First and Second Laws

  • Gianni Astarita


Thermodynamics, like any other science, makes use of a few primitive undefined concepts, and both body and state are such concepts. A body B is endowed with a fixed mass m t, and occupies some finite region of space of volume V t. In general, the volume V t will change in time, and so will the region of space occupied by the body considered. [The subscript t (for “total”) is used to indicate that the whole body is being considered.] In thermodynamics, a body is often referred to as a “closed system,” i.e., a system which does not exchange mass with its surrounding, as contrasted to an “open system” which is some region of space which may be occupied by different bodies at different instants in time. We will use the symbol ∂B to indicate the instantaneous external surface of the body.


Contact Force Body Force Thermal Engine Thermodynamic Theory Cold Source 
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  1. The formulation of the second law as given in equation (1.5.6) is usually called the Clausius-Duhem inequality. Admittedly, there is no general consensus on whether the second law should be formulated in general as the Clausius-Duhem inequality; see, e.g., A. E. Green and P. M. Naghdi, Proc. R. Soc. London, Ser. A 357, 253 (1977).CrossRefGoogle Scholar
  2. However, the Clausius-Duhem inequality has been very fruitfully employed in a number of recent analyses. A discussion of its logical foundations can be found in M. Feinberg and R. Lavine, “Foundations of the Clausius-Duhem inequality,” Appendix 2-A, in: C. A. Truesdell, Rational Thermodynamics, 2nd ed., p. 123, Springer-Verlag, New York (1984). This appendix also discusses the mathematical structure of thermodynamic theories. The book including this appendix is a good source for the foundations of the thermodynamic approach taken in this first part.CrossRefGoogle Scholar
  3. It is difficult to pinpoint exactly where the Clausius-Duhem inequality was first stated explicitly; the best source is B. D. Coleman and W. Noll, Arch. Ratl. Mech. Anal 13, 167 (1963).CrossRefGoogle Scholar
  4. Historically, the Clausius-Duhem inequality has developed as follows. The work of Planck in 1887 implied that, in general, the second law could be written as dUdw + TdS. If the differentials are interpreted as rates, this corresponds to U °w + TS ° which has been called the Clausius-Planck inequality, and it corresponds to omitting the last term on the left-hand side of equation (1.6.7) (or, equivalently, it corresponds to the requirement that Z M is nonnegative). If temperature is uniform in space, the Clausius-Duhem and Clausius-Planck inequalities are equivalent, but they are not if grad T is different from zero. If equation (1.6.6) is assumed to be universally true (i.e., Z T is required to always be nonnegative), then the Clausius-Duhem inequality implies the Clausius-Planck inequality. The Clausius-Duhem inequality by itself requires the sum Z M + Z T to be nonnegative, without requiring the two addends to be separately nonnegative.Google Scholar
  5. There has been an argument in the literature as to whether the temperature scale appearing in the denominator of equations (1.5.4) and (1.5.5) should be the same; see, e.g., M. E. Gurtin and W. O. Williams, Z. Angew. Math. Phys. 17, 626 (1966).CrossRefGoogle Scholar
  6. A mild global assumption, however, has been shown to be sufficient to imply that indeed the same temperature scale should be used; see M. E. Gurtin, Z. Angew. Math. Phys. 27, 775 (1976).CrossRefGoogle Scholar
  7. There is general consensus on the formulation of the first law, although in most textbooks it is not expressed in terms of rates as we have chosen to do. The validity of equation (1.3.8) for rigid-body motions is discussed in any standard textbook on the mechanics of rigid bodies.Google Scholar
  8. Thermal engines are discussed in detail in most textbooks of thermodynamics for mechanical engineers. A less known type of thermal engine is the rubber engine, which makes use of the properties of entropic elasticity exhibited by rubbers (see Sections 5.4 and 8.1). Rubber engines are discussed by R. J. Farris, Polym. Eng. Sci. 17, 737 (1977).CrossRefGoogle Scholar
  9. A procedure which turns out to be illuminating, but is nonetheless seldom followed, is to read the classics. Carnot’s work, Reflexions sur la Puissance Motrice du Feu, was published by Bachelier in Paris, 1824; an English translation can be found in W. F. Magie (ed.), The Second Law of Thermodynamics, Harper and Brothers, London (1899). The first Clausius paper on thermodynamics, “Ueber die bewegende Kraft der Warme, und die Gesetzte, welche sich daraus für die Warmelehre selbst ableissen lassen,” is in Ann. Phys. (3) 19, 368, 500 (1856). This is cited byGoogle Scholar
  10. J. W. Gibbs in his obituary of Clausius, Proc. Am. Acad., Vol. xvi, 458 (1889). Clausius published in 1897 a book, Theorie Mechanique de la Chaleur, Monceaux, Bruxelles, where his thermodynamic work is summarized. So did Joule in 1872, Das Mechanische Warmeaequivalent, F. Wieweg, Braunschwig; his original paper, “On Matter, Living Force, and Heat,” appeared in the Manchester and Lancashire General Advertiser, May 5–12, 1847.Google Scholar
  11. Berthelot’s contributions are well discussed in his two books, Thermochimie and Calorimetrie Chimique, published by Gauthier-Villars in Paris in 1897 and 1905, respectively.Google Scholar
  12. Duhem’s technical work is summarized in his books Mechanique Chimique, A. Hermann, Paris (1897) and Energetique, Gauthier-Villars, Paris (1911), and his philosophical views on science in La Theorie Physique, M. Riviere, Paris (1914).Google Scholar
  13. Finally, Gibbs’s fundamental thermodynamic work is presented in Trans. Conn. Acad. 2, 309 (1873); 2, 382 (1873); 3, 108 (1876); 3, 343 (1878).Google Scholar

Copyright information

© Springer Science+Business Media New York 1989

Authors and Affiliations

  • Gianni Astarita
    • 1
    • 2
  1. 1.University of DelawareNewarkUSA
  2. 2.University of NaplesNaplesItaly

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