References
355
42. Carry out the steps to obtain Eqs. (6.1.4) from Eq. (6.1.10a).
43. If the rotational and vibrational motions are assumed to be independent for
ground electronic state Cl 2 and HCl molecules, so that int = vib + rot , how
many rotational energy levels of the ground vibrational state can be found in the
energy gap between the ground vibrational energy 0 and the energy of the first
vibrational state, i.e., in the interval ((( vib ) 0→1 ? Assume that the vibrational
energy gap for Cl 2 is 554.4 cm −1 and that for HCl is 2885.2 cm −1 . Assume
also that both Cl 2 and HCl may be treated as rigid rotors, with B e (Cl 2 ) =
0.240 cm −1 and B e (HCl) = 10.593 cm −1 .
44. In Example 6.3, it was mentioned that the molecular oxygen isotopologue
17 O 2 (X 3 −
g ), unlike either 16 O 2 (X 3 −
g ) or 18 O 2 (X 3 −
g ), which have only
odd rotational states, has both even and odd rotational states. Explain why this
is so. What effect will this have on the pure rotational Raman spectrum of
17 O 2 (X 3 −
g )? How would you expect the pure rotational Raman spectrum of
17 O 2 (X 3 −
g ) to differ from the pure rotational Raman spectra of the homonuclear 16 O and 18 O diatomic isotopologues or from that of 16 O 18 O(X 3 −
g )?
References
1. G. Herzberg, Molecular Spectra and Molecular Structure. I. Diatomic Molecules, 2nd edn.
(Van Nostrand, New York, 1950), p. 92
2. J.E. Mayer, M. Goeppert-Mayer, Statistical Mechanics (Wiley, New York, 1940)
3. K.P. Huber, G. Herzberg, Molecular Spectra and Molecular Structure. IV. Constants of
Diatomic Molecules (Van Nostrand, New York, 1979). See also NIST Chemistry WebBook,
https://webbook.nist.gov/
4. L.S. Kassel, Chem. Rev. 18, 277 (1936)
5. D.A. McQuarrie, Mathematical Methods for Scientists and Engineers (University Science
Books, Sausalito, 2003), pp. 107–112
6. Mulholland was the first to obtain this result, although via a quite different mathematical
strategem: see H.P. Mulholland, Proc. Camb. Phil. Soc. 24, 280 (1928)
7. R.S. McDowell, J. Chem. Phys. 88, 356 (1988)
8. F.M. Fernández, R.H. Tipping, Spectrochim. Acta 48A, 1283 (1992)
9. R.S. McDowell, J. Chem. Phys. 39, 526 (1963)
10. W. Pauli, Phys. Rev. 58, 716 (1940). The Pauli Exclusion Principle that ‘no two electrons
can have all four quantum numbers n, , m , m s the same’ was enunciated by Pauli in 1925
[W. Pauli, Z. f. Physik 31, 765 (1925)], while the more general (symmetry) version was not
introduced until 1940
11. A. Eucken, Sitzber. Preuss. Akad. Wiss., 1912, 41
12. A. Eucken, R. Hiller, Z. f. Phys. Chem. B, 4, 142 (1929)
13. K. Scheel, W. Heuse, Ann d. Phys. 40, 473 (1913)
14. I.H. Brinkworth, Proc. R. Soc. A 107, 510 (1925)
15. R.E. Cornish, E.D. Eastman, J. Am. Chem. Soc. 50, 627 (1928)
16. K. Klusius, K. Hiller, Z. f. Phys. Chem. B 4, 158 (1929)
17. D.M. Dennison, Proc. R. Soc. A 115, 483 (1927)
18. K.F. Bonhöffer, P. Harteck, Z. f. Phys. Chem. B 4, 113 (1929)
19. J.E. Kilpatrick, Y. Fukuda, S.Y. Larsen, J. Chem. Phys. 43, 430 (1965)
20. A.R. Gordon, C. Barnes, J. Chem. Phys. 1, 297 (1933)
355
42. Carry out the steps to obtain Eqs. (6.1.4) from Eq. (6.1.10a).
43. If the rotational and vibrational motions are assumed to be independent for
ground electronic state Cl 2 and HCl molecules, so that int = vib + rot , how
many rotational energy levels of the ground vibrational state can be found in the
energy gap between the ground vibrational energy 0 and the energy of the first
vibrational state, i.e., in the interval ((( vib ) 0→1 ? Assume that the vibrational
energy gap for Cl 2 is 554.4 cm −1 and that for HCl is 2885.2 cm −1 . Assume
also that both Cl 2 and HCl may be treated as rigid rotors, with B e (Cl 2 ) =
0.240 cm −1 and B e (HCl) = 10.593 cm −1 .
44. In Example 6.3, it was mentioned that the molecular oxygen isotopologue
17 O 2 (X 3 −
g ), unlike either 16 O 2 (X 3 −
g ) or 18 O 2 (X 3 −
g ), which have only
odd rotational states, has both even and odd rotational states. Explain why this
is so. What effect will this have on the pure rotational Raman spectrum of
17 O 2 (X 3 −
g )? How would you expect the pure rotational Raman spectrum of
17 O 2 (X 3 −
g ) to differ from the pure rotational Raman spectra of the homonuclear 16 O and 18 O diatomic isotopologues or from that of 16 O 18 O(X 3 −
g )?
References
1. G. Herzberg, Molecular Spectra and Molecular Structure. I. Diatomic Molecules, 2nd edn.
(Van Nostrand, New York, 1950), p. 92
2. J.E. Mayer, M. Goeppert-Mayer, Statistical Mechanics (Wiley, New York, 1940)
3. K.P. Huber, G. Herzberg, Molecular Spectra and Molecular Structure. IV. Constants of
Diatomic Molecules (Van Nostrand, New York, 1979). See also NIST Chemistry WebBook,
https://webbook.nist.gov/
4. L.S. Kassel, Chem. Rev. 18, 277 (1936)
5. D.A. McQuarrie, Mathematical Methods for Scientists and Engineers (University Science
Books, Sausalito, 2003), pp. 107–112
6. Mulholland was the first to obtain this result, although via a quite different mathematical
strategem: see H.P. Mulholland, Proc. Camb. Phil. Soc. 24, 280 (1928)
7. R.S. McDowell, J. Chem. Phys. 88, 356 (1988)
8. F.M. Fernández, R.H. Tipping, Spectrochim. Acta 48A, 1283 (1992)
9. R.S. McDowell, J. Chem. Phys. 39, 526 (1963)
10. W. Pauli, Phys. Rev. 58, 716 (1940). The Pauli Exclusion Principle that ‘no two electrons
can have all four quantum numbers n, , m , m s the same’ was enunciated by Pauli in 1925
[W. Pauli, Z. f. Physik 31, 765 (1925)], while the more general (symmetry) version was not
introduced until 1940
11. A. Eucken, Sitzber. Preuss. Akad. Wiss., 1912, 41
12. A. Eucken, R. Hiller, Z. f. Phys. Chem. B, 4, 142 (1929)
13. K. Scheel, W. Heuse, Ann d. Phys. 40, 473 (1913)
14. I.H. Brinkworth, Proc. R. Soc. A 107, 510 (1925)
15. R.E. Cornish, E.D. Eastman, J. Am. Chem. Soc. 50, 627 (1928)
16. K. Klusius, K. Hiller, Z. f. Phys. Chem. B 4, 158 (1929)
17. D.M. Dennison, Proc. R. Soc. A 115, 483 (1927)
18. K.F. Bonhöffer, P. Harteck, Z. f. Phys. Chem. B 4, 113 (1929)
19. J.E. Kilpatrick, Y. Fukuda, S.Y. Larsen, J. Chem. Phys. 43, 430 (1965)
20. A.R. Gordon, C. Barnes, J. Chem. Phys. 1, 297 (1933)