The condition, Equation 9 or 12, of Sliepcevich and Finn says simply that if you symmetrize the phenomenological coefficients they will be symmetric. A condition given by LeLvis, Randall, which resembles work by Li (5), says Pitzer, and Erewer (4), that symmetric coefficients can arise if the matrix of the coefficients can be diagonalized. I n any event, all such macroscopic conditions represent assumptions which depend upon either experiment or statistical mechanics for verification. Since the reciprocal relations themselves also depend on experimental or theoretical verification, one must expect macroscopic approaches to be of limited value a t best.
Science Foundation grant, S S F G20725. and from a n hlfred P. Sloan Fellowship. literature Cited
Acknowledgment
(1) de Groot. S. R., Mazur, P., ‘.Non-Equilihrium Thermodvnamics.” North Holland Publishing Co.. .Amsterdam. 1962. (2) ‘Houlliwegue, I,., J . Phys. 5 , 5 3 (189;). (3) Landau. L. D.. Lifshitz, E. M.. ‘.Fluid Mechanics.” pp. . . 222-8. Pergamon Press. London, 1959. (4) Lewis, G. N.: Randall, M.. Pitzer. K. S.. Brewer: L.. “Thermodynamics.” pp. 454-6, McGraw-Hill, New York. 1961 15) Li. J . C. M.. J . ChPm. Phvs. 29. 747 119581. Miller, D. G.. Chern. Rer;. 60, 1 5 (1$60). ’ (7) (Imager, I,.. Phys. Rer,. 37, 405 (1931) ; 38, 2265 (1931). (8) Sliepcevich, C. M., Finn, Don, IND. ENG. C H E MFVSDA~, M E N T ~ L S2, 249 (1963)
T h e author acknowledges valuable discussions with R . B. Bird and C. F. Curtiss on the subject of this note. T h e author also acknowledges partial financial support from n’ational
Theoretical Chemislry Institute L‘niterritj of TVisconsin Madison, LVis.
CORRESPONDENCE
MACROSCOPIC APPROACH T O IRREVERSIBLE THERMODYNAMICS
S I R : .411 of the criticisms received to date focus on two major points:
I*
1, T h a t the Sliepcevich-Finn (SF) macroscopic development of the Onsager relations is trivial and is a result of being defined arbitrarily into existence. T h e critics offer as proof that SF have only shovm that
L12
+ LZ1 - + L2l
= CllY12
+ (Clr + CY1)YlYZ +
CS?Ii22
(5)
taking the second order cross-partials of Equation 3,
- L12
2
2
but that they have not shown L12
=
L21
2. T h a t the SF fluxes have no physical significance and therefore are invalid. or a t best highly undesirable. \\.’e shall attempt to reply to these two points briefly, since it is obvious from the unanimity among the reviewers that the
claritv \\ith LLhich these concepts were presented in the paper left much to be desired. Reciprocal Relations. From well-established concepts in classical (macroscopic) thermodynamics, SF postulate that lli’ is a homogeneous function of the second degree in the thermodynamic variables YIand Y2. They then derive the following 1 dlW t*=--y 2 dY1
1 dlW 1+--Yz 2 dY2
Defining (4)
276
and combining Equations 1 to 3,
I&EC FUNDAMENTALS
Equation 6, standing by itself, tells nothing about the relative values of CiZ and CZ1. However, by referring to Equation 4, one immediately sees that the second-order, cross partial derivatives are equal so that ClZ = cz1 (7) I t appears that all of the reviewers have essentially used Equation 5 as their starting point and have ignored Equations 1 to 4 from which it was derived. Consequently. they are confronted with Equation 6. T o establish Equation 7 : they must either revert to Equation 4 or to invoke some other independent relationship such as Onsager’s reciprocal relations based on microscopic reversibility. Apparently. all the reviewers prefer the latter alternative. We have carefully avoided using Equation 7 as a starting point since our development was to be independent of all statistical mechanics as well as the theorem of microscopic reversibility. Onsager derived Equation 7 from statistical mechanics. At this point, it is important to emphasize that our linear relations, Equations 2 and 3, are derived from Equation 1 ; therefore they are dependent on Equation 1 and are subject to the equivalence of the second-order. cross partial derivatives., I n the following, the development attributed to Onsager will be summarized briefly from his second paper [Physical Review, 38, 2265-79 (1931)l. His nomenclature \vi11 be used although it should be noted that his u and CY correspond to lw and 1’. respectively, as used above. The equation numbers also refer to his designations. Quoting from page 2273 : “LVe
must assume that the entropy. a>can be expressed by a multiple po\\.er series: and that the abridged Taylor development a =
3% ~
doc 12
ff12
+
b2U
+ b+ bffz fflbff, b2U
b2U
.
fflff2
bff?dffl
c ~ 1 c ~ 2
2-
CY?
(2.9)
x
’
1 2
= -
(-)aw
bci,
where
X, [x2
+
=
L’~,&,/T,
=
L/,,Q/T,
+ L’ZYM
where (2,lO) Equations 12 can be transformed without destroying the symmetry of the coefficient matrix to give
and
(2.11)
Onsager then proceeds to establish independently (page 2275) that the time derivatives, drl? “can be described by a set of linear differential equations of the form
(4 4) ’
Now it is a t this point that Onsager introduces his famous reciprocal relations which he derived from microscopic reversibility, namely that
Gii
=
Gzi
(4 8)
Now that the macroscopic and microscopic developments have been summarized, it is left for the reader to compare them and to draw his own conclusions. Choice of Fluxes and Forces. T h e authors decided to select the differences in thermodynamic potentials, so given b) Equation 10 in the SF paper, lw
=
($ +
J ~ M )AT
+ M(Ap)
(SF-10)
as the forces and then proceeded to derive the proper or conjugate fluxes, which result in being defined by Equations 2 and 3 above. However, strong objections have been raised by the reviewers because these fluxes are lacking in physical significance even though the corresponding forces are easily visualized. \Ye conclude from these criticisms that it is preferable to be able to attach physical significance to the fluxes and to accept the resulting conjugate forces, whatever they might be, as demonstrated by the alternate form of Equation 10 in the SF paper,
(SF-10) Therefore, if one decides to concentrate on physically significant fluxes he should start with W I
=
iw(6,M)
Then by proceeding Lvith the macroscopic development outlined in the SF paper he will obtain a similar set of relations corresponding to Equations 1 through 4 for which the fluxes, J , forces, X , and coefficients, L , are given in terms of thermodynamic, extensive variables, a, by
T o summarize, one can preselect the desired force or flux a t will, but not both, since the combination must satisfy the so-called conjugate criteria. T h e macroscopic development not only provides the means for deriving Equations 1 through 4 but also guarantees the conjugate requirement. I t really makes no difference from the standpoint of the validity of the resulting equations which choice is made, since by algebraic manipulations it is possible to convert from one choice to another. Essentially one can select a n easily visualized or physically meaningful flux, such as Equation 11. a t the expense of a more complex force, such as Equation 12, or vice versa (see Equations 17 to 19 of the SF paper). I t might be added, however, that despite the complexity of the forces in Equation 12, Equations 11 to 13 are consistent with
6,
~~- I
IZi, =
T,
T
+ lkjlp)T
(SF-10 )
Conclusion. Because of space limitations, it will have to suffice to say we are not willing to accept the blanket statements by some of the reviewers that even if our macroscopic development is valid, it can never predict those cases in which L , , = - L a ) . Such an avokved negation demands a proof. Judging from the nature and content of the many written criticisms that have been received (none of which to date, incidentally, have accepted the macroscopic development proposed by SF) we conclude that the point of argument goes much deeper than the question of mathematical uniqueness of the choice of fluxes and forces. T h e source of argument is more conceptual in nature, namely whether one prefers to adhere rigorously to the classical (macroscopic) definitions of thermodynamic quantities such as heat and mass transfers. and above all a precise definition of the system, or whether one wished to alter these basic definitions to suit the purposes of irreversible thermodynamics based on the microscopic vieLvpoint. Until this fundamental difference is clearly recognized, further arguments regarding the validity of the macroscopic development are futile. We still believe that the techniques described in the original SF paper are useful not only in obtaining the basic equations of irreversible thermodynamics but also in deriving other relationships in terms of physically meaningful and measurable variables as demonstrated above. If our arguments are still unconvincing, then we plead the notation shown below as the ast straw short of capitulation.
S H’ H
FI
-_
C. .M.Sliepcprich Don Finn Hndi Hiistipmi Michnrl Hrymnnn
L%iuersity of Oklahoma Norman, Okla. VOL. 3
NO. 3
AUGUST 1964
277
