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On the possible role of the reaction H+H2O→H2+OH in the radiolysis of water at high temperatures
ARTICLE IN PRESS
Radiation Physics and Chemistry 74 (2005) 210–219 www.elsevier.com/locate/radphyschem
On the possible role of the reaction Hd þH2 O ! H2 þ dOH in the radiolysis of water at high temperatures Dorota Swiatla-Wojcika,, George V. Buxtonb a
Institute of Applied Radiation Chemistry, Technical University of Lodz, Zeromskiego 116, 90-924 Lodz, Poland b School of Chemistry, University of Leeds, Leeds LS2 9JT, UK
Abstract It is proposed that oxidation of water by the hydrogen atom may play an important role in the high-temperature radiolysis of water at low linear energy transfer (LET). The observed increase of the primary yield of H2, g(H2), in the radiolysis of water at high temperatures is shown to be consistent with the occurrence of the reaction Hd þH2 O ! H2 þ d OH. The temperature dependence of the rate constant for this reaction has been determined by diffusion-kinetic modelling of spur processes in neutral water. Based on a rate constant of 0.086 M1 s1 at 25 1C estimated from thermodynamic data, and literature values of g(H2) as a function of temperature, a corresponding activation energy of 66.370.8 kJ mol1 has been calculated over the temperature range 20–300 1C. The deterministic d d calculations also indicate the important role of the reaction e aq þH2 O ! H þOH in the production of the H atom at elevated temperatures in agreement with stochastic studies. r 2005 Elsevier Ltd. All rights reserved. Keywords: Radiolysis of water; High temperature; Hydrogen chemistry; Diffusion-kinetic modelling
An understanding of the radiolysis of water at elevated temperatures has been the subject of several experimental (Buxton, 2001; Elliot, 1994; Christensen and Sehested, 1980) and theoretical investigations (LaVerne and Pimblott, 1993; Swiatla-Wojcik and Buxton, 1995; Swiatla-Wojcik and Buxton, 1998; Herve du Penhoat et al., 2000; Swiatla-Wojcik and Buxton, 2000; Meesungnoen et al., 2001). Of basic interest is knowledge of the effect of temperature on the rate constants of the relevant reactions as well as on the primary yields (g-values) of the radiolysis products: e aq, Hd, dOH, H2, and H2O2. The primary or escape yields
are the radiation chemical yields1 at the end of the chemical stage of water radiolysis (Buxton, 1987). The chemical stage is controlled by diffusion and reaction of non-homogeneously distributed reactants; it begins at ca. 1012 s after the absorption of energy when the initial products have reached thermal equilibrium with their surroundings and ends with the decay of their spatial correlations at ca. 107–106 s, depending on the temperature. The most complete and reliable data on the temperature dependence of the primary yields are available for low linear energy transfer (LET) radiolysis. d d The measured g-values of e OH and H2 aq , H , continuously increase, whilst g(H2O2) decreases, with increasing temperature. The effect of temperature on the
Corresponding author. Tel.: +4842 6313193; fax.: +4842 6365008. E-mail address: [email protected] (D. Swiatla-Wojcik).
1 The radiation chemical yields are given in the units of molecules per 100 eV of absorbed energy, where 1 molecule per 100 eV corresponds to 1.0364 107 mol J1.
1. Introduction
0969-806X/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.radphyschem.2005.04.014
ARTICLE IN PRESS D. Swiatla-Wojcik, G.V. Buxton / Radiation Physics and Chemistry 74 (2005) 210–219 d d primary yields of e OH, and H2O2 is well aq, H , understood within the generally accepted diffusionkinetic model of spurs (Buxton, 1987) and has been reproduced satisfactorily by deterministic (SwiatlaWojcik and Buxton, 1995; Swiatla-Wojcik and Buxton, 1998; Swiatla-Wojcik and Buxton, 2000) and stochastic (Herve du Penhoat et al., 2000; Meesungnoen et al., 2001) calculations. Some difficulties, however, appear in matching the experimentally observed monotonic increase of the molecular hydrogen yield (Elliot et al., 1993; Sunaryo et al., 1995). Although these calculations have explained an increase of g(H2) up to ca. 200 1C as resulting from the bimolecular reaction of e aq (SwiatlaWojcik and Buxton, 1995; Herve du Penhoat et al., 2000), above 200 1C the computed g(H2) tends to decrease with increasing temperature. Various explanations of the temperature dependence of g(H2) behaviour have been proposed recently and tested by deterministic (Swiatla-Wojcik and Buxton, 1995; Swiatla-Wojcik and Buxton, 2000) and stochastic (Herve du Penhoat et al., 2000) calculations. In the extended spur diffusion model (Swiatla-Wojcik and Buxton, 1995; Swiatla-Wojcik and Buxton, 2000) we postulated that an increasing contribution of the part of unscavengeable H2 comes from channel (2) of the
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dissociative decomposition of excited water molecules. H2 O ! Hd þ d OH;
(1)
H2 O ! H2 þ d Oð1 DÞ ! H2 þ 2d OH:
(2)
Good agreement with experiment was obtained with the total contributions of channels (1) and (2) being independent of temperature but with their ratio decreasing over the range 200–300 1C from ca. 21:1 to ca. 4:1 (Swiatla-Wojcik and Buxton, 1995). Such a result is consistent with the efficiency of channel (2) in water vapour and seems reasonable in view of the diminution of hydrogen bonding in liquid water at high temperatures. Two alternative concepts have been explored through Monte Carlo simulations (Herve du Penhoat et al., 2000). The first one is linked to an increase in the scattering cross section of sub-excitation electrons and assumes a continuous decrease of their thermalisation distance with increasing temperature. The second one allows for a possible screening of Coulomb forces between two e aq and assumes that the bimolecular reaction R4 (Table 1) is fully diffusion controlled. Both of these concepts led to an increased formation of H2 in
Table 1 Reaction scheme and temperature dependence of the rate constants up to 573 K Symbol
R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 R14 R15 R16 R17 R18 a
reactiona
Temperature dependence
þ d e aq þ Haq ! H d OH ! OH e þ aq Hþ aq þ OH ! H2 O d e aq þ eaq ! H2 þ 2 OH eaq þ H ! H2 þ OH Hd þ Hd ! H2 Hd þ d OH ! H2 O d OH þ d OH ! H O 2 2 d e aq þ H2 O2 ! OH þ OH d d H þ H2 O ! OH þ H2 d OH þ H ! d H þ H O 2
2
e aq þ H2 O ! H þ OH d H þ OH ! e þ H O aq
2
Hd þ H2 O2 ! d OH þ H2 O d OH þ H O ! HO þH O 2 2 2 2 þ Hd ! e aq þ Haq d OH þ OH ! O þ H O 2 O þ H2 O ! d OH þ OH
Parameters for kact in Eq. (5) kact(298 K) (1010 M1 s1]
Ea [kJ mol1]
N
Noyes relationship
2.37
10
0
Noyes relationshipb Diffusion controlled Noyes relationship Diffusion controlledc Diffusion controlled Noyes relationship Noyes relationship Noyes relationship Arrhenius (this work) Arrhenius up to 473 Kd Elliot (1994) Elliot (1994)
8.2 — 1.35 — — 3.58 1.1 1.8 — — — —
0 — 28 — — 0 0 15.5 — — — —
1 — 0 — — 1 1 0 — — — —
Noyes relationship Elliot (1994) equilibrium eqn. and kR1(T)
0.005 — —
16.6 — —
0 — —
Elliot (1994) Elliot (1994)
— —
— —
— —
Reactions R11, R13–R18 can be neglected in modelling low LET radiolysis. Fitted to the experimental data (Elliot et al., 1994) corrected for changes in dosimetry and extinction coefficient (Stuart et al., 2002). c The experimental data of Christensen et al. (1994) neglect the temperature dependence of the absorption coefficient of e aq. d Arrhenius fit to the experimental data up to 473 K (Christensen et al., 1983) and numerical interpolation of the rate constants reported for 473–623 K (Marin et al., 2003). b
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R4 and somewhat suppressed a decrease of the scavengeable yield of molecular hydrogen. In the present work, we focus on the reaction Hd þH2 O ! H2 þ d OH as a possible source of molecular hydrogen. This reaction has not been considered previously in either the deterministic or the stochastic models of the chemical stage of water radiolysis. On the other hand, as Jay-Gerin and co-workers have recently shown (Herve du Penhoat et al., 2000; Meesungnoen et al., 2001), the contribution of the hydrated electron reaction with water rapidly increases with temperature and its cumulative yield at the end of the chemical stage at 300 1C, 0.1 ms after irradiation, is substantial. Examination of any role of the reaction Hd þH2 O ! H2 þ d OH is, however, not straightforward. This reaction has been studied in the gas phase (Baulch et al., 1992) but its rate constant in liquid water is not well established at any temperature. The temperature dependence of the rate constant for oxidation of water by the Hd atom is necessary not only for a fuller understanding of hightemperature radiolysis, it is also of basic interest for nuclear power engineers to estimate the critical hydrogen concentration in water-cooled nuclear reactor loops. The importance of the reaction Hd þH2 O ! H2 þ d OH for water and hydrogen chemistry in nuclear power reactors has been indicated by Sunaryo et al. (1995); Ishigure et al. (1987), and Shiraishi et al. (1994). Moreover, our recent simulation of the coolant behaviour proved to be very sensitive to a value of the rate constant for Hd þH2 O ! H2 þ d OH (Swiatla-Wojcik and Buxton, 2003). In this paper, the primary hydrogen and hydrogen atom chemistry is re-investigated for the low LET radiolysis of water up to 300 1C. First, the room temperature value of k(Hd þH2 O ! H2 þ d OH) is calculated from the available electrochemical data. Then the temperature dependence of k is simulated by diffusion-kinetic modelling to obtain consistency between the calculated and the measured primary yields of H2, Hd, and dOH. Finally, implications of the derived temperature dependence are discussed.
2. Method Computation of the effect of temperature on the primary yields in the low LET radiolysis of water has been based on the diffusion-kinetic modelling of the decay of an average spur. This approach leads to a set of coupled differential equations describing the temporal and spatial evolution of the concentration ci of the reactive species i at a given temperature T: X XX @ci ¼ Di ðTÞr2 ci kij ðTÞci cj þ kjk ðTÞcj ck , @t j j k d d þ ð3Þ i; j ¼ e aq ; Haq ; H ; OH; H2 ; H2 O2 ; OH ,
where each equation consists of terms representing diffusion of species i, and reactions removing i and producing i, respectively. Numerical solution of (3) was obtained using the FACSIMILE program (Chance et al., 1977) for temperatures 273pTp573 K. The input data comprised diffusion coefficients Di and reaction rate constants kij. The diffusion coefficient of the hydrated electron, and its temperature dependence have been taken from Schmidt et al., 1995. The values of Di for other species are those used previously (SwiatlaWojcik and Buxton, 1998). The assumed reaction scheme and methods for describing rate constants kij ðTÞ as a function of temperature are summarised in Table 1. The temperature dependence kij ðTÞ for reactions R1, R2, R4, and R7–R9 has been described using the Noyes relationship (cf. Elliot et al., 1990; Buxton, 2001) 1 1 k1 obs ¼ kdiff þ kact ,
(4)
where the reciprocal of the measured value, kobs, is represented by the sum of the reciprocals of the diffusion part, kdiff, and the activation part, kact. The temperature dependence of kdiff has been calculated on the basis of the Debye–Smoluchowski equation using the reaction radii and the statistical spin factors given by SwiatlaWojcik and Buxton (1995). The temperature dependence of the activation part was obtained from kact ðTÞ ¼ AT n expðE a =RTÞ.
(5)
The parameters used to evaluate kact are listed in Table 1. The Noyes relationship deals specifically with the rate coefficients of reactions in the liquid phase that are limited by the rate of diffusion at low temperatures and which become limited by the rate of the chemical step at high temperatures. Recently, Ghandi and Percival (2003) have suggested that at high temperatures and pressures, kact in the Noyes equation should be multiplied by an efficiency factor which takes into account the reactive orientation and the number of collisions between reactants over the duration of their encounter. This modified Noyes relationship predicts a broad maximum in the Arrhenius plot for bimolecular reaction rate constants at high temperatures due to the decreasing number of collisions during the encounter. Ghandi and Percival (2003) used the modified Noyes equation to extrapolate to higher temperature the reaction rate constants of the hydroxyl radical measured by Elliot and co-workers (Elliot, et al., 1990; Buxton and Elliot, 1993; Elliot and Ouellette, 1994) up to 200 1C. Such extrapolation shows the downward bending at ca. 300 1C. Recently, Lundstrom et al. (2002) have extended experimental studies on the reaction of dOH with Hd up to 233 1C, as the authors stated, their results neither confirmed nor disproved the downward bending of kR7 ðTÞ.
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As in our previous studies (Swiatla-Wojcik and Buxton, 1995; Swiatla-Wojcik and Buxton, 2000) the temperature dependences for dOH reaction rate constants kR2, kR7, and kR8 are based on the Noyes relationship with reactions R2, R7, and R8 assumed to have no activation energy so that from Eq. (5) kact ¼ AT.
(6)
Our calculated values for kR2, kR7 and kR8 at 300 1C are higher by 54%, 18%, and 14%, respectively, compared to those predicted by Ghandi and Percival (2003). It should be noted that Ghandi and Percival (2003) based their extrapolation for kR2 on experimental data (Elliot and Ouellette, 1994) which has been recalculated recently to reflect the revised dosimetry values, extinction coefficient for e aq and rate constants for the self-reaction of the hydrated electron (Stuart et al., 2002). The revised rate constants are slightly higher than those given by Elliot and Ouellette (1994). However, as shown later, the differences in kR2, kR7 and kR8 have a relatively small effect on the calculated gvalues at high temperatures. According to the extended spur diffusion model (Swiatla-Wojcik and Buxton, 1995; Swiatla-Wojcik and Buxton, 2000) initial conditions for the set of differential equations (3) can be defined by four o independent yield parameters: Goex, G o ðe aq Þ, G 1 ðH 2 Þ o o o and G 2 ðH 2 Þ and two size parameters: se , s . These parameters refer to the beginning of the radiation chemical stage, 1 ps after irradiation. Go ðe aq Þ represents o the initial yield of e aq, G ex is the initial yield of excited molecules undergoing fragmentation in processes (1) and (2), with the contribution of the latter defined by Go2 ðH 2 Þ. G o1 ðH 2 Þ specifies that part of the prompt yield of H2 which is formed in the dissociative attachment of sub-excitation electrons, e sub: d d e sub þ H2 O ! OH þ H ! OH þ H2 þ OH .
(7)
In our model, 80% of the prompt yield of H2 comes from (7) and only 20% from the dissociation channel (2). The significant role of sub-excitation electrons in the formation of H2 is in fair agreement with the experimental studies on scavenging of the precursors to e aq (Pastina et al., 1999). The size parameters soe 4so are standard deviations of two Gaussian functions which describe the initial nonhomogeneous distribution of the species within the spur depending on the length of their thermalisation path. The wider distribution characterises the initial allocation of e and that fraction of dOH and H2 which aq, OH comes from the dissociative attachment (7). The calculations have been performed assuming various number densities of the primary species in the average spur. The room temperature values of the corresponding Goi, soe and so parameters have been given previously (Swiatla-Wojcik and Buxton, 2000). The
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o o initial yields Goex, Go ðe aq Þ, G 1 ðH 2 Þ and G 2 ðH 2 Þ have been assumed to be independent of temperature and the parameters soe , so have been scaled with temperature according to the density of water (Swiatla-Wojcik and Buxton, 1995), with the result that they increased by no more than 11.5% up to 300 1C. A possible decrease of the thermalisation distance due to energy dissipation through interaction with more energetic vibrational modes of H2O at high temperatures was estimated to be between 1.1 and 2.8% (Swiatla-Wojcik and Buxton, 2000). The effect of LET has been investigated in terms of the diffusion-kinetic model of the decay of an average cylindrical track (Swiatla and Buxton, 1998, 2000) using the reaction scheme from Table 1. As in the spherical o o case the parameters Goex, Go ðe aq Þ, G 1 ðH 2 Þ and G 2 ðH 2 Þ have been assumed to be independent of temperature.
3. Results and discussion 3.1. Estimation of k(Hd þH2 O ! H2 þ d OH) at room temperature There are no direct measurements of kR10 at any temperature. The published estimates at room temperature vary from 3:6 105 M1 s1 reported by Shiraishi et al. (1994) to 10 2 M1 s1 derived by Hartig and Getoff (1982) from a photolysis study of hydrogen peroxide solution containing hydrogen. Here, we estimate kR10 by considering reactions R10 and R11 as an equilibrium: kR10 =kR11 ¼ K ¼ exp½-Dr G0 =ðRTÞ :
(8) o
To obtain the free energy change Dr G for R10, we used the values of Eo at 25 1C reported by Stanbury (1989) for the half-cell reactions (9)–(11) that make up reaction R10: Haq ! Hþ aq þ e;
2:31 V;
H2 O ! OHaq þ Hþ aq þ e; e þ Hþ aq ! 1=2 H2 ;
(9) 2:72 V;
0 V:
(10) (11)
Substitution of the obtained value Dr G o ¼ 39:6 kJ mol1 in Eq. (8) gives K ¼ 1:15 107 at 25 1C. The thermodynamic equilibrium constant expressed in terms of activities takes the form K¼
aH2 aOH ½H2 ½OH ¼ , aH aH2 O ½H
(12)
where the activities can be replaced by concentrations, except for water, which by definition has the activity a ¼ 1. In terms of rates, we have at equilibrium kR10 ½H ½H2 O ¼ kR11 ½OH ½H2
(13)
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so that kR10 ¼ kR11 K=½H2 O .
(14)
Reaction R11 was studied in liquid water for the temperature range 15–230 1C by Christensen and Sehested (1983) and for 200–350 1C by Marin et al. (2003) who observed a decrease of kR11 above 275 1C. From the Arrhenius fit to the data of Christensen and Sehested (1983) Elliot (1994) obtained kR11 ¼ 4.2 107 M1 s1 at 25 1C. Substituting this value in Eq. (14) with [H2O] ¼ 55.56 M, we have kR10 ¼ 0.086 M1 s1 at 25 1C. 3.2. Derivation of the temperature dependence for k(Hd þH2 O ! H2 þ d OH) The temperature dependence of kR10 in solution is not well established. Two estimates of the rate constant kR10 at high temperatures have been reported. At 250 1C, Shiraishi et al. (1994) estimated kR10 2.5 102 M1 s1, and Sunaryo et al. (1995) obtained kR10 20 M1 s1, but no details of the calculations were given in either case. To obtain an expression for kR10 ðTÞ, we assumed that it is analogous to that for the reaction of e aq with water, R12. Schwarz (1992) obtained an Arrhenius dependence for R12 with Ea ¼ 31.4 kJ mol1 over the temperature range 4–65 1C, whilst Elliot (1994) calculated kR12 ðTÞ as kR12 ¼ kR13 K W =ðK H ½H2 O Þ
(15)
using the data for the back reaction R13 (Han and Bartels, 1990) and the temperature dependence of the ionic dissociation constants KW and KH for H2O and Hd, respectively. We calculated that kR12 ðTÞ resulting from Eq. (15) shows an Arrhenius dependence with A ¼ 9.2 106 M1 s1 and Ea ¼ 33.4 kJ mol1. By analogy with R12, and noting that the Noyes relationship is
not applicable to slow reactions that are not diffusion limited at low temperatures, we assumed an Arrhenius dependence for kR10 ðTÞ. As a rough approximation, the activation energy for reaction R10, E aðR10Þ , should be at least equal to Dr Go þ E aðR11Þ which is 60 kJ mol1 when taking E aðR11Þ ¼ 19 kJ mol1 as determined by Christensen and Sehested (1983) from measurements of kR11 in the temperature range 15–230 1C. Using the derived room temperature value of kR10, an influence of E aðR10Þ on the primary yields has been investigated. The computed primary yields g(H2), g(Hd), and g(dOH) turn out to be sensitive to kR10 ðTÞ at high temperatures. In Fig. 1, the primary yields calculated in the absence of R10 (solid curves) are compared with the g-values obtained using kR10 ðTÞ with the activation energy varying from E aðR10Þ ¼ 60 kJ mol1 (broken curves) to E aðR10Þ ¼ 80 kJ mol1 (dotted curves). The broken and dotted curves indicate a substantial difference between the g-values of H2, Hd, and dOH. An increase of gðH2 Þ due to R10 is compensated by a decrease of g(Hd) with the sum g(H2)+g(Hd) being insensitive to the values of E aðR10Þ . Reaction R10 does not affect the temperature dependence of either gðe aq Þ or gðH2 O2 Þ. As seen from Fig. 1, oxidation of water by the d H atom may explain the observed temperature effect on gðH2 Þ. The primary yield of molecular hydrogen is the most reliable measured yield and the relevant experimental data are available up to 300 1C (Kent and Sims, 1992; Elliot et al., 1993; Sunaryo et al., 1995). Kent and Sims (1992) made all their g-radiolysis 3 measurements with [NO M to scavenge dOH 2 ] ¼ 10 and prevent reaction R11 which could destroy H2 formed in the radiolysis. The data reported by Elliot et al. (1993) were obtained using the concentration range of potassium nitrite (1–500) 103 M and the measured g(H2) was extrapolated to [NO 2 ] ¼ 0 on a plot of g(H2)
Fig. 1. The g-values gðH2 Þ, gðH2 Þ+gðHÞ, gðOHÞ, gðHÞ versus temperature calculated for 107 s assuming: Ea(R10) ¼ 60 kJ mol1 (broken curves), Ea(R10) ¼ 80 kJ mol1(dotted curves). Solid lines illustrate the temperature dependence of the g-values in the absence of R10. The experimental points are from (Kent and Sims, 1992; Elliot et al., 1993; Sunaryo et al., 1995).
ARTICLE IN PRESS D. Swiatla-Wojcik, G.V. Buxton / Radiation Physics and Chemistry 74 (2005) 210–219 1/3 vs [NO so that the scavenging power was effectively 2] zero in the spur. The data of Elliot et al. (1993) seem more appropriate for comparison with the values of g(H2) calculated at 107 s after irradiation. Calculation of the values of kR10 that could reproduce the measured yield of H2 at high temperatures has been performed for the three sets of spur parameters representing various initial densities of the species within the spur (Swiatla and Buxton, 2000). An effect of the uncertainty dkRi in the rate constants of reactions Ri, ia10 listed in Table 1 has been estimated as follows. A change, DGR10 , in the cumulative contribution of R10 at 107 s due to the relative uncertainty dkRi in the rate constant of reaction Ri was calculated and then DG R10 was reversed to a relative uncertainty in kR10. The largest dkR10 corresponds to reactions R1, R2, R7 and R12. At 300 1C dkRi ¼ 10% results in the relative uncertainty in kR10 of 3%, 2%, 0.7%, and 0.7%, respectively, and in the overall uncertainty of 7%. The lowest and the highest values of kR10 that reproduce the measured radiation chemical yield of H2 within the reported experimental error have been calculated for each set of parameters. They are marked in Fig. 2A by (x). Linear regression applied to these points results in the Arrhenius parameters AðR10Þ ¼ ð3:5 0:8Þ 1010 M1 s1 and E aðR10Þ ¼ 66:3 0:8 kJ mol1. The obtained Arrhenius dependence is shown by the solid line. Marin et al. (2003) observed the downward bending of kR11 ðTÞ in the range 250–350 1C. This observation may suggest that kR10 ðTÞ is not Arrhenius at 250–300 1C. To examine this point, we invoke the van’t Hoff equation (Atkins, 1998) describing the temperature dependence of the equilibrium constant K defined in Eq. (12):
dln K Dr H o . ¼ dT RT 2
(16)
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Eq. (16) allows us to estimate the standard enthalpy of reaction R10, Dr H o , from a plot ln K vs 1/(RT), assuming that the effect of temperature on Dr H o is negligible. Taking the experimental data for k11 (Christensen and Sehested, 1983), the derived values for k10 and the density of water (cf. Elliot et al., 1993) to obtain [H2O], the equilibrium constant K was calculated up to 180 1C and linear regression was performed for ln K. The resulting temperature dependence of ln K with Dr H o ¼ 47:34 kJ mol1 and the experimental data for k11 reported by Marin et al. (2003) was then employed to obtain k10 above 180 1C. These values are shown in Fig. 2A by solid circles. As can be seen, there is no reason to invalidate the Arrhenius dependence for k10 up to 300 1C. At 300 1C, the point 2:22 104 M1 s1 compares well with ð3:18 1:25Þ 104 M1 s1 obtained from the derived Arrhenius parameters. The temperature dependence of the primary yield gðH2 Þ is shown in Fig. 2B by the solid curve. The dotted lines demarcate the limits obtained on the basis of the 95% prediction interval of the derived function kR10 ðTÞ. All the sets of experimental data show the increase with temperature and the flattening at 200 1C. These features are reproduced by the computed temperature dependence and the agreement with the data of Elliot et al. (1993) and of Sunaryo et al. (1995) is good. There is more uncertainty in the experimental points of Kent and Sims (1992) which are slightly above the computed values of gðH2 Þ. 3.3. The possible role of the reaction Hd þH2 O in the radiolysis of water Using the derived Arrhenius parameters for R10, one obtains kR10 ¼ ð3:18 1:25Þ 104 M1 s1 at 300 1C. Although the reaction of the hydrogen atom with water
Fig. 2. A. The calculated values of kR10: (x) values of kR10 that reproduce the measured primary yield of H2 and the Arrhenius fit to (x) (solid line); () kR10 obtained from the thermodynamic equilibrium constant K defined in Eq. (12) and the experimental data for k11 reported by (Marin, et al., 2003). B. Calculated primary yield of molecular hydrogen as a function of temperature (solid line). The dotted lines demarcate the limits obtained on the basis of the 95% prediction interval for the derived function kR10(T). Experimental data are from: Elliot et al. (1993) (’), Kent and Sims (1992) (K), and Sunaryo et al. (1995) (m). The dashed curve shows the results of Monte Carlo simulation (Herve du Penhoat et al., 2000).
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does not affect the chemistry in the spur at temperatures below 180 1C, because of its high activation energy it becomes significant at higher temperatures. Its contribution to gðH2 Þ, expressed in molecules per 100 eV, increases rapidly with temperature from ca. 0.005 at 200 1C to ca. 0.1 at 300 1C. The negative contribution of the back reaction R11 does not exceed 0.001 molecules per 100 eV. The extents DGi of the reactions contributing to gðH2 Þ are presented in Fig. 3. At 25 1C, 68% of gðH2 Þ is produced via processes (2) and (7) during the physicochemical stage, 1015–1012 s after irradiation, and less than one-third is formed within spurs in R4 and R5. The role of the chemical stage increases with temperature. At 300 1C the total contribution of processes (7) and (2) is less than 50%, whilst 24% of gðH2 Þ comes from R4, 16% from R10, 11% from R5, and 1% from R6. The rapid increase of the cumulative contribution of R10 is responsible for the increase of gðH2 Þ at high temperatures seen in Fig. 2B. On the other hand Monte Carlo calculations predict a slight decrease of gðH2 Þ above 220 1C (Herve du Penhoat et al., 2000). The result of Herve du Penhoat et al. (2000) is illustrated by the dashed curve in Fig. 2B. Verification of the computed temperature effect on the yield of H2 requires more experimental data, particularly at temperatures even higher than 300 1C. It is pertinent to note that Elliot et al. (1993) reported very high values of gðHd Þ þ gðH2 Þ in H2O, and gðDd Þ þ gðD2 Þ in D2O, above 2001C for solutions containing 103 mol dm3 acetone and 101 mol dm3 methanol to scavenge e aq and hydroxyl radical, respectively. At 300 1C, gðHd or Dd Þ+g(H2 or D2) was ca. 2.7 molecules per 100 eV in both liquids. Moreover, they showed that the excess deuterium at 300 1C came predominantly from the heavy water, which is qualitatively consistent with the mechanism proposed here. Diffusion-kinetic
Fig. 3. Cumulative contributions, DGi ðH2 Þ, of reactions Ri and processes to gðH2 Þ in low-LET radiolysis of water as a function of temperature.
modelling of the D2 formation in heavy water is in progress. To show whether or not R10 contributes to the hydrogen formation, one could measure gðH2 Þ and gðHDÞ in H2O containing a deuterated scavenger for Hd to form HD and look at the ratio of gðHDÞ to gðH2 Þ as a function of temperature at scavenger concentrations where no spur scavenging occurs. Thus, gðHDÞ would be a measure of H escaping from the spurs and gðH2 Þ would be a measure of the spur reactions forming H2 plus any H2 coming from R10. Our present calculations show an important contribution of reaction R12 to the production of the Hd atom at high temperatures in agreement with the stochastic modelling (Herve du Penhoat et al., 2000; Meesungnoen et al., 2001). The proportion of Hd formed in R12, which is 25–30% depending on the assumed density of the species within the spur (Swiatla and Buxton, 2000), agrees well with 25% obtained from the stochastic study of low-LET radiolysis (Herve du Penhoat et al., 2000). The loss of Hd in reaction R13 is negligible. Compared to the previous calculations (Swiatla and Buxton, 2000), incorporation of R12 results in a slight increase of gðHd þH2 Þ and in a slight decrease of gðe aq Þ at 300 1C. Considerable effects due to reactions R10 and R12 can be expected at times longer than 107 s. Time dependences of the computed radical and molecular yields are presented in Fig. 4 for 100, 200, and 300 1C. A 7 6 rapid decrease of gðe s aq Þ in the time interval 10 –10 caused by a decay of e due to its reaction with water aq has been also obtained in the stochastic calculations for low LET (Meesungnoen et al., 2001). A rapid increase of the yield of H2 and of dOH resulting from oxidation of water by the Hd atom is seen in Figs. 4f and 4b, respectively. Reaction R10 does not, however, affect the formation of H2O2. The time dependences of g(Hd) presented in Fig. 4e reflect competition between the formation of the Hd atom in R12 and its decay in R10. A significant increase of g(Hd) is observed at 200 1C but the developing contribution of R10 leads to the maximum exhibited by the time profile computed for 300 1C. As can be expected, reactions R10 and R12 have no effect on the temporal profiles calculated for the sum d gðe aq Þ þ gðH Þ þ gðH2 Þ presented in Fig. 4d. The dotted lines in Fig. 4 show time dependences of the g-values calculated for 300 1C using the values of k for the partially diffusion-controlled reactions R2, R7, and R8 suggested by Ghandi and Percival (2003) which, as discussed in the previous section, are lower than the rate constants used in our modelling. At high temperatures, these partially diffusion-controlled reactions are much less important than at room temperature. For example, reaction R2 is responsible for 58% of the loss of e aq at 25 1C and for only 28% at 300 1C (SwiatlaWojcik and Buxton, 2001). In consequence, the g-values at the end of the chemical stage, 107 s after irradiation at 300 1C, are not significantly altered.
ARTICLE IN PRESS D. Swiatla-Wojcik, G.V. Buxton / Radiation Physics and Chemistry 74 (2005) 210–219
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Fig. 4. Time dependences of the g-values calculated for temperatures: 100 1C (dash), 200 1C (dash-dot), and 300 1C using the values of kR2, kR7 and kR8 from (Swiatla and Buxton, 2000), solid lines, and from (Ghandi and Percival, 2003), dotted lines (see text).
Fig. 5. Cumulative contributions DGi of reactions R10 and R12 (solid symbols) and their back reactions R11 and R13 (open symbols), respectively versus LET calculated for 300 1C and 0.1 ms (left panel), 1 ms (right panel). Solid lines have been added to guide the eye.
Under typical conditions used in pulse radiolysis to measure kR7, the introduction of reaction R10 has a negligible effect on the kinetics of the decay of dOH up to 200 1C. Our preliminary simulation shows that R10 is not responsible for any curvature in the plots of kR7 vs 1/T up to 200 1C. The difference in the first half-life of d OH when R10 is included, of 5 108 s, is not detectable in an experiment. The simulation shows, however, a two-fold increase of the first half-life of dOH at 300 1C
The reactions of radicals with water can be expected to become less important with increasing LET. Fig. 5 presents cumulative contributions DG Ri calculated up to 107 s (left panel) and up to 106 s (right panel) for R10, R12 and the corresponding back reactions R11, R13 as a function of LET at 300 1C. Calculations have been performed for the average LET of 0.2, 20, 40, 60 and d 80 keV mm1. As can be seen, the decay of e aq and H in reactions with water become more limited with increasing LET whilst their reformation in the back reactions is
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more significant. For 80 keV mm1 the net contributions DGR12 DGR13 and DGR10 DGR11 at 107 s decrease to 0.005 and 0.07 molecules per 100 eV, respectively, but at 106 s DGR12 DGR13 ¼ 0:14 and DGR10 DGR11 ¼ 0.46 molecules per 100 eV. This result indicates that reactions R10 and R12 may play an important role in water chemistry of water-cooled nuclear reactors (PWRs and BWRs). They should also be taken into account when assessing the radiation chemistry in supercritical water-cooled reactors. In summary, the results of the calculations presented d here indicate that the reactions of e aq and H with water may contribute significantly to the chemical stage of water radiolysis at high temperatures, especially at low LET. Moreover, the reaction Hd þ H2 O ! H2 þ d OH could be responsible for the observed increase of molecular hydrogen yield at high temperatures. The computed Arrhenius dependence for the rate constant of this reaction in liquid water shows an increase of kR10 from 0.086 M1 s1 at 25 1C to (3.1871.25) 104 M1 s1 at 300 1C. Experimental measurements to test the validity of both the suggested role of the reaction Hd þ H2 O ! H2 þ d OH and the derived temperature dependence of its rate constant are an important requirement.
Acknowledgment D. S.-W. thanks the Technical University of Lodz for financial support.
References Atkins, P.W., 1998. Physical Chemistry, sixth ed. Oxford University Press, Oxford, Melbourne, Tokyo. Baulch, D.L., Cobos, C.J., Cox, R.A., Esser, C., Frank, P., Just, T., Kerr, J.A., Pilling, M.J., Troe, J., Walker, R.W., Warnatz, J., 1992. Evaluated kinetic data for combustion modeling. J. Phys. Chem. Ref. Data 21, 411–734. Buxton, G.V., 1987. Radiation Chemistry of the Liquid State: (1) Water and homogeneous aqueous solutions. In: Farhataziz, Rodgers, M.A.J. (Eds.), Radiation Chemistry: Principles and Applications. VCH Publishers, New York, pp. 321–349. Buxton, G.V., 2001. High Temperature Water Radiolysis. In: Jonah, C.D., Madhava Rao, B.S. (Eds.), Radiation Chemistry: Present Status and Future Trends. Elsevier, Amsterdam, pp. 145–162. Buxton, G.V., Elliot, A.J., 1993. Temperature dependence of the rate constant for the reaction H+OH in liquid water up to 200 1C. J. Chem. Soc., Faraday Trans. 89, 485–488. Chance, E.M., Curtis, A.R., Jones, T.P., Kirby, C.R., 1977. AERE Report AERE R8775. H.M.S.O., London. Christensen, H., Sehested, K., 1980. Pulse radiolysis at high temperatures and pressures. Radiat. Phys. Chem. 16, 183–186.
Christensen, H., Sehested, K., 1983. Reactions of hydroxyl radicals with hydrogen at elevated temperatures. Determination of the activation energy. J. Phys. Chem. 87, 118–120. Christensen, H., Sehested, K., Logager, T., 1994. Temperature dependence of the rate constants for reactions of hydrated electrons with H, OH and H2O2. Radiat. Phys. Chem. 43, 527–531. Elliot, A.J., 1994. Rate constants and g-values for the simulation of the radiolysis of light water over the range 0–300 1C. Report AECL-11073. Chalk River Laboratories, Canada. Elliot, A.J., Ouellette, D.C., 1994. Temperature dependence of the rate constant for the reaction e aq +OH in water up to to 150 1C. J. Chem. Soc., Faraday Trans. 90, 837–841. Elliot, A.J., McCracken, D.R., Buxton, G.V., Wood, N.D., 1990. Estimation of rate constants for near-diffusioncontrolled reactions in water at high temperatures. J. Chem. Soc., Faraday Trans. 86, 1539–1547. Elliot, A.J., Chenier, M.P., Oullette, D.C., 1993. Temperature dependence of g-values for H2O and D2O irradiated with low linear energy transfer radiation. J. Chem. Soc., Faraday Trans. 89, 1193–1197. Ghandi, K., Percival, P.W., 2003. Prediction of rate constants for reactions of the hydroxyl radical in water at high temperatures and pressures. J. Phys. Chem. A. 107, 3005–3008. Han, P., Bartels, D.M., 1990. Reevaluation of Arrhenius d parameters for H +OH-(e)aq+H2O and the enthalpy and entropy of hydrated electrons. J. Phys. Chem. 94, 7294–7299. Hartig, K.J., Getoff, N., 1982. Reactivity of hydrogen atoms with liquid water. J. Photochem. 18, 29–38. Herve du Penhoat, M.-A., Goulet, T., Frongillo, Y., Fraser, M.-J., Bernat, P., Jay-Gerin, J.-P., 2000. Radiolysis of liquid water at temperatures up to 300 1C: A Monte Carlo simulation study. J. Phys. Chem. A. 104, 11757–11770. Ishigure, K., Takagi, J., Shiraishi, H., 1987. Hydrogen injection in BWR and related radiation chemistry. Radiat. Phys. Chem. 29, 195–199. Kent, M. C., Sims, H. E., 1992. The yield of g-radiolysis products from water at temperatures up to 300 1C. In: sixth International Conference on Water Chemistry of Nuclear Reactor Systems. British Nuclear Energy Society, London, vol. 2, p. 153–158. LaVerne, J.A., Pimblott, S.M., 1993. Diffusion-kinetic modeling of the electron radiolysis of water at elevated temperatures. J. Phys. Chem. 97, 3291–3297. Lundstrom, T., Christensen, H., Sehested, K., 2002. The reaction of OH with H at elevated temperatures. Radiat. Phys. Chem. 64, 29–33. Marin, T.W., Jonah, C.D., Bartels, D.M., 2003. Reaction of OH radicals with H2 in sub-critical water. Chem. Phys. Lett. 371, 144–149. Meesungnoen, J., Jay-Gerin, J.-P., Filali-Mouhim, A., Mankhetkorn, S., 2001. Monte-Carlo calculation of the primary H atom yield in liquid water radiolysis: effect of radiation type and temperature. Chem. Phys. Lett. 335, 458–464. Pastina, B., LaVerne, J.A., Pimblott, S.M., 1999. Dependence of molecular hydrogen formation in water on scavengers of the precursor to the hydrated electron. J. Phys. Chem. A 103, 5841–5846.
ARTICLE IN PRESS D. Swiatla-Wojcik, G.V. Buxton / Radiation Physics and Chemistry 74 (2005) 210–219 Schmidt, K.H., Han, P., Bartels, D.M., 1995. Radiolytic yields of the hydrated electron from transient conductivity. Improved calculation of the hydrated electron diffusion coefficient and analysis of some diffusion-limited e aq reaction rates. J. Phys. Chem. 99, 10530–10539. Schwarz, H.A., 1992. Reaction of the hydrated electron with water. J. Phys. Chem. 96, 8937–8941. Shiraishi, H., Sunaryo, G.R., Ishigure, K., 1994. Temperature dependence of equilibrium and rate constants of reactions inducing conversion between hydrated electron and atomic hydrogen. J. Phys. Chem. 98, 5164–5173. Stanbury, D.M., 1989. Reduction potentials involving inorganic free radicals in aqueous solution. Adv. Inorg. Chem. 33, 69–138. Stuart, C. R., Ouellette, D.C., Elliot, A. J., 2002. Pulse radiolysis studies of liquid heavy water at temperatures up to 250 1C. Atomic Energy of Canada Limited Report, AECL-12107, September 2002. Sunaryo, G.R., Katsumura, Y., Ishigure, K., 1995. Radiolysis of water at elevated temperatures –III. Simulation of
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radiolytic products at 25 and 250 1C under the irradiation with g-rays and fast neutrons. Radiat. Phys. Chem. 45, 703–714. Swiatla-Wojcik, D., Buxton, G.V., 1995. Modeling of radiation spur processes in water at temperatures up to 300 1C. J. Phys. Chem. 99, 11464–11471. Swiatla-Wojcik, D., Buxton, G.V., 1998. Modelling of LET effect on the track core processes in water at temperatures up to 300 1C. J. Chem. Soc., Faraday Trans. 94, 2135–2141. Swiatla-Wojcik, D., Buxton, G.V., 2000. Diffusionkinetic modelling of temperature effect on the radiation chemistry in heavy water. Phys. Chem. Chem. Phys. 2, 5113–5119. Swiatla-Wojcik, D., Buxton, G.V., 2001. Isotopic effect in the radiolysis of water. Diffusion-kinetic modelling up to 300 1C. Res. Chem. Intermed. 27, 875–889. Swiatla-Wojcik, D., Buxton, G. V., 2003. Modelling radiolysis of water under operating conditions, 23rd Miller Conference in Radiation Chemistry, Bialowieza, Poland.
Radiation Physics and Chemistry 74 (2005) 210–219 www.elsevier.com/locate/radphyschem
On the possible role of the reaction Hd þH2 O ! H2 þ dOH in the radiolysis of water at high temperatures Dorota Swiatla-Wojcika,, George V. Buxtonb a
Institute of Applied Radiation Chemistry, Technical University of Lodz, Zeromskiego 116, 90-924 Lodz, Poland b School of Chemistry, University of Leeds, Leeds LS2 9JT, UK
Abstract It is proposed that oxidation of water by the hydrogen atom may play an important role in the high-temperature radiolysis of water at low linear energy transfer (LET). The observed increase of the primary yield of H2, g(H2), in the radiolysis of water at high temperatures is shown to be consistent with the occurrence of the reaction Hd þH2 O ! H2 þ d OH. The temperature dependence of the rate constant for this reaction has been determined by diffusion-kinetic modelling of spur processes in neutral water. Based on a rate constant of 0.086 M1 s1 at 25 1C estimated from thermodynamic data, and literature values of g(H2) as a function of temperature, a corresponding activation energy of 66.370.8 kJ mol1 has been calculated over the temperature range 20–300 1C. The deterministic d d calculations also indicate the important role of the reaction e aq þH2 O ! H þOH in the production of the H atom at elevated temperatures in agreement with stochastic studies. r 2005 Elsevier Ltd. All rights reserved. Keywords: Radiolysis of water; High temperature; Hydrogen chemistry; Diffusion-kinetic modelling
An understanding of the radiolysis of water at elevated temperatures has been the subject of several experimental (Buxton, 2001; Elliot, 1994; Christensen and Sehested, 1980) and theoretical investigations (LaVerne and Pimblott, 1993; Swiatla-Wojcik and Buxton, 1995; Swiatla-Wojcik and Buxton, 1998; Herve du Penhoat et al., 2000; Swiatla-Wojcik and Buxton, 2000; Meesungnoen et al., 2001). Of basic interest is knowledge of the effect of temperature on the rate constants of the relevant reactions as well as on the primary yields (g-values) of the radiolysis products: e aq, Hd, dOH, H2, and H2O2. The primary or escape yields
are the radiation chemical yields1 at the end of the chemical stage of water radiolysis (Buxton, 1987). The chemical stage is controlled by diffusion and reaction of non-homogeneously distributed reactants; it begins at ca. 1012 s after the absorption of energy when the initial products have reached thermal equilibrium with their surroundings and ends with the decay of their spatial correlations at ca. 107–106 s, depending on the temperature. The most complete and reliable data on the temperature dependence of the primary yields are available for low linear energy transfer (LET) radiolysis. d d The measured g-values of e OH and H2 aq , H , continuously increase, whilst g(H2O2) decreases, with increasing temperature. The effect of temperature on the
Corresponding author. Tel.: +4842 6313193; fax.: +4842 6365008. E-mail address: [email protected] (D. Swiatla-Wojcik).
1 The radiation chemical yields are given in the units of molecules per 100 eV of absorbed energy, where 1 molecule per 100 eV corresponds to 1.0364 107 mol J1.
1. Introduction
0969-806X/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.radphyschem.2005.04.014
ARTICLE IN PRESS D. Swiatla-Wojcik, G.V. Buxton / Radiation Physics and Chemistry 74 (2005) 210–219 d d primary yields of e OH, and H2O2 is well aq, H , understood within the generally accepted diffusionkinetic model of spurs (Buxton, 1987) and has been reproduced satisfactorily by deterministic (SwiatlaWojcik and Buxton, 1995; Swiatla-Wojcik and Buxton, 1998; Swiatla-Wojcik and Buxton, 2000) and stochastic (Herve du Penhoat et al., 2000; Meesungnoen et al., 2001) calculations. Some difficulties, however, appear in matching the experimentally observed monotonic increase of the molecular hydrogen yield (Elliot et al., 1993; Sunaryo et al., 1995). Although these calculations have explained an increase of g(H2) up to ca. 200 1C as resulting from the bimolecular reaction of e aq (SwiatlaWojcik and Buxton, 1995; Herve du Penhoat et al., 2000), above 200 1C the computed g(H2) tends to decrease with increasing temperature. Various explanations of the temperature dependence of g(H2) behaviour have been proposed recently and tested by deterministic (Swiatla-Wojcik and Buxton, 1995; Swiatla-Wojcik and Buxton, 2000) and stochastic (Herve du Penhoat et al., 2000) calculations. In the extended spur diffusion model (Swiatla-Wojcik and Buxton, 1995; Swiatla-Wojcik and Buxton, 2000) we postulated that an increasing contribution of the part of unscavengeable H2 comes from channel (2) of the
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dissociative decomposition of excited water molecules. H2 O ! Hd þ d OH;
(1)
H2 O ! H2 þ d Oð1 DÞ ! H2 þ 2d OH:
(2)
Good agreement with experiment was obtained with the total contributions of channels (1) and (2) being independent of temperature but with their ratio decreasing over the range 200–300 1C from ca. 21:1 to ca. 4:1 (Swiatla-Wojcik and Buxton, 1995). Such a result is consistent with the efficiency of channel (2) in water vapour and seems reasonable in view of the diminution of hydrogen bonding in liquid water at high temperatures. Two alternative concepts have been explored through Monte Carlo simulations (Herve du Penhoat et al., 2000). The first one is linked to an increase in the scattering cross section of sub-excitation electrons and assumes a continuous decrease of their thermalisation distance with increasing temperature. The second one allows for a possible screening of Coulomb forces between two e aq and assumes that the bimolecular reaction R4 (Table 1) is fully diffusion controlled. Both of these concepts led to an increased formation of H2 in
Table 1 Reaction scheme and temperature dependence of the rate constants up to 573 K Symbol
R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 R14 R15 R16 R17 R18 a
reactiona
Temperature dependence
þ d e aq þ Haq ! H d OH ! OH e þ aq Hþ aq þ OH ! H2 O d e aq þ eaq ! H2 þ 2 OH eaq þ H ! H2 þ OH Hd þ Hd ! H2 Hd þ d OH ! H2 O d OH þ d OH ! H O 2 2 d e aq þ H2 O2 ! OH þ OH d d H þ H2 O ! OH þ H2 d OH þ H ! d H þ H O 2
2
e aq þ H2 O ! H þ OH d H þ OH ! e þ H O aq
2
Hd þ H2 O2 ! d OH þ H2 O d OH þ H O ! HO þH O 2 2 2 2 þ Hd ! e aq þ Haq d OH þ OH ! O þ H O 2 O þ H2 O ! d OH þ OH
Parameters for kact in Eq. (5) kact(298 K) (1010 M1 s1]
Ea [kJ mol1]
N
Noyes relationship
2.37
10
0
Noyes relationshipb Diffusion controlled Noyes relationship Diffusion controlledc Diffusion controlled Noyes relationship Noyes relationship Noyes relationship Arrhenius (this work) Arrhenius up to 473 Kd Elliot (1994) Elliot (1994)
8.2 — 1.35 — — 3.58 1.1 1.8 — — — —
0 — 28 — — 0 0 15.5 — — — —
1 — 0 — — 1 1 0 — — — —
Noyes relationship Elliot (1994) equilibrium eqn. and kR1(T)
0.005 — —
16.6 — —
0 — —
Elliot (1994) Elliot (1994)
— —
— —
— —
Reactions R11, R13–R18 can be neglected in modelling low LET radiolysis. Fitted to the experimental data (Elliot et al., 1994) corrected for changes in dosimetry and extinction coefficient (Stuart et al., 2002). c The experimental data of Christensen et al. (1994) neglect the temperature dependence of the absorption coefficient of e aq. d Arrhenius fit to the experimental data up to 473 K (Christensen et al., 1983) and numerical interpolation of the rate constants reported for 473–623 K (Marin et al., 2003). b
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R4 and somewhat suppressed a decrease of the scavengeable yield of molecular hydrogen. In the present work, we focus on the reaction Hd þH2 O ! H2 þ d OH as a possible source of molecular hydrogen. This reaction has not been considered previously in either the deterministic or the stochastic models of the chemical stage of water radiolysis. On the other hand, as Jay-Gerin and co-workers have recently shown (Herve du Penhoat et al., 2000; Meesungnoen et al., 2001), the contribution of the hydrated electron reaction with water rapidly increases with temperature and its cumulative yield at the end of the chemical stage at 300 1C, 0.1 ms after irradiation, is substantial. Examination of any role of the reaction Hd þH2 O ! H2 þ d OH is, however, not straightforward. This reaction has been studied in the gas phase (Baulch et al., 1992) but its rate constant in liquid water is not well established at any temperature. The temperature dependence of the rate constant for oxidation of water by the Hd atom is necessary not only for a fuller understanding of hightemperature radiolysis, it is also of basic interest for nuclear power engineers to estimate the critical hydrogen concentration in water-cooled nuclear reactor loops. The importance of the reaction Hd þH2 O ! H2 þ d OH for water and hydrogen chemistry in nuclear power reactors has been indicated by Sunaryo et al. (1995); Ishigure et al. (1987), and Shiraishi et al. (1994). Moreover, our recent simulation of the coolant behaviour proved to be very sensitive to a value of the rate constant for Hd þH2 O ! H2 þ d OH (Swiatla-Wojcik and Buxton, 2003). In this paper, the primary hydrogen and hydrogen atom chemistry is re-investigated for the low LET radiolysis of water up to 300 1C. First, the room temperature value of k(Hd þH2 O ! H2 þ d OH) is calculated from the available electrochemical data. Then the temperature dependence of k is simulated by diffusion-kinetic modelling to obtain consistency between the calculated and the measured primary yields of H2, Hd, and dOH. Finally, implications of the derived temperature dependence are discussed.
2. Method Computation of the effect of temperature on the primary yields in the low LET radiolysis of water has been based on the diffusion-kinetic modelling of the decay of an average spur. This approach leads to a set of coupled differential equations describing the temporal and spatial evolution of the concentration ci of the reactive species i at a given temperature T: X XX @ci ¼ Di ðTÞr2 ci kij ðTÞci cj þ kjk ðTÞcj ck , @t j j k d d þ ð3Þ i; j ¼ e aq ; Haq ; H ; OH; H2 ; H2 O2 ; OH ,
where each equation consists of terms representing diffusion of species i, and reactions removing i and producing i, respectively. Numerical solution of (3) was obtained using the FACSIMILE program (Chance et al., 1977) for temperatures 273pTp573 K. The input data comprised diffusion coefficients Di and reaction rate constants kij. The diffusion coefficient of the hydrated electron, and its temperature dependence have been taken from Schmidt et al., 1995. The values of Di for other species are those used previously (SwiatlaWojcik and Buxton, 1998). The assumed reaction scheme and methods for describing rate constants kij ðTÞ as a function of temperature are summarised in Table 1. The temperature dependence kij ðTÞ for reactions R1, R2, R4, and R7–R9 has been described using the Noyes relationship (cf. Elliot et al., 1990; Buxton, 2001) 1 1 k1 obs ¼ kdiff þ kact ,
(4)
where the reciprocal of the measured value, kobs, is represented by the sum of the reciprocals of the diffusion part, kdiff, and the activation part, kact. The temperature dependence of kdiff has been calculated on the basis of the Debye–Smoluchowski equation using the reaction radii and the statistical spin factors given by SwiatlaWojcik and Buxton (1995). The temperature dependence of the activation part was obtained from kact ðTÞ ¼ AT n expðE a =RTÞ.
(5)
The parameters used to evaluate kact are listed in Table 1. The Noyes relationship deals specifically with the rate coefficients of reactions in the liquid phase that are limited by the rate of diffusion at low temperatures and which become limited by the rate of the chemical step at high temperatures. Recently, Ghandi and Percival (2003) have suggested that at high temperatures and pressures, kact in the Noyes equation should be multiplied by an efficiency factor which takes into account the reactive orientation and the number of collisions between reactants over the duration of their encounter. This modified Noyes relationship predicts a broad maximum in the Arrhenius plot for bimolecular reaction rate constants at high temperatures due to the decreasing number of collisions during the encounter. Ghandi and Percival (2003) used the modified Noyes equation to extrapolate to higher temperature the reaction rate constants of the hydroxyl radical measured by Elliot and co-workers (Elliot, et al., 1990; Buxton and Elliot, 1993; Elliot and Ouellette, 1994) up to 200 1C. Such extrapolation shows the downward bending at ca. 300 1C. Recently, Lundstrom et al. (2002) have extended experimental studies on the reaction of dOH with Hd up to 233 1C, as the authors stated, their results neither confirmed nor disproved the downward bending of kR7 ðTÞ.
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As in our previous studies (Swiatla-Wojcik and Buxton, 1995; Swiatla-Wojcik and Buxton, 2000) the temperature dependences for dOH reaction rate constants kR2, kR7, and kR8 are based on the Noyes relationship with reactions R2, R7, and R8 assumed to have no activation energy so that from Eq. (5) kact ¼ AT.
(6)
Our calculated values for kR2, kR7 and kR8 at 300 1C are higher by 54%, 18%, and 14%, respectively, compared to those predicted by Ghandi and Percival (2003). It should be noted that Ghandi and Percival (2003) based their extrapolation for kR2 on experimental data (Elliot and Ouellette, 1994) which has been recalculated recently to reflect the revised dosimetry values, extinction coefficient for e aq and rate constants for the self-reaction of the hydrated electron (Stuart et al., 2002). The revised rate constants are slightly higher than those given by Elliot and Ouellette (1994). However, as shown later, the differences in kR2, kR7 and kR8 have a relatively small effect on the calculated gvalues at high temperatures. According to the extended spur diffusion model (Swiatla-Wojcik and Buxton, 1995; Swiatla-Wojcik and Buxton, 2000) initial conditions for the set of differential equations (3) can be defined by four o independent yield parameters: Goex, G o ðe aq Þ, G 1 ðH 2 Þ o o o and G 2 ðH 2 Þ and two size parameters: se , s . These parameters refer to the beginning of the radiation chemical stage, 1 ps after irradiation. Go ðe aq Þ represents o the initial yield of e aq, G ex is the initial yield of excited molecules undergoing fragmentation in processes (1) and (2), with the contribution of the latter defined by Go2 ðH 2 Þ. G o1 ðH 2 Þ specifies that part of the prompt yield of H2 which is formed in the dissociative attachment of sub-excitation electrons, e sub: d d e sub þ H2 O ! OH þ H ! OH þ H2 þ OH .
(7)
In our model, 80% of the prompt yield of H2 comes from (7) and only 20% from the dissociation channel (2). The significant role of sub-excitation electrons in the formation of H2 is in fair agreement with the experimental studies on scavenging of the precursors to e aq (Pastina et al., 1999). The size parameters soe 4so are standard deviations of two Gaussian functions which describe the initial nonhomogeneous distribution of the species within the spur depending on the length of their thermalisation path. The wider distribution characterises the initial allocation of e and that fraction of dOH and H2 which aq, OH comes from the dissociative attachment (7). The calculations have been performed assuming various number densities of the primary species in the average spur. The room temperature values of the corresponding Goi, soe and so parameters have been given previously (Swiatla-Wojcik and Buxton, 2000). The
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o o initial yields Goex, Go ðe aq Þ, G 1 ðH 2 Þ and G 2 ðH 2 Þ have been assumed to be independent of temperature and the parameters soe , so have been scaled with temperature according to the density of water (Swiatla-Wojcik and Buxton, 1995), with the result that they increased by no more than 11.5% up to 300 1C. A possible decrease of the thermalisation distance due to energy dissipation through interaction with more energetic vibrational modes of H2O at high temperatures was estimated to be between 1.1 and 2.8% (Swiatla-Wojcik and Buxton, 2000). The effect of LET has been investigated in terms of the diffusion-kinetic model of the decay of an average cylindrical track (Swiatla and Buxton, 1998, 2000) using the reaction scheme from Table 1. As in the spherical o o case the parameters Goex, Go ðe aq Þ, G 1 ðH 2 Þ and G 2 ðH 2 Þ have been assumed to be independent of temperature.
3. Results and discussion 3.1. Estimation of k(Hd þH2 O ! H2 þ d OH) at room temperature There are no direct measurements of kR10 at any temperature. The published estimates at room temperature vary from 3:6 105 M1 s1 reported by Shiraishi et al. (1994) to 10 2 M1 s1 derived by Hartig and Getoff (1982) from a photolysis study of hydrogen peroxide solution containing hydrogen. Here, we estimate kR10 by considering reactions R10 and R11 as an equilibrium: kR10 =kR11 ¼ K ¼ exp½-Dr G0 =ðRTÞ :
(8) o
To obtain the free energy change Dr G for R10, we used the values of Eo at 25 1C reported by Stanbury (1989) for the half-cell reactions (9)–(11) that make up reaction R10: Haq ! Hþ aq þ e;
2:31 V;
H2 O ! OHaq þ Hþ aq þ e; e þ Hþ aq ! 1=2 H2 ;
(9) 2:72 V;
0 V:
(10) (11)
Substitution of the obtained value Dr G o ¼ 39:6 kJ mol1 in Eq. (8) gives K ¼ 1:15 107 at 25 1C. The thermodynamic equilibrium constant expressed in terms of activities takes the form K¼
aH2 aOH ½H2 ½OH ¼ , aH aH2 O ½H
(12)
where the activities can be replaced by concentrations, except for water, which by definition has the activity a ¼ 1. In terms of rates, we have at equilibrium kR10 ½H ½H2 O ¼ kR11 ½OH ½H2
(13)
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so that kR10 ¼ kR11 K=½H2 O .
(14)
Reaction R11 was studied in liquid water for the temperature range 15–230 1C by Christensen and Sehested (1983) and for 200–350 1C by Marin et al. (2003) who observed a decrease of kR11 above 275 1C. From the Arrhenius fit to the data of Christensen and Sehested (1983) Elliot (1994) obtained kR11 ¼ 4.2 107 M1 s1 at 25 1C. Substituting this value in Eq. (14) with [H2O] ¼ 55.56 M, we have kR10 ¼ 0.086 M1 s1 at 25 1C. 3.2. Derivation of the temperature dependence for k(Hd þH2 O ! H2 þ d OH) The temperature dependence of kR10 in solution is not well established. Two estimates of the rate constant kR10 at high temperatures have been reported. At 250 1C, Shiraishi et al. (1994) estimated kR10 2.5 102 M1 s1, and Sunaryo et al. (1995) obtained kR10 20 M1 s1, but no details of the calculations were given in either case. To obtain an expression for kR10 ðTÞ, we assumed that it is analogous to that for the reaction of e aq with water, R12. Schwarz (1992) obtained an Arrhenius dependence for R12 with Ea ¼ 31.4 kJ mol1 over the temperature range 4–65 1C, whilst Elliot (1994) calculated kR12 ðTÞ as kR12 ¼ kR13 K W =ðK H ½H2 O Þ
(15)
using the data for the back reaction R13 (Han and Bartels, 1990) and the temperature dependence of the ionic dissociation constants KW and KH for H2O and Hd, respectively. We calculated that kR12 ðTÞ resulting from Eq. (15) shows an Arrhenius dependence with A ¼ 9.2 106 M1 s1 and Ea ¼ 33.4 kJ mol1. By analogy with R12, and noting that the Noyes relationship is
not applicable to slow reactions that are not diffusion limited at low temperatures, we assumed an Arrhenius dependence for kR10 ðTÞ. As a rough approximation, the activation energy for reaction R10, E aðR10Þ , should be at least equal to Dr Go þ E aðR11Þ which is 60 kJ mol1 when taking E aðR11Þ ¼ 19 kJ mol1 as determined by Christensen and Sehested (1983) from measurements of kR11 in the temperature range 15–230 1C. Using the derived room temperature value of kR10, an influence of E aðR10Þ on the primary yields has been investigated. The computed primary yields g(H2), g(Hd), and g(dOH) turn out to be sensitive to kR10 ðTÞ at high temperatures. In Fig. 1, the primary yields calculated in the absence of R10 (solid curves) are compared with the g-values obtained using kR10 ðTÞ with the activation energy varying from E aðR10Þ ¼ 60 kJ mol1 (broken curves) to E aðR10Þ ¼ 80 kJ mol1 (dotted curves). The broken and dotted curves indicate a substantial difference between the g-values of H2, Hd, and dOH. An increase of gðH2 Þ due to R10 is compensated by a decrease of g(Hd) with the sum g(H2)+g(Hd) being insensitive to the values of E aðR10Þ . Reaction R10 does not affect the temperature dependence of either gðe aq Þ or gðH2 O2 Þ. As seen from Fig. 1, oxidation of water by the d H atom may explain the observed temperature effect on gðH2 Þ. The primary yield of molecular hydrogen is the most reliable measured yield and the relevant experimental data are available up to 300 1C (Kent and Sims, 1992; Elliot et al., 1993; Sunaryo et al., 1995). Kent and Sims (1992) made all their g-radiolysis 3 measurements with [NO M to scavenge dOH 2 ] ¼ 10 and prevent reaction R11 which could destroy H2 formed in the radiolysis. The data reported by Elliot et al. (1993) were obtained using the concentration range of potassium nitrite (1–500) 103 M and the measured g(H2) was extrapolated to [NO 2 ] ¼ 0 on a plot of g(H2)
Fig. 1. The g-values gðH2 Þ, gðH2 Þ+gðHÞ, gðOHÞ, gðHÞ versus temperature calculated for 107 s assuming: Ea(R10) ¼ 60 kJ mol1 (broken curves), Ea(R10) ¼ 80 kJ mol1(dotted curves). Solid lines illustrate the temperature dependence of the g-values in the absence of R10. The experimental points are from (Kent and Sims, 1992; Elliot et al., 1993; Sunaryo et al., 1995).
ARTICLE IN PRESS D. Swiatla-Wojcik, G.V. Buxton / Radiation Physics and Chemistry 74 (2005) 210–219 1/3 vs [NO so that the scavenging power was effectively 2] zero in the spur. The data of Elliot et al. (1993) seem more appropriate for comparison with the values of g(H2) calculated at 107 s after irradiation. Calculation of the values of kR10 that could reproduce the measured yield of H2 at high temperatures has been performed for the three sets of spur parameters representing various initial densities of the species within the spur (Swiatla and Buxton, 2000). An effect of the uncertainty dkRi in the rate constants of reactions Ri, ia10 listed in Table 1 has been estimated as follows. A change, DGR10 , in the cumulative contribution of R10 at 107 s due to the relative uncertainty dkRi in the rate constant of reaction Ri was calculated and then DG R10 was reversed to a relative uncertainty in kR10. The largest dkR10 corresponds to reactions R1, R2, R7 and R12. At 300 1C dkRi ¼ 10% results in the relative uncertainty in kR10 of 3%, 2%, 0.7%, and 0.7%, respectively, and in the overall uncertainty of 7%. The lowest and the highest values of kR10 that reproduce the measured radiation chemical yield of H2 within the reported experimental error have been calculated for each set of parameters. They are marked in Fig. 2A by (x). Linear regression applied to these points results in the Arrhenius parameters AðR10Þ ¼ ð3:5 0:8Þ 1010 M1 s1 and E aðR10Þ ¼ 66:3 0:8 kJ mol1. The obtained Arrhenius dependence is shown by the solid line. Marin et al. (2003) observed the downward bending of kR11 ðTÞ in the range 250–350 1C. This observation may suggest that kR10 ðTÞ is not Arrhenius at 250–300 1C. To examine this point, we invoke the van’t Hoff equation (Atkins, 1998) describing the temperature dependence of the equilibrium constant K defined in Eq. (12):
dln K Dr H o . ¼ dT RT 2
(16)
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Eq. (16) allows us to estimate the standard enthalpy of reaction R10, Dr H o , from a plot ln K vs 1/(RT), assuming that the effect of temperature on Dr H o is negligible. Taking the experimental data for k11 (Christensen and Sehested, 1983), the derived values for k10 and the density of water (cf. Elliot et al., 1993) to obtain [H2O], the equilibrium constant K was calculated up to 180 1C and linear regression was performed for ln K. The resulting temperature dependence of ln K with Dr H o ¼ 47:34 kJ mol1 and the experimental data for k11 reported by Marin et al. (2003) was then employed to obtain k10 above 180 1C. These values are shown in Fig. 2A by solid circles. As can be seen, there is no reason to invalidate the Arrhenius dependence for k10 up to 300 1C. At 300 1C, the point 2:22 104 M1 s1 compares well with ð3:18 1:25Þ 104 M1 s1 obtained from the derived Arrhenius parameters. The temperature dependence of the primary yield gðH2 Þ is shown in Fig. 2B by the solid curve. The dotted lines demarcate the limits obtained on the basis of the 95% prediction interval of the derived function kR10 ðTÞ. All the sets of experimental data show the increase with temperature and the flattening at 200 1C. These features are reproduced by the computed temperature dependence and the agreement with the data of Elliot et al. (1993) and of Sunaryo et al. (1995) is good. There is more uncertainty in the experimental points of Kent and Sims (1992) which are slightly above the computed values of gðH2 Þ. 3.3. The possible role of the reaction Hd þH2 O in the radiolysis of water Using the derived Arrhenius parameters for R10, one obtains kR10 ¼ ð3:18 1:25Þ 104 M1 s1 at 300 1C. Although the reaction of the hydrogen atom with water
Fig. 2. A. The calculated values of kR10: (x) values of kR10 that reproduce the measured primary yield of H2 and the Arrhenius fit to (x) (solid line); () kR10 obtained from the thermodynamic equilibrium constant K defined in Eq. (12) and the experimental data for k11 reported by (Marin, et al., 2003). B. Calculated primary yield of molecular hydrogen as a function of temperature (solid line). The dotted lines demarcate the limits obtained on the basis of the 95% prediction interval for the derived function kR10(T). Experimental data are from: Elliot et al. (1993) (’), Kent and Sims (1992) (K), and Sunaryo et al. (1995) (m). The dashed curve shows the results of Monte Carlo simulation (Herve du Penhoat et al., 2000).
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does not affect the chemistry in the spur at temperatures below 180 1C, because of its high activation energy it becomes significant at higher temperatures. Its contribution to gðH2 Þ, expressed in molecules per 100 eV, increases rapidly with temperature from ca. 0.005 at 200 1C to ca. 0.1 at 300 1C. The negative contribution of the back reaction R11 does not exceed 0.001 molecules per 100 eV. The extents DGi of the reactions contributing to gðH2 Þ are presented in Fig. 3. At 25 1C, 68% of gðH2 Þ is produced via processes (2) and (7) during the physicochemical stage, 1015–1012 s after irradiation, and less than one-third is formed within spurs in R4 and R5. The role of the chemical stage increases with temperature. At 300 1C the total contribution of processes (7) and (2) is less than 50%, whilst 24% of gðH2 Þ comes from R4, 16% from R10, 11% from R5, and 1% from R6. The rapid increase of the cumulative contribution of R10 is responsible for the increase of gðH2 Þ at high temperatures seen in Fig. 2B. On the other hand Monte Carlo calculations predict a slight decrease of gðH2 Þ above 220 1C (Herve du Penhoat et al., 2000). The result of Herve du Penhoat et al. (2000) is illustrated by the dashed curve in Fig. 2B. Verification of the computed temperature effect on the yield of H2 requires more experimental data, particularly at temperatures even higher than 300 1C. It is pertinent to note that Elliot et al. (1993) reported very high values of gðHd Þ þ gðH2 Þ in H2O, and gðDd Þ þ gðD2 Þ in D2O, above 2001C for solutions containing 103 mol dm3 acetone and 101 mol dm3 methanol to scavenge e aq and hydroxyl radical, respectively. At 300 1C, gðHd or Dd Þ+g(H2 or D2) was ca. 2.7 molecules per 100 eV in both liquids. Moreover, they showed that the excess deuterium at 300 1C came predominantly from the heavy water, which is qualitatively consistent with the mechanism proposed here. Diffusion-kinetic
Fig. 3. Cumulative contributions, DGi ðH2 Þ, of reactions Ri and processes to gðH2 Þ in low-LET radiolysis of water as a function of temperature.
modelling of the D2 formation in heavy water is in progress. To show whether or not R10 contributes to the hydrogen formation, one could measure gðH2 Þ and gðHDÞ in H2O containing a deuterated scavenger for Hd to form HD and look at the ratio of gðHDÞ to gðH2 Þ as a function of temperature at scavenger concentrations where no spur scavenging occurs. Thus, gðHDÞ would be a measure of H escaping from the spurs and gðH2 Þ would be a measure of the spur reactions forming H2 plus any H2 coming from R10. Our present calculations show an important contribution of reaction R12 to the production of the Hd atom at high temperatures in agreement with the stochastic modelling (Herve du Penhoat et al., 2000; Meesungnoen et al., 2001). The proportion of Hd formed in R12, which is 25–30% depending on the assumed density of the species within the spur (Swiatla and Buxton, 2000), agrees well with 25% obtained from the stochastic study of low-LET radiolysis (Herve du Penhoat et al., 2000). The loss of Hd in reaction R13 is negligible. Compared to the previous calculations (Swiatla and Buxton, 2000), incorporation of R12 results in a slight increase of gðHd þH2 Þ and in a slight decrease of gðe aq Þ at 300 1C. Considerable effects due to reactions R10 and R12 can be expected at times longer than 107 s. Time dependences of the computed radical and molecular yields are presented in Fig. 4 for 100, 200, and 300 1C. A 7 6 rapid decrease of gðe s aq Þ in the time interval 10 –10 caused by a decay of e due to its reaction with water aq has been also obtained in the stochastic calculations for low LET (Meesungnoen et al., 2001). A rapid increase of the yield of H2 and of dOH resulting from oxidation of water by the Hd atom is seen in Figs. 4f and 4b, respectively. Reaction R10 does not, however, affect the formation of H2O2. The time dependences of g(Hd) presented in Fig. 4e reflect competition between the formation of the Hd atom in R12 and its decay in R10. A significant increase of g(Hd) is observed at 200 1C but the developing contribution of R10 leads to the maximum exhibited by the time profile computed for 300 1C. As can be expected, reactions R10 and R12 have no effect on the temporal profiles calculated for the sum d gðe aq Þ þ gðH Þ þ gðH2 Þ presented in Fig. 4d. The dotted lines in Fig. 4 show time dependences of the g-values calculated for 300 1C using the values of k for the partially diffusion-controlled reactions R2, R7, and R8 suggested by Ghandi and Percival (2003) which, as discussed in the previous section, are lower than the rate constants used in our modelling. At high temperatures, these partially diffusion-controlled reactions are much less important than at room temperature. For example, reaction R2 is responsible for 58% of the loss of e aq at 25 1C and for only 28% at 300 1C (SwiatlaWojcik and Buxton, 2001). In consequence, the g-values at the end of the chemical stage, 107 s after irradiation at 300 1C, are not significantly altered.
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Fig. 4. Time dependences of the g-values calculated for temperatures: 100 1C (dash), 200 1C (dash-dot), and 300 1C using the values of kR2, kR7 and kR8 from (Swiatla and Buxton, 2000), solid lines, and from (Ghandi and Percival, 2003), dotted lines (see text).
Fig. 5. Cumulative contributions DGi of reactions R10 and R12 (solid symbols) and their back reactions R11 and R13 (open symbols), respectively versus LET calculated for 300 1C and 0.1 ms (left panel), 1 ms (right panel). Solid lines have been added to guide the eye.
Under typical conditions used in pulse radiolysis to measure kR7, the introduction of reaction R10 has a negligible effect on the kinetics of the decay of dOH up to 200 1C. Our preliminary simulation shows that R10 is not responsible for any curvature in the plots of kR7 vs 1/T up to 200 1C. The difference in the first half-life of d OH when R10 is included, of 5 108 s, is not detectable in an experiment. The simulation shows, however, a two-fold increase of the first half-life of dOH at 300 1C
The reactions of radicals with water can be expected to become less important with increasing LET. Fig. 5 presents cumulative contributions DG Ri calculated up to 107 s (left panel) and up to 106 s (right panel) for R10, R12 and the corresponding back reactions R11, R13 as a function of LET at 300 1C. Calculations have been performed for the average LET of 0.2, 20, 40, 60 and d 80 keV mm1. As can be seen, the decay of e aq and H in reactions with water become more limited with increasing LET whilst their reformation in the back reactions is
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more significant. For 80 keV mm1 the net contributions DGR12 DGR13 and DGR10 DGR11 at 107 s decrease to 0.005 and 0.07 molecules per 100 eV, respectively, but at 106 s DGR12 DGR13 ¼ 0:14 and DGR10 DGR11 ¼ 0.46 molecules per 100 eV. This result indicates that reactions R10 and R12 may play an important role in water chemistry of water-cooled nuclear reactors (PWRs and BWRs). They should also be taken into account when assessing the radiation chemistry in supercritical water-cooled reactors. In summary, the results of the calculations presented d here indicate that the reactions of e aq and H with water may contribute significantly to the chemical stage of water radiolysis at high temperatures, especially at low LET. Moreover, the reaction Hd þ H2 O ! H2 þ d OH could be responsible for the observed increase of molecular hydrogen yield at high temperatures. The computed Arrhenius dependence for the rate constant of this reaction in liquid water shows an increase of kR10 from 0.086 M1 s1 at 25 1C to (3.1871.25) 104 M1 s1 at 300 1C. Experimental measurements to test the validity of both the suggested role of the reaction Hd þ H2 O ! H2 þ d OH and the derived temperature dependence of its rate constant are an important requirement.
Acknowledgment D. S.-W. thanks the Technical University of Lodz for financial support.
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