Cooper H. Langford
Carleton University Ottawa 1, Canada
I
-
lcaunters and Slow Reactions
The explicit subject of this paperwill be the mechanisms and kinetics of some reactions of metal ions in solution. However, these reactions are reviewed here to focus attention on the attitude which has been developing in this field over the past ten years as a consequence of the development of tools (largely in the laboratories of IN.Eigen in Gijttingen) for the study of very fast reactions in solution. The removal of upper limits on the accessible time scale for the experimental study of solution reactions has had the effect of resolving many "simple" reactions into mnltistep processes. The resolution highlights the role of the fundamental diffusional (1) step in all solution processes. The new attention to the part played by diffusional processes has led to the description of reactions involving the formation of a coordination compound by the reaction of a metal ion and a ligand in a mechanistic language as much derived from the collision theory as from transition state theory. This paper will review this mechanistic viewpoint and try to show that it has useful applications to a variety of kinetic problems not limited to the field of coordination chemistry. Let's begin with the reaction leading to the formation of a very simple coordination compound
+ SOF(aq) = MgS04
Mg4+(aq)
(1)
At low concentration of reagents, the application of a "medium fast" kinetic technique (such as temperature jump if the analytical parameters of the system would allow) would reveal a second-order reaction for the formation of MgSO1 with a rate constant of approximately 2.5 X lo6 At-' sec-I. But a classic figure from Tamm's work (3)(Fig. 1) reveals that this result is an oversimplification! The figure shows a function of the excess ultrasonic absorption of a MgSO1 solution plotted against the frequency of the ultrasonic wave. Without lingering over the subtleties we see that there are two peaks in the curve and that these are to be linked
with two "relaxation times" associated with our reaction (1). We must identify two separate kinetic steps. The faster process is shown to be second-order by its concentration dependence. The slower process is firstorder. The rates associated with the faster step in which the second-order character of the overall reaction arises turn out to be very close to those calculated for diffusion of species in solution. The minimum mechanism for the reaction leading to MgSO, must now be written which is close to a mechanism proposed by Eigen (3). The first step of mechanism (2) is associated with the fast process revealed by ultrasonics. The product of this first step where the two are still separated by one (or a few) water molecules is called an encounter complex. It is the product of a diffusional step. I n solution, even two non-interacting partners which diffuse together do not immediately separate on the rebound from their first collision. They are trapped together as they are deflected back by the "cage" of surrounding solvent. Thus, a diffusional encounter lasts long enough for many collisions and efficient energy transfer with its surroundings. It is a good thermodynamic species. Now, it should he clear that it will be an excellent approximation to consider the encounter complex to he in equilibrium with the free ions throughout the course of the formation of MgS04. This product is formed in a much slower step. It follows that any "slow" himolecular reaction in solution may be written in at least two steps (perhaps more)
E
kl
Products
(3)
where the formation of the encounter complex, E, may be treated as an equilibrium process characterized by an equilibrium constant K E and the conversion of E to products as a firstorder reaction characterized by a rate constant kl (unless the concentrations of the reactants A and B are very small indeed). Nucleophilic Reactivity
Suppose that we are now interested in comparing the reactivity of a series of reagents with respect to some
FREQUENCY(CYCLES) Figure 1. A function of the excess ultrasonic absorption due to rolvts of M g S O l dissolved in w a k r plotted ogoinst the ultmsonic frequency.
This paper was presented in the symposium an chemical dynamics sponsored by the Division of Chemical Education at the 156th meeting of the American Chemical Society, Atlantic City, New Jersey, September, 1968. Volume 46, Number 9, September 1969
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substrate. If we are trying to learn about the ways these reagents attack the substrate, we are not greatly interested i n the differences i n their probabilities of encounter with the reagent. W e want to compare kl values. Reactions like that shown in eqn (1) are examples of nucleophilic substitution processes and the comparison of nucleophilic reactivities in reactions leading to the formation of coordination compounds will make an interesting example. Consider the reaction (4) Ni(OH,)sP+
+L
-
Ni(OH&L f HzO
(4)
Table 1 gives second-order rate constants, kz, for the formation of Ni(OH2),L compounds with various ligands L. The data comes from temperature jump studies in Table 1.
Reactions of Nickel Ion in Woter"
HPsOP 2 X ~ ~ 0 , ~ - u x 5 X SCNHC20a5 X NHa 2.5 x
108 10' 10" loS lo3
a I)& from ref. (6) KE calculated according to ref. (4) with an assumed distance of closest approach of 5 1.
mostcases. The second-order rate constants suggest variable reactivity. But, it might be argued that the most important variable appears to be charge. An equilibrium constant for the encounter between two ions can be estimated theoretically from Fuoss' ion pairing (4) equation (which is based on an encounter model) if we can judiciously guess the distance of closest approach in the encounter complex. Choosing 5 A yields the values of K E shown in Table 1. The second-order rate constants are to he interpreted as k2 = K=kl SO the final column of Table 1 lists values of k2/KE. We see that k2/KEis nearly constant and conclude that there is very little difference in the reactivity of the various nucleophiles. The result illustrated here for Ni(I1) compounds turns out to be quite general for reactions leading to the formation of octahedral coordination compounds of a wide variety of metal ions in both aqueous and nonaqueous solutions (5). I t makes possible a fairly clear identification of the process leading to the transition state in these substitutions. Since we have seen that all entering ligands react at very nearly the same rate once correction is made for the variation in the probability of formation of an encounter complex, it is reasonable to assume that the entering ligand has no specific role in determining the energy of the transition state. Rather, that energy must be determined by the requirements for breaking the bond to the leaving group. Substitutions nou-selective with respect to entering group have been called dissociative. A striking contrast is presented by the reactions of square planar Pt(I1) w-bich have second-order rate constants very scnsit,ivc to t,he naturc of thc entering ligand. The cntcring group scnsit,ivit,ydoes not correlate a t all well with the t,hcoret,ical Kr: values and provide strong evidence for an associalive mode of substit,ution a t Pt(I1). 558
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Another case of octahedral substitution where K E values are experimentally estimated rather than calculated from admittedly somewhat crude theoretical considerations can enrich the picture. Consider reaction (5) which has been studied by Taube and his students (6) and by Langford and Muir (7).
+ L"-
Co(NHJ5OH2a+
-
C ~ ( N N J ) ~-L*)+ (~
+H
9 0
(5)
Kn is measured from rapid changes that occur in the ultraviolet spectrum of the solution and the rate of formation of C O ( N H ~ ) ~-L"I+ ' ~ from slow changes that occur in the visible spectrum. The value of kl has been extracted for four different ligands L-". Table 2 reports these values as compared to the value of k,,, the rate of exchange of water molecules between the Table 2.
Reactions of Cobalt(lll) Ions in Water*
L
k,lk,,
OH% SO4'c1SCNHPOdi
(1.00) 0.25 0.21 0.16 0.13
Data from ref. (7).
bulk solvent and Co(NH&OHZa+,obtained from 180 tracer experiments. Now, Ic, is a first-order rate constant because the Co(II1) species is in perpetual encounter with the solvent so it should be directly comparable to kl. The interesting point is that the small variations have a pattern. All kl values for all four L's are of the order of five times smaller than k.,. Figure 2 shows a plausible explanation of this. If one ligand L is in encounter with CO(NH&OH~~+, there are still several water molecules in encounter, too. If dissociation of the leaving water molecule is smoothly followed by entry of a specific entering group (e.g., the one adjacent to the leaving group), water exchange remains the statistically more probable reaction when the encounter complex contains one L and several water molecules randomly distributed. All reactions
INNER
OUTER SPHERE
w/
Figure 2. An impression of tho arrangement of the inner and outer sphere ligands for the example complex ColNHahOHlat, L where L is o non-solvent outer-sphere ligmd.
of non-solvent L's should be slower than solvent exchange by the statistical factor of number of solvent to lignnd when the value of kl is independent of the nature of the entering group. At this point,, it's reasonable to wonder about an alt,ernative form of the dissociative mechanism. Suppose that an intermediate CO(NH&~+, a species of five instead of six coordination, were formed. The operational significance of such an intermediate would be that it reacts with the entering ligand after the leaving group has left and the transition state for the reaction is passed. Thus, it would not require that the entering group were a member of the original encounter complex. The statistical relation between k, and k,, indicates that such an intermediate is not formed. The fact that the probability of entry is determined by the initial population of the encounter complex shows that the entering group enters before rearrangement of the encounter complex. There is no way to detect reactive intermediates other than to show that they can survive long enough for the product distribution to be controlled by something other than the population of the original encounter complex. (In cases like CO(CN):~-where there is evidence for the intermediate, /el = lc,, for several entering groups which arc more reactive toward the intermediate than water.) Solvolysis
Reactions with t.hc solvent are generally found to be first-order experimentally. This is sometimes attributed to the fact that the solvent concentration is very large. More fundamentally, it is a consequence of the fact that solute substrate is in perpetual encounter with the solvent. Often it is important to compare a solvolysis rate t,o the secorul-odeel. rate of reaction with some other reagent in order to understand the energetics and detailed mechanism in a family of reactions. The observations on the Co(NH&0HZ3+system suggest how such n comparison might be made. If the value of K Efor any non-solvent reagent can be extracted experimentally or estimated theoretically with satisfactory accuracy, theu a lcl m a y be calculated from k2. Now, lcl may be compared to the solvolysis rate after correction for the statistical factor dependent upon the "solvation number" of the substrate. This solvation number must be guessed in most cases but the guesses are much better than order of magnitude. Note that the procedure often followed of identifying the first-order rate constant for solvolysis as the product of a second-order rate constant and the bulk concentration of water (lc, = k2 [H,O]) will lead to an erroneous comparison. Taube's work on the reaction of Co(NH3)jOH23+with Sop2- illustrates this clearly (6). Above about 0.01 A! SO?, the rate at which the Co(II1) species is converted increases very little with increasing sulfate concentration because the value of K. is large and the encounter equilibrium is satuvated. I t would be quite wrong to try to calculate the reaction rate at, say, 1 Ai' Son2from the second-order rate constant found a t low concentration. Neither is it possible to find the second-order rate constant from an experiment in which the first-order rate constant is found when there is a large excess of sulfate over a reasonable conccutration of the Co(II1) species. There is not even evidence in Taube's result to support the
frequent contention that the bulk activity of a reagent will continue to influence the rate after the encounter equilibrium is saturated because "the encounter complex is in thermodynamic equilibrium with the bulk." Taube's reaction changes rate very little over a quite broad range of values of the sulfate bulk activity. The imagined effect of bulk activity presumes that the encounter complex (unlike, for example, a crystal) will change its intimate structure in response to changes in the bulk activity of the reagent. This is clearly not always true. The encounter complex may sometimes be as unalterable as a microscopic crystal. Thus, second-order rate constant cannot really be found for reaction with the solvent. Mixed Solvents
Mixed solvents, from a kineticists' point of view, may be fairly characterized as "troubled waters." Perhaps we may venture cautiously into them armed with the present, notions. The specific case will be the reaction Cr(NCS)e"
+ Ha0
-
+ SCN-
Cr(NCS)sOHsZ-
This reaction has been studied in acetonitrile-water mixtures. Acetonitrile is solvolytically unreactive. To understand the influence of solvent composition on the rate of the reaction with water, the m i n i m u m information required is the relative probability of encounter of the Cr(II1) substrate with water or acetonitrile. A paramagnetic solute like Cr(NCS)6a-will produce modifications of the nuclear relaxation times which determine the shape of the nmr absorption signal in the nmr spectrum of the solvent molecules: In the event of fast exchange such as occurs between the encounter complex and the bulk of the solvent, the observed nmr signal may be treated as an appropriate average between the spectrum in the encounter complex and the spectrum in the bulk. The division of sites for solvent int,o only two, encounter and bulk, is appropriate because the paramagnetic interactions influencing nuclear relaxation times are short ranged. A comparison of the nmr spectrum of one of the solvents containing a convenient concentration of Cr(II1) species with the spectrum of thesame solvent in a mixed solvent solution containing the same Cr(II1) concentration can (in favorable circumstances) be translated into the ratio of the solvent in encounter with the Cr(II1) species in the mixed solvent to amount of that solvent in encounter in the pure solvent solution (8, 9). That is, the ratio of the solvation number for a given solvent between mixed and pure solvent. The ratio is represented n/no where n is the number of molecules of a given solvent solvating a CI(NCS)~~-ion in the mixed solvent and no is the number solvating in the pure solvent. (An absolute value of no remains beyond the scope of the experiment.) A preferential solvation curve for Cr(NCS)B3- in CH3CN-H20mixtures is shown in Figure 3. Note the strong preference of this solute for CH,CN. If the solvolysis reaction (6) is describable as firstorder in water and second-order overall this should mean that the encounter complex requires one Cr(II1) substrate ion and one water molecule specifically and the observed first-order solvolysis rate constant for any mixture should be proportional to n/na for water. (A value of n/nofor water is derivable from a value of Volume 46, Number 9, September 1969
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559
"In, (yo) X~H$2~ in CHJCN-HnO Figure 3. The preferential rolvotion curve for Cr(NCSls' mixtures as derived fmm nmr. "/no represents h e fraction of soivotion shell lor outer sphere) sites occupied by CHsCN.
n/n, for CH&N by assuming a constant solvation number and writing n/no(HzO) = 1 - n/no (CHaCN). A plot of first-order rate constant for reaction (6) against n/no (H,O) is shown in Figure 4. I t is linear with an intercept indistinguishable from zero. A plot of these rate constants against any water bulk concentration or activity variables would be very far from linear. Especially, it is significant that the rate corrected for encounter probability is constant as the activity of water in the bulk varies. Let's consider the photochemical analog of reaction (6). The reactive photoexcited state would be expected to be either the lowest *T state of CI(NCS)~~or, if crossover is efficient, the 2A state. The first (and shorter lived) of these would be expected to have a lifetime near sec. I t lasts long enough that it could function as an intermediate. It's fate may be independent of the initial encounter situation in which it found itself when it was "excited." The black squares in Figure 4 show the relative quantum yields for photosolvolysis of Cr(NCS)8a- at several values of n/no (H,O). I n contrast to the thermal reaction, the photochemical reaction is independent of the solvent composition. No water molecule is specifically required prior to excitation. Clearly there is a reactive intermediate which cannot be deactivated to a solvolytically unreactive state by any process on a time scale comparable to the lifetime of an encounter comdex. (The intermediate on this high energy path might be Cr(NCS)sNCCHi-, instead.) Summary
In conclusion are offered a speculation and some propositions which seem to follow from the preceding discussion of the explicit separation of slow reactions in solution into encounter and slow firsborder steps. They may prove ~ f ' ~ e n e ruse a l in the analysis of reaction mechanism in solution. The theoretical treatment of kinetic salt effects is similar in spirit to the Fuoss treatment of ion encounter equilibria. Some of the debates over general versus specific salt effects might he clarified by a systematic encounter viewpoint,.
can be estimated.
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Figure 4. The rate d hydmlyrir of Cr(NCS)& thermally (open circles) and photochemical quontvm yields (closed circler) as o funclion of the composition of the solvation rhell.
2) The energetic role of a reagent in the transition state of a slow reaction should be messed from its influence on the firstorder rate at which the encounter complexis converted ta product. Rate constants for higher order reactions contain mOre or less
incidental encounter probability factors. 3) To compare solvent reactivity with the reactivity of nonsolvent reagents, rate constants for the reactions of the nonsolvent reagents should be reduced to firshorder as suggested in proposition 1. The reverse process of writing the solv'blytio reaction ss higher order using the bulk solvent concentration will surely be misleading. 4) Most standard ways to identify "reactive" intermediates which do not accumulate for direct observation amount to demonstration of the independence of the reaction rate of the encounter probability between the init,ial substrat,eand some reagent. This may constitute a fair operational definition of "reactive" intermediate.
As a last word, a warning. Reactions here have been resolved into two steps. This is a minimum. The firsborder part may well be a series of first-order steps for even comparatively simple reactions. Eigen suggests, for example, that the collapse of the encounter complex between Mg2+ and SO4%-ions proceeds in two firsborder stages, loss of a water molecule coordinated to SOIZ- and then loss of a water molecule coordinated to RIg2+. It is a pleasure to acknowledge what I have learned from working on reactions in solution with Professor T. R. Stengle, Dr. L. S. Frankel, Dr. Stefan Behrendt, and Mr. W. R. Muir. An Alfred P. Sloan research fellowship and the support of the National Research Council of Canada and the U.S. Air Force Office of Scientific Research are gratefully acknowledged. Literature Cited (1) D ~ Y EP., , Trans. Electrochem. Sac., 82,265 (1942). (2) TAMM,K., AND KURZII,G., Nature, 168, 346 (1961) and Acuslica, 3, 33 (1053). (3) EIGBN,M., AND TAMM, K., Z. Elektroehem. 66, 93 & 107 M., Proceedings of the VIZth International (1962); EIGIIN,
Conference on Coordination Chcmislry, But,larworths, h n don, 1963, p. 97. (4) Fuoss, R.. M., J. Am. Chem. Soc., 80,5059 (1058). (5) EIQEN, M., AND WIIKINS,11.. G., Adu. inchem., 49,55 (1965). , A., J. Am. Chem. SOC.,7 5 , 1463 (6) Tau~ni,H., AND P o s I , ~F. (1953). SCHMIDT, W., .%NoTAUIIN,II., Inorg. Chem.,
.
2.698 (1963). , (7) LANGFORD, C. IT., AND Mum, W. R., J. Am. Chem. SOC., 89,3141 (1967). (8) FRANKEL, L. S., STICNQLIK, T. 11., AND IJ.\NOFOIII), C . 11.) Chem. Comms., 373 (1965). C. H., A N D WHIT!,:, J. F., Can. J . C h ~ m . 45, , (0) LANGF~RD, 3049 (1967). BIIIIRIINDT. C. IT.. A N D . S... L.!NGPORD. FRANKEL, L. S., J. Am. Chem. Soe., in press.
