Rate-Controlling Steps k r the Water-Gas Shift Reaction
447
Finally it is of interest to examine the rate expression for individual particle growth d f ( l l / t l t = ( k ( 1 - l } f ( l- 1) - k ( l } f ( l } ) [ M ] (6) where k ( l } is the rate constant for monomer deposition (ie., condensatiora] on a particle of 1 monomer units and [MI is the monomen concentration. In general [MI is a function oft. From eq 3 it can be shown that df(l!'dt = (dN(tl/dt)(f(I}/N(t})
(7)
Combination of eq 6 and 7 leads to h 1
-
1) = ( h j l } + 6 ) f ( l } / f ( Z- 1)
If the steady-state hypothesis is applied to [MI, then
[MI= ( d n i t l / d t i / S ~ K ( l } i l Z }=~ (dn{tl/dt)/kN{tl Z (10) 0
where k is the average value of k [ l l for all 1. Then, by substitution into eq 9
6
XdN(t}/dn(t}
(11)
which from eq 4 becomes
6
(8)
where
=
=
sp3/6 = 4.0
IO 8~
(12)
(9)
In principle every k [ l ) can now be computed if any one kill is known.
Now hll], h(l - 11, and f ( l l / f ( l - 11 are independent o f t . Thus 6 must also be independent of t . Since it is also independent of 1, 8 is a characteristic positive constant of the system.
Acknowledgment. We wish to thank William Corbett for assistance in obtaining the electron photomicrographs, J. B. Bodkin for the NH4Cl analyses, and Oakley Crawford and Rosa de Pena for useful discussions.
6 "s r:dN(1)/dt)(l/[M]N(t!)
dentificaticean of Rate-Controlling Steps Ifor the Water-Gas Shift Reaction aver an Iron Oxide Catalyst Shoichi Oki and Department of Chemistry. Utsunomiya University. Utsunomiya. Japan
Reiji Mezaki" Department of Chemical Engineering. New York University. Bronx. New York
(Received August 9. 79721
'The mechanistic study of the water-gas shift reaction was performed utilizing the hydrogen-deuterium equilibration reaction which occurs simultaneously during the course of the water-gas shift reaction. Reaction temperatures ranged from 467 to 522" and the total pressure of reaction system was 80 mm. The catalyst employed was an iron oxide. It was observed that the atomic fraction of deuterium in the total. hydrogen adsorbed by the catalyst coincides with the atomic fraction of deuterium of water. Material balance equations of hydrogen arid deuterium provided the estimates of the free energy changes of steps 2H(a) s Hz and CQ ~1 CQ(a). These estimates were in good agreement, indicating that, on certain assuinptions, these steps are rate-controlling. In addition the free energy change of hydrogen-exchange pzth also agrees with that of the associative desorption of hydrogen. The results obtained in the present investigation verify the conclusions, which were drawn from the previous mechanistic investigations of the water-gas shift reaction.
Introduction known that the determination of It has been stoichiometric nunntier is one of a number of powerful tools for the identification of rate-controlling step or steps of heterogeneous catalytic reacti0ns.l Applications of' the technique can be found in several solid-catalyzed gaseous reaction systems. They include the catalytic synthesis and decompos~t~on of ammonia,2-6 the water-gas shift reaction over an iron oxide ~ a t a l y s t , ~and - ~ the catalytic oxi-
dation of sulfur dioxide.lOJ1 In those investigations the selection of the rate-determining step was based on the
(1) J Horiuti and M lkushima Proc Imp Acad Jap 15,39 (1939) (2) s Enornoto and J Horiuti J Res l n s i Catal Nokkaido Univ 2, 87 (1951-1953) (3) s Enomoto, J Horlutl, and H Kobayashi J Res / / ? S t Cafal HOkkaido U n w 3, 985 (1953-1955) (4) K Tanaka J Res inst &tal Nokkaido U n i ~ 1 3 . 119 (1965)
The Journal of Physical Chemistry, Val. 77,No. 4, 1973
Shoichi Oki and Reiji Mezaki
448
TABLE I : PossibBe Mechanism for the Water-Gas Shift Reaction (CO f H20 = C O P 4-
H2)
and the Theoretical
and the Apparent StaichBometric Number Aooarent stoichiometric number Mechanism IC
II'
Illd
IWe
Ve
Theoretical u
Elementary step
GO -+CO(a) H20-2W(a) $. O(a)
CO(a) + O(a) +COn(a)
(i) (ii) (iii)
CQz(8)
(iv)
21-l(a)-*H2 CO GO(a) H20 -OH (a) -b H (a) CO(a) I- OH(a) -HCOO(a) HCOO(a)--CC$(a) + H(a)
(VI
(iiib)
1 1 1 1 1 1 1 1 1
GO2 (a) c02 2W(a) H2 H20 --2H(a) f O(a) GO I- O(a) -GO2 2W(a) "H2
(iv)
(VI (i) (ii) (iii)
CO -* 60 (a) H& HzO(a) CO(a) I- HzO(a) -Con I+ CO GO (a) H 2 0 4- CO(a)- 6 O 2 f HZ
(i)
1 1 1 1 1 1 1 1
-
-
GO2
--
-
a Isotopic tracer used for the experiment.
+
(i) (ii)
(iiia)
(ii) (iii) (i) (ii)
* Isotope exchange path.
Da
i4ca
H20-Hib
CO.-COzb
i80a CO-COZ~
__-
HzO-C02*
m
1
1
m
1
m
m
1
m
1 1
1
4
m
1
1
1
m
m
m
W
1
1
m
1
m
m
1 1 1 1
m
1
m
'I
1 1
I
m
1
1
1
1 1
m
m
m
m
m
Reference 18.
m
'I
7
1 1
1
m
m
m
m
1
1
m
1 1
m
m
1 1 1
1
1 1
W
1
1
m
1
1
Reference 15. e Reference 7.
apparent stoichiometric number of the reaction, which was experimentally det,ermined utilizing only one hind of isotopic tracer It is true that the employment of an isotopic tracer is effective to select the rate-determining step in a given mechnn,isrn. However, this method does not give much assurance for the adequacy of the mechanism itself. On the biasin of !,be stoichiometric number concept, Oki and Kaneko7-!',12.-14 recently proposed a new methodology in which more than one appropriate tracer are used to identify reaction mechanisms as well as their rate-determining steps. This method, in essence, consists of the, cornbined use of the apparent stoichiometric numbers observed by employing more than one appropriate isotopic tracer. Their method was, for the very first time, applied for the mechanistic study of the water-gas shift reaction with an iron oxide catalyst. The over-all equation for the reaction is
was approximately 2 for each of isotopic transfer paths with the exception of HzO-COz path. On the basis of this observation, Oki concluded that in mechanism I and I1 of Table I both step i and v are probably rate-controlling. This implies that all the steps other than steps i and v are at equilibrium. A previous experiment7 in which deuterium was used as DzO showed that the half-deuterated hydrogen is formed in the course of the water-gas shift reaction. This fact suggests that the hydrogen-deuterium equilibration given by eq 2 proceeds uia step v during the reaction. It was found in the present study that measurements of reaction rate of Dz and production rate of HD offers a novel method of obtaining the individual estimates of Ag, and Ag,. The comparison of the estimates thus obtained constitutes a quantitative justification of the earlier conclusion that steps i and v of mechanism I or 11 would be rate controlling. If these estimates agree, steps i and v are, indeed, rate controlling.
For this reaction system a variety of reaction mechanisms have been propostd by a number of investigators.i5-17 Table I shows these mechanisms along with their elementary steps. 'TabIe I also shows the theoretical stoichiometric number of each if the elementary steps. The theoretical stoichiometric numbers of all elementary steps listed in Table 1 are unity regardless of the reaction mechanism. On the contrary the apparent stoichiometric number which is an experimentally obtained value of the stoichiometric number varies depending on the kind of tracer employed for the experiment. Oki and coworkers used deuterium,7 carbon-14,8.9 and oxygen-1812Jg arid observed that irrespective of the kind of isotopic tracers the apparent stoichiometric number of the water-gas shift reaction over an iron oxide catalyst
(5) K. Tanaka, J. Res. Inst. Catal.. Hokkaido Univ.. 14, 153 (1966). (6) K. Tanaka, J. Res. Inst. Catai.. Hokkaido Univ.. 19, 63, (1971). (7) Y. Kaneko and S. Oki, J Res. insf. Catal.. Hokkaido Univ.. 13, 55 (1965). (8) Y . Kaneko and S. Oki. J . Res. lnst. Cafai.. Hokkaido Univ., 13. 169 (1965). (9) Y . Kaneko and S. Oki, J . Res. lnst. Catai.. HoKkardo Univ.. 15, 185 (1967), (10) Y. Kaneko and H . Odanaka, J. Res. lnst Catai.. Hokkaldo U n i v . . 13, 29 (1965). (11) J. Happel, H. Odanaka, and P. Rosche, Chem. Eng. Progr.. Symp. Ser. N-775, 67, 60 (1971). (12) S . Oki, J. Happel, M. A. Hnatow, and Y . Kaneko, Int. Congr. Cafal 5th. (1972). (13) S. Oki, Shokubai. 10, 180 (1968). (14) S. Oki and Y. Kaneko, J. Res. Inst. Catal.. Hokkaido Unrv.. 18, 93 (1970). (15) G. G. Shchibrya, N. M. Morozov, and M. i . Temkin. Kinet Kafal.. 6, 1057 (1965). (16) J. Nakanishi and K. Tamaru, Trans FaradaySoc.. 59, 1470 (1963). (17) C. Wagner, Advan. Catal., 21, 323 (1970). (18) S. Oki, Y . Kaneko, Y. Arai, and M. Shimada, Shokubai, 11, 184 (1969).
The Journal of /Yry:;ical Chemistry, Vol. 77, No. 4, 1973
Rate-Controlling Steps f o r the Water-Gas Shift Reaction -5
449
D2 = 2HD
(2) It was also found that the use of deuterium enables one to calculate the free energy change of the hydrogen atom exchange path between HzO and Hz, which is designated by AglI. In the case where steps i and v are rate controlling, the experimental estimate of AgH should coincide with that of Ag,. Consequently the determination of hg, from experimental data obtained with deuterium furnishes an independent determination of the rate-controlling steps. The objectives of the present investigation are threefold: (a) to obtain the values of Ag, from experimental data gathered utilizing deuterium, (b) to calculate the values of &H from the data, and (c) to definitely iderkify the rate-controlling kltep or steps of the water-gas shift reaction on the basrij of the results of a and b.
Introduction of eq 10 into eq 8 and 9 gives
AG = Ag,
-+ Ag,
(11)
For a set of consecutive hydrogen-deuterium exchange paths of the reaction, that is, steps ii and v of mechanism I or steps ii, iii-a, iii-b, and v of mechanism 11, the total free energy change of the consecutive paths, Ag14, is expressed as =
A&
(12)
In the derivation of eq 12, needless to say, we assume that the relationship of' eq 10 is also fulfilled in the present case. When deuterium is employed with hydrogen in kinetic experiments of water-gas shift reaction, the values of both AgH and Ag, can be determined independently from material balance equations of hydrogen, deuterium, ~ ~ t h e A r n ~~ ~ ~~ ~ ~~ ~ y ~ ~ and ~half-deuterated hydrogen. The following describes mathematical procedures which enable one to obtain the The chemical affinity of water-gas shift reaction 1, experimental values of AgFt and Ag,. -&G, is given by Consider a closed circulating reaction system in which a gaseous mixture of carbon monoxide, water vapor, carbon dioxide, hydrogen, and deuterium is circulating through a where K , is the theImodynamic equilibrium constant of catalyst bed. The following relationship holds between the the reaction, ?' is the reaction temperature, and p(i) is the over-all reaction rate of the system, V? and the forward partial pressure of gaseous component i. The chemical afand backward reaction rate of step v, u + and ~ v -,' finity of a reaction for a single route is also expressed in V u + ~- u - ~== dp(H,j/dt (13) terms of the .stoichicimetric number of elementary step, that is In eq 13 p(H2) 'is the partial pressure o f hydrogen in the s AG = CY(l)A&, (4) reaction system and t is reaction time elapsed. The rates 1-1 of partial pressure changes of H2, 1)2? and IID are given by the equations where V ( i) and Jag, are, respectively, the stoichiometric number and the free energy change of elementary step i. For the water-gas shift reaction, as stated in the previous section, the stoichiometric number of each of elementary steps considered in h i s study is unity (see Table I). Hence eq 4 becomes S
AG
=
xAgl I=
(5)
1
The free energy change of the Ith elementary step, Ag , is given by dig1 i= -RT
In (u+,/
(6)
ti-,)
Substituting eq 6 into eq 5 , we obtain
d(p(HdY(HJ) / d t = (1 - Z(H(a)>)'ti+ - Y(H2)u-, (14a) d(p(H,)Y(HD))/dt = 2Z(H(a)1(1 - Z(H(a))u+, -
Y(HD)ti-, (14b) d(p(H,)Y(Dd)/ d t = (Z(H(a)Yu+, -. Y(K),)u-, (14c) where Y(H2), Y(Dz), and Y(HD) are respectively the mole fractions of Hz, DS, and HD in the reaction system and Z(H(a)) denotes the atomic fraction of deuterium in the total number of hydrogen atoms adsorbed on the catalyst surface. Note that only two of eq 34 are independent. Equations 13,14b, and 14c may be combined to give
(.Z(H(a,)* - u(Z(H(a))) 3- F
S
(3 = RX In n(u+,/ti-,)
(7)
Application of eg 5 LO mechanism I and I1 of Table I yields the following relationships
AG
Ag,
+ Ag,,
-i- Ag,,,
+ Ag,
(8)
-iAg,
Oki and coworkers1:!,18 investigated the reaction utilizing carbon 14 and oxygen-18 and found that all the steps ii through iv are in equilibrium. On the ground of this experimental result WE' assumed that the steps ii through iv are fast and are in equilibrium. The assumption leads to the following relatioriship
Ag,,, '= Ag,,,-,
= Ag,,,..b =
Ag,,
==
0
(15)
in which
u = 2(dY(Dz)/dt)/(dY(E-ID)/dtd- 2(dY(Dz)/dt)I b = {Y(HD)(dY(D,)/dt) -
I*(DA(dY(HD)/dc)!/ I(dY(ED>/dt) 3- 2(dY(D2)/dt)[
and
&,,
=
0
(LO)
Equation 15 should be solved to obtain the value of %(H(a)) with the condition that 0 5 Z(H(a)) 5 1. The value of Z(H(a)) so obtained may be substituted into eq 14 to calculate u I v and u..,. Application of eq 6, thus, gives the values of Ag, for experimental conditions. The forward and the backward unidirectional rate of hydrogen atom exchange path between HzO and Hz, u t H and U - H , are correlated with the over-all reaction rate V as follows
V
=
u.+>~- u - ~= dp(HL)/ dt
(16)
The Journal ot Physical Chemistry, Vo!. 77, No. 4, 1973
Shoichi Oki arid Reiji Mezaki
450
0.61
.E E I
ttme of reaction ( h r )
Figure 1. Time-dependent ~ ( H z ) Y(HD), , and Y(D2) for run 1
(reaction temperature 467').
o
I% 0
2
4
6
j
8
t l m e o f reaction i h r )
Figure 2. Z(H(a)),Z(H2), and Z(H2O) vs, reaction time for run 1 (reaction temperature 467').
If it is assumed that the concentration of hydrogen atom adsorbed is constant, then, the rate of increase of deuterium content in hydrogen is given by
where Z(Hp0Zi and Z(H2) are, respectively, the atomic fraction of deuterium atom in water vapor and in hydrogen. By combining eq 16 and 17, we get
monoxide, hydrogen, and deuterium was circulated many times through the catalytic bed until the equilibrium of the water-gas shift reaction was reached. Gas samples were drawn into gas sampling bulbs a t specified time intervals. The samples so collected were analyzed by Hitachi mass spectrometer, RMS-3B type, with a constant electron accelerating voltage of 80 V. From the gas analysis the composition of the mixture in the reaction system and the distribution of the isotopic tracers of Dz and HD were obtained.
Thus Ag, can be readily calculated from eq 6.
Experimental Seat ion Carbon monoxide, carbon dioxide, water vapor, and hydrogen were purified in the same manner as that described in earlier articles.7-9 Deuterium was purchased from Takachiho Chemical Co., Ltd. The purity of the deuterium was 99.99 atom 70D. It was used for the experiment without further treatment. An iron oxide catalyst (Fe3O4) was supplied, by Mitsubishi Kasei Co., Ltd. The physical properties of the catalyst were reported in an earlier arti~le.~-O The catalyst was crushed and screened to about 12 mesh. The amount of the catalyst used was 0.5 g. The reaction temperature ranged from 467 to 522". For all experiments feeds contained carbon monoxide, water vapor, carbon dioxide, and hydrogen. The concentrations of these component gases were varied rather widely. However, the total pree,sure of the reaction system was kept at about 80 mm. In the absence of the iron oxide catalyst, a series of runs was made by employing the same experimental conditions as those of runs in the presence of the catalyst. These blank runs showed no measurable conversion for the isotopic exchange reaction between water vapor and hydrogen, the hydrogen -deuterium equilibration reaction, and the water-gas shift reaction. Thus it was established that all measurable conversion was due to the iron oxide catalyst. A detailed description of the experimental apparatus and the procedurie utilized in this study was presented elsewhere?-9 A closed recycling type reactor was used. A gaseous mixture of carbon monoxide, water vapor, carbon The Journal of Physical Chemistry, Vol. 77, NO. 4, 1973
Results Table I1 summarizes the experimental data gathered in this investigation. Figure P shows, as an example, the time-dependent experimental points of p(Hz) and Y(HD) of run 1. For the determination of all the time derivatives which appear in eq 13 through 18, first, corresponding experimental data were fitted with a polynomial function of reaction time, and then, the derivatives were obtained analytically. In Figure 2, for the experimental data of run 1, the estimates of Z(H(a)), which were calculated with the help of eq 15 are presented in comparison with experimental values of Z(H2) and Z(H2O). It i s evident from Figure 2 that in a region far removed from equilibrium the value of Z(H2) is greater than those of both Z(H(a)) and Z(H20) and that the value of Z(H(a)) is in good agreement with that of Z(Hz0). For other sets of experimental data, similar results were obtained, though not shown in this article. Figure 3 correlates the values of AG with values of Agv calculated by eq 6, 13, 14, and 15. It can be seen from the figure that a linear relationship exists between AG and Ag, and that the slope of this relationship is approxirnately 2. In Figure 4, the values of Ag, were plotted against those of Ag,. The relation is represented by a linear diagonal line. This indicates that AgH is approximately equal to Agv under experimental conditions employed in this investigation. I t may be noticed that a slight trend exists in the relation between Ag, and AG (see Figure 2). The studies of unidirectional rate of u + and ~ u-, may explain
Rate-Controlling Steps for the Water-Gas Shift Reaction
451
TABLE II: Experimentel Data
Run 1. Reaction Temperature 467"
0 30
BO 90 120 165 225 270 330 435 540 630
20.8 17.4 15.3 14.9 13.8, 13.5 13.1 t2.9 12.8 13.0 12.9 12.9
19.9 16.6 14.5 14.0 13.1 12.8 12.0 12.7 12.1 12.2 12.0 12.2
6.04 7.23 8.10 9.04 9.73 '10.17 10.64 11.14 11.88 11.93 12.37 12.33
6.10 7.29 8.16 9.10 9.79 10.23 10.71 10.96 11.20 11.94 11.99 12.43 12.39
17.93 17.31 16.15 15 05 14.41 13.99 13.63 15.72 16.00 16.37 16.59 16.34 16.74 16.88 17.05 17 03 47 27
9.95 9.33 8.17 7.07 6.43 6.01 5.65 7.74 8.02 8.39 8.61 8.36 8.76 8.90 9.07 9.l5 9.29
19.50 19.00 48.40 17 87 17.65 17.31 16.78 16.58 16.47 16.23 15.93 15.71 15.46 15.04 14 37
21.10 20.60 20.30 19.47 l9.25 18.91 18.30 18.18 18.07 17.83 17.53 17.31 17.06 16.64 15.97
20.6 24.1 26.2 26.6 27.6 27.9 28.4 28.6 28.6 28.5 28.7 28.5
21.3 25.0 26.8 27.3 28.2 28.5 29.0 29.2 29.2 29.1 29.3 29.1
0.55'1
0.440 0.552 0.593 0.609 0.614 0.610 0.606 0.594 0.569 0.563 0.553 0.537
0.003 0.038 0.067 0.093 0.120 0.152 0.192 0.217 0.264 0.291 0.322 0.346
0.410 0.340 0.298 0.266 0.;!38 0.202 0.?89 0.;67 0.944 0.125 0.117
0.455 0.463 0.460 0.474 0.479 0.470 0.470 0.465 0.465 0.454 0.454 0.436 0.427
0.002 0.015 0.028 0.041 0.054 0.068 0.079 0.101 0.123 0.146 0.166 0.208 0.238
0.542 0.520 0.51'1 0.483 0.466 0.460 0.450 0.432 0.410 0.399 0.379 0.354 0.333
0.250 0.285 0.356 0.394 0.407 0.409 0.408 0.554 0.560 0.561 0.561 0.558 0.550 0.448 0.545 0.536 0.527
0.003 0.018 0.045 0.088 0.121 0.153 0.197 0.198 0.206 0.220 0.233 0.253 0.273 0.288 0.306 0.332 0.353
0.750 0.696 0.597 0.519 0.471 0.436 0.393 0.246 0.232 0.219 0.204 0.188 0.776 0.163 0.147 0.131
0.720 0.718 0.729 0.741 0.743 0.745 0.743 0.750 0.751 0.764 0.754 0.757 0.757 0.750 0.757
0.002 0.005 0.008 0.009 0.012 0.017 0.021 0.031 0.037 0.046 0.054 0.061 0.069 0.093 0.097
0.276 0.276 0.262 0.249 0.244 0.237 0.234 0.218 0.210 0.789
Run 2. Reaction Temperature 489"
0 10 20 30 40 50 60 75 90 105 120 150 180
10,90
36.96 35.07 34.81 33.96 33.27 32.83 32.36 32.10 31.86 31.12 31.07 30.63 30.67
33.60 32.41 31.40 30.60 29.91 29.67 29.00 28.74 28.50 27.76 27.71 27.27 27.31
Run 3. Reaction Temperature 522"
0 5 18 35 47 60 80 0
IO 25 40 60
230
9.55 10.13 11.33 12.43 13.07 13.49 13.85 25.46 25.18 24.81 24.59 24.84 24.44 24.30 24.13 24.05 23.91
14.68 15.26 16.46 17.56 18.20 18.63 18.98 30.59 30.31 29.94 29.72 29.97 29.57 29.43 29.26 29.15 29.04
0.119
Run 4. Reaction Temperature 51 1"
, o 10 25 35 45 55 70 90 110 t 35 155 175 195 255 285
20.24 20.74 21.34 21.87 22.09 22.43 22.74 23.06 23.27 23.51 23.81 24.03 24.28 24.70 25.37
29.65 30.15 30.75 31.28 31.50 31.80 32.45 32.57 32.68 32.92 33.22 33.44 33.69 34.11 34.78
0.190
0.180 0.172 0.156 0.143
The Journal of Physical Chemistry, Vol. 77, No. 4, 1973
Shoichi Oki and Reiji Mezaki
452
3
/
-2
-27
-I
CFp
0
1
0 Run I
at467'C
0 Run 2 a 0 R u n 3 Q 5 2 2°C.
0 R u n 4 a 5 I 1°C.
2
//
I 0
Run I a t 467'C
0 Run 2 (IV 489% @ Run 3 at 522'C
0
3 Figure 3. Gibb!j free energy change of the reaction and that of
Figure 4. Gibbs free energy change of hydrogen exchange path and that of step v.
the cause of the trend. A comprehensive investigation has been in progress to clarify this matter.
A number of investigators studied the rate of the watergas shift reaction over an iron oxide catalyst and correlated their rate data by power function models. Their reaction rates were almost consistently first order with respect to the partial pressure of carbon monoxide. This implies that steps related to physical and chemical changes of carbon monoxide are rather significant far the reaction and that these steps could be rate controlling. Recently Nakanishi and Tamarule measured the amount of hydrogen which was adsorbed in the course of the reverse reaction of eq 1 and concluded that the reduction process of iron oxide catalyst by hydrogen would be rate determining. More recently Temkin and coworkers15 reported that two rate-controlling steps, perhaps the reaction of adsorbed water vapor with gaseous carbon monoxide and the desorption of hydrogen and carbon dioxide, may exist in the course of the water-gas shift reaction. On the ground of the results of these investigations in conjunction with those of Oki's studies, it would be adequate to propose that the adsorption of carbon monoxide, step i, and the associative desorption of hydrogen, step v, are the rate-controlling steps of the water-gas shift reaction with the iron oxide catalyst for the experimental conditions employed in this investigation.
step v.
Discussion As shown in Figure 3, -AG increases linearly with inMoreover, the slope of this linear relation creasing -&. is about 2. Because the values of both - A G and -Agv should be exactly zero at equilibrium and because eq 11 is assumed to be satisfied for the present system, we readily obtain AG = %Ag, (19) and
Equation 20 indicates that steps i and v are rate controlling. This conclusion coincides with those drawn from earlier experiments12,113 conducted utilizing carbon-14 and oxygen-18. Figure S shows that total free energy change of hydrogen ;Itoms exchange between HzO and Hz is approximately equal to the free energy change of step v and that the condition of eq 12 is, indeed, fulfilled. The result also supports the validity of previously obtained experimental r e s u l t @ J ~shown by eq 10.
The Journal of Physkal Chemistry, Vo/. 77,
No. 4, 1973
