9900
J. Phys. Chem. 1994, 98, 9900-9902
Electrochemical Experiments on Thermodynamics at Nonequilibrium Steady States A. Hjelmfelt and J. ROSS* Department of Chemistry, Stanford University, Stanford, Califomia 94305 Received: June 11, 1994; In Final Form: July 18, 1994@
We present experiments which test for a nonequilibrium component to the electromotive force (emf) for a half-reaction involving species generated by a nonlinear reaction in a nonequilibrium steady state. The existence of this effect has been predicted by Keizer. The minimal bromate oscillatory reaction is run in a continuousflow stirred tank reactor under conditions which result in bistability. The reaction forms Ce(1V) from Ce(II1) in a nonequilibrium steady state. This redox couple generates a voltage at a Pt electrode, relative to a reference electrode. Knowing the concentrations of the cerium species, this potential can be compared with that given by those concentrations of Ce(II1) and Ce(1V) under equilibrium conditions, the Nemst equation. In the nonequilibrium stationary state we find deviations. We search for the source of these deviations. We do not find evidence for such explanations as mixed potentials but are unable to entirely rule them out. In the absence of such effects, the results are consistent with the existence of a non-Nemstian component of the emf for nonlinear systems in nonequilibrium steady states.
Introduction The thermodynamics of systems at equilibrium was established by the tum of the century by Gibbs, Boltzmann, Nemst, and others. Less well understood are the thermodynamics of systems in nonequilibrium states. The concept of local eq~ilibriuml-~ is often used to generalize equilibrium thermodynamics to nonequilibrium systems. Local equilibrium assumes that all intemal degrees of freedom have relaxed to their equilibrium distributions resulting in a well-defined temperature. Essentially, only the degrees of freedom which are explicitly held away from equilibrium do not have equilibrium distributions. Thus, with that assumption the functional form of the equations of equilibrium thermodynamics are valid away from equilibrium. Other theories of nonequilibrium thermodynamics do not use the local equilibrium hypothesis as a starting p ~ i n t . ~ - 'The ~ existence of deviations of thermodynamic functions from their local equilibrium values has been predicted by Keizer,5-8 and we present experiments which test for these deviations. Despite the existence of a large body of theoretical work, very few experimental measurements of thermodynamic properties of nonequilibrium systems have been made.5J1-14 ManY of the experimental techniques applied to equilibrium systems are not readily applied to open systems. Electrochemical techniques, however, have been applied to reactive systems.5 When chemical species come into equilibrium with an electrode in an open circuit, the potential between this electrode and a reference electrode is related to the chemical potential difference of the half-reaction occurring at the electrode. If no other reactions are occurring, then this potential is also related to the Gibbs free energy difference of the half-reaction at the electrode. If other reactions are occurring, then the species may be in nonequilibrium states, even though they are in equilibrium with the electrode, and the potential of the electrode is that of a nonequilibrium steady state. If local equilibrium holds, then this potential is just the classical Gibbs free energy difference. If local equilibrium does not hold, then deviations from the Gibbs free energy may occur: AG = AG, AG,,, and emf = emfl, emf,,,. To our knowledge, only one experiment has been performed to test the adequacy of the local equilibrium hypothesis in
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Abstract published in Advance ACS Abstracts, September 1, 1994.
stationary states. Keizer and Changs measured the potential at a Pt electrode due to Fe(II)/Fe(III) for two conditions: in equilibrium in which there is no reaction involving the iron species occurring in the bulk solution; and nonequilibriumwhen there is a nonequilibrium steady state where Fe(II1) is formed by the oxidation of Fe(I1) by persulfate in the bulk solution. For given concentrations of both Fe(I1) and Fe(III), they find differences between the two conditions that they interpret as evidence for a nonequilibrium component to the chemical potential. The kinetics of the reaction used by Keizer and Chang is first order in Fe(I1) and persulfate. We perform similar experiments using an autocatalytic reaction system, the minimal bromate ~ s c i l l a t o r . ' ~We ~ ' ~ use this system in the bistable regime, and thus its kinetics are essentially third order. The minimal oscillator is the inorganic skeleton of the BelousovZhabotinskii oscillator and is one of the mechanistically best understood oscillatory systems. The net reaction is the oxidation of Ce(II1) to Ce(1V) by bromate. The NFT" mechanism of this reaction has seven elementary steps containing a variety of bromate derived species: Br-, Br2, HBrO2, HOBr, BrO2 radical, cerium as either Ce(1II) or Ce(1V) ions. Although it does not show the range of behavior of the B-Z reaction, a small oscillatory and a very large bistable region exist in constraint space. We use the system in the bistable regime. A state where virtually no reaction occurs coexists with a state where a percentage of the Ce(II1) is oxidized to Ce(1V). We simultaneously measure the optical density which gives concentrations of Ce(1V) by Beer's law and Ce(II1) by a conservation constraint and the emf of a Pt electrode which at equilibrium follows the Nemst equation. In the previous work5 these measurements were made sequentially. Under nonequilibrium conditions we find deviations from the local equilibrium predictions. We search for the source of such deviations. Although we are not able to rule out entirely such explanations as mixed potentials, the tests yield results consistent with the existence of a non-Nemstian component to the emf due to nonequilibrium.
Experimental Section The basis of our experiments is the measurement of the emf of the Ce(III)/Ce(IV) half-reaction at a redox electrode under equilibrium and stationary nonequilibrium conditions. To isolate the source of any deviations between the results for the two
0022-3654/94/2098-9900$04.50/0 0 1994 American Chemical Society
J. Phys. Chem., Vol. 98, No. 39, 1994 9901
Thermodynamics at Nonequilibrium Steady States experiments, measurements are made such that conditions are as similar as possible under both sets of conditions: only the compositions of the feed streams are different for equilibrium and nonequilibrium conditions. Under equilibrium conditions, the feed streams deliver a nonreactive mixture of Ce(1V) and Ce(II1). Under nonequilibrium conditions the feed streams deliver the reagents for the minimal bromate reaction in which Ce(III) is partially converted to Ce(IV) to attain a nonequilibrium steady state. The contents of three reservoirs are pumped by a threechannel peristaltic pump into a 12 mL quartz spectrophotometric cell, and the residence time is 200 s. The contents of the cell are stirred at 1000 rpm to ensure homogeneity, and an aspirator removes the excess solution from the cell. We use a combination of Pt-Ag/AgCl redox microelectrode to measure the potential developed due to the chemical potential difference of Ce(1V) and Ce(II1). The potential was measured with a Fisher Accumet pH meter which reads to 0.1 mV. Because a Pt electrode is nonspecific, we performed experiments described in the next section to determine if other species are active at this electrode. The concentration of Ce(1V) was measured at 400 nm with a UV/vis spectrometer; Ce(III), bromate, bromide, and sulfuric acid do not absorb at this wavelength. In all the experiments, reservoir 1 contains Ce2(S04)3*8Hz0 and (NH&Ce( S04)4*2H20such that the total concentration of cerium ions is 0.004 500 M (in the reactor, the total concentration of cerium ions is 0.001 500 M due to the other inflows). For the minimal bromate system reservoir 2 contains 1.00 x M NaBr and reservoir 3 contains 0.0100 M NaBrOs. For the equilibrium experiments reservoirs 2 and 3 contain a sulfuric acid buffer. All the solutions contain sulfuric acid at 0.72 M. Apart from being necessary for the minimal bromate reaction, this concentration of sulfuric acid ensures that the ionic strengths of all of the solutions are virtually identical since it is the major component. In a typical experiment, six different calibration points are taken at equilibrium to verify consistency with the Nemst equation and Beer’s law. This involves using six different Ce(III)/Ce(IV) solutions in reservoir 1, while reservoirs 2 and 3 contain sulfuric acid buffers. Once a calibration is established, the minimal bromate system is used. We start this system on the flow branch which is characterized by a very low Ce(1V) concentration and low emf. Unfortunately, the Ce(IV) concentration is too low for analytical work, so measurements are made only on the thermodynamic branch. The thermodynamic branch is obtained by tuming off the pump for a few minutes. Once this branch is achieved, the pumping is resumed, and after the system stabilizes the absorption and emf measurements are taken. The absorption measurement allows us to determine the Ce(1V) concentration, and by the conservation constraint we thus know the Ce(II1) concentration. Knowing these values, the Nemst equation gives the voltage that would be measured if the system were at equilibrium. This calculated voltage can be compared with the measured voltage that occurs under nonequilibrium conditions. If a difference is found, it then becomes necessary to search for the source of such deviations. The experiment is concluded by remeasuring one or two calibration points to estimate the size of any experimental drift.
Experimental Results The results of calibrations at equilibrium are shown in Figure 1 for one of the experiments. Six different solutions were used to establish the calibration lines, and two of the measurements were repeated after the minimal bromate system was measured. The empirical values for Beer’s law and the Nernst equation obtained from the data in Figure 1 are
’240
I
1160 1140
[ c e (b ) ]
Figure 1. Typical calibration lines obtained under equilibrium conditions. The linearity verifies the applicability of (a) the Nemst equation and (b) Beer’s law. The empirical values of the slopes, intercepts, and correlation coefficients for this data set are given by eqs 1 and 2. The data in Figure 1 was obtained at the same time as the experiments as that in Table 1.
emf = 1231.9
+ 25.37 ln([Ce(IV)]/[Ce(III)]) r = 0.9989 (1)
ABS = 0.01 19
+ 1674[Ce(IV)]
r = 0.9998
(2)
where r is the correlation coefficient. The theoretical value of the slope in the Nemst equation is 25.7 mV. The measurements under nonequilibrium conditions are made on the thermodynamic branch of the minimal bromate oscillator in its bistable regime. The concentration of Ce(1V) and the emf on this branch were measured for several different feed stream concentrations of Ce(II1) (and thus Ce(IV)), and the results are summarized in Table 1. The typical minimal bromate recipe includes no Ce(1V) in the feed stream, and during the reaction, Ce(II1) is oxidized to Ce(IV), a net product. As we add more Ce(1V) to the feed stream (and remove more Ce(II1) from the feed stream to keep the total cerium ion concentration constant), the extent of reaction decreases. The NFT mechanism15 contains two mechanistic steps involving cerium ions. Simulations of this mechanism indicate that the rate of one of these steps (reaction 6 in ref 15) is always very small. The rate of the other step (reaction 5 in ref 15) decreases as the percentage of Ce(II1) in the feed stream decreases, and thus the extent of the reaction of cerium ions also decreases. Table 1 shows that deviations exist between the measured and calculated emfs. The deviation decreases as the concentration of Ce(II1) in the feed stream decreases (i.e., as the extent of reaction decreases). The maximum deviation is on the order of +10 mV and decreases to 0 mV as the extent of reaction decreases. At even lower concentrations of Ce(II1) in the feed stream, negative deviations are not found; the deviation is nearly zero. The extent of the reverse reaction of Ce(IV) to Ce(II1) is always very small, even when the feed stream is composed mostly of Ce(1V). The size of the maximum deviation can be
9902 J. Phys. Chem., Vol. 98, No. 39, 1994 TABLE 1: Typical Results from a Minimal Bromate Experiment with Various Concentrations of Ce(II1) in the Combined Feed Streams, [Ce(III)lo [Ce(IIIIlo(MI 1.500 x 1.397 x 1.297 x 8.333 x 10-4
[Ce(III)lsS (MI 1.393 x 1.361 x 1.277 x 8.700 x 10-4
IE emf (mV)
M emf (mV)
1167.0 1176.0 1187.5 1223.7
1180.2 1183.0 1189.7 1223.8
The concentration of Ce(II1) in the steady state, [Ce(III)lss,is calculated using Beer's law, eq 1, and the conservation constraint. The local equilibrium emf, LE emf, is calculated from [Ce(III)lSsand [Ce(IV)lSsusing the Nemst equation, eq 2. The measured emf, M emf, is that actually measured by the Pt electrode.
2.0
,
I
lj
- 4
j
:0
compared with the spacing between the two stable steady states: the maximum deviation is about +10 mV, while the spacing is about -100 mV. Also, the deviation can be compared with RTIF which is about 26 mV. The next set of experiments are designed to help determine the origin of the observed deviations. If any other species were absorbing at 400 nm it would lead to an overestimate of the Ce(1V) concentration, and this would mean that the calculated emf would be greater than the measured emf. Thus, we can disregard this possibility. Another explanation is that cerium is present in a form that is inert at the electrode so that the conservation constraint used in calculating the Ce(II1) concentration is invalid. On the basis of the NFI' mechanism,15 this possibility is doubtful, and to bring the emf's into agreement, approximately 25% of the cerium ions would have to be inert at the electrode. Another possibility is that electrocatalysis is occurring at the electrode and that a mixed potential is measured. Although, there must be a redox couple which is more oxidizing (Le., has a higher potential) than Ce(III)/Ce(IV) in order to oxidize Ce(II1) in the CSTR, we find no indication that this other couple is involved in electrocatalysis. Although we are unable to remove entirely the possibility of a mixed potential, the following experiments do not support this explanation of the deviation, thus raising the question of the existence of a nonequilibrium component of the chemical potential. To investigate the possibility that the mixed potential is due to bromate or bromide, we deliver them either alone or together into the reactor while the other feed stream(s) deliver the sulfuric acid buffer. In all cases the potentials are rather unstable and always less than the steady-state potential. The bromide potential is over 100 mV less, the bromidehromate potential is about 50 mV less, and the bromate potential is about 10-20 mV less than the steady-state potential of the thermodynamic branch. Another experiment to investigate the effects of bromate and bromide on the emf is to run the minimal bromate experiment, as described previously, but to keep the system on the flow branch as we increase the Ce(1V) concentration in the cerium feed. As long as the Ce(1V) concentration in the feed stream is sufficiently small, there is essentially no reaction of Ce(II1) to form Ce(1V) on that branch. If the Ce(III)/Ce(IV) mixture is fed into the reactor with the NaBr solution, then the potential always drops by about 5 mV with no noticeable change in the concentrations of the cerium species. If the Ce(III)/Ce(IV) solution is fed with either bromate or bromate and bromide, then the potential varies by less than 1 mV, usually falling by a few tenths of a millivolt, again with no noticeable change in the concentrations of the cerium species. In some experiments the system jumps spontaneously to the high Ce(1V) state after an induction period. During the induction period there is virtually no conversion of Ce(II1) to Ce(1V) and the emf is essentially that of the equilibrium mixture. To examine the activity of various species at the surface of the electrode we used cyclic voltametry of the NaBr (3.33 x
Hjelmfelt and Ross
1
1
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e c \/G:age { ' / )
'.3
1.4
1 5
Figure 2. Cyclic voltamagrams obtained from the reagents of the minimal bromate system and the equilibrium composition of the minimal bromate system. The scan rate was 5 mVls. The increase in current at high potentials is due to the activity of water. lo-' M), NaBrO3 (0.003 33 M), and Ce(III)/Ce(IV) (0.0015 M), and the equilibrium mixture that results from the minimal bromate wastes. The results are shown in Figure 2. At the concentrations used in the experiment the bromide and bromate reagents are indistinguishable from each other and from the sulfuric acid media. The Ce(III)/Ce(IV) voltamagram and the voltamagram of the equilibrium mixture are quite similar implying that there is no other redox couple that is active at the electrode at this potential. Conclusion In this paper we have presented measurements which imply a possible nonequilibrium component of a half-cell emf. Although we are unable to eliminate all other possible sources of the observed deviations, the results are consistent with the existence of nonequilibrium effects. The results raise intriguing possibilities, and we believe further experiments are well warranted. Despite the difficulties encountered, electrochemical measurements are one of the few methods available for determining thermodynamic properties of species in solution under nonequilibrium conditions. The use of ion-specific electrodes would eliminate the possibility of measuring mixed potentials and thus are an improvement over a nonspecific Ptredox electrode, as we have used. Of course, the concentration of the ion must be measurable by an independent method not based on its thermodynamic properties. Acknowledgment. We thank Profs. Paul Hunt, Katherine L. C. Hunt, Christopher E. D. Chidsy, and Dr. Christopher Doona for helpful discussions. This work was supported in part by the Department of Energymasic Energy Sciences. References and Notes (1) Ross, J.; Mazur, P. J . Chem. Phys. 1961, 35, 19. (2) Prigogine, I. Physica 1944, 15, 271. (3) Prigogine, I.; Xhrouet, E. Physica 1949, 15, 913. (4) Glansdorff, P.; Prigogine, I. Thermodynamic Theory of Structure, Stabiliry, and Fluctuations; Wiley-Interscience: London, 1971. ( 5 ) Keizer, J.; Chang, 0. J . Chem. Phys. 1987, 87, 4064. (6) Keizer, J. J . Chem. Phys. 1987, 87, 4074. (7) Keizer, J. J . Phys. Chem. 1989, 93, 6939. (8) Keizer, J. J . Chem. Phys. 1985, 82, 2751. (9) Mou, C.; Luo, J.; Nicolis, G. J . Chem. Phys. 1986, 84, 7011. (10) Li, R . 4 . J . Chem. Phys. 1989, 90, 899. (11) Roelofs, M. G. J. Chem. Phys. 1988, 88, 5516. (12) Lamprecht, I.; Schaarschmidt, B. Thermochim. Acta 1978,22,257. (13) Koros, E.; Orban, M.; Nagy, Zs. Acta Chim. Acad. Sci. Hung. 1979, 100, 449. (14) Koros, E.; Orban, M.; Nagy, Zs. J . Phys. Chem. 1973, 77, 3122. (15) Geiseler, W.; Bar-Eli, K. J . Phys. Chem. 1981, 85, 908. (16) Geiseler, W. Ber. Bunsen-Ges. Phys. Chem. 1982, 86, 721. (17) Noyes, R. M.; Field, R. J.; Thompson, R. C. J . Am. Chem. SOC. 1971, 93, 7315.
