THERMODYNAMICS A N D KINETICS IN
NEAR-EQUILIBRIUM
SYSTEMS
31
On the Applicdtion of Thermodynamics and Kinetics to Some Near-Equilibrium Systems
by Milton Manes Pithburgh Chemical Company, Research and Development Department, Neville laland, I’ittaburgh ,576, Pennsylvania (Received May 3, 1963)
Near-equilibrium systcms, including steady-state and nonisothermal systems, appear to be amenable to analysis in terms of classical thermodynamics arid kinetics, provided that the actively equilibratirtg processes in such systems are kinetically independent. In B number of cases the c*lassicalanalysis (in the linear approximation) has some advantages over the phenomenological approach; it leads to specific suggestions for experiments and to straightforward (*riteria for judging whether the assumption of equilibrium may be properly applied to a given system. The classid analysis is applied to elcctrochemical systems with internal leaks, thcrmocdls, thermocouplt:s, electrokinetic systems, and thermal transpiration. I t turns out that the nature of the rionequilibrium constraint is less impo.rtant than its relation to the nature of the leakage processes; nonisothermal systems may be handled with no particular diffivulty.
I, Introduction I t has been shown previously’ that the behavior of near-cquilibrium multireactiori systems may he understood in terms of classical thermodynamics and kinetics. I’or each chemical reaction, one uses the fact that for a single reaction, in the linear approximation, the observable rate v is proportional to the thermodynamic force, A , (e.g., free energy, affinity, etc.), i.e. v = TA
(1)
The proportionality factor T can be explicitly related to the equilibration velocity, or “gross” reaction rate, and thereby to the kinetics of the opposing reactioiq2 and is independent of the manner in which the system is perturbed from equilibrium. In extending eq. 1 to simultaneous chemical reactions, one can write, for example VI
=
~1.4,
(2)
affinity in the linear approximation. It has been shown‘ that the identity of the actively equilibrating reactions must in general be determined by what are esscntially kinetic investigations. We shall here be concerned with the extension of eq. 2 to multiprocess two-phase systems in which the transport of heat, matter, and/or electricity takes place between two distinct phases, and we shall place special emphasis on systems that come to a nonequilibrium steady state undcr some constant external constraint. As long as one can write a thermodynamic force function4 for each single process, we can extend eq. 2 to two-phase and multiprocess systems without requiring any new disciplines or any postulates other than the rather modest oncs thus far stated. The approach to be presented amounts to the introduction of kinetics on an equal standing with thermodynamics iti the study of thc noncquilibrium steady state arid a reluctance to apply myst,erious torminology before cxhaustitig the possibilit.ios of a straightfo~~ward
vz = T z A ~ etc: ,
provided that the velocities and affinities refer to the actively equilibrating reactions and that the system conforms to the postulate of Li3 in that the rate of each active reaction is completely determined by its own
M. hlanes, *J. Clrem. Phua., 39, 45G (lg(i3). (2) iM.Manes, 1,. J . E. ITofcr, aiid S.Wcllw, ibid,, IS, 1365 (1950). (3) J. C. M.Li, ibid., 29, 747 (1958). (4) K. G . Denbigh, ”The Thermodyrinmics of the Stcsdy State,” John Wiley and Sons, Iuc., New York, N. Y . , 1951. (1)
Volume 68,Number 1 Januarlj, 1964
32
analysis. The advantages of this approach, aside from its simplicity, is that it leads to suggestions for further experiments and to straightforward criteria for determining whether or not the Thomson hypothesis6 (i.e., the arbitrary separation into “reversible” and “irreversible” processes and the assumption of equilibrium for the “reversible” process) properly applies to a given system. The application of the suggested approach will be illustrated on a highly simplified example, after which its possible applications to other systems will be discussed. The analysis to be presented resembles that of Thomson,6 Eastrnaq6 and Wagner’ in the use of classical thermodynamics, but differs in the explicit use of kinetics. It resembles an earlier analysis by Rice8 in the use of both thermodynamics and kinetics, but emphasizes processes rather than the “transfer units” of Rice. It uses the idea of “independent” processes as presented by Li,3 with some modifications,’ differing in the definition of “independence” and in emphasizing the use of near-equilibrium observations and the linear approximation. Finally, although we shall use the matrix language of the phenomenological theory of irreversible thermodynamic^,^*^ it is emphasized that this language is used for convenience, without any implication that a separate discipline is being used. The discussion to follow will require several definitions. The term “process” will always be understood as the direct analog of a chemical reaction, in the sense that all of the changes brought about by a single process are understood to be expressible in terms of a single progress variable or a single velocity. Thus, we shall distinguish between pure electrical conductivity, which may exist as a separate process, as against the passage of electrical current when it is one of the effects of an electrochemical reaction. In the latter case the electrochemical reaction is the process, and its electrical effect has the same standing as the appearance or disappearance of each chemical constituent. We shall later see how failure to observe this sort of distinction has led to confusion. It will be convenicnt to distinguish between those processes that involve more than one state variable (e.g., chemical reaction, osmosis, Donnan equilibrium) and those processes that involve only a single variable (e.g., pure electrical conduction). We shall refer to the former as “internally coupled” processes and to the latter as “dissipative” or “leakage” processes. For leakage processes, eq. 1 takes a particularly simple form; for example, it becomes Ohm’s law for pure electrical conductivity. The utility of the distinction and the justification for expressing a relation as simple The Journal of Phyaicd Chemietry
R~ILTON MANES
as Ohm’s law in the form of eq. 1 will become apparent later. Finally, we note that the term “independent processes” will refer to sets of processes that are kinetically independent, here defined in the sense that one can in principle vary the rate of each process without varying the rates of any of the othersS3 One can assumc the simultaneous existence of an infinite number of such processes without violating either thermodynamics or microscopic reversibility. Therefore, we shall differ from Li3 in setting no criterion of linear dependence on the active processes; their number as well as their identity must in general be determined by direct observation. although the set of active processes may be linearly independent in many cases, this must be recognized as a fortuitous consequence of the limited number of available reaction paths in a given system rather than the result of a thermodynamic limitation.
11. Kinetic Approach to Simultaneous Near-Equilibrium Processes Before considering examples, we shall first consider how the active processes may be disclosed, using a modification of a method presented earlier.’ In the experimental methods suggested by Li3 and by Manes’ for finding the real processes in a given system, one expresses the observations in terms of the rates of some set of assumed “processes.” The experimental information is then used to find the linear transformation that relates the real set to the assumed set. An alternative method, presented here, is to consider the rates of change of each of the extensive variables as the “flows” and the corresponding intensive variables as the “forces,” without any prior assumptions as to the nature of the processes. In this language the real processes may be determined by appropriate modification of the equilibration velocities, as described in an earlier article’ ; the present approach, however, is somewhat simpler, as shown in the following illustration. Assume a homogeneous chemical systcm a t constant temperature and pressure containing u constituents, Q,, , . ., Qc, in which the equations for m actively equilibrating reactions (processes) arc I
2v i j ~ i
i=l
=
0; j
=
1, , . ., m
(3
’I’lioiiisori. Mnth. PhuR. Papers, I , 232 (1882). E. D . Eastman, J. Am. Chem. Soe., 48, 1482 (1926); 50, 283
W.
(1928). C. Wagner, Ann. Phws., 3, 629 (1929); 6, 370 (1930). 0. K. Rice, 1.Phys. Chem., 61, 622 (1957). S. R. DeGroot, “Thermodynamics of Irrcvenible Processes,” Interscience Publishers, Inc., New York, N. Y., 1952.
THEHMOVYNAMICS ANI) KINETICS IN ?;EAR-EQUILIURIUM SYSTEMS
where Vij is the stoichiometric coefficient of Qi in the jth reaction (positive for reactants, negative for products) and where the number m and the identity of the active processes remain to be determined.’ We can consider u i j as -dni/d& where ni is the number of moles of & i and 6, is the progress variable of the jth reaction. The afhiity A , ( = - AIf’j) of the jth reaction may be written as
5
A j =
2
vijpi=
uijspi
i=1
i=l
(4)
where - AI> r2. This would occur as the bore of the capillary is reduced, since the diffusion rate decreases as the square of the diameter and the leakage rate as the third to fourth power, depending on pressure. To subject thc preceding picture to an experimental test, one could determine the value of r1 for a series of capillaries by measuring the rate of pumping when the system is operated as a thermomechanical pump, with no pressure gradient. The value of (rl r z ) could then be determined on the same capillaries by measuring the combined rates of leakage and diffusion under a pressure difference a t comparable pressure, in the absence of a thermal gradient. The determination of r1 and r2 should now account for 6Pl6T over a range of capillary bores in terms of measurable quantities.
+
IV. Conclusion Application of classical analysis to near-equilibrium systems requires that thermodynamics and kinctics be assigned approximately equal weight. I t is important to recognize that the actively equilibrating processes are the natural units in which to describe the system,a and to make a clear distinct,ion between the active procesws and their stoichiometric equivalents. Once this is recognized, the ‘(cross-effect~~~ of phenomenological theory lose their mystery. I t then becomes importantr to note tha.t the number and identity of the active processes can be determined only by direct observation, and that they can be found through kinetics experiments. Once the idea of a process is properly clarified, w e are able to consider leakage processes as distinct entities and, by specific consider: ation of their rates, to disentangle their effects from those of the internally coupled processes. The recognition of heat leaks as separate processes helps to remove some of the hindrances to considering nonisothermal systems on the same basis as constrained isothermal systmns. Finally, although it is not’ yet certain that the classical approach will be applicable to all systems, some optimism on its further applicability appears to be justified. (18) See ref. 9, p. 26. (19) See ref. 17, p. 71.
