1. Glasser University of the Witwatersrand Johannesburg, south Africa
The Nonequilibrium Thermodynamics of Chemical Processes
In a recent issue, THIS JOURNAL published an excellent introduction to nonequilibrium thermodynamics by H. J. M. Hanley (1) who emphor sized the importance of this topic in modern physics and chemistry. However, Hanley stopped short of the application of the principles to chemical processes. I t is t.he purpose of this article to indicate the function of nonequilihrium thermodynamics in the study of chemical processes, and the sort of information which may he gained by its application. The material which will be reviewed here is dealt wilh in the hooks of Denhigh (g), D e Gront (S), Fitts (4),Prigogine (5), and van Rysselberghe (G). The l~rescntdiscussion follows on from Hanley's paper (I), t,o which reference should he made for the pnstulatory background. For convenience, a full list of symbols used and their definit,ions is appended.' The symholism follows that of Hanley to simplify reference to his paper. The process of analysis of an irreversible process, to which the noncquilibrium thermodynamics is to be applied, may be summarized as follows: (1) Cdculation of the equation ior the rate of entropy production (d&/dt) from classical thermodynamic considerations. (2) Selection of conjugat,ed t,hermodynamic forces (Xi)and fluxes (JJ from the equat,iou for entropy production. (3) Study of the consequent linear phenomenological equa~ ) areciprocal n d relations (Lu = Lji). tions ( J t = x ~ ~ ~ ~Onsager I
The Second Law as an Equality
I n the study of irreversible processes, the inequality statement of the Second Law of Thermodynamics is unwieldy and is better used 3s an equality. For an 'List of symbols: A, chemical affinity; E, total internal energy; G , Gibbs free energy; Ji, generalized t,hermodynamir flux; Lij, phenomenologicsl coefficient; n,, and n,', number of moles of participant in a chemical react,ion, the prime indicates t,hzi. Ihe qunntit,ies of all pilrticipants ot,her than n, are under ronsiderxt,io~t;P, power oi irreversibility, T(dS./dt); ,p, pressure; a&, inlini1,esimd heat transferred during an irreversrhle process; r, 1,ho ,lumber in a set of simultsneaus reactions; S, total entropy; rLS., entropy change due to heat interchange wit,h surroundiugs; dS,, internal entropy produced during an irreversible process; 'Iy,absolute t,emperature; t, time; V , volume; v , rate of reaction (De lfonder natation); Xi,generalised thermodynamic force (or dfinity); y, rumling number, indicating one of the chemical 0 for a spontaneous process. :.P
=
LnAIa
+ 2LrzA,A?+
L41'
>0
on substituting the phenomenological relations for vl and ua, and using the Onsager reciprocal relation, Llz = Lzl. If both Al and A2 are zero, P is zero, and the process is a t equilibrium. If Al alone is zero, then P is positive only if Ln is positive, and the same can be shown for Lll. It is necessary that both L,, he positive otherwise the independent reactions, whose rates have been given as vl and v2, will proceed in directions opposite to those indicated by their affinities in the absence of mutual influence (i.e. when L,, = 0); this is clearly impossible. Consider now the case when neither A1 nor A2 is zero. We may divide through by Az2(which is necessarily positive) to obtain: LdAt/AdP 2L4ArIAJ hz > 0 (9) Tbis is the equation of a parabola with variable (AI/A2). For the quadratic function to be always positive, however (Al/Az) varies, the parabola must be above the abscissa [(A1/A2) axis], convex downwards. The quadratic funct,ion has then no real roots, a condition
+
+
a The derivation which follows exemplifies the approach used in, and the type of results obtainable from, s. study of the phenomenological relations.
Thus, the square of the mutual influence coefficients must be smaller than the product of the self-influence coefficients for the reactions to produce entropy and so proceed in the direction written. Limitation on Coupling
An important aspect of the results shown here is the fact that coupling can occur between otherwise independent processes. Other instances of coupling occur in wide varieties of situations, such as the thermomolecular pressure effect and thermoelectricity discussed by Hanley. However, coupling is not unlimited, and not all types of properties may couple in isotropic systems; in particular processes whose tensorial characters differ by an odd integer may not couple (Curie's theorem). Thus, scalar properties (zeroth rank tensor) may couple with scalars, vectors (first rank tensors) with vectors, scalars with dyadics (second rank tensors), and so forth. But, scalars and vectors may not couple. In terms of this theorem, then, it is not possible to obtain coupling, for instance, between a chemical reaction (scalar) and diffusion (vector) although, as Prigogine and van Rysselberghe show, there can be an indirect control of diffusion by a chemical reaction. The Status of the Theory
It has been emphasized (S), (10) that the linear phenomenological relations for chemical reactions are valid only over small deviations from equilibrium, which limits the applicability of the theory considerably. Furthermore, an overall chemical reaction can always be described by a number of different reaction mechanisms, and it has been shown that thermodynamic coupling can always he introduced into the results by a suitable choice of mechanism (11); this means that the true mechanism of the reaction must he used in the calculations to give significance to any coupling found-a requirement with which compliance cannot be guaranteed! Tbis might seem to make a study of the irreversible thermodynamics of chemical reactions a fruitless task. However, Prigogine (5)shows that the equations of thermodynamics can be applied outside the region where the linear laws and Onsager's reciprocity relations hold, and where phenomenological coefficientsare no longer constant. The recent literature (10) is much concerned with this extension of the theory to nonlinear situations, which augments the domain of application considerably. An important application of irreversible thermodynamics which has not been touched on is the study of stationary or steady states, characterized by a minimum in the rate of entropy production. The study of such states is dealt with in the books referred to a t the beginning of this article. Finally, it may be mentioned that there would appear Volume 42, Number I I, November 1965
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599
to be scope for a study of the action of masers and lasers, including the newly developed chemical lasers, by the methods of irreversible thermodynamics, for here there seems to be a change from a nonequilibrinm initial state towards a final equilibrium state, in one degree of freedom only. The treatment should be especially suitable for continuous laser discharges. Discussion, with Miss J. Hardwich of this University, of mathematical niceties relevant to this paper is acknowledged. Literature Cited (1) HANLEY,H. J. M., J. CAEM.EDUC.,41,647 (1964). (2) DENBIGA,K. G., "The Thermodynamics of the Steady State," Methuen and Co. LM., London, 1951.
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Journal of Chemical Education
(3) DE GROOT,S. R., 'LTherrnodynamicsof Irreversible Proeesses," North-Holland Publishing Co., Amsterdam, 1952. (4) FITTS,D. D., "Nonequilibrinm Thermodynamics," McGraw-Hill Book Ca. Inc., New York, 1962. (5) PRIGOGINE,I., "Introduction to Thermodynmnics of Irreversible Processes," 2nd ed., Interscience Publishers, New York, 1961. P., "Thermodynamics of Irreversible (6) VANRYSBELBERGHE, Processes," Hemsnn, Paris, 1963. (7) HnoAEs, D. O., AND LATEAM, J. L.,E ~ u cin . Chem., 2,36 (1965). (8) MILLER,D. G., C h m . Rev., 60, 15 (1960). (9) COLEMAN, B. D., AND TRUESDELL, C., J. Chem.Phys., 33.28 (1960). C., Brit.Chem. Eng., 9,164 (1964). (10) STOREY, (11) KOBNIG, F. O., HORNE,F. H., AND MOAILNER, D. M., J . Am. C h m . Soe., 83,1029 (1961).
