A MACROSCOPIC APPROACH T O
IRREVERSIBLE THERMODYNAMICS C. M. SLlEPCEVlCH
AND DON FINN
University of Oklahoma, Norman, Okla.
The basic equations of irreversible thermodynamics are derived completely from the macroscopic viewpoint without recourtse to statistical mechanics. It i s shown that the theorem of microscopic reversiblity i s not essential to establishing the equivalence of the cross phenomenological coefficients, commonly referred to as the Onsager reciprocal relations. By means of the macroscopic derivation, the precise physical significance of the phenomenological coefficients, l ’ s , and the fluxes, l ’ s , becomes readily apparent.
HE C U R R E N T LITERATURE. on irreversible thermodynamics Tcontinues to emphasize that the Onsager reciprocal relationship constitutes a new law or axiom which is not deducible from the laws of classic,al thermodynamics. The purpose of this article is to propose a macroscopic derivation. Aside from serving this purpose, it is hoped that this paper will stimulate interest and discussion among thermodynamicists regarding the validity of the postulates set forth. T o facilitate comparison, the Onsager development-as it is usually presented i n the literature-will be summarized first, followed by a rnacroscopic derivation that is in no way dependent on the properly of microscopic reversibility or on statistical mechanics.
Irreversible Thermodynaimics of Onsager
Experimental observations of transport processes indicate that certain forces or gradients, Xi,in the intensive thermodynamic properties or potentials such as temperature, T ; pressure, P; chemical potential, p ; etc., give rise to fluxes of heat, mass, or other forims of energy, Ji. If these gradients are not too large, it is a fundamental postulate of irreversible thermodynamics that the fluxes, Ji,are linear: homogeneous functions of the gradients, Xi. Thus,
where the phenomenological coefficients, L i k , depend only on the state of the system, as, determined by the local values of the intensive properties, and are assumed to be independent of the gradients in these properties. The coefficients, Lis, ( i = k) are ”related” to the heat conductivity, ordinary diffusion coefficient, electrical conductivity, chemical drag coefficient, etc. The coefficients Lik (z’ # k) are connected to interference (cross) phenomena such as the thermal diffusion coefficient or its inverse, the Dufour coefficient. For the case of a system undergoing a simultaneous transfer of heat, J I >and mass, J z :under the influence of a temperature gradient, X l , and chemic,al potential gradient, X Z : Jl =
LIIX’1
Jz
L2Xi
=
+ + L22Xz
L12X2
(la) (1b)
Another basic postulate of irreversible thermodynamics is that for a system undergoing a n irreversible process, all thermo-
dynamic functions of state exist and are the same functions of the local state variables as the corresponding equilibrium quantities. I t therefore follows that the time rate of entropy production can be expressed as
where S, = entropy production; 6 = time. Equation 2 in terms of two forces takes the form,
The third basic postulate is referred to as Onsager’s fundamental theorem: If a “proper” choice is made for the conjugated fluxes (J,)and forces ( X , ) in Equation 2, the matrix of the phenomenological coefficients, Lik, in Equation 1 is symmetrical-Le., Lzk =
Lki
(i, k = 1, 2 , . . ., n )
(3)
These identities are called the Onsager reciprocal relations. They express a connection between tlvo reciprocal phenomena -for example, the simultaneous transfer of heat and mass as given by Equations l a and 1b. Equation 3 is the essence of the theorem of microscopic reversibility which Onsager derived by statistical mechanics. Equations 1, 2, and 3 constitute the basic equations of irreversible thermodynamics. Onsager conceived his ideas from consideration of two special cases : heat conduction in anisotropic crystals and a chemical, monomolecular triangle reaction which he extrapolated to a general, but a priori, result. Although his theory is based upon assumptions regarding the microscopic behavior of physical systems, the result is macroscopic in that it conforms to limited experimental evidence. For this reason, his derivation is not important if one is willing to accept his theorem as a “new” principle of thermodynamics or a general axiom. I t is beyond the scope of this paper to present further details regarding Onsager’s development ( 7 , 2). There are three pertinent comments regarding Onsager’s approach to irreversible thermodynamics : I t leaves the erroneous impression that the reciprocal relations are a product of statistical mechanics and cannot be derived from macroscopic thermodynamics. I t gives no physical insight as to the nature of the phenomenological coefficients, LEk. V O L 2 NO. 4 NOVEMBER 1 9 6 3
249
I t is not immediately obvious what constitutes “properly” selected conjugate fluxes and forces such that Equations 1, 2, and 3, are valid. Potential Concept of Lost Work
To simplify the following development and to emphasize its practical significance, a particular example involving the simultaneous (one-dimensional) transfer of heat and mass will be used. I t is to be clearly understood, however, that this simplification introduces no loss in generality; it can be readily extended to three-dimensional space under the influence of more than two potential differences. Consider a system of mass, m, which is identified by the uniform, intensive properties : temperature, T , and chemical potential, p . A quantity of mass, dml, having a specific enthalpy, hi, is transferred into the system. The quantity of heat transferred across the boundary of the system a t Tt is S Q while no work is done. Neglecting kinetic and potential energy the following equations apply:
+ d(um) = S Q
Energy balance: -hidmi Entropy balance: d(sm)
6
=
(X) +
(4)
dmi
+7 61W
Mass balance: dmi = dm Gibbs equation: du = T d s - Pdv
Defining equation:
p =
h
-
(g +
sidm) ( T i
T)
(7)
-
Ts
(8)
+ dm(pi -
p)
(9)
replacing the potential differences by A’s and introducing the time derivative, such that I3 =
(4
f si.6f)AT
ae
=
(’), Equation 9 becomes
+ h;r(Ap)
=
Q AT
~~
Assuming Homogeneous Function of Second Degree. According to Euler’s theorem, Equation 10 becomes for a homogeneous function of degree 2,
+ hi(Ap)r
Expanding the partial differential coefficients of Equation 11 in a MacLaurin series about the origin.
Equation 12 is precise and involves no truncation? regardless of the magnitude of A T and Ap since for a second degree function the third order partial derivatives themselves are equal to zero, The only rentriction is the extent to which the partial differential coefficients are defined and remain continuous a t the origin for large potential differences in the system. Since
(10)
(2%) aAT
where
( A p ) r 3 (si A T
~
and similarly for mass transfer, lW = K A (Ap)?. 2. Assume that lw is a n arbitrary, analytical function (not necessarily homogeneous) of some degree higher than 2.
(6)
T s = u f Pu
-
1. Assume that lw is a homogeneous function (admittedly, a predilection prompted by the fact that thermodynamics deals extensively with homogeneous functions) of the second degree i n the potential differences. This assumption does not seem unreasonable in most cases since, for example, luj due to temperature differences is given by UA(AT) UA(AT)? /$E$(AT) = 7 ( A T ) = T
(5)
where, u, s, u are the specific internal energy, entropy, and volume, respectively, and p is the chemical potentia’. Combining these equations yields 61w =
If one is willing to accept these four conditions as basic postulates, then none of the preceding development leading to Equation 10 is essential to the remaining derivations. Equation 10 serves only to establish the precise form of the Iw function. The derivation of the Onsager equations will be demonstrated by two methods each of which imposes certain limitations :
4- A F )
This equation has been derived in a variety of ways by others. For example, De Groot ( 7 ) used the differential energy, mass, and momentum balances along with the Gibbs equation to arrive a t a n entropy balance and a n equation similar to Equation 10. Equation 10 defines 1W for a system under the influence of temperature and chemial potential differences only since all other potential differences were assumed to be nonexistent for the sake of simplicity but without loss of generality.
=o o . ~
Equation 12 becomes
similarly,
Now, substituting Equations 14 and 15 into Equation 11,
Macroscopic Derivation of Onsager Equations
By inspection of Equation 10, coupled with well-established thermodynamic concepts, four conditions, which are pertinent to the subsequent derivations, can be inferred : lw = lw ( A T , A p ) . This condition stems from the fact that in Equation 10, Q is a function of A T and Ak is a function of A p . I t is assumed that lw and its derivatives are continuous functions of A T and A p . lw ( 0 , 0) = 0. I n other words if 4 T and Ap are equal to zero so is lw. a‘w(o, O ) = 0,and = 0 , since ~w is always aAT a- -,..h positive and is a continuous, even function of AT and Ap-i.e., lw ( A T ) = lw ( - A T ) , and lw ( A p ) = lw ( - A p ) . _ aZlw _ _ _ --_ _ _ The equivalence of the cross partials aApaAT aATaAp’ follows immediately from the first condition. 250
l&EC FUNDAMENTALS
Assuming Arbitrary Analytic Functions of Higher O r d e r Than 2. Expanding 1W directly in a MacLaurin series about the origin, truncating all terms containing third order terms in the potential differences and higher, since it is assumed that 4 T and hu are small, and noting that IW (0,O)= 0, and
($$)o,O
=
0 =
(g)o,o, -
I t is important to re-emphasize the differences between Equations 1G and 16a.
Equation 16 expresses lzi: precisely subject to the validity of the assumption of a homogeneous function of the second degree. T o this extent it imposes no restrictions on the magnitude of the potential differences. Equation 16a. based on a n arbitrary, analytic function, expresses ZLL only approximatel> since it was necessary to restrict its derivation to small values of the potential differences.
NOW, by comparing the similarity of Equations 16 and 16a, there appears to be a n interdependence of the limitations imposed by each. The assumption of a homogeneous function of degree 2 seems to be valid only for “small” potential differences. Note. however, that in the derivation of Equation 16a the expansion \vas truncated a t the terms involving the potential differences to the second order. Qualitatively speaking, the restrictions implied rn both equations are for ..small” potential differences, rathrr than ‘.very small.” Returning now to E,quation 16 and defining the terms in the brackets on the right-hand side as J 1 and Jz,respectively, and replacing the partial differential coefficients bv L’s in accordance v i t h
Although the derivation was restricted to differences in A T and Ap, it is obvious from the above that systems under more than two potential differences can be treated in a similar manner in which case additional reciprocal relations must be satisfied (the resulting equations become quite cumbersome). Even for the case of two potential differences, Equations 17 and 18 demonstrate that the phenomenological coefficients are third order (including time) partial derivatives, which present difficulties in seeking direct experimental confirmation. Finally, the nature or physical aspects of the phenomenological coefficients can be readily identified by Equations 17 and 18 in terms of the lost work or irreversibility (or entropy production) and the thermodynamic potentials. There is nothing for example in Equation 17 which “proves” that the classical thermodynamic definition of heat (as that form of energy in transition across a temperature interval) is inadequate. Throughout the macroscopic development, this definition of Q was retained; its presence in Equations 4 through 10 is a result of the first and second laws of thermodynamics which demand one, and only one, precise definition of Q. Conclusions
and
Sow for the sake of comparing the macroscopic development with the Onsager microscopic development, presented earlier, noting that.
The conventional Onsager reciprocal relations are a direct consequence of the first and second laws of macroscopic thermodynamics. Although the relations were ostensibly derived for irreversible processes, similar relations can be developed for reversible processes. Furthermore, the generalized mass, energy. and entropy balances in conjunction with the Gibbs equation and the momentum balance are completely adequate to derive the equations which are associated with so-called modern, irreversible thermodynamics without having to introduce new terminology. Many other important inferences can be made based on the macroscopic development, which, however, are beyond the scope of the present paper. From the viewpoint presented in this paper. it is questionable whether the Onsager development can be regarded as a new theorem or axiom in thermodynamics. Hoirever, the reader is again reminded that the most authoritative sources have repeatedly emphasized that the Onsager reciprocal relationship is a new axiom or law not deducible from the laws of classical thermodynamics, which-if true-would negate the validity of the macroscopic development presented herein. Acknowledgment
and setting,
The authors are indebted to the graduate students in chemical engineering thermodynamics over the past 10 years for many stimulating discussions on irreversible thermodynamics. As a result of their solicitation and encouragement from colleagues, this paper was prepared for publication. The contributions by R . Babcock, W. Brigham, H. Hashemi, M. Heymann, and W. Orthwein are particularly noted. The inspiration for this paper is the teaching and writing of the late George Granger Brown of the University of Michigan. literature Cited
Nowhere in the entire macroscopic development was it necessary to introduce the theorem of microscopic reversibility nor to resort to statistical mechanics.
(1) De Groot, S. R., “Thermodynamics of Irreversible Processes,” Interscience, New York, 1951. (2) Onsager, L., Phys. Rev. 37,405 (1931) ; 38,2265 (1931). RECEIVED for review May 13, 1963 ACCEPTED August 16, 1963
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