International Journal of Thermophysics

, Volume 35, Issue 9–10, pp 1785–1802 | Cite as

The Ideal Gas and Real Gas Heat Capacity of Sodium Atoms

  • Louis Biolsi


The ideal gas heat capacity of sodium atoms in the vapor phase is calculated to high temperatures using statistical mechanics. Since there are, in principle, an infinite number of atomic energy levels, the partition function and the heat capacity will grow very large unless the summation over energy levels is constrained as temperature increases. At higher temperatures, the increasing size of the atoms, which is a consequence of the increased population of highly excited energy levels, is used as a mechanism for limiting the summation over energy levels. The “\( {IP-kT}\)” and “Bethe” procedures for cutting off the summation over energy levels will be discussed, and the results obtained using the two methods will be compared. In addition, although experimental information is available about lower atomic energy levels and some theoretical calculations are available for excited energy levels, information is lacking for most individual atomic states associated with highly excited energy levels. A “fill” procedure for approximating the energy of the unknown states will be discussed. Sodium vapor will also be considered to be a real gas that obeys the virial equation of state. The first non-ideal term in the power series expansion of the heat capacity in terms of virial coefficients involves the second virial coefficient, \(B(T)\). This depends on the interaction potential energy between two sodium atoms, i.e., the potential energy curves for the sodium dimer. Accurate interaction potential energies can be obtained from either experimental or theoretical information for the lowest ten electronic states of the sodium dimer. These are used to calculate \(B(T)\) for each state, and the averaged value of \(B(T)\) for all ten states is used to calculate the non-ideal contribution to the heat capacity of sodium atoms as a function of temperature.


Heat capacity Partition function cut-off Second virial coefficient Sodium atoms 


  1. 1.
    M.W. Chase Jr. (ed.), NIST-JANAF Thermochemical Tables, Part 2, 4th edn. (NIST, Washington, DC, 1998), p. 1647Google Scholar
  2. 2.
    R.C. Weast (ed.), CRC Handbook of Physics and Chemistry, 51st edn. (The Chemical Rubber Co., Cleveland, 1971), pp. D-412, D-176Google Scholar
  3. 3.
    Y.A. Ralchenko, A.E. Kramida, J. Reader, NIST ASD Team, NIST Atomic Spectra Database, version 3.1.5. (National Institute of Standards and Technology, Gaithersburg, MD, 2008). Accessed 15 May 2008
  4. 4.
    J.R. Downey Jr., AFOSR-TR-78-1960 (AD-A054854) (Dow Chemical Company, Midland, MI, 1978)Google Scholar
  5. 5.
    B.J. McBride, S. Heimel, J.G. Ehlers, S. Gordon, NASA Rep. No. SP-3001 (Washington, DC, 1963)Google Scholar
  6. 6.
    B.J. McBride, S. Gordon, NASA TN D-4097 (Lewis Research Center, Cleveland, OH, 1967)Google Scholar
  7. 7.
    S. Gordon, B.J. McBride, NASA/TP-1999-208523 (Cleveland, OH, 1999)Google Scholar
  8. 8.
    C. Moore, Atomic Energy Levels, Circular 467, vol. 1 (National Bureau of Standards, Gaithersburg, MD, 1949), pp. 89–91Google Scholar
  9. 9.
    L.V. Gurich, I.V. Veyts, C.B. Alcock (eds.), Thermodynamic Properties of Individual Substances, vol. 1, Part 1 (Hemisphere Publishing, New York, 1989), pp. 15–19Google Scholar
  10. 10.
    M. McChesney, Can. J. Phys. 42, 2473 (1964)CrossRefMATHADSGoogle Scholar
  11. 11.
    S.J. Strickler, J. Chem. Educ. 43, 364 (1966)CrossRefGoogle Scholar
  12. 12.
    H.R. Griem, Phys. Rev. 128, 997 (1962)CrossRefADSGoogle Scholar
  13. 13.
    M. McChesney, AIAA J. 1, 1666 (1963)CrossRefGoogle Scholar
  14. 14.
    H.R. Griem, Plasma Spectroscopy (McGraw-Hill, New York, 1964), pp. 137–142Google Scholar
  15. 15.
    H. Krempl, in Physical Chemistry. An Advanced Treatise, Volume I/Thermodynamics, ed. by W. Jost (Academic, New York, 1971), pp. 551–555Google Scholar
  16. 16.
    L. Alexandre, C. Villani, Ann. Inst. H. Poincaré Anal. Non Lineaire 21, 61 (2004)MathSciNetMATHADSGoogle Scholar
  17. 17.
    B.J. McBride, S. Gordon, M.A. Reno, NASA TP-3287 (1993)Google Scholar
  18. 18.
    B.J. McBride, S. Gordon, M.A. Reno, NASA TM-4513 (1993)Google Scholar
  19. 19.
    H. Myers, J.H. Buss, S.W. Benson, Planet. Space Sci. 3, 257 (1961)CrossRefADSGoogle Scholar
  20. 20.
    E.A. McLean, C.E. Faneuff, A.C. Kolb, H.R. Griem, Phys. Fluids 3, 843 (1960)CrossRefADSGoogle Scholar
  21. 21.
    K. Fuchs, J.G. Kynch, R. Peierls, Los Alamos Rept. BM-83 (1942)Google Scholar
  22. 22.
    J.W. Bond Jr., Phys. Rev. 105, 1683 (1957)Google Scholar
  23. 23.
    H.N. Olsen, Phys. Fluids 2, 614 (1959)CrossRefADSGoogle Scholar
  24. 24.
    H. Bethe, Office Sci. Res. Dev. Rept. 369 (1942)Google Scholar
  25. 25.
    D.A. McQuarrie, J.D. Simon, Physical Chemistry: A Molecular Approach (University Science Books, Sausalito, 1997), pp. 20, 658Google Scholar
  26. 26.
    O. Sinanoglu, M.S. Vardya, E.M. Mortensen, W.C. Johnson Jr., Phys. Fluids 5, 665 (1962)Google Scholar
  27. 27.
    P. Calaminici, K. Jug, A.M. Koster, J. Chem. Phys. 111, 4613 (1999)CrossRefADSGoogle Scholar
  28. 28.
    I.A. Solov’yov, A.V. Solov’yov, W. Grener, Phys. Rev. A 65, 053203 (2002)CrossRefADSGoogle Scholar
  29. 29.
    J.O. Hirschfelder, C.F. Curtiss, R.B. Bird, Molecular Theory of Gases and Liquids (Wiley, New York, 1954), pp. 30, 150Google Scholar
  30. 30.
    M.E. Boyd, S.Y. Larsen, J.E. Kilpatrick, J. Chem. Phys. 50, 4034 (1969)CrossRefADSGoogle Scholar
  31. 31.
    G. Herzberg, Molecular Spectra and Molecular Structure I. Spectra of Diatomic Molecules, 2nd edn. (Van Nostrand, New York, 1950), pp. 318–321Google Scholar
  32. 32.
    L. Biolsi, P.M. Holland, Int. J. Thermophys. 31, 831 (2010)CrossRefADSGoogle Scholar
  33. 33.
    P. Kusch, M.M. Hessel, J. Chem. Phys. 68, 2591 (1978)CrossRefADSGoogle Scholar
  34. 34.
    K.P. Huber, G. Herzberg, Molecular Spectra and Molecular Structure IV. Constants of Diatomic Molecules (Van Nostrand Reinhold, New York, 1979), pp. 432–433Google Scholar
  35. 35.
    C. Effantin, J. d’Incan, A.J. Ross, R.F. Barrow, J. Verges, J. Phys. B 17, 1515 (1984)CrossRefADSGoogle Scholar
  36. 36.
    L. Liu, S.F. Rice, R.W. Fields, J. Chem. Phys. 82, 1178 (1985)CrossRefADSGoogle Scholar
  37. 37.
    C. Effantin, O. Babaky, K. Hussein, J. d’Incan, R.F. Barrow, J. Phys. B 18, 4077 (1985)CrossRefADSGoogle Scholar
  38. 38.
    H.M. Hulburt, J.O. Hirschfelder, J. Chem. Phys. 9, 61 (1941)CrossRefADSGoogle Scholar
  39. 39.
    H.M. Hulburt, J.O. Hirschfelder, J. Chem. Phys. 35, 1901 (1961)CrossRefADSGoogle Scholar
  40. 40.
    D.D. Konowalow, M.E. Rosenkrantz, M.L. Olson, J. Chem. Phys. 72, 2612 (1980)CrossRefADSGoogle Scholar
  41. 41.
    G. Jeung, J. Phys. B 16, 4289 (1983)CrossRefADSGoogle Scholar
  42. 42.
    J.T. Vanderslice, E.A. Mason, W.G. Maisch, J. Chem. Phys. 32, 515 (1960)CrossRefADSGoogle Scholar
  43. 43.
    D. Steele, E.R. Lippincott, J.T. Vanderslice, Rev. Mod. Phys. 34, 239 (1962)CrossRefADSGoogle Scholar
  44. 44.
    G. Das, A.C. Wahl, J. Chem. Phys. 44, 87 (1966)CrossRefADSGoogle Scholar
  45. 45.
    P.H. Krupenie, J. Phys. Chem. Ref. Data 1, 423 (1972)CrossRefADSGoogle Scholar
  46. 46.
    G.C. Lie, E. Clementi, J. Chem. Phys. 60, 1288 (1974)CrossRefADSGoogle Scholar
  47. 47.
    L. Biolsi, J.C. Rainwater, P.M. Holland, J. Chem. Phys. 77, 448 (1982)CrossRefADSGoogle Scholar
  48. 48.
    D. Klein, Z. Phys. 76, 226 (1932)CrossRefMATHADSGoogle Scholar
  49. 49.
    R. Rydberg, Z. Phys. 80, 514 (1933)CrossRefADSGoogle Scholar
  50. 50.
    A.L.G. Rees, Proc. Phys. Soc. Lond. 59, 998 (1947)CrossRefMATHADSGoogle Scholar
  51. 51.
    A.E. Sherwood, E.A. Mason, Phys. Fluids 8, 1577 (1965)CrossRefADSGoogle Scholar
  52. 52.
    O. Sinanoglu, K.S. Pitzer, J. Chem. Phys. 31, 960 (1959)CrossRefADSGoogle Scholar
  53. 53.
    M. Klein, in Physical Chemistry. An Advanced Treatise, Volume I/Thermodynamics, ed. by W. Jost (Academic, New York, 1971), pp. 508–510Google Scholar
  54. 54.
    G. Herzberg, Atomic Spectra and Atomic Structure (Dover, New York, 1944), pp. 13, 55Google Scholar
  55. 55.
    W.T. Zemke, W.C. Stwalley, J. Chem. Phys. 100, 2661 (1994)CrossRefADSGoogle Scholar
  56. 56.
    K.M. Jones, S. Maleki, S. Bize, P.D. Lett, C.W. Williams, H. Richling, H. Knoeckel, E. Tiemann, H. Wang, P.L. Gould, W.C. Stwalley, Phys. Rev. A 54, R1006 (1996)CrossRefADSGoogle Scholar
  57. 57.
    N. Matsunaga, A.A. Zavitsas, J. Chem. Phys. 120, 5624 (2004)CrossRefADSGoogle Scholar
  58. 58.
    O. Babaky, K. Hussein, Can. J. Phys. 67, 912 (1989)CrossRefADSGoogle Scholar
  59. 59.
    R.F. Barrow, J. Verges, C. Effantin, K. Hussein, J. d’Incan, Chem. Phys. Lett. 104, 179 (1984)CrossRefADSGoogle Scholar
  60. 60.
    M.E. Kaminsky, J. Chem. Phys. 66, 4951 (1977)CrossRefADSGoogle Scholar
  61. 61.
    P. Qi, J. Bai, E. Ahmed, A.M. Lyyra, S. Kototchigova, A.J. Ross, C. Effantin, P. Zalicki, J. Vigue, G. Chawla, R.W. Field, T.J. Whang, W.C. Stwalley, H. Knockel, E. Tiemann, J. Shang, L. Li, T. Bergeman, J. Chem. Phys. 127, 044301 (2007)CrossRefADSGoogle Scholar
  62. 62.
    C. Tsai, J.T. Bahns, W.C. Stwalley, J. Chem. Phys. 99, 7417 (1993)CrossRefADSGoogle Scholar
  63. 63.
    Y. Liu, J.L. Dieyan, L. Li, K.M. Jones, R.J. Le Roy, J. Chem. Phys. 111, 3494 (1999)CrossRefADSGoogle Scholar
  64. 64.
    P.H. Krupenie, E.A. Mason, J.T. Vanderslice, J. Chem. Phys. 39, 2399 (1963)CrossRefADSGoogle Scholar
  65. 65.
    J. Verges, C. Effantin, J. d’Incan, D.L. Cooper, R.F. Barrow, Phys. Rev. Lett. 53, 46 (1984)CrossRefADSGoogle Scholar
  66. 66.
    A. Bousheri, L.A. Viehland, E.A. Mason, Physica 91A, 424 (1978)CrossRefADSGoogle Scholar
  67. 67.
    P. Clancy, D.W. Gough, G.P. Matthews, E.B. Smith, W.A. Wakeham, Mol. Phys. 30, 1397 (1975)CrossRefADSGoogle Scholar

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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Chemistry DepartmentMissouri University of Science and TechnologyRollaUSA

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