Miscellaneous Problems in Specific Heats

  • E. S. R. Gopal
Part of the The International Cryogenics Monograph Series book series (INCMS)


In the previous chapters, various aspects of specific heats of solids, liquids, and gases have been discussed. It is a common experience to find that two phases can coexist over a range of pressure and temperature. Consider, for instance, water and its vapor contained in a vessel of volume V. If the temperature is raised slightly, a small quantity of water is converted into steam, absorbing latent heat in the process, and a new equilibrium pressure is established. In a P-T plane (Fig. 8.1), this will be represented as an equilibrium curve. Quantities such as the density, specific heat, and compressibility remain finite but different in the two phases. An interesting relation among the thermodynamic quantities at such an equilibrium curve is furnished by the Clausius-Clapeyron equation. To derive this, apply Maxwell’s relation (∂P/∂T) υ = (∂S/∂V) T [equation (1.11)] to the system. The latent heat L12 is equal to T dS at the phase boundary, and so
$$\frac{{DP}}{{DT}} = \frac{{{S_2} - {S_1}}}{{{V_2} - {V_1}}} = \frac{{{L_{12}}}}{{T({V_2} - {V_1})}}$$
where D/DT stands for the derivative along the equilibrium curve. This simple equation, in which all the quantities can be determined experimentally, forms a rigorous practical test of the first and second laws of thermodynamics.


Latent Heat Natural Rubber Equilibrium Curve Vitreous Silica Debye Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    H. N. V. Temperley, Changes of States, Cleaver-Hume, London, 1956.Google Scholar
  2. 1a.
    A. B. Pippard, Elements of Classical Thermodynamics, Cambridge University Press, Cambridge, 1957, chapters 8 and 9.Google Scholar
  3. 2.
    A. J. Hughes and A. W. Lawson, J. Chem. Phys. 36, 2098 (1962).CrossRefGoogle Scholar
  4. 3.
    C. W. Garland and J. S. Jones, J. Chem. Phys. 39, 2874 (1963).CrossRefGoogle Scholar
  5. 4.
    R. Viswanathan and E. S. R. Gopal, Physica 27, 765, 981 (1961);CrossRefGoogle Scholar
  6. 4a.
    R. Viswanathan and E. S. R. Gopal, Physica 29, 18 (1963).CrossRefGoogle Scholar
  7. 4b.
    C. W. Garland, J. Chem. Phys. 41, 1005 (1964).CrossRefGoogle Scholar
  8. 4c.
    M. P. Mokhnatkin, Soviet Phys.-Solid State 5, 1495 (1964).Google Scholar
  9. 5.
    D. Turnbull, Solid State Phys. 3, 225 (1956).CrossRefGoogle Scholar
  10. 6.
    J. S. Rowlinson, Liquids and Liquid Mixtures, Butterworth, London, 1959, p.40.Google Scholar
  11. 7.
    J. J. Markham, R. T. Beyer, and R. B. Lindsay, Rev. Mod. Phys. 23, 353 (1951).CrossRefGoogle Scholar
  12. 7a.
    R. O. Davies and J. Lamb, Quart. Rev. Chem. Soc. (London) 11, 134 (1957).CrossRefGoogle Scholar
  13. 7b.
    G. S. Verma, Rev. Mod. Phys. 31, 1052 (1959).Google Scholar
  14. 7c.
    K. F. Herzfeld and T. A. Litovitz, Absorption and Dispersion of Ultrasonic Waves, Academic, New York, 1959.Google Scholar
  15. 8.
    P. Flubacher, A. J. Leadbetter, and J. A. Morrison, Proc. Phys. Soc. (London) 78, 1449 (1961).CrossRefGoogle Scholar
  16. 8a.
    R. H. Beaumont, H. Chihara, and J. A. Morrison, Proc. Phys. Soc. (London) 78, 1462 (1961).CrossRefGoogle Scholar
  17. 9.
    E. C. Heltemes and C. A. Swenson, Phys. Rev. Letters, 7, 363 (1961);CrossRefGoogle Scholar
  18. 9a.
    E. C. Heltemes and C. A. Swenson, Phys. Rev. 128, 1512 (1962).CrossRefGoogle Scholar
  19. 9b.
    D. O. Edwards, A. S. McWilliams, and J. G. Daunt, Phys. Letters 1, 218 (1962).CrossRefGoogle Scholar
  20. 9c.
    F. W. de Wette, Phys. Rev. 129, 1160 (1963).CrossRefGoogle Scholar
  21. 10.
    G. Borelius, Solid State Phys. 15, 1 (1963).CrossRefGoogle Scholar
  22. 11.
    A. R. Ubbelohde, Quart. Rev. Chem. Soc. (London) 4, 356 (1950).CrossRefGoogle Scholar
  23. 12.
    J. D. Hoffman and B. F. Decker, J. Phys. Chem. 57, 520 (1953).CrossRefGoogle Scholar
  24. 12a.
    R. E. Meyer and N. H. Nachtrieb, J. Chem. Phys. 23, 405 (1955).CrossRefGoogle Scholar
  25. 13.
    A. B. Lidiard, Handbuch der Physik, XX (II), 246 (1957).CrossRefGoogle Scholar
  26. 14.
    A. J. E. Foreman and A. B. Lidiard, Phil. Mag. 8, 97 (1963).CrossRefGoogle Scholar
  27. 14a.
    G. F. Nardelli and N. Terzi, J. Phys. Chem. Solids 25, 815 (1964).CrossRefGoogle Scholar
  28. 15.
    A. Eucken and H. Werth, Z. anorg. allgem. Chem. 188, 152 (1930).CrossRefGoogle Scholar
  29. 15a.
    C. G. Maier and C. T. Anderson, J. Chem. Phys. 2, 513 (1934).CrossRefGoogle Scholar
  30. 15b.
    D. L. Martin, Can. J. Phys. 38, 17 (1960).CrossRefGoogle Scholar
  31. 15c.
    F. A. Otter and D. E. Mapother, Phys. Rev. 125, 1171 (1962).CrossRefGoogle Scholar
  32. 16.
    P. H. Keesom, K. Lark-Horovitz, and N. Pearlman, Science 116, 630 (1952).CrossRefGoogle Scholar
  33. 16a.
    W. DeSorbo and W. W. Tyler, J. Chem. Phys. 26, 244 (1957).CrossRefGoogle Scholar
  34. 16b.
    B. B. Goodman, L. Montpetit, and L. Weil, Compt. rend. acad, sci (Paris) 248, 956 (1959).Google Scholar
  35. 17.
    W. H. Lien and N. E. Phillips, J. Chem. Phys. 29, 1415 (1958).CrossRefGoogle Scholar
  36. 18.
    M. Dupuis, R. Mazo, and L. Onsager, J. Chem. Phys. 33, 1452, (1960).CrossRefGoogle Scholar
  37. 18a.
    R. Stratton, J. Chem. Phys. 37, 2972 (1962).CrossRefGoogle Scholar
  38. 19.
    A. A. Maradudin, E. W. Montroll, and G. H. Weiss, Theory of Lattice Dynamics in the Harmonic Approximation, Academic, New York, 1963, chapter 6.Google Scholar

Copyright information

© Plenum Press 1966

Authors and Affiliations

  • E. S. R. Gopal
    • 1
  1. 1.Department of PhysicsIndian Institute of ScienceBangaloreIndia

Personalised recommendations