Langmuir-Rideal Mechanism for Atom Recombination
The Journal
of Physical Chemistry,
Vol. 82, No. 25, 1978 2715
Kinetics of the Langmuir-Rideal Mechanism for Heterogeneous Atom Recombinationt W. Brennen" and M. E. Shuman Department of Chemistry and Laboratory for Research on the Structure of Matter, University of Pennsylvania, Philadelphia, Pennsylvania 19104 (Received February 13, 1978; Revised Manuscript Received September 8, 1978) Publication costs assisted by the National Science Foundation
The behavior of numerical solutions to the rate equations governing the Langmuir-Rideal adsorption-abstraction mechanism of heterogeneous atom recombination is summarized. The results of two approximate treatments of the kinetics are compared to the exact solutions. The concept of surface recombination coefficient is examined in the context of this mechanism.
Introduction The Langmuir-Rideal mechanism1 for the heterocatalytic recombination of atoms at active surface sites may be written, using nitrogen atoms to illustrate N(g) N(g) + F
+E h,
kl
Results and Discussion Integration of the Rate Equations. The rate equations governing the kinetics of the mechanism in reactions 1and 2 are
F
k-1
N2(g) + E
Reaction 1 represents the reversible adsorption of a gaseous atom on an empty surface site, E, to give a filled surface site, F. In reaction 2 a gaseous atom successfully encounters an adsorbed one to form the diatomic molecule, which is assumed to desorb immediately leaving behind an empty site capable of participating in reaction 1 again. We have been led to investigate the non-steady-state behavior of this mechanism in trying to interpret our experimental results on the kinetics of heterocatalytic nitrogen atom recombination on glass.2 Until relatively recently, quantitative discussions of the kinetics and mechanisms of processes occurring a t the gas-solid interface were based on the quasi-steady-state approximation or related approximations. The application to the investigation of of modulated molecular surface processes has made it necessary to develop methods for evaluating the transient response of a system governed by an assumed mechanism in order to make comparisons between theory and experiment possible. In a previous paper6 we examined the kinetic properties of the special case of the Langmuir-Rideal mechanism which arises when kl = 0. This simplification allowed us to obtain accurate reactant and intermediate time histories using a combination of analytic and numerical techniques. In the present paper we examine the behavior of this mechanism when the reverse of reaction 1 is taken into account. This complication is inescapable whenever the adatom is sufficiently weakly bound to the surface, Le., when the mean desorption time is comparable to or shorter than the pseudo-first-order lifetime of a filled site with respect to abstractive denudation. Unfortunately, allowing k-l to be nonzero makes it necessary to resort to machine computation from the outset to obtain accurate solutions to the kinetic equations for all times. A disadvantage of having only computed solutions is that general properties of the system are less conveniently and surely discerned and presented. 'Supported in part by the National Science Foundation MRL Program under Grant No. DMR-76-80994. 0022-3654/78/2082-27 15$0 1.OO/O
(4) where [N] represents atom concentration in units cm-3 and [E] and [F] represent surface species concentrations in units cm-2. V and A are the volume and surface area of the system. Both rates are expressed as fluxes which allows k l and k2 to have the dimensions of a second-order rate constant, cm3 s-l, and kl to have the dimensions of a first-order rate constant, s-l. We shall use the unbracketed letters N, F, and E to represent the numbers of atoms, filled sites, and empty sites, respectively. Let S be the total number of sites, empty and filled, so that S = E F. It is convenient for numerical work to !se reduced, $imensionlessyariables. Accordingly, let N = (N/N,),E = ( E / S ) ,and F = (F/S),where No is the initial number of atoms. Also, let R = @/No),p = ( k 2 / k 1 ) b', = kl[Nlot, and K = K[N],, where K = (kl/k-J is the equiiibrium constant for reaction 1. We note in passing that K is the reduced lifetime of an adsorbed atom with respect to desorption. Using these definitions, eq 3 and 4 may be rewritten in the dimensionless form
+
(1
+ p)N&
For the special case in which h_l = 0 the third term on the right side of each of eq 5 and 6 is absent, and the equations are then equivalent to eq 5 and 6 of ref 6. Solutions of these coupled differential equations may be computed using a predictor-corrector method7 which requires possessing values of each of N and F a t two values-of b' to start the computation. At 0 = 0 we know that N(0) = 1 by definition. If we choose the sites to ,be fille_d initially to a fractional extent, Fo, then we have F(0) = F , In all our computations for this paper we have taken F, = 0, corresponding to initially bare sites. Values of N and F a t a later, sufficiently short time, 0 = 7, may be predicted using Taylor exp_ansionsabout 0 = 0. To terms in T~ these expansions for Fo = 0 are R(7)= 1 - RT + 7'/2{R2 + R ( l - p ) + R P I J ( 7 ) P(T)= 7 - T ~ / ~ -k( R(1 p ) 4(8)
+
0 1978 American Chemical Society
2716
The Journal of Physical Chemistry, Vol. 82,
li
Pm5.0
d
R=05
O
i;
06i-
I
O2
\
No. 25, 1978
-
\
0
t
1
I
0
k -
0
0
W. Brennen and M. E. Shuman
IO
20
30
50
40
.
2
q
P
60
6
B
10
6
B
10
t
0
Figure 1. Typical form of the functionsAd(8)and p(O), computed for the parameter set p = 5, R = 1, and K = 0.5.
We encountered no computational instabilities so long as the reduced time interval, T , was chosen to be somewhat shorter than the value which, on the basis of eq 8, just ensured a positive value of F(7). Due caution was exercised to ensure accuracy in the face of sometime large first and second derivatives of F with respect to 8 in the interval before and in the vicinity of its maximum, Employing the start-up values of N and F , the program generates reduced time histories of N and F for any particu1ar:hoice of the fundamental trio of parameters: p , R, and K. In all cases F , which is here taken to be zero to start, passes through a single paximum and approaches zero monotonically thereafter. N starts a t a value of unity and decreases monotonically to zero. A single instance of this general behavior is illustrated in Fig>re 1which shows the computed time histories of N and F for p = 5, R = 1, and K = 0.5. It is difficult to summarize the detailed dynamic behavior of a three-parameter kinetic system compactly. We shall focus attention on certain features which may serve t o enhance intuitive grasp of the situation in the present case. A coarse measure of the rate of atom depletion is the reduced first half-life, Bob, which is the value of 8 at which N has reached the value 0.5. Figures 2a-c show graphs of as a function of p and K for three values of R , the number of sites per atom initially present. The qualitative behavior within each figure is the same, and a comparison of the figures shows clearly that increasing R decreases the reduced time required for half the atoms initially present to disappear. The decisive control exercised by R over the initial rate also follows directly from eq 5 which reveals that - dN/dBls=o= R , since F ( 0 ) = 0 and N(0) = 1. It is important to bear in mind, however, that the control which R has over the initial rate of atom disappearance need not persist throughout the reaction. Tor example, Figure 3 shows semilogarithmic graphs of N(B) for p = K = 1 for the values of R used in Figure 2 . It is clear that late in the reaction the rate of atom disappearance is no longer strongly dependent on R. The Quasi-Steady-State Approximation (QSSA). The QSSA, or Bodenstein approximation? has served long and well as a tool for the analysis and testing of kinetic mechanisms. The range of its validity, methods for systematically refining it, and comparisons to accurate solutions of systems of kinetic equations have been the subjects of several investigation^.^-^' Since we possess accurate solutions to the system of eq 5 and 6, we may ~
0
2
4
p
Figure 2. Graphs of the reduced half-time, B,o,s, at which d reaches half its initial value, as a function of p and K f o r three values of R.
examine the performance of this approximation. Designating filled sites to play the role of intermediate, we may set the right side of eq 6 equal to zero in the customary way and solve for the "steady-state" reduced population of filled sites
F, =
A
(1
+ p)A 4- R-1
(9)
By evakuating &, from eq 9 a t various times using computed N values, we may compare these approximate intermediate populations to the compytej ones a t corresponding times by forming the ratio F I F , and noting the extent to which it deviates from unity. The results are shown in Figure 4a for p = 2, K = 2 , and the three values of R used in Figure 2. Consultation of these figures shows that for-R_= 5 and R = 1the reaction is already haif over before F I F , has even reached a maximum. While F I F , is
The Journal of Physical Chemistry, Vol. 82,No. 25, 1978 2717
Langmuir-Rideal Mechanism for Atom Recombination
The integration of eq 10 resulis in the following implicit, transcendental equation for Ns(0):
P.I.0
ol s
Ll.0
0
1
\
I
(1
R.02
1
Fpe=
O 5 I
5
IO
IO
25
20
,I
(11)
Although one might solve eq 11for fiBnumerically, we have instead found it convenient to use another predictorcorrector code to integrate eq 10 numerically. Having thus obtained Y,(B),it is interesting to make comparisons to the true N(8) computed directly from eq 5 and 6. T o fgcil$ate global comparisons we have calculated the ratio NIN, for each history at the reduced timeAOo.:,and graphed this ratio for various values of p , R, and K. These graphs are shown in Figures 5a-c. In terms of this test the QSSh does best for large p and, less critically, small 4 and K. These are the very conditions which ensure that F is small throughout the reaction, and this is precisely the situation in which the QSSA is usually purported to be a reasonable approximation. The PreequilibriumApproximation (PEA). In reaction schemes containing one or more reversible reactions producing or consuming intermediates, an alternative to the QSSA is frequently used. This scheme, which we designate PEA, is based on assuming the reversible reactions to be rapid compared to any irreversible step which converts intermediates to products. In this situation a quasi-equilibrium pool of intermediate is established quickly, and the product formation rate is determined by relatively slow reactive leakage from this pool. Applying this scheme in the present case, reaction 1 is assumed to stand in equilibrium, which yields the following PEA expression for the reduced population of filled sites: N
N
0
+ p ) In (NS)+ k l ( l- Nc1)= 2pRO
,I
I
0
10
15
eo
25
35
30
Equation 12, which may be compared to eq 9, is the Langmuir adsorption isotherm. It is instructive to calculate Fpe.using accurate N values as j function of time and examine the behavior of the ratio FIF,. This is done in Figur? 4b analogously to Figure 4a. For R = 1 and R = 5, F / F p e is clpser to unity for a greater fraction of the reaction than F I F , is. A PEA rate law may be obtained by setting the overall rate of disappearance of atoms to be just twice the rate of reaction 2, using eq 12 for the population of filled sites. The result is
Figure 4. (a) Graphspf the ratio vs. 8 in which pwas computed from eq 5and 6 and f , from eq 9 for three values of Rand equal values of p and K. (b)Graphspf the ratio f / f Pvs. 0 in which Fwas computed from eq 5 and 6 and f pe from eq 12 for three values of Rand equal values of p and K .
seen to approach unity a t long times for each of the three cases, only in the case R = 0.2 is it consistently near unity for a significant fraction of the reaction. In some instances the QSSA may be pressed beyond deriving expressions for concentrations of intermediates in terms of conceptrations of reactants and products. The expression for F, in eq 9 may be substituted into eq 5 to obtain the approximate rate law for the disappearance of atoms 2pRRs2 --dR, - dB (1 p)R8 R-1
+
-dRpe
--
do
-
2pRNpe2
Npe+ K - 1
As in the case of eq 10, this approximate rate law may be used to compute time histories of Npe.Figures 6a-c show
(10)
+
i9 + K - 1 ~
t
4 17102
0
0
0
I/.. 0
2
4
6
8
1
0
P
Figure 5. Graphs of the ratio
0
2
4
6
0
1
0
P
f i / f i s evaluated at the reduced half-time, Bas, as a function
0
2
.... . ....,*
6
4
I......~.. ....f 6
0
IO
P
of p and k f o r three values of R.
't
2718
The Journal of Physical Chemistry, Vol. 82, No. 25, 7978
W. Brennen and M. E. Shuman
0
O
0
0
O
Z
4
S
L
I
I
0
O
2
4
6
8
1
0
0
2
4
6
Figure 6. Graphs of the ratio
fi/fipeevaluated at the
+ wall
kw
'12N2 + wall
1
0
reduced half-time, O,,,, as a function of p and k f o r three values of R
the behavior of fi/fip eJaluated at as a function of the parameters p , R, and K. In contrast to the QSSA case, PEA does well when all parameters are small. Small p ensures slow leakage of intermediate into the product channel, and small K keeps the intermediate concentration low. The Probability of Reaction per Encounter (7). The efficiency of a surface for removing atoms from a gaseous sample is called y and has been defined traditionally12as the probability per gas kinetic collision with the surface that an atom will be permanently removed from the system. Although some authors13 have offered evidence that in the case of nitrogen atoms, the atom removal reaction at the surface is kinetically second order, most14-16 have maintained that it is first order and described it schematically as N
8
P
P
P
I/
\I
(14)
On the basis of this nonmechanism and its implied first-order rate law for atom disappearance, y would by definition become
0 01
0
04
OB
I2
16
20
24
28
8
where C is the average speed of the atoms a t the system temperature. In the present case the rate of heterogeneous atom removal cannot generally be described in terms of order number at all. In view of eq 3, y for the Langmuir-Rideal mechanism becomes h,[NI[El - [FIG-, - M N I )
(16) j/4[N]F By converting to reduced, dimensionless variables we obtain
y=
y/yo = (1- P[1 - p
+ (flIl-71
(17) where yo =A4hlS/tA is the value of y a t 0 = 0 where by definition N = 1 and F = 0. To determine the behavior of y a t long times we must evaluate y m / y O= lim (1- P[1 - p (flK)-11= ] 8-m
+
lim (1- P(flI?)-l) (18) 8+m
Although we are unable to evaluate this limit ana!ytisally, our computed results indicate that the ratio F I N ap-
Flgure 1. Graphs of fi, i, and y/yo vs. 0 for two sets of parameters ( p , R , K)differing only in the value of K , showing qualitatively distinct forms of the graphpf yIyo. In part(b) of this figure the ordinate scale is appropriate for F ,whereas for Nand y/yo it must be multiplied by ten.
proaches the value K at long times, implying that the limit in eq 18 is zero. Thus, in terms of the present model, y starts at some value, yo, and chang_esin away determined by the time dependence of both N and F , inevitably approaching zero a t sufficiently long time. Figures 7a and 7b illustrate qualitatively different forms of the time dependence of y/yo. In the restricted Langmuiz-Rideal mechanism discussed previously,6 wherein kl = K-l = 0, we have in place of eq 17 7/70 = 1- P(1- P )
(19)
Since i? = 1- E , we may use eq 9 of ref 6 to obtain for this special case ?/yo = 2 / ( 1
+ P ) + (1-
+ P)I e x p H 1 + PPI (20)
where 2 = S t f i ( 0 ) d0 is a monotonically increasing function of time, which is zero a t 0 = 0 and approaches a
The Journal of Physical Chemistry, Vol. 82, No.
Electrochemistry of Semiconductor Electrodes in DMF
25, 1978 2719
References and Notes This mechanism is sometimes referred to in the current literature as the Eley-Rideal mechanism, presumably on account of D. D. Eley and E. K. Rideal, Nature(London), 146, 401 (1940). In the present paper and in ref 2 and 6 we have used the designation "Langmuir-Rideal" because of I. Langmuir, Trans. Faraday Soc., 17, 607, 621 (1921),and E. K. Rideai, R o c . Camb. Phil. Soc., 35, 130 (1939). It seems plain to us that Langmuir's name should be associated with this mechanism, whatever else is done. M. E. Shuman and W. Brennen, to be submitted for publication. J. A. Schwartz and R. Madix, Surface Sci., 46, 317 (1974). D. R. Olander and A. Ullman, Int. J . Chem. Kinet., 8, 625 (1976). H.4. Chang and W. H. Weinberg, Surface Sci., 65, 153 (1977);72, 617 (1978). W. Brennen and M. E. Shuman, J . fhys. Chem., 79, 741 (1975). R. W. Hamming, "Introduction to Applied Numerical Analysis", McGraw-Hill, New York, 1971, p 203 et seq. This reference describes the basic idea of predictor-corrector methods. We designed and carefully tested a program specifically for solving eq 5 and 6, which included an automatic, variable time-step feature to maintain accuracy during the interval from 19 = 0 to a time beyond the maximum in F and to minimize computer time usage thereafter. The program may be found in M. E. Shuman, Ph.D. Dissertation, University of Pennsylvania, 1978. M. Bodenstein and H. Lutkemeyer, 2.fhys. Chem., 114, 208 (1924). J. R. Bowen, A. Acrivos, and A. K. Oppenheim, Chem. Eng. Sci., 16, 177 (1963). L. A. Farrow and D. Edelson, Int. J . Chem. Kinet., 6, 787 (1974). L. A. Farrow and T. E. Graedel, J . Phys. Chem., 61, 2480 (1977). F. A. Paneth and K. F. Herzfeld, 2.Elektrochem., 37, 577 (1931). I. M. Campbell and B. A. Thrush, R o c . R . SOC.London, Ser. A , 296, 201 (1967). C. Mavrovannis and C. A. Winkler. Can. J. Chem.. 39. 1601 11961). J. T. Herion, J. L. Franklin, P. Bradt, and V. H. Dibeier, J . C h e i . Phys., 30, 879 (1959). R. A. Young, J. Chem. fhys., 34, 1292 (1961).
2 .o
,,. P = 2 ,
0
2
4
R: 10 8
6
1 0 1 2 1 4
6 Figure 8. Graphs of y/yo vs. 6 compujed for R = 1 and various values of p with K = m (solid curves) and K = 2,lO (dotted curves).
finite constant asymptotically at long time. Figure 8 shows the dependence of eq 20 on the choice of p for the case R = 1. Included as dotted curves are examples calculated from eq 17 for the unrestricted mechanism. Cases in which R # 1 are qualitatively similar to those shown. It is clear from this discussion that in the context of the Langmuir-Rideal mechanism it is not generally possible to view the quantity y as a constant attribute of the catalytic surface, since it depends not only on rate constants and active surface site density but also on the instantaneous values of fractional site coverage and atom concentration.
Electrochemistry of n-type CdS, Gap, and GaAs and p-type Ge Semiconductor Electrodes in N,N-Dimethylformamide Solutions Lun-Shu Ray Yeh Allied Chemical Go., Morristown, New Jersey 07960
and Norman Hackerman* Depatiment of Chemistiy, Rice University, Houston, Texas 7700 1 (Received January 9, 1978; Revised Manuscript Received June 5, 1978) fubllcation costs assisted by the Robert A. Welch Foundation
The electrochemical and photoelectrochemical behavior of p-Ge and n-type CdS, Gap, and GaAs single crystal semiconductors as electrodes for the reduction and reoxidation of anthracene, p-benzoquinone, p-nitroaniline, trans-stilbene, diethyl fumarate, and iodine in N,N-dimethylformamide solutions was investigated in the dark and under illumination. The redox potentials of the various compounds at these electrodes were compared with the reversible behavior at a platinum electrode. The redox potentials in solution were also correlated with the flat-band potentials of the semiconductors which were estimated by using photocurrent onset potential under continuous irradiation, chopped light experiments, and Schottky-Mott plots. Good agreement was obtained in determining the flat-band potential for CdS using these three methods. The electrode behavior of n-GaP and n-GaAs depends strongly on electrode pretreatment. The presence of invisible films or surface states after the first reduction or oxidation voltammetric scan also is influential. Redox potentials of various compounds at these semiconductorsgenerally occurred at higher negative potentials than at a platinum electrode. However, trans-stilbene reduced at lower potentials on n-GaP and n-GaAs than at Pt when the semiconductors were exposed to light. At illuminated n-CdS the oxidation of iodide to triiodide occurred at a potential more negative than that at Pt indicating the CdS was stabilized against photodecomposition.
Introduction General theory and models of reaction mechanisms a t semiconductor-solution interfaces have been given by Gerischer1g2and Myamlin and Pleakova3 Due to the low free carrier concentration of the semiconductor compared to an electrolyte the potential would drop within the 0022-3654/78/2082-2719$01.00/0
interior of the semiconductor, the space charge region, rather than a t the semiconductor-electrolyte interface. Because of this, polarizing the semiconductor electrode would result in bending of the conduction and valence bands. Even though there is a potential drop in the Helmholtz double layer by changes of the potential a t the 0 1978 American Chemical Society
