Final Exam of Physical Chemistry 2005

(120 minutes)

*department*： *register number*：

*name*： *score*：

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*This is a closed book exam. Use of a calculator and an English dictionary is permitted. Show all of your work and check your units carefully. ***Don’t give help to, or get help from others**. Thanks for your cooperation. **GOOD LUCK***！*

*Some useful constants and results are:*

(atomic unit)

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 C_{10}H_{8} 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 H_{3}^{+} 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 (2s^{2} 2p^{1}), 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-C_{4}H_{4} (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 C_{2}¢ 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-C_{4}H_{4} 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).