Your Name: Your Student Number:

110 Normal Points + 10 Bonus Points If you get 110 points out of 120 points, you will get the full 40% assigned to the midterm exam.

Your scores 1) 2) 3) 4) 5) 6) Total: out of 25 out of 20 out of 20 out of 15 out of 20 out of 20 out of 120

* All the problems are connected. In other words, to solve the problem, you may need the information and/or answers given in other problems. All necessary information is basically given. Also please consult the supplementary material handed out to you.

1) (25 pts) a) (6 pts) The translational energy states in a cubic container are given by the following equation.

Derive the following equation for the molecular translational partition function.

You will need the following integral relation.

0

e

n 2

dn 4

1/ 2

Answers) ( ) ∑ ∑

( )

(

)

∑

∑

∑

(

)

(∫

)

(

) (

√

( ( ) )

)

( a^3 =V

)

(

)

b) (3 pts) If the vibrational energy levels are given as follows, (b-1) where is the zero of vibrational energy? (b-2) What approximation has been made regarding the vibrational motion?

Answers) (b-1) The zero of vibrational energy is at the bottom of the internuclear potential well. (2 pts) (b-2) The vibrational motion is approximated as an harmonic oscillator. (1 pt)

c) (6 pts) Derive the following equation for the molecular vibrational partition function.

You will need the relation applicable when x is less than 1.

xn

0

1 1 x

Answers) ( ) ∑

( )

(

)

∑

(

)

∑

(

)

(

)

d) (2 pts) If the molecular electronic partition function can be approximated as follows, where is the zero of electronic energy?

Answers) The zero of the electronic energy is taken to be the separated atoms at rest in their electronic states (2 pts).

e) (3 pts) If the molecular rotational partition function is given by the following equation, what assumptions have been made?

Answers) 1) The molecule is approximated to be a rigid-rotor. (1 pt) 2) The temperature is much higher than the rotational temperature. (2 pts)

f) (5 pts, no partial points) Based on the answers and information given above, write down the equation for the molecular partition function of a diatomic molecule.

Answers) ( No partial points. ) ( )

2) (20 pts) a) (3 pts, no partial points) For a diatomic ideal gas, write down the relationship between the partition function (Q) and the molecular partition function (q).

Answers) 3pts. No partial points. ( )

(

)

b) (7 pts) Derive the following equation.

Answer) ∑ ( ) ( )( ) ∑ ( ( )

( )

)

(

)

(

(

) ) ∑

( ) (

)

( )

∑

(

(

)

)

)

(

∑

(

) ( )

(

)

(3 pts) (4 pts) )( ( ) ( ) (7 pts) )

( ( ( ) ) ( )

) (

c) (5 pts) A is given as follows.

Express S in terms of Q.

Answers)

Path1 (1 pt) (2 pts) ( ( ) ( 5 pts) )

Path2 ( ) ∏ ( ) ∑

∏

∑

(2 pts) ∑ ∑

∑

(3 pts)

∑ ∑ ∑ ∑ ∑ ⁄ ∑ (4 pts)

( )

∑

∑

(

)

(

)

∑

∑

(

)

(

)

(5 pts)

d) (5 pts) Express S in terms of q. Use Stirling’s approximation (ln N! = N ln N – N).

Answers) ( )

( (

) ) (2 pts)

(

)

[

(

)

]

(

)

(5 pts)

3) (20 pts) a) (10 pts) Calculate the standard molar entropy of Br2(g) at 298.15 K. (10 pts if the value and unit is correct. 3 pts deduction if the value is wrong but within 20 percent of the correct answer. 3 pts deduction whenever the unit is not written or incorrect. 7 pts deduction if the value is wrong and outside 20 percent of the correct answer.) Answers) For ( ) ( ) (3 pts)

Then,

(

(

)

)

(

)

( ( ( ( )

)

(

)

)

)

10 pts if the value and unit is correct. 3 pts deduction if the value is wrong but within 20 percent of the correct answer. 3 pts deduction whenever the unit is not...