6.7 Problems for This Chapter
353
34. Measurements of C P (T ) for O 2 have been reported by Lewis and von Elbe
[50]. A selection of their values is: C P (200) = 6.951, C P (500) = 7.434,
C P (800) = 8.072, C P (1250) = 8.621, C P (1500) = 8.840, C P (2000) =
9.208, C P (2500) = 9.500 and C P (3000) = 9.725; temperatures in K, heat
capacities in cal mol −1 K −1 [based upon R = 1.98714 cal mol −1 K −1 ].
From your knowledge of C P (T ) for an ideal gas, utilize the expression obtained
in Problem 33 to calculate values of C P at the experimental temperatures
of Lewis and von Elbe, and compare your results with their experimental
values. Comment on the quality of agreement that you obtain, and discuss
the significance of the role of the 1 g excited state of O 2 at the temperatures
concerned, bearing in mind the results that you obtained in the latter part of
Problem 33.
A slightly more recent (than 1933), and relatively complete calculation of
the thermodynamic functions for molecular oxygen in the ideal gas state was
carried out by Woolley [51]. Compare your values calculated using the rigidrotor-SHO approximation with the results obtained by Woolley (which go
beyond the SHO approximation) and displayed in his Tables 5 and 6. Comment
on the appropriateness of the rigid-rotor-SHO approximation that you have
employed for your calculations.
35. Just as the possession by the proton ( 1 H) of a nuclear spin of magnitude
1
2 has
significant ramifications for the spectroscopic and thermodynamic properties of
H 2 , so it has similar consequences for the polyatomic centrosymmetric linear
molecule acetylene, 12 C 2 H 2 . [Note that the 12 C nucleus has nuclear spin I =
0.]
Given that the electronic ground term of C 2 H 2 is 1 +
g , obtain an expression
that is applicable at low temperature for the combined rotational-nuclear spin
canonical partition function for the ortho and para spin modifications of
12 C 2 H 2 . Discuss the effect of the nuclear spin statistics on the Raman spectrum
of 12 C 2 H 2 . How would the Raman spectrum of 12 C 2 H 2 differ from those of its
isotopologues 12 C 2 D 2 and 12 C 2 HD? Determine a value for the residual entropy
for each acetylene isotopologue.
36. The near infrared spectrum of 12 C 2 H 2 has five fundamental absorption bands,
located at 611.7, 729.2, 1973.5, 3294.9, and 3372.5 cm −1 . These bands have
been identified as having symmetries π g , π u , σ +
g , σ +
u and σ +
g , respectively.
Note that vibrations of type π u/g are doubly degenerate (cf. individual π MO’s),
while vibrations of type σ
±
u/g are nondegenerate. The rotational constant B 0 of
12 C 2 H 2 has the value 1.1766 cm −1 . Calculate U
◦
298.15 − U
◦
0 , H
◦
298.15 − H
◦
0 ,
C V ,298.15 , and S
◦
298.15 for C 2 H 2 , and compare your values with the accepted
literature values [given in the JANAF tables of the American National Institute
of Standards and Technology (NIST)].
37. Consider the class of general spherical top molecules, such as silane, SiH 4 ,
with tetrahedral (T d ) symmetry, sulphur hexafluoride, SF 6 , with octahedral
(O h ) symmetry or buckminsterfullerene, C 60 , with icosahedral (I h ) symmetry.
Obtain an expression for the fractional population p j for a spherical top
353
34. Measurements of C P (T ) for O 2 have been reported by Lewis and von Elbe
[50]. A selection of their values is: C P (200) = 6.951, C P (500) = 7.434,
C P (800) = 8.072, C P (1250) = 8.621, C P (1500) = 8.840, C P (2000) =
9.208, C P (2500) = 9.500 and C P (3000) = 9.725; temperatures in K, heat
capacities in cal mol −1 K −1 [based upon R = 1.98714 cal mol −1 K −1 ].
From your knowledge of C P (T ) for an ideal gas, utilize the expression obtained
in Problem 33 to calculate values of C P at the experimental temperatures
of Lewis and von Elbe, and compare your results with their experimental
values. Comment on the quality of agreement that you obtain, and discuss
the significance of the role of the 1 g excited state of O 2 at the temperatures
concerned, bearing in mind the results that you obtained in the latter part of
Problem 33.
A slightly more recent (than 1933), and relatively complete calculation of
the thermodynamic functions for molecular oxygen in the ideal gas state was
carried out by Woolley [51]. Compare your values calculated using the rigidrotor-SHO approximation with the results obtained by Woolley (which go
beyond the SHO approximation) and displayed in his Tables 5 and 6. Comment
on the appropriateness of the rigid-rotor-SHO approximation that you have
employed for your calculations.
35. Just as the possession by the proton ( 1 H) of a nuclear spin of magnitude
1
2 has
significant ramifications for the spectroscopic and thermodynamic properties of
H 2 , so it has similar consequences for the polyatomic centrosymmetric linear
molecule acetylene, 12 C 2 H 2 . [Note that the 12 C nucleus has nuclear spin I =
0.]
Given that the electronic ground term of C 2 H 2 is 1 +
g , obtain an expression
that is applicable at low temperature for the combined rotational-nuclear spin
canonical partition function for the ortho and para spin modifications of
12 C 2 H 2 . Discuss the effect of the nuclear spin statistics on the Raman spectrum
of 12 C 2 H 2 . How would the Raman spectrum of 12 C 2 H 2 differ from those of its
isotopologues 12 C 2 D 2 and 12 C 2 HD? Determine a value for the residual entropy
for each acetylene isotopologue.
36. The near infrared spectrum of 12 C 2 H 2 has five fundamental absorption bands,
located at 611.7, 729.2, 1973.5, 3294.9, and 3372.5 cm −1 . These bands have
been identified as having symmetries π g , π u , σ +
g , σ +
u and σ +
g , respectively.
Note that vibrations of type π u/g are doubly degenerate (cf. individual π MO’s),
while vibrations of type σ
±
u/g are nondegenerate. The rotational constant B 0 of
12 C 2 H 2 has the value 1.1766 cm −1 . Calculate U
◦
298.15 − U
◦
0 , H
◦
298.15 − H
◦
0 ,
C V ,298.15 , and S
◦
298.15 for C 2 H 2 , and compare your values with the accepted
literature values [given in the JANAF tables of the American National Institute
of Standards and Technology (NIST)].
37. Consider the class of general spherical top molecules, such as silane, SiH 4 ,
with tetrahedral (T d ) symmetry, sulphur hexafluoride, SF 6 , with octahedral
(O h ) symmetry or buckminsterfullerene, C 60 , with icosahedral (I h ) symmetry.
Obtain an expression for the fractional population p j for a spherical top