difference in the order of numbering. This is especially helpful and timesaving in information retrieval when a large volume of data has been collected and stored on magnetic tape. For these reasons, Equation 1 should be prtferred over the corresponding asymmetrical form Nomenclature
measure of interaction between molecules i, j , constants defined by Equation 2 constants defined by Equation 6 constants defined by Equation 4 D i j = constants defined by Equation 3 E t j k c = constants defined by Equation 5 N = total moles of a system ni = moles of ith component P = total pressure R = universal gas constant T = absolute temperature Xi = mole fraction of ith component = = Cijk = Ciijk = ut1 Ail
,
. , etc.
= activity coefficient of ith component AGE = excess free energy
Yi
literature Cited
( I ) Marek. J.! Chem. Listy 47, 739 (1953); Collection Czech. Chem. Commun. 19, 1 (1954). (2) Maraules, M.. Sitzher. Akad. Wzrs. Wien. ‘Math. .Va/urzc. K1. 11. ’104. 1234 11895) (3) Sdatchard, G., Hamer. i V . J.. J . ‘4m. Chem. So(. 57, 1805 (1935). (4)’ Lran’Laar. J. J.. Z. Phys Chern. 72, 723 (1910). (5) Zhzd.. 185, 35 (1929). (6) \Vohl, K., Chem. Eng. Progr. 49, 218 (1953). (7) Wohl, K., Trans. Am. Inst. Chem. Engrs. 42, 215 (1946)
CHIEN-SHIH LU Y U I - L 0O N G \\’ANG Professional Services Group Burroughs Gorp. Pasadena, Calif. RECEIVED for review September 3. 1963 ACCEPTED February 28, 1964
CORRESPONDENCE
MACROSCOPIC APPROACH TO IRREVERSIBLE THERMODYNAMICS SIR: In a recent paper ( 1 4 , Sliepcevich and Finn presented a macroscopic derivation of the Onsager reciprocal relations. Since we are of the opinion that this derivation is incorrect and that there appears to exist some confusion regarding the reciprocal relations, we thought it would be useful if we not only discussed the derivation of these authors but also presented our interpretation of the current status of the validity of the reciprocity theorem of Onsager. T h e basic equation of nonequilibrium thermodynamics is the equation of change of entropy, which is derived from the familar conservation laws of continuum mechanics and from the so-called Gibbs equation. In the differential equation for the rate of change of entropy there appears a n important term commonly referred to as the entropy production and usually expressed (4, 5) as
where the tensor notation of McConnell (7) has been used to preserve the generality of the individual terms in this definition. Each of the terms in Equation 1 may be regarded as the product of a flux or flow quantity and of another quantity which is capable of giving rise to one or more fluxes; the latter parameter is usually referred to as a thermodynamic force. T h e assignment of the flux and force in each term of the entropv production is guided by experimental knowledge of the behavior of matter. Furthermore, one of the basic postulates of nonequilibrium thermod) namics is that. for systems close to equilibrium, each flux appearing i n the entropy production term is most generally a linear. homogeneous function of all the thermodynamic 272
l&EC FUNDAMENTALS
forces present in this term. If we restrict the discussion to isotropic systems, it follows from Curie’s theorem that the fluxes in the entropy production are functions only of the thermodynamic forces n i t h the same tensorial character. I t is interesting to note that Curie’s theorem \vas neither stated in its general form nor proved by Curie ( 3 ) :although it is almost always attributed to him; de Groot and hlazur ( d ) have subsequently established the validity of the Curie theorem. Consequently, for each flux in an isotropic system we may write
J=l
where JI and X J may both be scalars or the components of vectors’ or second-order tensors (tensorial indices have been suppressed) and where the scalar phenomenological coefficients, LIJ. are functions only of the temperature. pressure. and composition of the system. The goal of Onsager and others who have followed him has been to derive relationships among the phenomenological coefficients in the above linear laws in addition to the equations which necessarily result if not all of the forces and fluxes are independent. I t is evident that any equations relating the transport coefficients serve to reduce the experimental work needed to evaluate these parameters. I n general. it is in principle possible to derive equations similar or equivalent to the Onsager reciprocal relations by one of three methods. Perhaps the most appealing approach is a completely ‘microscopicdevelopment involving the application of the microscopic laws of motion and the principles of statistical mechanics. T h e validity of the Onsager relations could be tested in a straightforbvard manner b)- this method if a n explicit nonequilibrium theory of molecular distribution functions were available. However, since no such theorb- of a
general nature exim and since there is no satisfactory method of handling chemical reactions microscopically, it is not a t present possible to formulate a general. detailed proof of the reciprocal relations at thr molecular level. It is possible, however, to drvelop a prrturbation theory for nonequilibrium distribution f~inctioiisand, consequrntly. to formulate explicit expressions for thr phenomenological coefficients of the usual linear laws in term\ of molecular variables. Kirk\vood and Fitts (6) pursued such an approach for a multicomponent system of structureless molecules i n the absence of chemical reactions. They were able to sho\v that the matrix of the phenomenological coefficirnts which appear in the linear relations for the heat flux and dilfrision fluxes of the various components is symmetric. Similar reciprocal relations \vere derived by Mori ( 9 ) ,who used a quantum statistical-mechanical approach, and by de Groot and Mazur. Although a microscopic approach does not yet provide a general proof of the reciprocal relations, the above investigations do provide some evidence of the validity of the reciprocity theorem. 'I'he second approach to the derivation of the reciprocal relations is of the type used by Onsager himself in his celebrated papers (70. 7 7) over 30 years ago. His development and others of this type might best be classified as a combination of microscopic and macroscopic methods, since a theorem derived on a rigorous statistical-mechanical basis is applied to macroscopic postulates. Onsager essentially used the so-called principle of microscopic reversibility as a starting point in his derivation. This is not an assumption, since it has been shown by \Yigner (73) that microscopic reversibility is simply a consequence of the time reversal invariance of the microscopic equations of motion. By applying the microscopic reversibility property and by postulating that certain macroscopic variables obey linear first-order differential equations. Onsager was able to develop the reciprocal relations. In his derivation, Onsager showed that the first-order differential equations are equivalent to linear laws relating a specific class of forces and fluxes; his reciprocal relations apply, of course. to these linear law coefficients which are necessarily related to the coefficients of the original differential equations. It is very important to emphasize that the Onsager development places relatively severe restrictions on the types of fluxes and thermodynamic forces for which the reciprocal relations are valid. Onsager did not derive his reciprocity theorem for the fluxes and forces appearing in the entropy production term of the entropy balance but for a special class of fluxes and forces obeying certain requirements. Thus, Onsager proved only that the matrices of phenomenological coefficients which relate a certain class of fllixes and forces are symmetric. In other words, the fact that forces and fluxes are related by linear equations and that these quantities exhibit a bilinear form in the entiopy production term does not necessarily imply the validity of the reciprocity theorem for these particular forces and flu, es. Clear and complete statements of the reciprocity theorem as derived by Onsager are given by de Groot and Mazur and by Coleman and Truesdell (2). I t is evident that the phenomenological coefficients introduced by Onsager muSt be evaluated experimentally, since they were introduced into the development a t a macroscopic level. The derivation of Onsager requires, among other things, that the fluxes be time derivatives of thermodynamic state variables. Examination of the entropy production term defined by Equation 1 shows that the vectorial fluxes (heat flux, diffusion fluxes) and the tensorial flux (viscous stress tensor) d o not satisfy this requirement. Consequently. it does not follow
from the theorem of Onsager that any reciprocal relations a r e valid for the coefficients relating these fluxes to the thermodynamic forces. Casimir (7): who probably was the first t o point out the limitations of the Onsager derivation, attempted to extend the validity of the theorem to fluxes and forces appearing in the entropy production term by using the macroscopic equations of change to introduce fluxes and forces which obey the restrictions of the Onsager theorem. By proper manipulation, Casimir was able to verify that the validity of the Onsager relations for these fluxes and forces directly resulted in a reciprocity theorem for the original fluxes and forces. The proof of Casimir and several others of the same type are summarized by de Groot and Mazur. T h e reciprocal relations which result from these derivations are of limited validity because relatively stringent assumptions were introduced a t a macroscopic level to facilitate the developments. The third possible method of deriving reciprocal relations is the purely macroscopic approach involving only the principles of continuum mechanics and classical thermodynamics. This method is of course restricted to the concepts of the thermodynamics of continuous media, there being no attempt to explain any phenomenon in terms of the physics of particles. T o our knowledge, there exists no completely macroscopic derivation of the Onsager reciprocity theorem. Nor? perhaps, should it be expected that any relationships among the phenomenological coefficients can be developed on a purely macroscopic basis. Complete ignorance of the nature of the phenomenological coefficients is perhaps the penalty one must pay for asserting an unrealistically simple model of matternamely, that of a continuum. However, Sliepcevich and Finn have presented a macroscopic approach to irreversible thermodynamics, which they claim establishes the validity of the Onsager reciprocal relations. To examine their derivation, we must convert our Equation 1 to a form comparable to their Equation 10, which is the starting point of their development. If the effects of viscous flow, chemical reactions, and external force fields are not considered, Equation 1 reduces to
Furthermore, it is easy to show that this equation may equivalently be written as
(4) where N
and where
Finally, if thr discussion is restricted to a two-component system, there results the following expression for what Sliepcevich and Finn call the rate of change of lost work:
For the sake of comparison, this equation may be considered essentially equivalent to Equation 10 in the paper by Sliepcevich and Finn if the analvsis is restricted to one-dimensional VOL. 3
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1964
273
heat and mass transfer. I t is evident that this expression for the lost work contains the temperature explicitly in addition to a heat flux and a mass flux, which are linearly related to the thermodynamic forces in the expression by phenomenological coefficients. Consequently, it follows that the lost work is not only a function of the thermodynamic forces, the gradients of temperature, and chemical potential, but also of the temperature, pressure, and composition of the system. Thus, the first condition imposed by Sliepcevich and Finn on the behavior of the lost work term places a severe limitation on the generality of what follows. However, if this condition is simply regarded as a restriction on the derivation and if the development is further limited to one-dimensional processes. it is still possible to attempt a derivation of a limited version of the reciprocal relations. I t is evident that the derivation of Equation 16a by Sliepcevich and Finn is correct if their first four conditions are accepted. The problem then is to define the fluxes J1and JZ in terms of the partial derivatives of Equation 16a in a manner consistent with Equations 19 and 21. It follows by differentiating Equation 21 that the terms of Equation 16a must be grouped so that
(9)
In summary, it can be said that although there is much evidence in their favor, no completely general proof of the reciprocal relations exists for all irreversible phenomena at the present time. However, there is sufficient justification for accepting the Onsager reciprocity theorem as a basic postulate of nonequilibrium thermodynamics. Nomenclature
A J = chemical affinity of J t h reaction e,j = rate of strain tensor F f i = ith component of external force per unit mass acting orL component I hr = partial specific enthalpy of component Z j f i = ith component of mass diffusion flux of component I J, = flux of Zth process L I J = phenomenological coefficient relating Zth and J t h processes M = number of thermodynamic forces of same tensorial character N = number of chemical species in system qi = ith component of conductive heat flux r = number of chemical reactions RJ = chemical reaction rate of reaction J S f = partial specific entropy of component Z T = temperatuie Ti’ = viscous stress tensor Xi = zth ‘ coordinate variable X , = J t h thermod>-namicforce p f = chemical potential of component I = rate of entropy production per unit volume u literature Cited
+
These definitions for Lll*, LSz*,and L12* Lzl* are the only ones which dlow Equation 16a to be consistent with Equation 21. There is no indication as to what are the correct expressions for L12* and Lzl* individually. There is, therefore, no justification within the realm of macroscopic thermodynamics for allowing LIZ* and LZl* to be each equal to one half of the right-hand side of our Equation 10 as Sliepcevich and Finn have done. By stating that their Equation 22 defines LIZ* and Lzl* uniquely, these authors have necessarily assumed what they have set out to prove-namely, that L12* = LZ1*. The point is that it is not possible to go beyond Equation 10 above without either introducing an assumption or, as Mixon ( 8 ) has remarked, injecting moleculat concepts into the development.
(1) Casimir, H. B. G., Rez. .Mod. Phys. 17, 343 (1945). (2) Coleman, B. D., Truesdell. C.. J . Chem. Phys. 33, 28 (1960). 13) Curie. P.. “Oeuvres de Pierre Curie.” D. 129. Gauthier-Villars. Paris, 1908’. (4) de Groot, S. R., Mazur, P., “Non-Equilibrium Thermodynamics,” North-Holland Publishing Co., Amsterdam, 1962. (5) Fitts, D. D., “Non-Equilibrium Thermodynamics,” pp. 29-33, McGraw-Hill, New York: 1962. (6) Kirkwood, J. G., Fitts, D. D.: J . Chem. Phys. 33, 1317 (1960). (7) McConnell, A. J., “Applications of Tensor Analysis,” Dover Publications, New York. 1957. (8) Mixon, F. 0.. IND.ENG.CHEM.FUNDAMENTALS 2, 325 (1963). (9) Mori, H., Phys. Reu. 112, 1829 (1958). (10) Onsager. L., Ibid..37, 405 (1931). (11) Ibid.. 38. 2265 11931). ( l 2 j Sliepcevkh, C. M.,Finn, D.. IND.ENG.CHEM. FUNDAMENTALS 2, 249 (1963). (13) Wigner, E. P., J . Chem. Phys. 22, 1912 (1954). \
I
J. L. Duda J. S. Vrentas
The Dow Chemical Go. Midland, Mich.
CORRESPONDENCE MACROSCOPIC APPROACH T O IRREVERSIBLE THERMODYNAMICS SIR: In a recent paper ( 8 ) , Sliepcevich and Finn claim to derive the basic equations of irreversible thermodynamics, in particular the Onsager reciprocal relations, by strictly macroscopic arguments. Their arguments boil down to the following : Postulate I. .411 intensive thermodynamic functions of state exist and are related to each other in the same way as they would be a t equilibrium. Postulate 11. The entropy production per unit volume, d,s/dt. is a homogeneous function of second degree in a set of 274
l&EC FUNDAMENTALS
thermodynamic forces, { X } . It also depends, of course, on the intensive state functions, { Y ). If dds/dt is written as djs
_ _-- 1W dt
T
postulate 11 implies, by virtue ofEu]er’stheorem, that
I
