- Email: [email protected]
Muonium kinetics in sub- and supercritical water
Physica B 326 (2003) 55–60
Muonium kinetics in sub- and supercritical water K. Ghandi, B. Addison-Jones, J.C. Brodovitch, S. Kecman, I. McKenzie, P.W. Percival* Department of Chemistry and TRIUMF, Simon Fraser University, Burnaby, B.C. Canada V5A 1S6
Abstract Muonium is long-lived in pure water and has been studied over a very wide range of temperatures and pressures, from 51C to over 4001C and from 1 to 400 bar. We have determined rate constants for representative reactions of muonium in aqueous solution; equivalent data on H atom kinetics is sparse and stops well short of the maximum temperature and pressure attained in our experiments. The results show remarkable deviations from the predictions of standard reaction theories. In particular, rate constants pass through a maximum with temperature well below the critical point. This seems to be a general phenomenon, since we have observed it for spin-exchange and chemical reactions that are diffusion limited at low temperatures, as well as for activated reactions. We believe that a key factor in the drop of rate constants at high temperature is the cage effect, in particular the number of collisions between a pair of reactants over the duration of their encounter. Whatever the reason, the implications are profound for both the efficiency of supercritical water oxidation reactors and for the modelling of radiation chemistry in pressurized water nuclear reactors. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Muonium; Supercritical water; Chemical kinetics; Radiation chemistry
1. Introduction During the last decade, there has been substantial interest in the chemistry of near critical water (NCW) and supercritical water (SCW). Two important applications are hazardous waste destruction by supercritical water oxidation (SCWO) [1–3] and NCW or SCW for cooling and power generation in nuclear power plants [4,5]. Study of the reaction kinetics of transients in hot, pressurized water is essential for the design and optimization of future SCWO reactors [2,3] as well as the next generation of water-cooled nuclear reactors *Corresponding author. Tel.:+1-604-291-4477; fax: +1-604291-3765. E-mail address: [email protected] (P.W. Percival).
[5]. However, data on key transients, such as the H atom, is sparse and stops well short of the maximum temperature attained in experiments on Mu chemistry [6–9] (the critical point of water is 3741C and 220 bar). Due to the lack of experimental data, modelling of radiation chemistry in SCW has been based on extrapolation of low-temperature data [4,10,11] using the predictions of standard reaction rate theories. One aspect of our studies is to provide experimental data relevant to SCWO processes and the radiolysis of water in pressurized water nuclear reactors. Such data can be used to test the validity of models based on the extrapolation of rate coefficients. Another aspect of our studies is to investigate fundamental questions on chemical kinetics, e.g.: How does the solvent’s thermodynamic state affect
0921-4526/03/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 2 ) 0 1 5 7 2 - 7
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K. Ghandi et al. / Physica B 326 (2003) 55–60
chemical dynamics? The physicochemical properties of NCW and SCW change dramatically with temperature and pressure [1]. In particular, the extent of hydrogen bonding is reduced at higher temperatures and lower density. Molecular dynamics studies suggest significant changes in the interactions between water molecules and point to the possible disappearance of the solvent cage above B3301C [12]. Our strategy in studying Mu kinetics has been to factorize different effects. To probe transport and cage effects it was necessary to study a chemical system without an activation barrier, i.e. a diffusion-limited process. Spin exchange (SE) between Mu and a paramagnetic species is such a system, so we studied Mu kinetics in solutions of Ni2+ [9]. Next, we studied different kinds of chemical reactions with activation barriers. In this paper, we first review our results for diffusion-controlled reactions of muonium, and then present new data on the temperature dependence of rate constants for activation-controlled reactions.
2. Experimental procedure The experimental details have been described elsewhere [6,9]. Mu decay rates were measured at selected temperatures and pressures for oxygenfree deionised water and for solutions of one or two concentrations. The decay rates were corrected for the damping of the Mu signal in pure water, and then divided by the effective concentration of the solute (taking account of the density [9]) to give the second-order rate constant, kMu : The temperature and pressure conditions were taken in an irregular sequence to guard against possible systematic error. The uncertainty in temperature depends on the temperature gradient in the reaction vessel and varies from B21C below 3001C to as much as 61C at the highest temperatures. The pressure uncertainty is estimated to be 72 bar. In the worse case, close to the critical point, the uncertainty in temperature and pressure could result in an error in density and thence solute concentration of B30%. This translates directly into uncertainty in the calculated rate constant. Below 3001C the estimated errors in
temperature and pressure have negligible effect on the density, and the statistical error in the muonium decay rate is the dominant source of uncertainty. The error bars in the figures of this article are solely due to this latter source of uncertainty.
3. Results and discussion Our study of spin exchange between Mu and Ni2+ showed that over a low-temperature range the rate constant increases with temperature [9], consistent with expectations for a diffusion-controlled process. The second-order rate constant for a diffusion-controlled bimolecular reaction can be predicted by the Smoluchowski equation: kdiff ¼ 4000p ðD1 þ D2 Þ Reff NAv ;
ð1Þ
where D1 and D2 are the diffusion constants of the reactants and Reff is the effective distance at which they react. The Avogadro constant, NAv ; and the factor of 1000 are needed to express the rate constant in conventional chemical kinetics units, M1 s1. The Stokes–Einstein equation usually gives reasonable estimates for diffusion constants in NCW and SCW [13]: D¼
kB T ; 4pZR0
ð2Þ
where kB is the Boltzmann constant, T the temperature, Z the solvent viscosity and R0 the hydrodynamic radius of the diffusing species. It is the solvent viscosity which provides the principal source of temperature dependence to D and hence to kdiff : At high temperatures, our results revealed an unexpected decrease of the rate constant with temperature. It was already known that the rate constants of some reactions important in the radiolysis of water fall below the predictions of the Stokes–Einstein–Smoluchowski (SES) equations at high temperatures [10]. Buxton and Elliot rationalized this behaviour by means of the Noyes equation [14]: 1=kobs ¼ 1=kdiff þ 1=kreact ;
ð3Þ
K. Ghandi et al. / Physica B 326 (2003) 55–60
kreact ¼ A exp ðEa =RTÞ:
ð4Þ
It is possible to fit our spin-exchange data in the manner of Buxton and Elliot (Eqs. (3) and (4)), but only with unrealistic Arrhenius parameters: A ¼5 107 M1 s1 and Ea ¼ 30 kJ mol1. The temperature dependence of the rate constant makes more sense if one considers a reduction in efficiency of spin exchange as a result of a shift from strong exchange to weak exchange [9,15]: kex ¼ Pspin kdiff fJ ;
fJ ¼ J 2 t2enc =ð1 þ J 2 t2enc Þ;
ð5Þ
where kex is the spin exchange rate constant, Pspin is a spin statistical factor [15,16], fJ is the efficiency factor, J is the strength of the exchange interaction and tenc is the encounter time. It is only for long encounter times ðJ 2 t2enc b1; the strong exchange limit) that kex ¼ Pspin kdiff : A question that arises is whether the efficiency of chemical reactions also falls with temperature, as encounter times decrease. If so, this could have important consequences for the efficiency of chemical processes in SCWO or radiolysis in pressurized water nuclear reactors. To explore this important question we studied a variety of reactions. The reaction between muonium and hydroquinone was chosen as an example of a fast reaction in which the diffusion process is expected to affect the rate of reaction. We determined rate constants at various pressures and at temperatures from 51C to 3701C, and found similar behaviour to the spin exchange investigation, i.e. the rate constant rises with temperature, goes through a maximum, and then falls [9]. The mechanism of this reaction is addition of muonium to the benzene ring of hydroquinone to form a cyclohexadienyl-type radical [17]. To exclude the possibility that the unusual temperature dependence of the kinetics is connected with the hydroxyl groups of hydroquinone we have now studied the reaction between Mu and benzene itself. This is also an addition reaction, as evident by the detection of muoniated cyclohexadienyl in
supercritical aqueous solutions of benzene [8]. Another reason to study this reaction is because it is almost an order of magnitude slower than addition to hydroquinone, so whereas the latter is near diffusion-controlled at room temperature, the reaction with benzene is clearly activation controlled. Our results for the temperature dependence of the rate constants for Mu addition to benzene are presented in Fig. 1. Once again a falloff in rate constants at high temperature is evident. Addition reactions, such as those discussed above, typically have low activation energies, so it is desirable to investigate other types of reaction to test the generality of the fall-off of rate constants at high temperatures in water. We therefore studied reactions of muonium with methanol, the formate ion and the hydroxide ion: Mu þ CH3 OH-MuH þ CH2 OH;
ð6Þ
Mu þ HCOO -MuH þ COO ;
ð7Þ
Mu þ OH -e aq þ MuOH;
ð8Þ
all of which have large activation energies. Reaction (8) is effectively a muon transfer, and is the analogue of a key reaction of H in the radiolysis of water [4,10,11]. Reactions (6) and
A
10 10 kMu / M -1s-1
where kreact represents the rate constant for the ‘activation-controlled’ reaction, i.e. the situation where the rate-limiting step is chemical reaction rather than diffusion. It is usually assumed to have an Arrhenius temperature dependence:
57
R
10 9
0
100
200 300 Temperature / oC
400
500
Fig. 1. Rate constants for reaction of Mu with benzene in aqueous solution at 25075 bar. The triangles represent data from Refs. [24] and [25]. Curve A corresponds to the Arrhenius equation: A¼1011 M1 s1 and Ea¼9.8 kJ mol1. Curve R shows the prediction of the multiple collisions model with parameters A¼1.1 1011 M1 s1, Ea ¼9.7 kJ mol1 and P ¼ 0:3:
K. Ghandi et al. / Physica B 326 (2003) 55–60
(7) are hydrogen abstraction reactions and are relevant to SCWO [2,3]. Although their mechanisms are similar, the reactants have significant differences: methanol is neutral but formate is charged. Thus, the solute–solvent interaction in the former case is mainly hydrogen bonding, while in the second case it is dominated by the ion– dipole interaction. The activation energy for reaction (7) is significantly affected by solvent water molecules [18], in contrast to (6), whose activation energy is only slightly affected by the solvent [19]. Reaction (6) has not previously been studied for Mu, since it is too slow at low temperatures. Fig. 2a shows the rate constants for Mu+ CH3OH as a function of temperature at constant pressure. Similar plots for reaction of Mu with formate and hydroxide are presented in Figs. 2b and c. It is immediately obvious that regardless of the type of reaction and reactants there is a fall-off region under subcritical conditions. The Noyes equation does not predict a decrease of rate constants with temperature unless unphysical negative activation energies are assumed [10]. Complex reaction mechanisms have been invoked to rationalize the apparent negative activation energies in the case of reactions of the hydrated electron and hydroxyl radical [20]. In view of our findings, the underlying cause of the remarkable temperature dependence of rate constants could be more general. Troe and co-workers have also reported a falloff region in their studies of the density dependence of radical recombination reactions in simple fluids. Their explanation [21,22] is based on a modified Noyes equation in which kreact has been replaced with a pressure-dependent rate constant kgas which is proportional to the inverse diffusion coefficient of the moderator gas. The justification for such a procedure is that the collision frequency scales with the inverse diffusion constant. This model works for different reactions in a variety of media over a broad range of thermodynamic conditions [21,22]. However, it was not entirely successful in the case of reactions of Mu in NCW and SCW. This motivated us to develop a ‘multiple collisions’ model, which is based on the Noyes equation and borrows concepts from Troe’s work.
10 9
(a) 10 8 10 7 10 6 10 5 10 4
(b) 10
kMu / M-1s-1
58
10
10 9 10 8 10 7 10 6
(c)
10 11 10 10 10 9 10 8 10 7 0
100
200
300
400
500
o
Temperature / C Fig. 2. Rate constants for the reaction of Mu with (a) methanol, (b) formate, and (c) hydroxide in aqueous solution at B230–260 bar. Literature data are represented by the solid curve in (b) [18] and the triangles in (c) [26]. Arrhenius temperature dependence is denoted in each plot by the dashed curve.
The model is described in detail in Ref. [9], and only briefly summarized in the following. In common with Troe, Buxton and Elliot, we employ the SES equations to calculate kdiff : However, for kreact we use a modified Arrhenius equation which incorporates a variable frequency factor: kreact ¼ fR kArr ¼ fR AeEa =RT ;
ð9Þ
where fR is an efficiency factor similar in some respects to the spin exchange efficiency factor fJ :
K. Ghandi et al. / Physica B 326 (2003) 55–60
The latter decreases at high temperature because the encounter time falls, and this is also expected to affect the efficiency of a chemical reaction. However, for chemical reaction there is an additional factor, namely the number of bimolecular collisions during an encounter. In addition, reactions involving polyatomic molecules often depend on the relative orientation of the reactants; an increase in the collision rate implies shorter rotational correlation times and increased reaction probability. We express the reaction efficiency factor as 1 1 fR ¼ Pt1 ð10Þ coll = tenc þ Ptcoll ; where P is an orientation factor which determines the probability that a collision occurs in a reactive orientation, tenc is the encounter duration and tcoll is the time between collisions. fR reflects the competition between reaction and escape of unreacted collision partners from the solvent cage. Some predictions based on Eqs. (9) and (10) are shown in Fig. 3 (curves B and C) and compared with simple Arrhenius temperature dependence (curve A). For a reaction with a moderate activation energy under standard conditions, there is Arrhenius behaviour at low temperatures but a significant drop below the Arrhenius prediction at
10 10
kMu / M -1s-1
C
10
9
10
8
A B
10 7
0
100
200 300 Temperature / oC
400
500
Fig. 3. Comparison of rate constants calculated from Eqs. (9) and (10) (curves B and C) with the prediction of the Arrhenius equation (curve A). For all three curves the pre-exponential factor was set to A ¼1.0 1011 M1 s1; for A and C, the activation energy Ea ¼20 kJ mol1, and for B, Ea ¼10 kJ mol1. A value of P ¼ 0:11 was used for curves B and C.
59
higher temperatures. In particular, the rate constant is predicted to pass through a maximum at a temperature below the critical point, consistent with the experimental findings (Figs. 1 and 2, and Ref. [9]). The predicted maximum is broader for smaller values of Ea : A specific example of the predictions of the multiple collisions model is shown by the solid curve in Fig. 1. There is reasonable agreement except in the vicinity of the critical point, where reaction efficiency is considerably lower than predicted. One source of error is in the calculation of collision and encounter times [9], since the procedures for their estimation are unlikely to be accurate close to the critical point. Another possible factor, not included in our model, is the so-called ‘critical slowing down’ of transport properties close to the critical point [23], according to which there should be a sharp drop in the rate constant as the diffusion constant approaches zero. This phenomenon is distinct from the predicted rise in the rate constant at temperatures well above the critical point. In this regime, the solvent behaves more like a dense gas than a liquid. Unfortunately, this happens close to the working limit of our apparatus. Nevertheless, the data for all reactions presented in Figs. 1 and 2 show some indication of the predicted return to positive temperature dependence. Finally, it should be pointed out that the use of a fixed activation energy is a simplistic assumption, particularly when there are strong interactions between a reactant and the solvent. For example, solvent effects are probably the reason for different observed trends for reactions of Mu with OH– and HCOO–. Although both reactions show evidence of the cage effect under near critical conditions, the reaction of formate shows a large positive deviation from the Arrhenius extrapolation (Fig. 2b) while the reaction with OH– shows a negative deviation (Fig. 2c). Lossack et al. have calculated that the dipole moment of the transition state for the reaction between H and formate in water is significantly smaller than the dipole moment of the reactants [18]. Our preliminary modelling of the reaction between H and OH– in water shows the opposite situation.
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K. Ghandi et al. / Physica B 326 (2003) 55–60
4. Conclusions We have provided experimental data relevant to the chemistry of pressurized water nuclear reactors and SCWO. Our results suggest that published models of these processes are based on invalid extrapolations of rate coefficients. In particular, the use of simple Arrhenius behaviour in the Noyes equation [10,11] seriously overestimates rate constants in high temperature water. Our modelling of kinetics in NCW and SCW can be considered as extension of the work of Troe, and our experimental results show similar behaviour: the rate constants pass through a maximum at intermediate densities (temperatures) and then fall with increase of temperature. The common trends among the different reactions reported here supports the idea that for all reactions under near critical conditions the cage effect is important, namely the efficiency of the reactions depends on the number of collisions over the duration of the encounter. However, at lower temperatures, and when there are strong interactions with the solvent, electrostatic effects may be more important.
Acknowledgements We thank Syd Kreitzman and the staff of the TRIUMF mSR Facility for technical support. This research was financially supported by the Natural Sciences and Engineering Research Council of Canada and, through TRIUMF, by the National Research Council of Canada.
References [1] R.W. Shaw, T.B. Brill, A.A. Clifford, C.A. Eckert, E.U. Franck, Chem. Eng. News 69 (51) (1991) 1. [2] A.R. Katritzky, S.M. Allin, M. Siskin, Acc. Chem. Res. 29 (1996) 399. [3] P.E. Savage, Chem. Rev. 99 (1999) 603.
[4] D.R. McCracken, K.T. Tsang, P.J. Laughton, Aspects of the physics and chemistry of water radiolysis by fast neutrons and fast electrons in nuclear reactors, AECL Report, AECL-11895, 1998. [5] R.B. Duffey, W.T. Hancox, D.F. Torgerson, Phys. Can. 56 (2000) 295. [6] P.W. Percival, J.-C. Brodovitch, K. Ghandi, B. AddisonJones, J. Schuth, . D.M. Bartels, Phys. Chem. Chem. Phys. 1 (1999) 4999. [7] K. Ghandi, J.-C. Brodovitch, B. Addison-Jones, P.W. Percival, J. Schuth, . Physica B 289–290 (2000) 476. [8] P.W. Percival, K. Ghandi, J.-C. Brodovitch, B. AddisonJones, I. McKenzie, Phys. Chem. Chem. Phys. 2 (2000) 4717. [9] K. Ghandi, B. Addison-Jones, J.-C. Brodovitch, I. McKenzie, P.W. Percival, J. Schuth, . Phys. Chem. Chem. Phys. 4 (2002) 586. [10] A.J. Elliot, D.R. McCracken, G.V. Buxton, N.D. Wood, J. Chem. Soc. Faraday Trans. 86 (1990) 1539. [11] D. Swiatla-Wojcik, G.V. Buxton, J. Chem. Soc. Faraday Trans. 94 (1998) 2135; D. Swiatla-Wojcik, G.V. Buxton, Phys. Chem. Chem. Phys. 2 (2000) 5113. [12] N. Yoshii, H. Yoshie, S. Miura, S. Okazaki, J. Chem. Phys. 109 (1998) 4873. [13] C.-Y. Liu, S.R. Snyder, A.J. Bard, J. Phys. Chem. B 101 (1997) 1180. [14] M.J. Pilling, P.W. Seakins, Reaction Kinetics, Oxford University Press, Oxford, 1996. [15] Yu.N. Molin, K.M. Salikhov, K.I. Zamaraev, Spin Exchange, Springer, Berlin, 1980. [16] M. Senba, Phys. Rev. A 52 (1995) 4599. [17] S.K. Leung, Ph.D. Thesis, Simon Fraser University, 1991. [18] A.M. Lossack, E. Roduner, D.M. Bartels, Phys. Chem. Chem. Phys. 3 (2001) 2031. [19] A.M. Lossack, E. Roduner, D.M. Bartels, J. Phys. Chem. A 102 (1998) 7462. [20] G.V. Buxton, S.R. Mackenzie, J. Chem. Soc. Faraday Trans. 88 (1992) 2833. [21] J. Troe, Ber. Bunsen-Ges. Phys. Chem. 94 (1990) 1183. [22] J. Troe, V.G. Ushakov, Faraday Discuss. 119 (2001) 145. [23] S.C. Tucker, Chem. Rev. 99 (1999) 391. [24] E. Roduner, P.W. Louwrier, G.A. Brinkman, D.M. Garner, I.D. Reid, D.J. Arseneau, M. Senba, D.G. Fleming, Ber. Bunsen-Ges. Phys. Chem. 94 (1990) 1224. [25] E. Roduner, D.M. Bartels, Ber. Bunsen-Ges. Phys. Chem. 96 (1992) 1037. [26] B.W. Ng, J.M. Stadlbauer, D.C. Walker, J. Phys. Chem. 88 (1984) 857.
Muonium kinetics in sub- and supercritical water K. Ghandi, B. Addison-Jones, J.C. Brodovitch, S. Kecman, I. McKenzie, P.W. Percival* Department of Chemistry and TRIUMF, Simon Fraser University, Burnaby, B.C. Canada V5A 1S6
Abstract Muonium is long-lived in pure water and has been studied over a very wide range of temperatures and pressures, from 51C to over 4001C and from 1 to 400 bar. We have determined rate constants for representative reactions of muonium in aqueous solution; equivalent data on H atom kinetics is sparse and stops well short of the maximum temperature and pressure attained in our experiments. The results show remarkable deviations from the predictions of standard reaction theories. In particular, rate constants pass through a maximum with temperature well below the critical point. This seems to be a general phenomenon, since we have observed it for spin-exchange and chemical reactions that are diffusion limited at low temperatures, as well as for activated reactions. We believe that a key factor in the drop of rate constants at high temperature is the cage effect, in particular the number of collisions between a pair of reactants over the duration of their encounter. Whatever the reason, the implications are profound for both the efficiency of supercritical water oxidation reactors and for the modelling of radiation chemistry in pressurized water nuclear reactors. r 2002 Elsevier Science B.V. All rights reserved. Keywords: Muonium; Supercritical water; Chemical kinetics; Radiation chemistry
1. Introduction During the last decade, there has been substantial interest in the chemistry of near critical water (NCW) and supercritical water (SCW). Two important applications are hazardous waste destruction by supercritical water oxidation (SCWO) [1–3] and NCW or SCW for cooling and power generation in nuclear power plants [4,5]. Study of the reaction kinetics of transients in hot, pressurized water is essential for the design and optimization of future SCWO reactors [2,3] as well as the next generation of water-cooled nuclear reactors *Corresponding author. Tel.:+1-604-291-4477; fax: +1-604291-3765. E-mail address: [email protected] (P.W. Percival).
[5]. However, data on key transients, such as the H atom, is sparse and stops well short of the maximum temperature attained in experiments on Mu chemistry [6–9] (the critical point of water is 3741C and 220 bar). Due to the lack of experimental data, modelling of radiation chemistry in SCW has been based on extrapolation of low-temperature data [4,10,11] using the predictions of standard reaction rate theories. One aspect of our studies is to provide experimental data relevant to SCWO processes and the radiolysis of water in pressurized water nuclear reactors. Such data can be used to test the validity of models based on the extrapolation of rate coefficients. Another aspect of our studies is to investigate fundamental questions on chemical kinetics, e.g.: How does the solvent’s thermodynamic state affect
0921-4526/03/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 0 2 ) 0 1 5 7 2 - 7
56
K. Ghandi et al. / Physica B 326 (2003) 55–60
chemical dynamics? The physicochemical properties of NCW and SCW change dramatically with temperature and pressure [1]. In particular, the extent of hydrogen bonding is reduced at higher temperatures and lower density. Molecular dynamics studies suggest significant changes in the interactions between water molecules and point to the possible disappearance of the solvent cage above B3301C [12]. Our strategy in studying Mu kinetics has been to factorize different effects. To probe transport and cage effects it was necessary to study a chemical system without an activation barrier, i.e. a diffusion-limited process. Spin exchange (SE) between Mu and a paramagnetic species is such a system, so we studied Mu kinetics in solutions of Ni2+ [9]. Next, we studied different kinds of chemical reactions with activation barriers. In this paper, we first review our results for diffusion-controlled reactions of muonium, and then present new data on the temperature dependence of rate constants for activation-controlled reactions.
2. Experimental procedure The experimental details have been described elsewhere [6,9]. Mu decay rates were measured at selected temperatures and pressures for oxygenfree deionised water and for solutions of one or two concentrations. The decay rates were corrected for the damping of the Mu signal in pure water, and then divided by the effective concentration of the solute (taking account of the density [9]) to give the second-order rate constant, kMu : The temperature and pressure conditions were taken in an irregular sequence to guard against possible systematic error. The uncertainty in temperature depends on the temperature gradient in the reaction vessel and varies from B21C below 3001C to as much as 61C at the highest temperatures. The pressure uncertainty is estimated to be 72 bar. In the worse case, close to the critical point, the uncertainty in temperature and pressure could result in an error in density and thence solute concentration of B30%. This translates directly into uncertainty in the calculated rate constant. Below 3001C the estimated errors in
temperature and pressure have negligible effect on the density, and the statistical error in the muonium decay rate is the dominant source of uncertainty. The error bars in the figures of this article are solely due to this latter source of uncertainty.
3. Results and discussion Our study of spin exchange between Mu and Ni2+ showed that over a low-temperature range the rate constant increases with temperature [9], consistent with expectations for a diffusion-controlled process. The second-order rate constant for a diffusion-controlled bimolecular reaction can be predicted by the Smoluchowski equation: kdiff ¼ 4000p ðD1 þ D2 Þ Reff NAv ;
ð1Þ
where D1 and D2 are the diffusion constants of the reactants and Reff is the effective distance at which they react. The Avogadro constant, NAv ; and the factor of 1000 are needed to express the rate constant in conventional chemical kinetics units, M1 s1. The Stokes–Einstein equation usually gives reasonable estimates for diffusion constants in NCW and SCW [13]: D¼
kB T ; 4pZR0
ð2Þ
where kB is the Boltzmann constant, T the temperature, Z the solvent viscosity and R0 the hydrodynamic radius of the diffusing species. It is the solvent viscosity which provides the principal source of temperature dependence to D and hence to kdiff : At high temperatures, our results revealed an unexpected decrease of the rate constant with temperature. It was already known that the rate constants of some reactions important in the radiolysis of water fall below the predictions of the Stokes–Einstein–Smoluchowski (SES) equations at high temperatures [10]. Buxton and Elliot rationalized this behaviour by means of the Noyes equation [14]: 1=kobs ¼ 1=kdiff þ 1=kreact ;
ð3Þ
K. Ghandi et al. / Physica B 326 (2003) 55–60
kreact ¼ A exp ðEa =RTÞ:
ð4Þ
It is possible to fit our spin-exchange data in the manner of Buxton and Elliot (Eqs. (3) and (4)), but only with unrealistic Arrhenius parameters: A ¼5 107 M1 s1 and Ea ¼ 30 kJ mol1. The temperature dependence of the rate constant makes more sense if one considers a reduction in efficiency of spin exchange as a result of a shift from strong exchange to weak exchange [9,15]: kex ¼ Pspin kdiff fJ ;
fJ ¼ J 2 t2enc =ð1 þ J 2 t2enc Þ;
ð5Þ
where kex is the spin exchange rate constant, Pspin is a spin statistical factor [15,16], fJ is the efficiency factor, J is the strength of the exchange interaction and tenc is the encounter time. It is only for long encounter times ðJ 2 t2enc b1; the strong exchange limit) that kex ¼ Pspin kdiff : A question that arises is whether the efficiency of chemical reactions also falls with temperature, as encounter times decrease. If so, this could have important consequences for the efficiency of chemical processes in SCWO or radiolysis in pressurized water nuclear reactors. To explore this important question we studied a variety of reactions. The reaction between muonium and hydroquinone was chosen as an example of a fast reaction in which the diffusion process is expected to affect the rate of reaction. We determined rate constants at various pressures and at temperatures from 51C to 3701C, and found similar behaviour to the spin exchange investigation, i.e. the rate constant rises with temperature, goes through a maximum, and then falls [9]. The mechanism of this reaction is addition of muonium to the benzene ring of hydroquinone to form a cyclohexadienyl-type radical [17]. To exclude the possibility that the unusual temperature dependence of the kinetics is connected with the hydroxyl groups of hydroquinone we have now studied the reaction between Mu and benzene itself. This is also an addition reaction, as evident by the detection of muoniated cyclohexadienyl in
supercritical aqueous solutions of benzene [8]. Another reason to study this reaction is because it is almost an order of magnitude slower than addition to hydroquinone, so whereas the latter is near diffusion-controlled at room temperature, the reaction with benzene is clearly activation controlled. Our results for the temperature dependence of the rate constants for Mu addition to benzene are presented in Fig. 1. Once again a falloff in rate constants at high temperature is evident. Addition reactions, such as those discussed above, typically have low activation energies, so it is desirable to investigate other types of reaction to test the generality of the fall-off of rate constants at high temperatures in water. We therefore studied reactions of muonium with methanol, the formate ion and the hydroxide ion: Mu þ CH3 OH-MuH þ CH2 OH;
ð6Þ
Mu þ HCOO -MuH þ COO ;
ð7Þ
Mu þ OH -e aq þ MuOH;
ð8Þ
all of which have large activation energies. Reaction (8) is effectively a muon transfer, and is the analogue of a key reaction of H in the radiolysis of water [4,10,11]. Reactions (6) and
A
10 10 kMu / M -1s-1
where kreact represents the rate constant for the ‘activation-controlled’ reaction, i.e. the situation where the rate-limiting step is chemical reaction rather than diffusion. It is usually assumed to have an Arrhenius temperature dependence:
57
R
10 9
0
100
200 300 Temperature / oC
400
500
Fig. 1. Rate constants for reaction of Mu with benzene in aqueous solution at 25075 bar. The triangles represent data from Refs. [24] and [25]. Curve A corresponds to the Arrhenius equation: A¼1011 M1 s1 and Ea¼9.8 kJ mol1. Curve R shows the prediction of the multiple collisions model with parameters A¼1.1 1011 M1 s1, Ea ¼9.7 kJ mol1 and P ¼ 0:3:
K. Ghandi et al. / Physica B 326 (2003) 55–60
(7) are hydrogen abstraction reactions and are relevant to SCWO [2,3]. Although their mechanisms are similar, the reactants have significant differences: methanol is neutral but formate is charged. Thus, the solute–solvent interaction in the former case is mainly hydrogen bonding, while in the second case it is dominated by the ion– dipole interaction. The activation energy for reaction (7) is significantly affected by solvent water molecules [18], in contrast to (6), whose activation energy is only slightly affected by the solvent [19]. Reaction (6) has not previously been studied for Mu, since it is too slow at low temperatures. Fig. 2a shows the rate constants for Mu+ CH3OH as a function of temperature at constant pressure. Similar plots for reaction of Mu with formate and hydroxide are presented in Figs. 2b and c. It is immediately obvious that regardless of the type of reaction and reactants there is a fall-off region under subcritical conditions. The Noyes equation does not predict a decrease of rate constants with temperature unless unphysical negative activation energies are assumed [10]. Complex reaction mechanisms have been invoked to rationalize the apparent negative activation energies in the case of reactions of the hydrated electron and hydroxyl radical [20]. In view of our findings, the underlying cause of the remarkable temperature dependence of rate constants could be more general. Troe and co-workers have also reported a falloff region in their studies of the density dependence of radical recombination reactions in simple fluids. Their explanation [21,22] is based on a modified Noyes equation in which kreact has been replaced with a pressure-dependent rate constant kgas which is proportional to the inverse diffusion coefficient of the moderator gas. The justification for such a procedure is that the collision frequency scales with the inverse diffusion constant. This model works for different reactions in a variety of media over a broad range of thermodynamic conditions [21,22]. However, it was not entirely successful in the case of reactions of Mu in NCW and SCW. This motivated us to develop a ‘multiple collisions’ model, which is based on the Noyes equation and borrows concepts from Troe’s work.
10 9
(a) 10 8 10 7 10 6 10 5 10 4
(b) 10
kMu / M-1s-1
58
10
10 9 10 8 10 7 10 6
(c)
10 11 10 10 10 9 10 8 10 7 0
100
200
300
400
500
o
Temperature / C Fig. 2. Rate constants for the reaction of Mu with (a) methanol, (b) formate, and (c) hydroxide in aqueous solution at B230–260 bar. Literature data are represented by the solid curve in (b) [18] and the triangles in (c) [26]. Arrhenius temperature dependence is denoted in each plot by the dashed curve.
The model is described in detail in Ref. [9], and only briefly summarized in the following. In common with Troe, Buxton and Elliot, we employ the SES equations to calculate kdiff : However, for kreact we use a modified Arrhenius equation which incorporates a variable frequency factor: kreact ¼ fR kArr ¼ fR AeEa =RT ;
ð9Þ
where fR is an efficiency factor similar in some respects to the spin exchange efficiency factor fJ :
K. Ghandi et al. / Physica B 326 (2003) 55–60
The latter decreases at high temperature because the encounter time falls, and this is also expected to affect the efficiency of a chemical reaction. However, for chemical reaction there is an additional factor, namely the number of bimolecular collisions during an encounter. In addition, reactions involving polyatomic molecules often depend on the relative orientation of the reactants; an increase in the collision rate implies shorter rotational correlation times and increased reaction probability. We express the reaction efficiency factor as 1 1 fR ¼ Pt1 ð10Þ coll = tenc þ Ptcoll ; where P is an orientation factor which determines the probability that a collision occurs in a reactive orientation, tenc is the encounter duration and tcoll is the time between collisions. fR reflects the competition between reaction and escape of unreacted collision partners from the solvent cage. Some predictions based on Eqs. (9) and (10) are shown in Fig. 3 (curves B and C) and compared with simple Arrhenius temperature dependence (curve A). For a reaction with a moderate activation energy under standard conditions, there is Arrhenius behaviour at low temperatures but a significant drop below the Arrhenius prediction at
10 10
kMu / M -1s-1
C
10
9
10
8
A B
10 7
0
100
200 300 Temperature / oC
400
500
Fig. 3. Comparison of rate constants calculated from Eqs. (9) and (10) (curves B and C) with the prediction of the Arrhenius equation (curve A). For all three curves the pre-exponential factor was set to A ¼1.0 1011 M1 s1; for A and C, the activation energy Ea ¼20 kJ mol1, and for B, Ea ¼10 kJ mol1. A value of P ¼ 0:11 was used for curves B and C.
59
higher temperatures. In particular, the rate constant is predicted to pass through a maximum at a temperature below the critical point, consistent with the experimental findings (Figs. 1 and 2, and Ref. [9]). The predicted maximum is broader for smaller values of Ea : A specific example of the predictions of the multiple collisions model is shown by the solid curve in Fig. 1. There is reasonable agreement except in the vicinity of the critical point, where reaction efficiency is considerably lower than predicted. One source of error is in the calculation of collision and encounter times [9], since the procedures for their estimation are unlikely to be accurate close to the critical point. Another possible factor, not included in our model, is the so-called ‘critical slowing down’ of transport properties close to the critical point [23], according to which there should be a sharp drop in the rate constant as the diffusion constant approaches zero. This phenomenon is distinct from the predicted rise in the rate constant at temperatures well above the critical point. In this regime, the solvent behaves more like a dense gas than a liquid. Unfortunately, this happens close to the working limit of our apparatus. Nevertheless, the data for all reactions presented in Figs. 1 and 2 show some indication of the predicted return to positive temperature dependence. Finally, it should be pointed out that the use of a fixed activation energy is a simplistic assumption, particularly when there are strong interactions between a reactant and the solvent. For example, solvent effects are probably the reason for different observed trends for reactions of Mu with OH– and HCOO–. Although both reactions show evidence of the cage effect under near critical conditions, the reaction of formate shows a large positive deviation from the Arrhenius extrapolation (Fig. 2b) while the reaction with OH– shows a negative deviation (Fig. 2c). Lossack et al. have calculated that the dipole moment of the transition state for the reaction between H and formate in water is significantly smaller than the dipole moment of the reactants [18]. Our preliminary modelling of the reaction between H and OH– in water shows the opposite situation.
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4. Conclusions We have provided experimental data relevant to the chemistry of pressurized water nuclear reactors and SCWO. Our results suggest that published models of these processes are based on invalid extrapolations of rate coefficients. In particular, the use of simple Arrhenius behaviour in the Noyes equation [10,11] seriously overestimates rate constants in high temperature water. Our modelling of kinetics in NCW and SCW can be considered as extension of the work of Troe, and our experimental results show similar behaviour: the rate constants pass through a maximum at intermediate densities (temperatures) and then fall with increase of temperature. The common trends among the different reactions reported here supports the idea that for all reactions under near critical conditions the cage effect is important, namely the efficiency of the reactions depends on the number of collisions over the duration of the encounter. However, at lower temperatures, and when there are strong interactions with the solvent, electrostatic effects may be more important.
Acknowledgements We thank Syd Kreitzman and the staff of the TRIUMF mSR Facility for technical support. This research was financially supported by the Natural Sciences and Engineering Research Council of Canada and, through TRIUMF, by the National Research Council of Canada.
References [1] R.W. Shaw, T.B. Brill, A.A. Clifford, C.A. Eckert, E.U. Franck, Chem. Eng. News 69 (51) (1991) 1. [2] A.R. Katritzky, S.M. Allin, M. Siskin, Acc. Chem. Res. 29 (1996) 399. [3] P.E. Savage, Chem. Rev. 99 (1999) 603.
[4] D.R. McCracken, K.T. Tsang, P.J. Laughton, Aspects of the physics and chemistry of water radiolysis by fast neutrons and fast electrons in nuclear reactors, AECL Report, AECL-11895, 1998. [5] R.B. Duffey, W.T. Hancox, D.F. Torgerson, Phys. Can. 56 (2000) 295. [6] P.W. Percival, J.-C. Brodovitch, K. Ghandi, B. AddisonJones, J. Schuth, . D.M. Bartels, Phys. Chem. Chem. Phys. 1 (1999) 4999. [7] K. Ghandi, J.-C. Brodovitch, B. Addison-Jones, P.W. Percival, J. Schuth, . Physica B 289–290 (2000) 476. [8] P.W. Percival, K. Ghandi, J.-C. Brodovitch, B. AddisonJones, I. McKenzie, Phys. Chem. Chem. Phys. 2 (2000) 4717. [9] K. Ghandi, B. Addison-Jones, J.-C. Brodovitch, I. McKenzie, P.W. Percival, J. Schuth, . Phys. Chem. Chem. Phys. 4 (2002) 586. [10] A.J. Elliot, D.R. McCracken, G.V. Buxton, N.D. Wood, J. Chem. Soc. Faraday Trans. 86 (1990) 1539. [11] D. Swiatla-Wojcik, G.V. Buxton, J. Chem. Soc. Faraday Trans. 94 (1998) 2135; D. Swiatla-Wojcik, G.V. Buxton, Phys. Chem. Chem. Phys. 2 (2000) 5113. [12] N. Yoshii, H. Yoshie, S. Miura, S. Okazaki, J. Chem. Phys. 109 (1998) 4873. [13] C.-Y. Liu, S.R. Snyder, A.J. Bard, J. Phys. Chem. B 101 (1997) 1180. [14] M.J. Pilling, P.W. Seakins, Reaction Kinetics, Oxford University Press, Oxford, 1996. [15] Yu.N. Molin, K.M. Salikhov, K.I. Zamaraev, Spin Exchange, Springer, Berlin, 1980. [16] M. Senba, Phys. Rev. A 52 (1995) 4599. [17] S.K. Leung, Ph.D. Thesis, Simon Fraser University, 1991. [18] A.M. Lossack, E. Roduner, D.M. Bartels, Phys. Chem. Chem. Phys. 3 (2001) 2031. [19] A.M. Lossack, E. Roduner, D.M. Bartels, J. Phys. Chem. A 102 (1998) 7462. [20] G.V. Buxton, S.R. Mackenzie, J. Chem. Soc. Faraday Trans. 88 (1992) 2833. [21] J. Troe, Ber. Bunsen-Ges. Phys. Chem. 94 (1990) 1183. [22] J. Troe, V.G. Ushakov, Faraday Discuss. 119 (2001) 145. [23] S.C. Tucker, Chem. Rev. 99 (1999) 391. [24] E. Roduner, P.W. Louwrier, G.A. Brinkman, D.M. Garner, I.D. Reid, D.J. Arseneau, M. Senba, D.G. Fleming, Ber. Bunsen-Ges. Phys. Chem. 94 (1990) 1224. [25] E. Roduner, D.M. Bartels, Ber. Bunsen-Ges. Phys. Chem. 96 (1992) 1037. [26] B.W. Ng, J.M. Stadlbauer, D.C. Walker, J. Phys. Chem. 88 (1984) 857.