J . Phys. Chem. 1988, 92, 3017-3018 medium, ro is an initial distance of the pair, k is Boltzmann's constant, and T i s the absolute temperature. One must first note that, though in principle 4 corresponds to the survival probability at infinite time, this formula should be applied not to the discussion of G(e,-) at the end of the spur decay but rather to that of G(e,;) at a much earlier stage. Even at 250 OC the static dielectric constant of water is as large as 2745and hence 9 is small only for pairs with short distances (at 250 O C 4 = exp(-1) for ro of 2.1 nm). Most of the initial geminate recombination should, therefore, be finished within a very short period. In multipair spurs various intraspur reactions should then follow to cause further decrease of eaq-. Although the present data of the scavenging experiment are not sufficient to estimate C(e,,-) at the earliest stage, they are suggestive of a positive temperature coefficient also for this value. At least below 250 "C the result does not seem to be in accord with the view by Burns and Marsh. The following arguments may be made in favor of the presumed increasing tendency for the initial C(e,,-) just after the geminate recombination. Obviously, in eq 12 the effect of decrease in 6 at high temperature is partly compensated by the factor k T . In addition, an average value of thermalization distance, ro, may increase with temperature correspondingly to a decrease in the density. Another important point may be the time dependence of e. The dielectric relaxation is very fast in water, and the solvation time of an electron has been reported to be less than 0.3 ps even at room t e m p e r a t ~ r e . ~ ~ Yet, . ~ ' some part of the geminate recombination reaction may take place before the full relaxation of the field between a positive ion (probably a proton) and an electron. The radius of a reaction between an electron and an OH radical may also be larger for a presolvated electron. Since the major component of the dielectric relaxation in water it seems probable that has an activation energy of -20 kJ the efficiency of eaq- formation becomes larger at higher temperature owing to an increase in the rate of dielectric relaxation. Alternatively, the initial event of the solvation may be trapping of quasi-free or dry electron into preexisting sites, as discussed in the case of solvated electron in alcohols.49 If such a process is important in determining the probability of e,; formation, one may well expect an increase in the number density of trapping sites at high temperatures. Singh, Chase, and Hunt have discussed the contribution of excited states of water to ionization yield in liquid water.s0 A possibility may be envisaged that relaxation (45) Akerlof, G. C.; Oshry, H. J. J . Am. Chem. SOC.1950, 72, 2844. (46) Wiesenfeld, J.; Ippen, E. Chem. Phys. Lett. 1980, 73, 47. (47) Migus, A.; Gauduel, Y.; Martin, J. L.; Antonetti, A. Phys. Rev. Lett. 1987, 58, 1559. (48) Hasted, J. B. In Water: A Comprehensive Treatise; Franks, F., Ed.; Plenum: New York, 1972 Vol. 1 , p 300. (49) Kennev-Wallace. G. A.: Jonah. C. D. J . Phvs. Chem. 1982.86.2572. (SO) Singh;A.; Chase, W. J:; Hunt, J. W. Fariduy Discuss. Cheh. SOC. 1978, 63, 28.
3017
of some excited state leading to ionization is more efficient at higher temperatures. The results of the scavenging experiments in Figure 8 indicate that the slope of increase in G(Cd+) is approximately the same for the three temperatures. Although the previously discussed uncertainty exists in the concentration dependence of the net scavenging rate, this may imply that fractional decrease of ea,in spurs may be somewhat smaller at 200 or 250 "C than a t 20 OC. In other words, the relative amount of eaq-'s that diffuse out of spurs may be larger at the high temperatures. A possible reason for this trend might be the suggested decrease in the rate of bimolecular recombination of e,;, reaction 3, above 150 "C.I1 Summary In order to estimate temperature dependence of G(eaq-), two series of experiments were carried out by use of a pulse-radiolysis technique. These are direct observation of e,; absorption in pure water and a separate series of scavenging experiments with Cd2+. Both experiments contain a problem about,,e, the absorption coefficient at the maximum, at elevated temperatures, and we had to assume in each case that e,, varies proportionally to inverse of the spectral width. It is remarkable that the two series of experiments gave consistent results about the temperature dependence of G(eaq-). If the present assumption is valid, G(e,,-) at the end of the spur decay must increase from 2.65 at 20 OC to -3.5 and -3.8 respectively at 200 and 250 OC. One cannot give any definite argument about the situation above 250 OC,but it is noted that the present estimate up to 250 OC agrees fairly well with that by Jha, Ryan, and Freeman, who reported, on the basis of a y-radiolysis experiment with SF,, a further increase up to 300 OC.*O It is added that the present results are of Geagnot grossly inconsistent with the reported temperature dependence of the yield of reducing radicals in a low-pH solution as estimated with Fricke dosimeter ~ y s t e m s . ~ J * - ~ ~ Although no distinct spur decay was observable in pure water at temperatures above 150 OC, the scavenging experiment showed that G(Cd+) increases with Cd2+concentration as well at 200 and 250 "C. Both of these observations are consistent with the view that the spur decay at these high temperatures occurs in a shorter time scale, corresponding with an increase in the diffusion rate. Note Added in Proof. We are informed that in a recent study by Buxton, Wood, and Dyster ( J . Chem. SOC.,Faraday Trans. 1, in press) quite a similar degree of increase with temperature as inferred here has been suggested for the sum of G,- and GOH. Acknowledgment. We express our thanks to Prof. Y . Tabata, to Dr. S . Tagawa, and to Dr. H. Kobayashi for giving helpful advice and encouragement. We are also grateful to Dr. J. Takagi, to Y. Takeuchi, and to H. Okuda for their cooperation in the experimental work. Registry No. H,O, 7732-18-5; Cd2+,22537-48-0.
COMMENTS Comments on "The Kinetic Mass Action Law Revislted by Thermodynamics" S i c In a recent paper, Lebon et al. claim a derivation of the nonlinear kinetic mass action law (KMAL) by restoring to thermodynamic arguments only. Following the guidelines provided by extended irreversible thermodynamics (EIT), the authors assert that such a law can be derived by appealing to a minimum number of assumptions, implying by this that those used are only of a thermodynamic nature. This short paper is devoted, first, to argue (1) Lebon, G.; Torrisi, M.; Valenti, F. J . Phys. Chem. 1987, 91, 5103.
0022-3654/88/2092-3017$01.50/0
that such a general statement is untenable and, second, to prove that the derivation proposed by these authors is, at best, equivalent to the standard one given in several well-known monographs on the subject.*,) One must recall that the KMAL is an expression which summarizes the way in which molecular reactive collisions behave as a many-body system. It is therefore a consequence of the mechanistic model proposed for chemical reactions, so that its (2) de Groot, S . R.; Mazur, P. Non-equilibrium Thermodynamics;Dover: New York, 1984; Chapter 10. ( 3 ) Haase, R. Thermodynamics of Irreversible Processes; Addison-Wesley:
Reading, MA, 1971; Chapter 2.
0 1988 American Chemical Society
3018 The Journal of Physical Chemistry, Vol. 92, No. 10, 1988
mathematical structure must be intimately related to the nature of such a model. That this is so is clearly borne out in the many attempts that have been made in the past to derive this law from microscopic Hence, to assert that it may be obtained from purely thermodynamical reasoning is as futile as asserting that any constitutive equation, linear or nonlinear, may be derived solely from thermodynamic premises. Obvious examples are Fourier’s heat conduction equation, Fick’s equation for diffusion, and many others. On the other hand, in the derivation of the KM-AL using the full formalism of EIT recently given by Garcia-Colin et al.,1° and whose content gave rise to Lebon et a1.k work, this feature is clearly pointed out. Indeed, it is shown in that paper how from purely thermodynamic arguments only a very general relationship may be obtained for the chemical affinity A , the chemical flux J, and the temperature T (see eq 3.16 of ref 10). A close examination of this relationship shows immediately that it is of the same nature as the starting assumption of Lebon (eq 3.5, ref 1 ) . Furthermore, it is also emphasized in ref 10, that in order to extract from such a general expression the standard form of the KMAL, an additional assumption of a mechanistic nature is required. This is precisely the basis for the proposal of a nonnegative entropy-like production contained in the, certainly nonunique, form of eq 4.1 of ref 10. But the physical context of this assumption, granting of course that the entropy-like production of EIT is an extension of the ordinary nonnegative entropy production of linear irreversible thermodynamics,” is that such a requirement is directly connected to the mechanistic behavior of the system, namely, molecular collision^.^^^^* Thus, the well-known expression for KMAL cannot be obtained from thermodynamic reasoning alone. In ref 1 the authors indeed fail to obtain the KMAL, as may be readily seen by a close examination of their procedure. The main assumption is expressed by eq 3.5 stating that where is the degree of advancement of a homogeneous.chemical reaction measured with respect to its equilibrium value, = &$/at, and $( T,p,E) is an unknown function of temperature, pressure, and ,$ subject to certain conditions that are unnecessary for our argument. Notice that eq 1 is an assumption already containing the KMAL. In a second step, using the well-known relationship for the chemical potential of each chemical species involved in the reaction expressed in terms of the fugacity, as well as the definition of the chemical affinity A , and assuming that these expressions remain valid far away from the equilibrium state, one readily obtains an expression for the affinity in terms of E and other quantities (see eq 3.1 1 of ref 1). This expression, which is essentially in the form of the logarithm of a polynomial in $, is used to reexpress it in such a way that can be factored out conveniently. This is achieved by a well-defined sequence of algebraic steps which lead to an equation of the form
Comments Here, Y,, C:, and fa are the stoichiometric coefficient, equilibrium concentration, and fugacity of the species LY and K ( T , p ) is the equilibrium constant. The only reason for writing this clumsy expression for M( T,p,z) explicitly will become clear shortly. The third trivial step is to combine now eq 1 and 2 to obtain that (4)
Equation 4 is what Lebon et al. call the “the standard mass action law”. But being nothing else than eq 1 in which $ has been replaced by eq 2 and $( T,p,$)still remaining an undetermined function implies that (1) is the standard mass action law. Nothing has been proved yet. In order to identify eq 4 with the standard KMAL requires, as the reader may have guessed already, an additional assumption, and I will now show that it must be of a mechanistic nature. Since eq 4 has to be consistent with the principles of linear irreversible thermodynamics,2s8close to equilibrium the chemical flux E has to be proportional to the thermodynamic force A , so we may write, as usual, that
where I,, is the chemical transport coefficient. Since eq 5 is valid under the condition that A / R T < 1 , it follows also from eq 4 that
Therefore, compatibility among the two expressions requires that the so far arbitrary function $( T,p,$) must obey the following condition, namely $ ( T , P , ~= ) M(T,p,E)RI,, (7) On the other hand, in order that eq 5, which is strictly phenomenological in nature, be consistent with the ordinary mechanistic form of Guldberg and Waages’ form for the KMAL, it is required
n,
where kf stands for the forward rate constant and the is taken over the reactants. Therefore, eq 7 and 8 are at grips only when $(T,p,{) is so chosen that when divided by the clumsy expression for M , yields the correct value of lrr. In that case -
& = k f na ( C , ” f , ) u ~ ( e - A /-R1)T
(9)
which is the thermodynamic version of the standard KMAL.2.3 I wish to emphasize that in order to derive eq 9, the mechanistic condition expressed by eq 8 is required. The results of ref 1 are, at best, equivalent to those already given in the l i t e r a t ~ r e . ~ . ~ In short, this argument clearly shows that in order to recover the standard form of the KMAL as stated by Guldberg and Waage, an additional assumption of a mechanistic nature is required. Thermodynamics alone cannot lead to mechanistic results. ‘Also at El Colegio Nacional
(4) Greene, E. F.; Kuppermann, A. J . Chem. Edur. 1968, 45, 361. (5) Eliason, M.;Hirschfelder, J. 0. J . Chem. Phys. 1959, 30, 1426. (6) Yamamoto, T.J . Chem. Phys. 1960, 33, 261. (7) Ross, J.; Mazur, P. J . Chep. Phys. 1961, 35, 19. (8) Elerrondo, M.; Robles-Dominguez, J. A.; Garcia-Colin, L. S. J . Chem. Phys. 1976, 65, 1927. (9) For an excellent review of the subject, see: Laidler, K. J. Theories of Chemical Reacfion Rates; McGraw-Hill: New York. 1969; Chapters 3 and 8. (10) GarGa-CoIh, L. S . ; de la Selva, S. M. T.;Pifia, E. J . Phys. Chem. 1986, 90,953. (1 1) Garcia-Colin, L.S.; de la Selva, S. M. T.,submitted for publication in J . Phys. Chem.
Department of Physics Uniuersidad Autonoma Metropolitana- Iztapalapa P.O. Box 55-534. 09340 Mexico, D.F. Instituto de Investigaciones en Materiales Uniuersidad Nacional Autonoma de Mexico Circuit0 Exterior CU, 04510 Mexico, D.F.
L. S . Garcia-Colin+
Received: August 31, 1987: I n Final Form: February 10, 1988
