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
We may find dlzi., dXf simply by differentiating Equation 2 :
sz=cm, x d1I.i
then
b21W
X5
(3)
and Furthermore, these coefficients are clearly of the type of Equation 9. as may be seen by evaluating b212i’,‘dX$X, from Equation 11 :
from which it follows immediately that
(4) 1
For small X‘s. b l ~ d X i may be expanded about { X ) = 0, the fir\[ term of the expansion vanishing by virtue of Equation 4 :
3x1 2 1
This ma) be compared n i t h Equation 3, lrhich is exact. It means that for small values of the forces, the function b211.i, d X f b X , may just as well be evaluated a t { X ) = 0 as a t { X ) . If Equation 5 is now substituted into Equation 2, we obtain
Clearly. each term in Equation 6 is unchanged if i a n d j are interchanged. If one uses Equation 6 to define a flux J i conjugate to the force X i : then
IC
JJi
=
1
1 -
J
fl
