

 

12. Boron has a doublet ground state (2s² 2p¹), when it is coordinated with CO, the resulting BCO molecule has a quartet ground state, please give a brief discussion on the bonding mechanism (8 points).

 

13. Determine the point group of each of the following molecules. (8 points)

 

 

 

  

 

14. Taking the four p orbitals perpendicular to the molecular plane of cyc-C₄H₄ (D_4h) as bases.

1) Write out the characters of the representation; (4 points, the axis and plane involving two C atoms are labeled as C₂¢ and s_v)

2) Determine whether it is an irreducible or reducible representations, if it is a reducible representation, reduce it to irreducible representations; (4 points)

3) The four Pi molecular orbitals are listed below, work out which irreducible representations they belong to.

                                    (4 points)
4) Determine the relative stabilities of cyc-C₄H₄ cation, neutral and anion with the HMO approximate hypothesis (4 points)

5) Determine the spectral possibility from the ground state to the first excited state.(6 points).