354
6 Molecular Systems
molecule in a rotational level characterized by the rotational quantum number
j , and from it derive an expression for the value j max at which the fractional
population is maximal. What is the meaning of this expression? Obtain values
of j max at temperature 500 K for the CH 4 ( rot = 7.54 K), CF 4 ( rot =
0.276 K), and SF 6 ( rot = 0.131 K) molecules.
38. From spectral data acquired from the atmosphere of Titan, the largest moon
of Saturn, an astronomer observes a series of resolved rotational fine-structure
lines in an infrared absorption band located at approximately 3010 cm −1 that
she has identified as the ν 3 -fundamental band of CH 4 . She notes that, within
the accuracy of her astronomical spectral data, the rotational fine-structure
transitions show equal maximal intensity for transitions originating from j = 3
and j = 4. Use this observation to bracket the temperature in the atmosphere
of Titan. Are your values consistent with what is known about Titan and its
atmosphere?
39. The dimethyl acetylene molecule possesses, in addition to the three end-overend rotational degrees of freedom, an additional free relative internal rotation
of the two end methyl groups. Given that the relevant moment-of-inertia for
the relative internal rotation of the two methyl groups is 2.566 × 10 −47 kg m 2 ,
determine its contribution to the entropy of dimethyl acetylene at 300 K, if the
symmetry number associated with this internal rotational motion is σ = 3. As
far as the overall (end-over-end) rotational motion of the molecule is concerned,
dimethyl acetylene is a symmetric top molecule, with one of its three momentsof-inertia having the value 10.657×10 −47 kg m
2 , and the other two each having
the value 249.66×10 −47 kg m
2 . Construct a sketch showing the axes about
which these three rotations occur. What is the entropy contribution due to the
overall rotational motion, if the associated symmetry number is σ = 6?
40. The formaldehyde molecule has a 1 A 1 ground electronic state. Its infrared
spectrum gives the vibrational fundamental frequencies ν 1 (a 1 ) = 2766 cm −1 ,
ν 2 (a 1 ) = 1746 cm −1 , ν 3 (a 1 ) = 1510 cm −1 , ν 4 (b 1 ) = 2843 cm −1 , ν 5 (b 1 ) =
1251 cm −1 , and ν 6 (b 2 ) = 1167 cm −1 , while the principal moments-of-inertia
can be extracted from the far infrared spectrum as I A0 = 2.976 × 10 −47 ,
I B0 = 21.61 × 10 −47 and I C0 = 24.68 × 10 −47 , all in units of kg m 2 .
The notation a 1 , b 1 , b 2 accompanying the vibrational fundamental frequencies
identify the point-group symmetry species of the corresponding vibrational
motion (they are all nondegenerate). Calculate U
◦
298.15 − U
◦
0 , H
◦
298.15 − H
◦
0 ,
C V ,298.15 , and S
◦
298.15 for formaldehyde, and compare your values with the
accepted literature values (found in the NIST JANAF tables).
41. Give a quantitative explanation of the statement ‘At a temperature of 300 K,
the heat capacity C V of IBr has nearly its classical value, while that for F 2 does
not’. The rotational Raman spectrum for F 2 corresponds to a rotational constant
B = 0.8833 cm −1 and ν osc = 894.2 cm −1 , while the infrared spectrum of IBr
gives values B = 0.05673 cm −1 and ν osc = 267.01 cm −1 . By how much do the
heat capacities of these two molecular gases differ from the values that would
be expected classically?
6 Molecular Systems
molecule in a rotational level characterized by the rotational quantum number
j , and from it derive an expression for the value j max at which the fractional
population is maximal. What is the meaning of this expression? Obtain values
of j max at temperature 500 K for the CH 4 ( rot = 7.54 K), CF 4 ( rot =
0.276 K), and SF 6 ( rot = 0.131 K) molecules.
38. From spectral data acquired from the atmosphere of Titan, the largest moon
of Saturn, an astronomer observes a series of resolved rotational fine-structure
lines in an infrared absorption band located at approximately 3010 cm −1 that
she has identified as the ν 3 -fundamental band of CH 4 . She notes that, within
the accuracy of her astronomical spectral data, the rotational fine-structure
transitions show equal maximal intensity for transitions originating from j = 3
and j = 4. Use this observation to bracket the temperature in the atmosphere
of Titan. Are your values consistent with what is known about Titan and its
atmosphere?
39. The dimethyl acetylene molecule possesses, in addition to the three end-overend rotational degrees of freedom, an additional free relative internal rotation
of the two end methyl groups. Given that the relevant moment-of-inertia for
the relative internal rotation of the two methyl groups is 2.566 × 10 −47 kg m 2 ,
determine its contribution to the entropy of dimethyl acetylene at 300 K, if the
symmetry number associated with this internal rotational motion is σ = 3. As
far as the overall (end-over-end) rotational motion of the molecule is concerned,
dimethyl acetylene is a symmetric top molecule, with one of its three momentsof-inertia having the value 10.657×10 −47 kg m
2 , and the other two each having
the value 249.66×10 −47 kg m
2 . Construct a sketch showing the axes about
which these three rotations occur. What is the entropy contribution due to the
overall rotational motion, if the associated symmetry number is σ = 6?
40. The formaldehyde molecule has a 1 A 1 ground electronic state. Its infrared
spectrum gives the vibrational fundamental frequencies ν 1 (a 1 ) = 2766 cm −1 ,
ν 2 (a 1 ) = 1746 cm −1 , ν 3 (a 1 ) = 1510 cm −1 , ν 4 (b 1 ) = 2843 cm −1 , ν 5 (b 1 ) =
1251 cm −1 , and ν 6 (b 2 ) = 1167 cm −1 , while the principal moments-of-inertia
can be extracted from the far infrared spectrum as I A0 = 2.976 × 10 −47 ,
I B0 = 21.61 × 10 −47 and I C0 = 24.68 × 10 −47 , all in units of kg m 2 .
The notation a 1 , b 1 , b 2 accompanying the vibrational fundamental frequencies
identify the point-group symmetry species of the corresponding vibrational
motion (they are all nondegenerate). Calculate U
◦
298.15 − U
◦
0 , H
◦
298.15 − H
◦
0 ,
C V ,298.15 , and S
◦
298.15 for formaldehyde, and compare your values with the
accepted literature values (found in the NIST JANAF tables).
41. Give a quantitative explanation of the statement ‘At a temperature of 300 K,
the heat capacity C V of IBr has nearly its classical value, while that for F 2 does
not’. The rotational Raman spectrum for F 2 corresponds to a rotational constant
B = 0.8833 cm −1 and ν osc = 894.2 cm −1 , while the infrared spectrum of IBr
gives values B = 0.05673 cm −1 and ν osc = 267.01 cm −1 . By how much do the
heat capacities of these two molecular gases differ from the values that would
be expected classically?