6.7 Problems for This Chapter
349
and by discrete summation. Why should only odd values of j be included in
your discrete summation?
15. Both 12 C and 16 O are bosons, each with nuclear spin I a = 0. Obtain an
expression for the combined rotational-nuclear spin partition function for a
gas of 12 C 16 O 2 molecules. If the rotational constant for CO 2 is 0.390 cm −1 ,
calculate z rot at T = 298 K. Does this have any consequences for the pure
rotational Raman spectrum of the 12 C 16 O 2 molecule? If yes, what are they? If
no, why not?
16. In a vibration–rotation spectrum of H 35 Cl (I = 2.65 × 10 −47 kg m
2 ), the line
corresponding to the j = 4 to j = 5 transition is the most intense. At which
temperature has this spectrum been obtained? Which line of this spectrum
would have the greatest intensity for temperature 1000 K?
17. Determine S
◦
298 , C
◦
V ,298 , and C
◦
P ,298 for gaseous HF, given that ν osc (HF) = 3959
cm −1 and B(HF) = 20.56 cm −1 .
18. The I 2 molecule has ν osc = 6.395 × 10 12 s −1 , and an equilibrium internuclear
separation of 2.67 Å. By making appropriate assumptions, calculate the values
of U
◦
500 − U
◦
0 , H
◦
500 − U
◦
0 , S
◦
500 , and G
◦
500 − U
◦
0 for gaseous I 2 .
19. The iodine molecule, I 2 , has a moment-of-inertia I = 750 × 10 −47 kg m
2 and
a characteristic vibrational frequency ν osc = 6.40 × 10 12 s −1 . Calculate the
value of the heat capacity C V (T ) and the entropy for one mole of gaseous
iodine at 298 K and for a pressure of 1 bar. Calorimetric measurements extended
back towards 0 K lead to S
◦
298 = 116.7 J mol
−1 K −1 . Determine the value of
S
◦
298 for the sublimation of one mole of iodine at 298 K. Employ the value
0.305 mm Hg for the equilibrium pressure of iodine vapour over solid iodine at
T = 298 K to determine the value of G
◦
298 for the sublimation of one mole of
iodine at 298 K. From your results for S
◦
298 and G
◦
298 , obtain the standard
enthalpy of sublimation, H
◦
298 for one mole of iodine at 298 K and compare
your result with the accepted experimental value of 62.26 kJ mol
−1 .
20. For CO 2 the four fundamental vibrational frequencies are given by ν osc =
1388, 667, 667, and 2349 cm −1 ; the rotational constant is B = 0.390 cm −1 .
Compute a value for S
◦
298 for CO 2 .
21. A thermodynamics table gives (G
◦
T − H
◦
0 )/T = −257.7 J mol
−1 K −1 for
CH 3 OH(g) at 1000 K. Use this information to obtain a value for the canonical
partition function Z
◦ for CH 3 OH(g) at 1000 K, assuming ideal gas behaviour.
22. The NO molecule has one unpaired electron, and has an electronic orbital
angular momentum that has a projection = 1 along the NO figure axis,
so that its ground electronic term symbol is 2 . The total electronic angular
momentum J can be
3
2 or
1
2 . It happens that the ground electronic state is
the 2 1
2
state, but the first excited electronic 2 3
2
state lies only 119.82 cm −1
above it. As the 2 1
2
- 2 3
2
separation in energy is far less than the fundamental
oscillator frequency ν osc = 1876.014 cm −1 , and the characteristic rotational
temperatures are very similar for the two electronic states (for which B( 2 1
2
) =
1.6634 cm −1 and B( 2 3
2
) = 1.7111 cm −1 ), the canonical partition function
349
and by discrete summation. Why should only odd values of j be included in
your discrete summation?
15. Both 12 C and 16 O are bosons, each with nuclear spin I a = 0. Obtain an
expression for the combined rotational-nuclear spin partition function for a
gas of 12 C 16 O 2 molecules. If the rotational constant for CO 2 is 0.390 cm −1 ,
calculate z rot at T = 298 K. Does this have any consequences for the pure
rotational Raman spectrum of the 12 C 16 O 2 molecule? If yes, what are they? If
no, why not?
16. In a vibration–rotation spectrum of H 35 Cl (I = 2.65 × 10 −47 kg m
2 ), the line
corresponding to the j = 4 to j = 5 transition is the most intense. At which
temperature has this spectrum been obtained? Which line of this spectrum
would have the greatest intensity for temperature 1000 K?
17. Determine S
◦
298 , C
◦
V ,298 , and C
◦
P ,298 for gaseous HF, given that ν osc (HF) = 3959
cm −1 and B(HF) = 20.56 cm −1 .
18. The I 2 molecule has ν osc = 6.395 × 10 12 s −1 , and an equilibrium internuclear
separation of 2.67 Å. By making appropriate assumptions, calculate the values
of U
◦
500 − U
◦
0 , H
◦
500 − U
◦
0 , S
◦
500 , and G
◦
500 − U
◦
0 for gaseous I 2 .
19. The iodine molecule, I 2 , has a moment-of-inertia I = 750 × 10 −47 kg m
2 and
a characteristic vibrational frequency ν osc = 6.40 × 10 12 s −1 . Calculate the
value of the heat capacity C V (T ) and the entropy for one mole of gaseous
iodine at 298 K and for a pressure of 1 bar. Calorimetric measurements extended
back towards 0 K lead to S
◦
298 = 116.7 J mol
−1 K −1 . Determine the value of
S
◦
298 for the sublimation of one mole of iodine at 298 K. Employ the value
0.305 mm Hg for the equilibrium pressure of iodine vapour over solid iodine at
T = 298 K to determine the value of G
◦
298 for the sublimation of one mole of
iodine at 298 K. From your results for S
◦
298 and G
◦
298 , obtain the standard
enthalpy of sublimation, H
◦
298 for one mole of iodine at 298 K and compare
your result with the accepted experimental value of 62.26 kJ mol
−1 .
20. For CO 2 the four fundamental vibrational frequencies are given by ν osc =
1388, 667, 667, and 2349 cm −1 ; the rotational constant is B = 0.390 cm −1 .
Compute a value for S
◦
298 for CO 2 .
21. A thermodynamics table gives (G
◦
T − H
◦
0 )/T = −257.7 J mol
−1 K −1 for
CH 3 OH(g) at 1000 K. Use this information to obtain a value for the canonical
partition function Z
◦ for CH 3 OH(g) at 1000 K, assuming ideal gas behaviour.
22. The NO molecule has one unpaired electron, and has an electronic orbital
angular momentum that has a projection = 1 along the NO figure axis,
so that its ground electronic term symbol is 2 . The total electronic angular
momentum J can be
3
2 or
1
2 . It happens that the ground electronic state is
the 2 1
2
state, but the first excited electronic 2 3
2
state lies only 119.82 cm −1
above it. As the 2 1
2
- 2 3
2
separation in energy is far less than the fundamental
oscillator frequency ν osc = 1876.014 cm −1 , and the characteristic rotational
temperatures are very similar for the two electronic states (for which B( 2 1
2
) =
1.6634 cm −1 and B( 2 3
2
) = 1.7111 cm −1 ), the canonical partition function