J. Phys. Chem. 1988, 92, 2526-2536
2526
Computer Simulations of Photochemlcal Water Cleavage Systems. 2. Effects of Particle Size and Ion Concentration on the Production of Hydrogen from Water Mediated with a Colloidal Catalyst Pierre M. Lenoir, Richard E. Sassoon,*t and John J. Kozak* Department of Chemistry and Radiation Laboratory, University of Notre Dame, Notre Dame, Indiana 46556 (Received: July 6, 1987)
An in-depth analysis of the reactions of the reduced radical cation of methylviologen (MV") with a colloidal platinum catalyst (Pt,) has been performed using our computer simulation technique. Rate equations, containing both diffusion- and activation-controlled components, were derived for the individual stages of our reaction scheme. The electrostatic effects due to the charge carried by all the adsorbed species on the particle were treated by an approximate solution of the Poisson-Boltzmann equation for a charged sphere and the results incorporated into the value of the diffusion-controlled component of the rate constant. Optimization of the activation-controlled components was achieved by carrying out numerous computer simulations on the reaction system until good agreement was found with experimental results for the behavior of the MV" concentration profiles with time over the pH range 1-1 1, The significance of the values of the rate constants is discussed by considering the effects on the behavior of the overall reaction system realized by decreasing their values. Predictions are made concerning the dependence of the overall reaction rates on catalyst particle size in the range of radii from 15 to 100 A. A decrease in the rate of reaction on increasing the size of particle is generally observed and is attributed to a corresponding reduction in the total number of accessible reactive surface sites at constant Pt atom concentration. The variation of order with respect to platinum as a function of size and pH was also investigated by using our present treatment and a thorough analysis of the behavior of the system again revealed the importance of fractional coverage terms in explaining the behavior. Moreover, it was found that the order with respect to [Pt] also depends on whether the rate-controlling steps in the system are diffusionor activation-controlled. Finally salt effects on the system were investigated for added adsorbing and nonadsorbing ions. Only those ions which adsorb on the catalyst markedly affect the behavior of the system and the degree of this effect depends on the charge and concentration of the adsorbing ions and the pH of the system.
Introduction
Photoinduced-electron-transfer systems have been the subject of extensive investigation during the last decade because of their importance in solar energy conversion systems.'-3 The use of colloidal catalysts in mediating the production of useful chemical fuels from the photoredox products has also been well studied,',"' and in our previous studys we proposed a reaction mechanism to describe the colloidal platinum-mediated production of hydrogen from water using the electron-acceptor species, methylviologen, (MVz+) as described in eq a. This reaction has been studied in MV"
Pt + H+ e MV2+ + 1/2H2
considerable detail by using various experimental technique^.^-^' Computer simulations of pulse radiolysis investigations of this reaction yielded excellent agreement between the suggested reaction scheme and the experimental results over a wide range of pH. Moreover, the effects of varying the concentrations of MV2+, initial M Y + , and Pt found experimentally were also very well reproduced with our computer modelling technique. The value of performing computer simulations on a chemical system was demonstrated since it enabled us not only to probe the reaction intermediates which were identified as adsorbed electrons, H atoms, hydride ions, and H2 molecules but also provided an explanation for the greater-than-first-order dependencies on the concentration of platinum catalyst found experimentally for the reaction of MV" with Pt without invoking collisions between the p a r t i c l e ~ . ~ ~The l ~ Jreason ~ was attributed to the variation in the number of free sites on the catalyst available for adsorption by MV" which was considered in addition to the usual first-order dependence of rate of [Pt] expected for the reaction of MV'+ with Pt particles. The reaction system given in eq a thus provides a convenient model for the study of reaction kinetics at the colloid-solution interface, a field of considerable interest in itself, 11,19-23 Several aspects of the reaction system were, however, not treated rigorously in the previous investigation. For example, an arbitrary 'Present address: Vitreous State Laboratory, Catholic University of America, Washington, DC 20064.
0022-365418812092-2526$01SO10
parameter, E , was included in the rate equations in order to account for the buildup of charge on the catalyst during the course of reaction at different pH's. The value of this electrostatic factor was determined by optimizing the fit of the rate equations to the experimental data. However, the value of E was not allowed to vary during the course of the reaction even though it is clear that (1) Kalyanasundaram, K.; Gratzel, M. Coord. Chem. Reo. 1986, 69, 57. (2) Fendler, J. M. J. Phys. Chem. 1985, 89, 2730. (3) Claessan, S.; Engstrom, L. Solar Energy-Photochemical Conoersion and Storage; National Swedish Board for Energy Development: Stockholm,
Sweden, 1977. (4) Gratzel, M. Acc. Chem. Res. 1981, 14, 376. (5) Kiwi, J.; Kalyanasundaram, K.; Grltzel, M. Visible Lighr Induced Cleavage of Water into Hydrogen and Oxygen in Colloidal Microheterogeneous Systems; Springer-Verlag: Heidelberg, West Germany, 1981. Struct. Bonding (Berlin) 1982, 49, 37. ( 6 ) Photogeneration of Hydrogen; Harriman, A., West, M. A., Eds.; Academic: London, 1982. (7) Brandys, M.; Sassoon, R. E.; Rabani, J. J. Phys. Chem. 1987,91,953. ( 8 ) Sassoon, R. E.; Lenoir, P. M.; Kozak, J. J. J . Phys. Chem. 1986, 90, 4654. (9) Matheson, M. S.; Lee, P. C.; Meisel, D.; Pelizzetti, E. J . Phys. Chem. 1983, 87, 394. (10) Brandeis, M.; Nahor, G. S.; Rabani, J. J.Phys. Chem. 1984.88, 1615. (1 1) Albery, W. J.; Bartlett, P. N.; McMahon, A. J. J. Electroanal. Chem. Interfacial Electrochem. 1985, 182, 7. (12) (a) Miller, D.S.; Bard, A. J.; McLendon, G.; Fergusson, J. J . Am. Chem. SOC.1981, 103, 5336. (b) Miller, D. S.; McLendon, G. Ibid. 1981, 103, 6791. (13) (a) Keller, P.; Moradpour, A.; Amouyal, E.; Kagan, H. B. Nouu. J . Chim. 1980, 4, 377. (b) Amouyal, E.; Grand, D.; Moradpour, A,; Keller, P. Ibid. 1982, 6, 241. (14) (a) Kiwi, J.; Gratzel, M. J . Am. Chem. SOC.1979, 101, 7214. (b) Kalyanasundaram, K.; Gratzel, M.; Angew. Chem. 1979, 91, 759. ( 1 5) Ebbesen, T. W. J . Phys. Chem. 1984, 88, 41 3 1. (16) Venturi, M.; Mulazzani, Q.G.; Hoffman, M. 2. J . Phys. Chem. 1984. 88. 912. (17) NenadoviE, M. T.; MiEiE, 0. I.; AdziE, R. R. J . Chem. SOC.,Furaduy Trans. I 1982, 78, 1065. (18) Meisel, D.; Mulac, W. A.; Matheson, M. S. J . Phys. Chem. 1981,85, 179. . .
(19) Spiro, M. Faraday Discuss. Chem. SOC.1984, No. 77, 275. (20) Dung, M. H.; Kozak, J. J. J . Chem. Phys. 1982, 76, 984. (21) Astumian, R. D.; Schelly, Z. A. J. Am. Chem. SOC.1984, 106, 304. (22) Mysels, K. J. J . Phys. Chem. 1982, 86, 4648. (23) Albery, W. J.; Bartlett, P.N.; Wilde, C. P.; Darwent, J . R . J . Am. Chem. SOC.1985, 107, 1854.
0 1988 American Chemical Society
Production of Hydrogen on a Colloidal Catalyst such a variation does occur. Furthermore, the effects of adding to the reaction mixture salts which do not directly participate in the reaction scheme but which vary the ionic strength and which may or may not adsorb on the catalyst particle could not be treated. Finally, a complete dependence of rate of disappearance of MV" on the size of the platinum particle could not be achieved in our earlier treatment. In the present study we add further reactions to the previously proposed scheme in order to account for the adsorption of otherwise nonparticipating ions on the catalyst surface.24 The reactions are then described by a set of kinetic equations in which the overall charge and size of catalyst particle are intrinsically included so that our previous kinetic model is extended to include all sizes and adsorbed charges of catalyst particles. In generalizing our previous model we also discuss the relevance and importance of the numerical values which we assign to the rate constant in our optimization procedure. On carrying out the computer simulations described in this article it is now possible to apply our reaction model to any size of particleZ5and it may be used for predicting optimum experimental conditions for producing hydrogen from water in a colloidal metal, catalyst-mediated photochemical process.
The Reaction Scheme The mechanistic scheme proposed for the catalyst-mediated production of hydrogen is essentially the same as the one put forward in our first paper and is described in Scheme I. The species A and A- represent the electron acceptor, methylviologen, in its oxidized and reduced forms, respectively, and the subscripts sol and ads describe the species in solution or adsorbed on the particle, respectively.
SCHEME I
The Journal of Physical Chemistry, Vol. 92, No. 9, 1988 2521 hydrogen. These final adsorption reactions are included in our scheme here however since they provide a more complete model to describe the charge and salt effects which occur in the overall system. As noted in our previous article, other side reactions are either already corrected for in the experimental data used (e.g., catalytic reduction of the polymer stabilizerlo) or are considered negligible for a single pulse. Such reactions could include destructive hydrogenation of the acceptor specie^^^,^^ or the slow reaction of methylviologen with OH- which may be important at p H s above those treated in this These reactions should however be included in a simulation of any continuously illuminated steady-state reaction system.
The Rate Equations In order to produce a more general model in which catalyst particle size and charge effects are considered, a more detailed approach to the kinetic rate equations is necessary than that given in our previous study* and is presented below. The equilibria given in Scheme I may be divided into two types. Firstly, there are those reactions which occur solely on or within the particle and do not involve species in solution. These are equilibria 2, 5, and 9 and may be described in general by the equation
LL
Bads
Cads + Dads
(b)
where Bads,Cads, and Dadsrepresent the adsorbed reactants and products in equilibrium b. The equilibrium constant, Kb, is equal to kb/k-b. In such equilibria the general rate equations may be written as shown in eq c and d, where vb and VYbrepresent the rates of the forward and reverse reactions of equilibrium b in units of M s-l, n is the average number of Pt atoms per particle, and [PtJ is the total platinum particle concentration in solution. These vb
=
(c)
kb[Badsl
= k-btcadsl [Dads] /n[Ptcl
( 4 rate equations are identical with those used in the previous study8 and the size of the catalyst particle is accounted for in the rate equations by the denominator n[Pt,]. All other processes taking place in the chemical system involve reactions between species in solution, represented by Esol,and species adsorbed on the catalyst particle, represented by Fads, yielding an adsorbed product, Gads. These processes may thus be described by the general equilibrium equation
+ Fads Gads (e) Note that, in equilibria 1, 3, 7, 8, 10, 11, and 12, Fadsrepresents a set of one or more unoccupied platinum surface sites. Each of these equilibria in this second category may be divided into two stages:8 namely a diffusion-controlled stage in which E,, diffuses either toward or away from the particle, and an activation-controlled stage in which the actual chemical reaction (in which Gads is either produced or destroyed) occurs. This is represented in a general fashion in eq f. Here, k,d'ffand keAiffrepresent the rate The processes involved in producing H2 from the reduced acceptor A- include adsorption of A- onto the catalyst (eq 1) followed by transfer of its additional electron into the particle (eq 2) and then desorption of the original acceptor A (eq 3). Protonation of the electrons in the catalyst may occur with either H+ ions in solution (reaction 4) or adsorbed H' ions (reaction 9 following reaction 8). Adsorbed H atoms are produced which may react with an additional electron (reaction 5) to produce the adsorbed hydride ion and further protonation (eq 6) yields adsorbed H z molecules which may desorb via process 7 . Adsorption of OHions on the catalyst occurs at high pH (eq 10) while adsorption of other anions in solution, represented by X- in eq 11, or of other cations in solution, represented by Y+ in eq 12, may also take place without participating directly in the reaction sequence producing (24) Furlong, D. N.; Launikonis, A,; Sasse, W. H. F. J . Chem. SOC., Faraday Trans. 1 1984, 80, 571. (25) Keller, P.; Moradpour, A. J . Am. Chem. SOC.1980, 102, 7193.
Esol
+ Fads
keW Er=rRe
-k Fads
-
k.'h"
' Gads
(0
constants for diffusion of E,, toward and away from the surface of the Pt particle, respectively. E,=,, describes the species E at the particle surface, and k,"hm and ke4""' are the actual chemical rate constants for the forward and reverse reactions between E on the particle surface and Fadsto produce Gads. It is readily shown28 that the overall rate equations for the forward and reverse processes in these type of equilibria may be described by the expressions given in eq g and h. Here, V, and (26) (a) Johansen, 0.; Launikonis, A.; Loder, J. W.; Mau, A. W.-H.; Sasse, W. H. F.; Swift, J. D.; Wells, D. Ausr. J. Chem. 1981, 34, 981. (b) Johansen, 0.;Launikonis, A.; Loder, J. W.; Mau, A. W.-H.; Sasse, W. H. F.; Swift, J. D.; Wells, D. Ibid. 1981, 34, 2347. (27) Novakovik, V.; Hoffman, M. Z. J . Am. Chem. SOC.1987,109,2341. (28) Levine, I. N. Physical Chemistry; McGraw-Hill: New York, 1978; pp 115-777.
2528
The Journal of Physical Chemistry, Vol. 92, No. 9, 1988
Lenoir et al. TABLE I: Diffusion Coefficients of Species under Investigation 1O'O X
V, represent the velocities of the overall forward and reverse reactions in equilibrium e, respectively, OT is the fraction of sites on the particle occupied by all adsorbed species, and a is the number of free sites occupied or liberated in the equilibrium process. Note that, in equilibria 1, 3, 7, 8, 10, 11, and 12, [Fads]is given by nJPt,], where n, is the number of surface Pt atoms per particle, while in equilibrium 4 it is given by fs[e-ads] since the adsorbed electron, in contrast to all the other adsorbed species, is assumed to be present throughout the whole bulk of the particle.8 The term fs is the fraction of Pt atoms in the particle which are present on the surface and is given by n,/n. The value of a is taken as 4 for the reduced and oxidized acceptors, M V " and MV2+,in equilibria and is 2 for equilibrium 7 where adsorbed 1 and 3, respectively,8*18 H2 is assumed to occupy two sites. In all the other equilibria, the number of platinum sites occupied by adsorbing species is assumed to be 1. Separation of the rate constants for the diffusion-controlled and chemically controlled stages for these processes allows us to consider in this manner each stage separately with respect to particle size and charge effects and, hence, the overall theoretical treatment is greatly facilitated. Optimization Method for Determining Rate Constants In order to fit the model described above to the experimental data, numerical values must be assigned to the rate constants of the reactions in Scheme I. In the present simulation of the methylviologen-platinum system, adsorption of cations, as described in equilibrium 12, was not considered, although adsorption of anions, as given by eq 11, is important even in the absence of added salts due to the presence of chloride counterions of methylviologen. Hence, rate constants for equilibrium 11 were determined for the chloride anion in the optimization procedure. The rate constants for equilibria 2, 5, and 9 are identical with those used in our previous study while the other equilibria require determination of both the diffusion and the chemical components of the rate constants. The values of kediff and kediffwere determined (see text below) such that their values would change during the course of reaction as the total charge of adsorbed species on the particle varied. The time-independent Debye-Smoluchowski equation was used to determine kediffand Thus, kew is given by the equation kediff =
4x
1
0
3
q
+~D ~~ , ) (9
I
where L is Avogadro's number, DE and Dptcare the diffusion coefficients of species E and the platinum particle, respectively, and I is given by the equation
I =
J-;
exP(ezEwr)/kTl dr r2
6)
where e is the electronic charge, zE is the charge of species E, \k(r) is the potential at a distance r from the center of the platinum particle, k is Boltzmann's constant, T is the absolute temperature, and rp,, is the radius of the catalyst particle. kediffis given by the same expression except that the sign of the potential, Jr(r), is reversed. The values of the diffusion coefficients for the various species in the chemical system under study are given in Table I. The evaluation of g ( r ) is carried out according to the method of Bentz30 in which an approximate solution of the Poisson-
diffusion coeff,
1010X
species
m2 s-I
MV'+"
5.0
€3 za
MV2+a
5.0 93.3 52.7
Pt,".'
H'b OH-b
diffusion coeff,
species
c1-
36.0
20.3 0.9
Determined from molecular dimensions using the Stokes-Einstein relation. Determined form ionic mobilities. Moore, W. J. Physical Chemistry, 5th ed.; Longman: London; p 435. 'Radius of particle is 25
A.
Boltzmann equation for a negatively charged sphere is given for a surrounding electrolyte containing mixed monovalent and divalent ions. The solution of \k(r) for a positively charged sphere is determined in an analogous fashion except the unique solution for expf-e*(r=r&)/kfi is that for which 0 C exp(-e\k(r=r&)/k'l)
1 / ~ as the surface potential does not exceed 50-75 mV.30 The integrals, I, are then evaluated on integrating from r = rR,to (r = rpt 10000 A) by using Simpson's rule with intervals of 0.5 A. The integrals in eq j were evaluated for every 0.2 charge per particle over a wide enough range of particle charge to treat all particle charges of all adsorbed species in the simulations to be carried out. Linear interpolation of the values of the integrals was performed in order to determine their values for intermediate average particle charges. With knowledge of values of I , kediff and kediffwere calculated at each step of the computer simulation (via eq i) and hence their values vary with time during the course of reaction as the particle charge due to adsorbed species varies. Estimates of kechemand kc*hemcould be made by comparison of computer-simulated fits obtained in the previous study8 with those obtained in the present investigation. Simulations were performed under the conditions: pH 8, [Pt] = 4 X lo4 M, [MV2+] = 4 X lo4 M, initial [MV"] produced per pulse = 13.3 hM, and rpt = 25 A, using both the kinetic analysis given in the previous paper with E = 1 and the kinetic analysis given the present study M and no adsorption of Y'. with [X-] = [Cl-] = Values of kechemand ke-chemwere estimated in the following manner in the present kinetic treatment in order to obtain the optimum agreement between the decays of [MV"] observed for the two treatments. Since MV2+ was found to be the major adsorbing species at pH 8 in the previous study, the rates of the C1- adsorption and desorption reactions were introduced and adjusted together with those of the MV2+ adsorption and desorption reactions in order to fulfill two requirements. Firstly, the total fractional coverage of the particle including that by C1using the present kinetic treatment should be approximately the same as that in the previous investigation. Secondly, the total overall charge on the catalyst should be, on average, about equal to zero (corresponding to E = 1 in our previous investigation) during the course of reaction. Then, assuming zero charge on the particle, the values of kediffand keAiffmay easily be calculated by using eq i for all the equilibria in the mechanistic scheme involving diffusion to or away from the catalyst particle. Estimates of kechemand ke*hemcould then be evaluated by solving the simultaneous eq k and 1, using the rate constants ke and k, from the previous study and approximately average values of [Fads]and OT determined from analysis of results from the previous investigation. k,diffk chem e k, = (k) kediff k echem[FadsI(l- &)e/[ptcI
+
+
~
(29) Capellos, C , Bielski, B H J In Marhemarical Descriprion o/ Chemical Kinetics i n Solution, Kreiger, R E , E d , Kinetic Systems Huntington, NY, 1984, pp 106-1 10
m2 s-l
(30) Bentz, J. J . Colloid Interface Sci. 1981, 80, 179.
The Journal of Physical Chemistry, Vol. 92, No. 9, 1988 2529
Production of Hydrogen on a Colloidal Catalyst a
b I5
I
I
a
lo
z
.> I
5 -
0
2
iio C
I
d
time ( m i l
t
*O
time (SI PH Figure 1. Three-dimensional surfaces showing computer simulations of the concentration changes of MV" as a function of time and pH over the H range 1-1 1 following pulsed irradiation of a solution containing 0.2 M 2-propano1, 4 X lo4 M MV2+ and 4 X lo4 M Pt (radius of colloid = 15 The concentration of MV" initially produced is 13.3 pM. (a) View of surface over the short time scale (0-25 ms) rotated 45O clockwise. (b) View of surface over the short time scale rotated 45O anticlockwise. (c) View of surface over the long time scale (0.025-2.525 s) rotated 45O clockwise. (d) View of surface over the long time scale rotated 45' anticlockwise.
1).
TABLE 11: Optimized Chemical Rate Constants eauilibrium k-chcm,oM-' s-I 1 3 4 6 7 8 10 11
1.32 x 1.00 x 5.78 x 1.61 X 1.17 X 5.00 x 2.70 x 1.66 x
109 107 109 lo8 lo8 103 109 107
k.-chcm 1.34 5.05 1.13 5.03 7.02 2.50 3.39 2.00
(1
s-I
x 105 X lo2 lo6 10' lo1 10' x 104 x 104 X X X X
See text for details concerning the determination of these rate constants.
The obtained estimated values of kechemand ke-chem were then optimized to yield more accurate values of these rate constants by performing numerous simulations over the whole pH range studied experimentally (as described in the previous study) in order to produce the values for the optimum chemical rate constants given in Table 11. The program used in order to carry out the modelling described in this work was developed at the Lawrence Livermore Laboratory31 and the simulations were performed on a V A X 11/780 superminicomputer.
The Model Chemical System The chemical system chosen to be modelled is essentially the same as the one given in our previous study and is based on the pulse radiolysis investigations carried out by Brandeis et al.IO The reaction system usually contains MV2+and Pt, both at concentrations of 4 X 1O4 M, and the irradiating pulse would normally produce an initial concentration of 13.3 pM MV'+. Note that (31) (a) Numerical Solution of Ordinary Differential Equations; UCRL-75652;Lawrence Livermore Laboratory: Livermore, CA, 1974. (b) Hindmarsh, A. C. Assoc. Comput. Mach., Trans. Math. Software 1975, I , 71.
unless otherwise stated, the concentration of Pt is given in terms of atoms rather than particles. The reactions leading to the rapid production of MV" are given in our previous article. In the present study, several sizes of catalyst particle were investigated and the parameters associated with their sizes and diffusion properties are given in Table 111. It is readily seen that, as the size of the particle is increased, although both the number of atoms per particle and the number of surface atoms on the particle also increase, the fraction of platinum atoms which are present as surface atoms actually decreases with increasing size of particle. Furthermore, the diffusion coefficient of the particle decreases with increasing size and the consequences of such variations with particle size on our chemical system will be examined later in the paper. The ionic strength of a typical chemical system was set a t 2 X M, in which the added salt is assumed to be the necessary buffer in order to ensure a constant p H throughout the decay. The buffer is assumed to be made up of ions which do not adsorb onto the catalyst particles. At very low pH's, where the ionic strength is clearly greater than this value, higher values of ionic strength were used in our model with the added acid being the predominant component in determining this property. It should also be noted that, when buffer is present in our system, the time required for the buffer system to restore the pH following the pulse is assumed to be very short as compared to the other concentration changes to be monitored. Hence, it is assumed that the pH is effectively constant throughout the reaction sequence at all the pH's studied in this investigation.
General Description of the Reaction System The decays and formation of [MV"] following pulse irradiation of the reaction system vary considerably with pH and size of platinum particle, and the results of computer simulations of the chemical system are given in Figures 1 and 2 for particles of radius
2530
The Journal of Physical Chemistry, Vol. 92, No. 9, 1988
Lenoir et al. b
1
0
C
d
20
15
I to 3
-5:
5
6 12
3
0
1
Figure 2. Similar three-dimensional surfaces as in Figure 1 for identical experimental conditions except that the radius of the Pt colloid = 100 A. (a and b) Short time scale is 0-125 ms. (c and d) Long time scale is 0.125-6.125 s.
TABLE III: Size and Diffusion Parameters for Several Colloidal Platinum Particle Sizes rh: A n:b na fib 10ioDpt: m2s-I 15 430 935 0.459 1.47 4330 0.298 0.88 25 1290 0.158 0.44 50 5460 34600 100 22500 277000 0.08 1 0.22 "See text for explanation of symbols. bCalculated according to the expression: f, = n,/n = I - (1 - 2rR/rpt)3 where rR is the radius of a Pt atom. See ref 11. CDiffusioncoefficient for the platinum particles calculated according to the Stokes-Einstein relation.
15 and 100 A, respectively. Parts a and b of the figures show the time profiles of [MV'+,,,] over short time scales from two different views in order to reveal the decays of the total concentration of MV" at every pH. Parts c and d of the figures show the corresponding changes in [MV'+to,]on longer time scales until equilibrium of the reaction system is reached. The values of the first half-lives for the decay or formation of [MV'+,,,] in the fast and slow processes, together with their [MV",,] values at the end of these processes, obtained by our computer simulation technique, are summarized in Table IV for three different sizes of platinum particles. At low pH's, a single decay in [MV'+,,,] to zero is observed which generally becomes faster as the pH is raised, except at intermediate p H s where this fast decay occurs at an approximately constant rate. Furthermore, beginning at around neutral pH's, only a plateau is reached after the first decay in which its concentration value increases with increasing pH. A second slower decay of [MV'+,,,] is observed to take place following the fast process in this pH range. At the more alkaline pH's, a smaller fraction of [MV'+,o,] decays via the fast process; the slow process changes from a decay to a buildup in MV'*,, concentration. A detailed description of the system as revealed by our computer modelling technique was given in the previous study for a catalyst of 25 A radius together with an in-depth analysis of the behavior of all the intermediates on the catalyst particle.6 Since the results
obtained from computer simulations in this work are virtually identical for this size of catalyst, this discussion will not be repeated. The same general description of intermediate formation and disappearance applies also to the other sized particles. Regarding intermediate formation on the catalyst particle, it should be noted here that further experimental evidence has been obtained in a pulse radiolysis investigation using both optical and conductivity monitoring techniques32 in which some of the predictions of our previous computer simulation results have been confirmed. In particular it w a s found that the charge injection process by the reduced acceptor species to the catalyst particle yielding the adsorbed electron detected optically is followed by a slower protonation process to yield adsorbed protonated reduced species (such as H-ads) detected by comparison of conductivity and optical results at around pH 9.5. This could be seen in the results of our previous investigation where the rate of H-adsformation at this pH is considerably slower than the rate of formtion of e-ads on the short-time scale. Furthermore, several pulse radiolysis s t ~ d i e shave ~ ~ provided ,~~ additional evidence showing the importance of adsorption of MV" and MV2+,given in equilibria 1 and 3, respectively, in Scheme I, in determining the orders with respect to [Pt] for the fast decay of MV" in the reaction system. In our present treatment of the catalytic system for hydrogen generation from water on a platinum colloidal particle we have included the adsorption on the catalyst particle of the counterions of MV*+, namely C1-, in our reaction scheme. The consequences of considering this reaction are revealed in the results of our present computer simulations where it is found that MVZ+adsand C1-ads are the predominantly adsorbed species, particularly at intermediate pH's, and their relative degrees of adsorption are such that the overall charge on the catalyst remains small. Adsorption by MV" is usually followed by rapid reaction such that the steady-state concentrations of MV'+adsnever reach significantly high values. However, a t low pH's, HfadSreplaces MV2+adsas (32) Rabani, J.; Fessenden, R. W.; Sassoon, R. E. J . Phys. Chem., in press. (33) Slama-Schwok, A,; Rabani, J. J . Phys. Chem. 1987, 91, 4394.
The Journal of Physical Chemistry, Vol. 92, No. 9, 1988 2531
Production of Hydrogen on a Colloidal Catalyst
TABLE I V Effect of pH on the Computer-Simulated Changes of [MY'] with Time for Various Sizes of Colloidal Platinum Particles' rpt = 25 A rRc = 100 A rRc = 15 A tl/2fp!
pH 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0
ms 30.4 8.2 2.7 1.5 1.2 1.1 1.0 1.0 1.0 1.0 1.0 1.6 1.6 0.8 0.7 0.7 0.6 0.6 0.6 0.6 0.5
[ M V ' + ] e ~ p ~t l j l l p ? N ms
2.4 3.2 4.2 6.1 8.2 10.1 11.5 12.3
20 50 110 195 270 225 150 105
[MV.'IcsptC clM
0.01 0.03 0.08 0.17 0.33 0.61 1.20 2.51 5.20 9.23 12.92 15.18 16.40
tl/2fp,b
ms 58.0 16.5 5.9 3.3 2.6 2.4 2.3 2.2 2.2 2.2 2.2f 2.2f 2.1f 2.6 1.3 1.1 1.1 1.o 0.9 0.9 0.8
[MV'+lefp: PM
4.1 5.0 6.5 8.4 10.2 11.6 12.4
t i / ~ 8 p ~[MVDtIesp,C ti/2fp?
ms
60 115 170 305 240 150 115
PM
0.11 0.23 0.43 0.80 1.54 3.11 6.06 9.94 13.22 15.29 16.46
ms 689 200 77.9 39.2 29.0 26.8 26.1 25.4 24.2 22.7f 21.6 20.6 18.2f 9.3 6.4 4.1 5.49 5.0s 4.5 4.0 3.2
[MV'+Iefp,c PM
6.6 8.0 9.3 8.76 10.06 11.11 11.91 12.52
ms
[MV"lup,( WM
135 190 105 635 495 310 230 191
0.01 0.04 0.13 0.37 0.97 2.08 3.51 5.15 7.17 9.55 11.94 14.00 15.55 16.56
Z1/2SP,d
'Conditions of experiment as in Figures 1 and 2. bFirst half-lifetime for the fast decaying process. [MV",,,] at the end of the fast decaying process. dFirst half-lifetime for the second slower process (decay or growth). e[M*t,ot]at the end of the slow process. fFirst half-lifetimes for the total decay since at these pH's the fast and slow processes are not easily distinguished. $Here the fast process is made up to two decays which may be distinguished from one another. r,/;P represents the apparent first half-lifetime for the total fast process. At pH 9.0 [MV'+,,,] decays in a very fast process with a first half-life to 2.8 ms down to a plateau of 10.3 pM which is followed by a second decay within the overall fast process with a first half-life of about 45 ms before it reaches the plateau of 8.76 pM. At p H 9.5 similar behavior is observed for the overall fast process which is made UP of two decays with first half-lives of 3.0 and about 20 ms with an intermediate plateau observed for [MV'+,,,] at about 10.8 pM. An explanaiion for these iesults is given in the text.
the predominantly adsorbed positively charged species, such that at pH 1 the concentration of H+adsis always about 2 orders of magnitude larger than that of MV2+,&during the decay of M Y + . At alkaline pH's, OH-,d, becomes the predominantly adsorbed, negatively charged species and already at pH 8 the concentration of OH, is greater than that of Cl,* for all the sizes of catalyst studied. By pH 11 the concentration of Cl-ad, is negligible and the adsorption of negatively charged species on the catalyst is dominated by OH-,d,. The description of adsorption given above has consequences on the variation of both fractional coverage and total charge of the particle with pH. The fractional coverage of the particle by all the adsorbed species becomes larger at both low and high pH's due to the additional concentrations of H+ and OH- in solution, with respectively. The resulting changes in the values of (1 pH markedly affect the behavior of the system at different pH's as will be seen later. The total charge on the particle due to adsorbed species, however, although not usually very large, does change from positive values at low p H s to negative values at high pH's. For a 25-A particle, for example, the point of zero charge on the particle due to adsorbed species in the chemical system occurs between pH 9.5 and 10. This variation of charge with pH and during the course of reaction also affects the chemical system by virtue of its effect on the diffusion-controlled rate constants of the reaction scheme. This will also be discussed later.
Significance of the Optimized Chemical Rate Constants in the Reaction Scheme Before describing the predictions made using our computer simulation technique concerning various aspects of the system, it is useful to know both the sensitivity of the chemical system to the values attributed to the rate constants and the importance of the individual reactions in determining the rates of the concentration changes of MV'+ at different stages of the overall chemical change at various pH's. We therefore studied the effect of reducing the values of the chemical rate constants of each equilibria given in Scheme I by a factor of 10 for the system containing 4 X lo4 M of both MV2+ and Pt in which the radius of the platinum particle is 25 A. In this way, the value of the equilibrium constant remains the same
but the rate at which that equilibrium is established is reduced. For reactions which involve diffusion of species to or from the catalyst particle, only the rate constants for the activation-controlled stages were reduced since the rate constants for the diffusion-controlled stages of these equilibria are determined just by the diffusion constants, sizes, charges, and temperature of the species involved. For reactions which are totally activation-controlled, decreasing the values of kcchemand ke*hcm for a particular equilibrium by an order of magnitude will decrease the rate of establishment of that equilibrium also by a factor of 10, but this may not be observed in the overall concentration change of MV" if the reaction under investigation is not a rate-controlling stage in the overall process. For reactions which are diffusion-controlled, decreasing the values of kechcmand ke-chemfor such equilibria will only have an effect on the overall system if such a reduction causes the reaction to be no longer diffusion-controlled and again, this will only be detected experimentally if the reaction is important in controlling the overall rate of change of MV". The results of computer simulations on the reaction system in which the rate constants of the forward and back reactions of each individual equilibria were reduced by an order of magnitude, are presented in Table V for the three pH's, 1.5, 8.0, and 11.0. The results at all the three pH's studied appear to demonstrate that the establishment of equilibria 1-3 is very important in determining the overall rate of disappearance of MV" in the fast decay, since decreasing their values by a factor of 10 leads to a considerable reduction in the rate of disappearance of MV". At pH 1.5, the only other equilibrium which, when its rate constants are decreased, reduces the rate of overall decay, is equilibrium 8 suggesting that the desorption of also contributes to a small degree in determining the overall rate of decay of MV'' at this pH. The lifetime of decay of MV" is little affected by reducing the rate constants for the other equilibria at pH 1.5 and, hence, the values of the chemical or activation-controlled rate constants are probably unimportant in determining the overall behavior of the system at this pH. At pH 8, the fast and slow decay of MV" also appears to be quite sensitive to the values of the rate constants in equilibria 10 and 11, suggesting that these adsorption4esorption equilibria may
2532 The Journal of Physical Chemistry, Vol. 92, No. 9, 1988
Lenoir et al.
-
TABLE V Effect of Decreasing Values of Ootimized Chemical Rate Constants on the Com~uter-SimulatedResults' pH 8.0 pH 11.0
1 2 3 4 5 6 7 8 9
IO I1
d
40.8 49.7 29.0 16.1 15.4 16.4 14.1 23.4 17.1 16.5 16.5 16.5
2.4 2.9 3.9 1.2 1.2 1.3 1.2 1.2 4.2 1.4 1.4 1.3
4.3 4.5 4.1 4.2 4.2 4.2 4.2 4.2 4.2 3.5 3.5 4.1
51 45 57 440 59 50 55 55 56 90 90 58
1.54 1.54 1.54 1.54 1.54 1.54 1.54 1.54 1.54 I .54 1.54 1.54
3.3 4.0 1.3 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8 0.8
12.46 12.50 12.40 12.37 12.38 12.37 12.37 12.38 12.38 12.38 12.38 12.38
115 I15 110
625 140 350 195 115 115 115 115 115
16.46 16.46 16.46 16.46 16.46 16.46 16.46 16.46 16.46 16.46 16.46 16.46
'Experimental conditions as in Table IV for a Pt catalyst of radius 25 A. bThe identification of the equilibria given in Scheme I whose values of rate constants for the forward and back reactions were both reduced by a factor of 10. In the case of reactions involving diffusion of species to or from the catalyst particle, only the values of the activation-controlledcomponents of the rate constants were decreased. See text for details. 'See Table IV for explanations of terms. dComputer-simulatedresults taken from Table IV using the optimized values of the chemical rate constants. partially control the rate of disappearance of MV'+ (in an indirect manner) via their effect on the overall charge carried by the catalyst, which determines the diffusion-controlled components of the rate constants in equilibria 1-3. Since at this pH, the concentration of OH-,& is very close to that of Cl-,* in the course of reaction, this accounts for the similarity in behavior observed on reducing the rate of either reaction. The values of the concentration of MV'+ at the end of the fast decay a t the higher pH's are consequently also slightly affected by changing the values of the rate constants in equilibria 1-3, and a t pH 8 also in equilibria 10 and 11. At pH 8, it is clear that the slow process is controlled by the rate of establishment of equilibrium 4, namely the protonation of e-ads,where a decrease in the rate of the forward and reverse reactions by a factor of 10 causes a decrease in the rate of the slow decay by a factor of about 7. The total charge on the catalyst must also play some part in determining the overall rate of this process, as may be seen by the effect of reducing the rate of reactions 10 and 11 on slowing down the rate of the slow process. At pH 11, a buildup in [MV"] is observed in the slower process due to reduction of MV2+ by hydrogen produced by the pulse. The results in Table V for this pH demonstrate that the establishment of equilibria in the stages involved in adsorbing H2onto the catalyst followed by the subsequent deprotonation to yield e-ad, (equilibria 4-7) all contribute to determining the overall rate of formation of MV". The final values of [MV"] at the end of the overall process always remain the same at all pH's in Table V since the variation of rate constants performed here will not change the position of equilibrium but only its rate of attainment. Finally, it should be noted that the results given in Table V are very important in determining at which pH's and stages of the overall process optimization of these rate constants should be carried out. It is also clear that nearly all the values of the rate and equilibrium constants given for the equilibria in Scheme I have significance in determining the overall behavior of MV'+ at some particular times and pH's.
Dependence of the Behavior of the System on the Size of the Catalyst Particle The present investigation affords predictions to be made concerning the effects of size of catalyst particle on the rates of MV" disappearance or formation. Three sizes of colloid of radii 15, 25, and 100 8, were treated by using our computer modelling technique on the reaction system for which the concentration of Pt atoms was set at 4 X M. The results of our study are presented in Table IV. The essential features of the variations of concentration of MV" with time remain virtually the same for the different sizes of particle studied although, of course, the lifetimes and amounts of MV" participating in them vary quite considerably with size.
The rate of the fast process is seen to decrease with increasing size of particle for all pH's with the first half-life for the fast over the acid process being approximately proportional to rptC1.' pH range up to about pH 5,after which the first half-life becomes more linearly dependent on rR,. At pH 11, the first half-life is (nearly) directly proportional to rk. The previous discussion has demonstrated the importance of equilibria 1-3 in determining the rate of fast decay and several reasons are now suggested for the dependence of rate on particle radius. Increasing the size of the particle leads to a smaller fraction of platinum atom concentration available as surface sites, as may be observed in Table 111, and this should retard adsorption of MV" via reaction 1. Sincef, is seen to possess an approximate inverse dependence on radius, this could account for the linear variation of the first half-life of the fast decay on particle size. However, the rate of reaction is also affected by the value of (1 which decreases at equivalent times, in the fast decay with increasing size of catalyst. Although the values of (1 - 6T)4 are largest at neutral pH's, the on size is greatest at pH 1 where, for example, variation of (1 at the first half-life is smaller by a factor the value of (1 of about 6 for a 100-8, radius catalyst particle as compared to its value for a 15-A radius particle at the same point in the decay, and smallest at pH 11 where this factor is only about 1.5. This probably accounts for the dependence of rate on a decreasing power of rpt, as the pH is raised. The difference in variation of (1 - 8T).4 with size at different pH's is probably related to the change in the major, adsorbed, positively charged species on the catalyst from H+ads.at low pH (which occupies one surface site), to MV2+adsat higher pH's (which occupies four surface sites). The end of the fast process, which can be distinguished from a second slower process at intermediate and alkaline pH's, occurs at slightly higher MV" concentrations as the size of catalyst is increased. The difference in these concentration values for the catalyst sizes investigated is more pronounced at intermediate p H s where the end of the fast process represents partial protonation of e-adsto give some H-,ds. This is because the concentration of MV'+ remaining after equilibria 1-5 have been established, which may be calculated by manipulation of the rate constants for these equilibria, is proportional to the inverse of the product off, and (1 - 0,) both of which decrease with increasing size of particle. When the pH is raised, the dependence of [MV'+efp]on these terms becomes negligible since only equilibria 1-3 are established and hence its value becomes less dependent on size. The only additional factor affecting [MV'+efp]is the charge density of the catalyst particle, which is found to decrease with increasing size, and thus may account for the small differences in [MV'+ef,] for different particle radii at the very high pH's. The behavior of the subsequent slow process, which involves establishment of equilibrium for all the reactions described in Scheme I, and which changes from a decay of [MV"] at intermediate pH's to a buildup in [MV'+] at more alkaline pH's, also
Production of Hydrogen on a Colloidal Catalyst depends on the size of catalyst used. Indeed at pH 9 and 9.5 for a 100-8, platinum particle, both of these phenomena are observed following the fast process. At these pH's, the fast process represents establishment of equilibria 1-3 producing e-ads, the inand termediate decay represents the formation of H-adsfrom e, the final slow buildup represents the overall establishment of equilibrium of the whole chemical system with hydrogen produced from the pulse. The rates of the slow processes do decrease with increasing size of catalyst by factors varying from close to 1 up to around 7 and may again be explained by the complex dependencies onfs, (1 - &), and particle charge density for the kinetics of reactions 4-1 1 in Scheme I for different particle sizes. The values of [MV'+,,,] are also observed to increase with particle radius even though final equilibrium has been reached. This is due to the smaller total number of surface sites for the intermediate species adsorbed on the catalyst and a higher concentration of final products and reactants present in solution when overall equilibrium is established.
Dependence of the Behavior of the System on the Concentration of Reactants and Products The effects of varying [MV2+]and the initial concentration of MV", [MV",], on the chemical system were described and explained in our previous article* and are virtually identical with the results obtained in this study. A slightly better agreement with experimental observations was found for the dependence of the decaying fraction of [MV"] on pulse intensity and hence [MV'+,] at pH's 8 and 9. The fraction decaying is now observed to become larger by up to about 5% as [MV",] is lowered, although with the present model it is still not possible to fully reproduce the experimental data for this effect.', In this study we have also attempted to simulate the effects of multiple pulses on the system in which hydrogen produced from the pulse and in the reaction system can accumulate and thus affect the position of final equilibrium. We could not make a quantitative comparison of our simulation results with the experimentally determined data since insufficient information was given in the experimental studies to indicate whether full equilibration of the system was allowed to occur between individual pulses.1° Good qualitative agreement is observed, however, in that the rate of disappearance of MV'+ at pH 1.5 is found to decrease sharply with pulse number, although after many pulses our calculated rate constants become considerably smaller than those measured experimentally. This is attributed to the following two reasons. Firstly, in our simulations we assume that the system reaches complete equilibrium between each pulse, whereas this probably does not reflect the actual experimental conditions. Secondly, a side reaction involving partial irreversible hydrogenation of methylviologen occurs under such conditions,26 and this also was not be considered in our model. From our simulation data, however, it is clear that the reason for the sharp drop in rate with increasing number of pulses is due to the buildup in concentration of H2& such that the total fractional coverage increases. This causes the value of the term (1 - 8T)4 to decrease, and hence the rate of adsorption of MV" on the catalyst also decreases, as may be seen from the rate equation for reaction 1. The variation of the rate of the fast decay of MV'+ with Pt concentration was considered in some detail in the previous investigation and an explanation for the deviations from first-order dependence of the rate of reaction on [Pt] was given based on the coverage of the particle [and hence on the dependence of rate on the term (1 - &)4]. The results of computer simulations carried out using our present model reproduce reasonably closely the previous findings particularly at pH 1.5 and we have now been able to extend these investigations to include different sizes of catalyst. Plots of the logarithm of the measured first-order rate constant for the fast decay versus the logarithm of the platinum concentration are given in Figures 3 and 4 for catalyst particles of radii 15, 25, and 100 8, at pH's 1.5 and 8, respectively. It is immediately apparent that there is some curvature in the plots given in Figure 3 for the data at pH 1.5. However, the best straight lines have been drawn through the points, and the apparent
The Journal of Physical Chemistry, Vol. 92, No. 9, 1988 2533
-2
-3
-4
Log ( [ P t ] / M )
Figure 3. Plots of the logarithm of the computer-simulated rate constants for the fast decay as a function of the logarithm of [Pt]. [2-propanol] = 0.2 M; [MV**] = 4 X lo4 M; initial [MV''] = 13.3 FM; pH 1.5. ( 0 ) Radius of Pt colloid = 15 A;(m) radius of Pt colloid = 25 A; (A)radius of Pt colloid = 100 A.
V -4
J -3
-2
L o g ([Pt] / M )
Figure 4. Plots as in Figure 3 except pH is 8
orders with respect to [Pt] calculated from the gradients of these plots are given in Table VI. At pH 1.5, the gradients of the graphs yield apparent orders with respect to [Pt] of about 1.74 for 15- and 25-A-radius particles, a value which is virtually identical with that from the previous study, although for a 100-&radius particle this order falls to 1.40. At pH 8, however, the orders with respect to [Pt] all lie very close to 1.0 with very little variation with size. These results agree well with those given in the literature where close to second-order dependence on [Pt] was observed at low pH's9*'Owhile orders closer to 1.O were found at near-neutral pH's.'' It should be noted that the orders, which are obtained from computer simulations, depend somewhat on the platinum concentration range used due to the curvature of the plots. For example, at the lowest concentrations of platinum, given in Figure 3 at pH 1.5 for a 15-A-radius catalyst particle, the apparent order with respect to [Pt] does indeed become 2 and this is in excellent agreement with the results of Matheson et aL9 obtained over a similar concentration range of M) for a 16-A;radius platinum catplatinum ((1.5-4.5) X alyst. The explanation for the order given earlier' is based on the fact that, as the platinum concentration is increased, the total concentration of free platinum surface sites available for reaction increases by a factor greater than that of the platinum concentration itself, since there will be fewer adsorbed species per particle at the higher platinum concentrations. Consequently, both of the terms [Pt] and (1 increase in the rate equations for equilibria 1 and 3 with increasing [Pt], thus leading to orders of greater than one with respect to [Pt]. Such an explanation is clearly valid for the results for the smaller sized particles at pH 1.5, as is evident from study of Table VI. The value for the term log {(I - 19T)~hlph[pt]/(l - &)410w(Pt]}, where e T is calculated at the half-life of the decay, for a con-
2534
The Journal of Physical Chemistry, Vol. 92, No. 9, 1988
Lenoir et al.
TABLE VI: Dependence of Apparent Order with Respect to [Pt] and Rate Constants for the Fast Decay of MV'+ on pH and the Size, Charge, and Coverage of the Catalyst Particles" rR apparentb [pt], tIi2lp, log { ( I - oT)4(high[pt])/ 10-9kl"iff,d 10m9k 1 n ( l - 6T)4,d pH A'' order mM ms 104a',d 102(i - o T ) 4 d ( I - e,)4(iOw [ ~ t l ) p M-I s-l M-f s-I 1.5
1.5
1.5
15
25
100
1.76
1.71
1.40
8
15
0.98
8
25
0.98
8
100
1.03
1 .o 0.1 0.4 0.04 2.0 0.2 1 .o 0.1 10.0 1 .o 4.0 0.4 2.0 0.2 4.0 0.4 20.0 2.0
2.1 74.2 8.1 534.0 1.7 48.2 4.5 192.1 2.9 41.2 7.6 192.5 0.14 1.13 0.16 1.25 0.40 3.60
33.5 31.0 32.7 30.6 12.9 11.7 12.7 11.4 2.07 1.82 1.95 1.77 9.26 8.35 8.66 8.32 0.62 0.59
1.93 0.41 1.33 0.10 2.34 0.61 1.68 0.20 3.38 0.97 2.24 0.23 5.25 3.12 5.84 3.44 6.81 4.06
0.67 1.12 0.58 0.92 0.54 0.99 0.23 0.23 0.22
20.0 18.3 19.6 18.1 30.3 28.2 29.9 27.8 106.6 100.8 103.9 99.5 11.3 10.3 16.6 15.6 58.3 54.2
11.0 2.3 7.6 0.6 39.8 10.4 28.6 3.3 1010 288 667 66.8 29.9 17.7 99.6 58.6 2030 1200
"Results computed from simulations of the fast decay of [MV'+,,,] with experimental conditions as described in Figures 3 and 4. *Calculated from the slopes of the best straight lines drawn through the points given in Figures 3 and 4.
