1972
1. 2. 3. 4. 0.
(j.
7. 8. 9. 10.
0. K. RICE
Matrix
System Random numbers 4 dyes-A.,h 4 dyes-D,.h 3 dyes--A,,A 3 dyes-D,.A 2 dyes--A,.h 2 dyes--D,,h
Cytochrome oxidase-D,, Hb/OaHb-A..X Hb/OrHb-Da,h
h
4 4 4 3 3 4 4 4 3 3
x 8 x 10 x 10 x 4 x 4 x 5 x 5 x 11 x 7 x 7
-s-
TABLEIV OF R A N K DETERMINATION
Obsd. Calcd. 27.1 2.4 2.0 12.6 8.2 4.6 0.14 3.1 1.5 0.11
1'01. 65
12.3 6.3 8.4 4.5 2.7 1.4 0.91 4.5 1.5 0.33
S -nObsd. 184 12.6 7.8 47.9 3.7 0.50 0.01 1.0 0.12
Calcd. 140 48.8 55.1 49.2 47.4 17.8
...
101 21.8
..... and Harboe,12have shown that the light absorption of a partially saturated hemoglobin solution a t a given wave length is a linear function of its percentage saturation. It seemed worthwhile to check this result by the present method since it is independent of determinations of percentage saturation. Table IV shows that F A = 2, thus confirming that the four haems of the hemoglobin molecule act independently in the absorption of light. Chlorella Difference Spectra.-Coleman nnd RabinowitchlYhave presented difference spectra in (12) h1. Ifarboe, Scand. J . Clin. and Lab. Inoest., 11, 66 (1959). (13) J W. Coleman a n d E. Rabinowitch, J . Phya. Chem., 6 3 , 30 ( 1959 1.
-s4-
Obsd.
2000 21.1 5.1
..... .....
0.06 ,032 .82
.....
--&/Sa7-Ss/Sr Calcd. Obsd. Calcd. nd Obsd. Calcd. 2060 462 1050
10.9 1.7 0.65
..
14.7 9 . 5 0 53 19.0 0.53
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . . 810
0.S2
8.3
. . . . . . . . . . . . . .
..
0.8 5.1 3.9 3.8 0.45
..
0.80
.I1
11.4 7.5 6.6 10.9 17.3 12.5
. . . .
.5l
22.5
.OS 1 5 . 0
na
Rank
.. 0.42 .42 .46 .40 .30
..
.GO .28
Q-l :3
4 3
1 1
1" 2 2
....... .. I? which the light absorption of a suspension of Chlorella kept in the dark is compared with that of an identical suspension exposed to one of a range of light intensities. These spectra have been examined by the present method. Unfortunately, the available data are not extensive enough to reach reliable conclusions. However, matrix analysis, in principle, should permit enumeration of the species contributing to the various bands of the difference spectra. Acknowledgment.-The author gratefully ncknowledges the advice and criticism of Dr. G. Weber.
ON THE RELATION BETWEEN AX' EQUILIBRIUM CONSTANT AND THE XONEQUILIBR1UR.I: RATE COXSTANTS OF DIRECT AND REVERSE REACTIONS1 BY 0. K. RICE Department o j Chemistry, University of North Carolina, Chapel Hill, A'orth Carolina Recewed April 86, 1961
It is kr own that the rat2 of dissociation of a small niolecule activated by an inert substanre is affected by the fact that some of the excited states with energies close to the dissociation energy have less than their equilibrium population. It is shown that, in spite of this, the actual observed rates of dissociation and association will bear the same relation to the equilibrium constant as if equilibrium in all intermediate states were completely maintained. A similar conclusion can be extended to other types of reaction.
Recent calculations* have indicated that the rate of dissociation of small molecules activated by inert
molecules may be affected, probably not by a very large factor but still appreciably, due to t,he fact that the energy states leading up t)odissociation do not have their equilibrium population. A complementary effect JTould of course occur in the reverse association of the fragments in the presence of the same third body. Since the rates which are thus observed are not true equilibrium rates, it has been sugqested by Nikit'in and Sokolov,2 by Pritchardl2 by Widoiq2and, by implication, by Ross and Mazur3 that it is not correct to set kobs/ka,oba
= K
(1)
where kd,oba is the observed rate constant for the dissociation, say, (1) Work assisted by the National Science Foundation. (2) E. E. Nikitin and N. D. Sokolov, J . Chern. Phya., 3 1 , 1.571 (1959); J . C. Polanpi. ibid., 31, 1338 (1959); H. 0. Pritohard. J . P h o n . Chem., 66, 504 (1061): B. Widorn, 139th meeting, Am. Chern. Sac.. Paper No. 3, Div. of I'hys. Chern. (March, 1961). (3) J . 1 b s s and P. Rlazur, J . Chem. Phys., 35, 19 (1961).
AB
+ M --+ A + B + BI
with no A or B present, ka,obs is the observed rate constant for the association A
+ 13 + PII +AB + $1
with no AB present, and K is the equilibrium constant for A B Z A + B
This is of some practical importance in connection with experiments on the association of atoms and the dissociation of diatomic molecules. For the most part experiments a t low temperatures (usually using flash photolysis) give the rate of the association reaction, while experiments a t high temperatures (usually using shock waves) give the rate of the dissociation reaction. It has been suggested that these experiments are not strictly comparable, because it has been supposed that, one cannot, calculate one rate from the other via the equilibrium constant. In order to decide whether such a conclusion is
SOT’.,
RELATIOS BETWEEN EQ~ILIBRII-N CONSTANT ASD RATECONSTASTS
1961
correct or not it is necessary to consider the conditions under which eq. 1 might possibly be expected to hold. In the first place, a rather trivial condition is that the reactions must really be those written. This means that the concentration of M should be large compared to those of AB or A and B. This will avoid side reactions, such as AB + A B
-+
A
+ B + AB
In practice it may be possible to make corrections for the presence of other third bodies, including the reacting species themselves, which is equivalent. One matter which is of considerable importance is the relation between the relaxation time for redistribution of molecules AB among the various energy levels as compared to the time required for an appreciable amount of reaction to occur. This might be a matter of some concern in shock tubes, where the temperature becomes very high and the reaction becomes very fast. Smiley and Winkler4 have investigated relaxation times for the vibrational energy of Ch a t atmospheric pressure. These times decrease rapidly with temperature, and become of the order of see. at 1500OK. This probahly may be fairly compared with the rate of dissociation of Brz, which is about5 5 X lo8 cc. mole-’ sec.-l a t 1300°K. At one atmosphere the concentration is ahout 8 X mole cc.-’, and the time of the reaction about 2.5 X sec. It thus would seem reasonable io infer that the relaxation time would always be negligible. Palmer and Hornig also cited internal evidence that the relaxation time for Br2 is negligible in the Brg dissociation. Indeed, this mould he inferred from the fact that activation and reactio? are in a sense the culmination of the relaxation process. (If the third body hl is not relaxed, then, of course, the association and dissociation must he compared under similar conditions). The fact :hat the relaxation time is so small has a number of important consequences. Let us fix our attention oil the great bulk of the molecules in low energy levels which are much more highly populated than are those excited levels which are close to the transition :;tate for dissociation, There will be relatively little attrition of these lorn levels due to reaction in the time of relaxation; therefore, they will be esseiitially in equilibrium. Only the energy levels near the transition state will depart appreciably from their equilibrium quota. Thus, as I have s’lown prev~ouslyq6 the proper condition is set up for tlle population of all the levels to be in a stead2 qtate (it is Irue that a somewhat simplified mechanism was considered, as it would be impossible to treat the actual case in its full complexity). This being true, the ac2ual rate a t which the reaction will occur, regardless of the departure from equilibrium conditions in any of the energy levels, will be uniquely determined by the concentration and the temperature. I n view of the shortness of the relaxation time compared to the reaction time, w e expect the elapsed time of the process of activation and reaction (or, inversely, for. deactivation) to be small compared to the time required for any appreciable amount of F Smiley and E. H. Winkler, J Chem Phys.. 22, 2018 (1454). ( 5 ) II R. Palrnei and D. F Hornig zbzd.. 26, 98 (1957). (6) 0 Iea t which the AB’S decompose and that a t which A and B associate must be equal, we must have
Yol. 65
the experimental kineticist can determine the latter, for it is impossible to observe the elementary processes in which the molecules we have called AB* are formed or destroyed; only the accumulated increase of products or decrease in the reactants can be measured, but the observed rate constants will obey eq. 1. It is, of course, possible to imagine special cases in which special manipulations are made, such as removing all molecules which appear in certain energies (say all energy levels belonging to AB, including those of AB*), which upset the applicability of eq. 1, but, lacking a Maxwell demon, it is hard to see hen- this could be done. One also could imagine a situation in which the relaxation time ITas compah,ohs(AB)r(hf)6t = ka,ohdA)rdBhi(bI)6t (3) rable to the reaction time, but this would require Equation 1follows immediately from eq. 2 and 3. the activation energy of the dissociation, compared When A, B and AB are preser-t in equilibrium to k T , to be rather low, and would probably prequantities, every energy level must have its equi- clude the possibility of making a measurement. librium population. If we include in the rate of de- If one could make a measurement, he would, as composition the total contribution from every en- noted above, find an induction period of the order ergy level of AB, regardless of its history (that is the of magnitude of the reaction time, and the reaction contribution from every AB whose representative rate constant would lose its significance. The actipoint crosses a certain surface in configuration vation energy for the association can be low, because space regardless of whether this system has just the relaxation depends on binary collisions while the been formed from a pair of A and B), we mill get a association depends on the much less frequent different rate constant, which we may call kd,eq. ternary collisions. This is the rate constant which one calculates from Our discussion can be considered to be, in a transition-state theory, unless a specific correction certain sense, an application of the law of microis made for states not in equilibrium. If, in the asso- scopic reversibilitg, though a little more is involved. ciation reaction, we count all crossings of the sur- If only the dissociation is taking place, and some of face in the configuration space regardless of whether the energy lerels do not have their equilibrium or not they are soon reversed, we get the correspond- quota, then the rate a t which a given transition to a ing constant ka,eq and again we have, when (AB), specific level is taking place may not be equal to the (-1) and (B) are equilibrium concentrations rate at which the reverse transition is taking place. kd.cdAB)(M)Gt h3eq(-4)(B)(hf)6t (4) This is not a violation of the law of microscopic rerersibility, since equilibrium is not established. and But when the reaction is occurring in the direction kdseqlks.eq = K (5) of association also, the products being present in In this formulation we have specifically included in equilibrium amounts, the difference in the direct the rate of dissociation the redissociation of the and reverse transition rates is made up by contrimolecules AB* which we have mentioned above, butions arising from molecules formed in the assoand have also included the rate of their formation in ciation reaction, these being independent of the conthe rate of association. Since, in the system we are tributions arising from the dissociation reaction. considering, the fate of a single AB molecule or a It is the independence of these contributions, demsingle pair of A and B molecules does not depend on onstrated above, which is the element needed to the molecules present other than M’s, it is legiti- supplement the law of microscopic rerersibility. mate to say that the observed rate of reaction (the The law of microscopic rerersibility says that for rate which would actually be measured in an experi- any portion of kd,eq which is excluded for purposes ment in which no products were present) would dif- of calculating the “actual rate constant” for the fer from the equilibrium rate by an amount which is reaction, there 11411 be an exactly corresponding and proportional to (AB) for the dissociation and pro- balancing set of inverse processes which will not be portional to the concentration of pairs or (A) (B), included in the “actual rate Constant” of the inverse for the association. This is consistent with the ex- reaction. The “law of independence or noninterpression obtained by subtracting eq. 3 from eq. 4, ference” says that these “actual rate constants” and shows that eq. 3 has the correct form. The can be identified with kd,obs and k a o b s , as deterchange in the rate constant of the direct reaction mined when the system is not in equilibrium. due to lack of equilibrium is exactly compensated by In the preceding discussion we have confined the change in the reverse reaction. to an association-dissociation reaction in It is of interest to note that near equilibrium the ourselres the gas phase. With the conditions described a t rate a t which equilibrium is being approached will the beginning of the article, the same conclusion can he given by the absolute value of kd obs (-4B) (hr) be extended to other cases. In particular, it is -1CaobP (-4) (€3). Thus, even in the immediate that it can be applied to other stoichiometries. neighborhood of equilibrium the apparent observed clear It will be of especial interest to consider the case rate of reaction will be measured by the “observed” of an association-dissociation reaction of solutes rate constants rather than by the “equilibrium” A B S A + B rate constants. There will be no method by which
RELATIOS BETTVEES
s o v . , 1961
E Q U I L I B R I U h l C O K S T A S T hS1)
in solution. Assuming first that the solution is dilute we may note that in this case it is in the association reaction, rather than in the dissociation reaction, that departures from equilibrium are likely to occur. In general, a pair of atoms which did not react would make several collisions in a short time interval before drifting apart. If they react on almost every collision, however, the pair of atoms will disappear a t the first collision, its place being taken by a molecule. Thus there is a dearth of atoms close together. This results in a lowering of ka,obs-the well-known cage effect. But kd,obs mill be lowered in the complementary manner-a molecule once dissociated may associate again in a relatively short time compared to the time of reaction. I n general the behavior of any pair of atoms will be independent, in dilute solution, of the pressence of AB molecules. Likewise the behavior of an AB molecule will be independent of A and B atoms, the probability being small that one of the atoms formed by the AB will react with another atom in the time it mould take the pair to reassociate or to definitely dissociate. Also the number of A and B atoms in the intermediate state between association and dissociation will be small, and will not appreciably affect an observed rate. So exactly the same argument as given above will show that the equilibrium constant is given by K = kd,obs/ka,obs. If the concentration of atoms is sufficiently large so that an atom does not in general hare one nearest partner, and thus there is no cage effect, or a reduced one, a similar argument may be applied, a t least in certain cases. Suppose the concentration of AB in equilibrium with A B was small compared to the concentrations of A and B. We then can imagine a case I1 in which AB was absent, and another case I in which AB mas present in a concentration considerably greater than the equilibrium concentration, without appreciably changing the environment if the concentrations of A and B remained k e d . In case I we would observe the reaction AB A B proceeding at the rate kd,obs (AB); in case I1 we observe A 4- B AB proceeding at the rate ka,obs (A) (B). The behavior of a given AB would be unaffected by the other AB’S present. An atom, once formed, might react with other atoms already there, but this would simply merge in with the back B reaction. Furthermore, the fate of any pair A mould be unaffected by the AB’Spresent. We have thus fulfilled the condition that we always measure the same reactions. Therefore, at the equilibrium state obtained by properly k i n g the concentration of AB, we again will have ka,obs IA)(B) = (AB), where ka,obs and kd,obs have the same values as before, and the usual relation between equilibrium and rate constants follows. JJ7e will close by considering in some detail a special schematic mechanism suggested by Widom,’ which may be used to illustrate the assumptions made and the points involved. We suppose we have two substances X and B, which may exist in a total of four states, and that transitions may take place only hetween adjacent states
+
-+
+
-+
+
.1,
Az
Ba r ’B4
(7) B. Widom, private comunication.
RATEC 0 S S T . l S T S
1975
We suppose that A:! and Bo are activated states, which are present only in small numbers. The chemist will measure the quantities of AI and B, and will be essentially unaware of Az and B3, If the system is not in equilibrium the latter will not be present in their equilibrium quotas, but we can assume that they are in a steady state.8 If the whole system is in equilibrium we mill have ~ = k3: (B3),and k34 (B3) = JcdA1) = k q l (A,) , k : ! (A?) k4, (B4),where k,, is the rate constant for the transition from state i to state j . For the equilibrium constant, then, we have K (&)/(Ai) = [(R,)/iB,)l[iB,)/(A,)l[(h,I/(Ail = h l>k?lk34/k?lk3?k43
Xow assuming that initially we have rlq = 0, we can write for the forward rate constant kf,obs
=
-Ai-’(dAi/dt)
and the usual treatment of a steady state gives kf.ohs
ki2k2&34/(kdzk21
+ kdzi +
kaak23)
For the hack reaction we have A1 = 0 and kb,ots =
-Ba-’(dBd/dt)
Simply by interchanging 1 and 4,and 2 and 3, in the expression for kf,obsJ we obtain kb
ob8
k4ikd21/(kn3ks4
+ knikw + k a k a )
Therefore K
=
ki,oba/kb.obs
Let us now examine what has gone into this. In the first place the assumption that we are dealing always with the same reaction is embodied in the supposition that the IC,, are truly constant. The assumption that the relaxation time is small compared to the reaction time is implied in the assumption that, in a state of equilibrium, (A?) and (B3)are very small compared t o either (A,) or (B4). The time required for emptying of the state A2 into AI, if the reverse reaction did not take place, would be of the order of k y ‘ . About the same amount of time would be required to fill the state AP if it started empty. Thus kzl-l can be taken as the relaxation time, and me wish to show that knl >> kf,obs. But it is obvious from the expression for kf&s that it is less than k12. A!d~o since AI)^ >> (A& we know that kql >> klz. Thus the necessary condition is fulfilled. We may compare kf,obs with lqeq. At equilibrium the rate of “crossing of the surface in the phase space,” would be just equal to k ~ ( - b ) ~ qk~
[(-42Ieq/(
Ai) ](AI)
(kz&iz/kzi)(Ai)
Thus ki,,q
= kzakiz/ka
kb.eq
= ka?kra/kaa
and, similarly
It is seen that K =
kf,eq/kb.eg
though k i e q and ICb,eq are quite different from kf,oba and kb-obs. The general conclusions of this paper may, I (8) Since writing the original version of this article, I have been informed by Dr. H. 0. Pritchard t h a t Dr. Norman Davidson demonstrated to him that an equation like eq. 1 would hold if all intermediate levels were in a steady state. This certainly is a necessary condition as we noted above, and we will use i t directly in this example.
1976
BERNARD J. Woon
think, be fairly summarized by the statement that if unambiguous reaction rate constants can be found, then the quotient of the experimentally determined constants will give the equilibrium constant.
AXD
HEXRYWISE
T'ol. 63
Without committing them to my views, I wish tjo thank Drs. Wendell Forst, Benjamin Widom, John Ross, and H. 0. Pritchard for discussions and correspondence. In the light of their criticisms, the presentation has been considerably revised.
KIKETICS OF HYDROGEN ATOM RECOMBINATION ON SURFACES1 BY BERNARD J. Woon
AND
HENRYWISE
Division of Chemical Physics. Stanford Research Institute, Menlo Park, California Receiced April 17, 1961
The catalytic efficiencies of various solid surfaces for hydrogen atom recombination have been determined as a function of surface temperature. The values were derived from experiments in which atom concentration profiles in a cylindrical tube of finite length were measured with a movable catalytic probe. The experimental data were interpreted by means of a theoretical analysis which accounts for both radial and longitudinal atom diffusion. The applicability of the theory to the experimental arrangement and the limitations of the technique are evaluated. For such highly exothermic reactions M atom recombination, an interpretation of the catalytic properties must take into account energy transfer and dissipation in the solid. The effect of dissolved hydrogen on the catalytic activity of a metal is discussed.
Introduction Xumerous measurements of the catalytic efficiency of surfaces for heterogeneous atom recombination have been carried out. The methods employed varied widely in detail, but basically they may be differentiated according to the manner in which the gaseous atom density was estimated. Probe techniques involved measurements of the temperature rise2+ or t'he heat input6S7-l1to a surface situated in a tube through which atoms migrated by diffusion or forced convection (or both) in order to estimate t'heir concentration. Related techniques included the use of Wrede gages to estimate the change in concentration of hydrogen atoms adong a cylindrical tube12 and the photometric determination of the reduction of a molybdenum oxide layer by gaseous atomic hydrogen.I3 More recently, electron paramagnetic resonance spectrometry has been applied to evaluate atom concentrations in a tube. 11~14,15 Many investigators saturated the molecular gas with water before admitting it to the discharge tube (1) This work was eponsored by Project Squid, which is supported by the Office of Naval Research, Department of the Navy, under Contract Nonr 1858(25) NR-008-038. Reproduction in full or in part is permitted for aqy use of the United States Government. (2) (a) W. V. Smith, J. Chem. Phys., 11, 110 (1943): (b) V. V. Vowndski and G. K. Lavrovskaya. Dokladu Akad. Nauk, S.S.S.R.. 63, 151 (1918). (3) S. Katz, G. B. IZistiakowsky and R. F. Steiner, J . A m . Chem. Sw,71, 22.58 (1949). (4) K. Nakada. S. Sat0 and S. Shida, Proc. Japan Acad., 31, 449 (1R55). ( 5 ) J. W. Fos, A . C. 11. SmithandE..J. Smith, PTOC. Phgs. Soc., 73,
553 (1969). (6) K. Nakarla. Bull. Chem. Sac. Japan, 32, 809 (1959). (7) S. Roarinsky and A. Schechter, Acta Physicochim. U.R.S.S., 1, 318 (1034). 18) R . Siihrrnann and H. Cseqch, 2. physik. Chem.. B28, 215 (1935). (9) H. G. Poole, Proc. Roy. Sac. (London). 8163,404 (1937). (10) S. Snto. J . Chem. Soc. Japan, 76, 1308 (1955). ( 1 1 ) T. If. Shaw. J . Chem. Phys.. 30, 1366 (1959). (12) S.Sztn. J . Chem. Sac. Japan, 77, 940 (1956). (1.7) €1. W. hlelville and J. C. Robb, P ~ o cRou. . Sac. (London),A196, 479 (1949). (14) S. Krongelb and M. W . P . Strandberg, J . Chem. Phys.. 31,1196
(1959).
(15) D.IIacker. S.A . Marshall and M. Steinberg, "Catalytio Surface Recombination of Atomic Oxygen." Quarterly Report ARF4203-3 I .4rmour Research Foundatinn. Chicago, 111.).
employed for the production of hydrogen atoms. Some workers who used catalytic probes ignored the effect of this atom-measuring device upon the diffusional flow of atoms. Both of these factors may profoundly affect the results of the experiment and the interpretation of the data. A comprehensive theoretical analysis of the diffusional flow of atoms in a cylindrical tube, as a result of heterogeneous recombination, has been developed. l6.I7 The theoretical model consists of a plane source of atoms situated a t one end of the tube which is terminated a t the opposite end by a plane closure. The wall of the tube has a uniform catalytic activity which may be identical with or different from that of the closure. The solutions to the steady-state diffusion equation yield the fractional atom concentrations to be expected at the closed ends of cylinders of various lengths, radii, and catalytic surface activities. By means of this analysis it is possible to use the experimental technique of Smithza to evaluate absolute atom recombination coefficients for the tube wall and the closure. Experimental Procedure A horizontal, water-jacketed Pyrex cylinder, closed by a flanged glass cap a t one end, was joined at the open end to an electrodeless discharge tube. The effective length of the cylinder available to atomic species could be varied by means of a movable catalytic probe which effectively approximated a closure for the cylinder. Figure 1 iE a schematic diagram of the apparatus. Gas was pumped through the discharge tube, which was at right angles to the axis of the cylinder, so that the flow of atoms within the cylinder was purely diffusional. The gas in the discharge tube was dissociated by means of the 17-megacycle signal from the radio transmitter. The probe (Fig. 2) was a helix wound from fine wire of the metal whose catalytic activity was to be determined. I t was affixed to a glass assembly which supported the filament in the center of the tube and which incorporated a ylass-enclosed iron armature that permitted axial ositioning of the probe from outside the tuhe by means o p a large magnet. The filament was connected to flexible lead mires which did not interfere with the positioning of the probe, but which communicated with the outside through the cap a t the end of the cylinder. The decrease in electrical power from a battery required to maintain the filament a t a constant resistance (and therefore (16) H. Wise and C.M. Ablow. J . Chem. Phyn. 29, 631 (1958). (17) I€. Mots and H. Wise. rhzd 38, 1893 (1960)
