J. Phys. Chem. 1985,89, 1748-1752
1748
HF-C3H6complex. These observations support a structure in which the H F is complexed adjacent to the methyl substitution site. For the complexes of trans-1,2-DMCP and cis-l,2-DMCP, very little structural information can be determined, from the combined difficulties of low symmetry and low product yield. By analogy only, one might suggest that the coordination is to the carboncarbon bond connecting one of the substituted carbons with the unsubstituted carbon, but no direct evidence was observed to support this suggestion. Table 111 presents a correlation of the shifts in the hydrogen halide stretching frequency with the different substituted cyclopropanes studied vs. the dipole moment of the hydrogen halide and the proton affinity of the corresponding halide anion. The magnitude of shift has been correlated numerous times28with the strength of interaction and indicates that the H F complexes are most strongly bound. This is in agreement with the gas-phase studies on cyclopropane-HX complexes and correlates with the dipole moment trend within the hydrogen halides. This suggests that, for these rather weakly bound complexes, electrostatic interactions are more important than covalent contributions, which have been shown to be significant for more strongly bound complexes. Trends within the set of methyl-substituted cyclopropanes are less apparent; Table I11 demonstrates that complexes of cyclopropane are slightly less strongly bound than those of the methyl-substituted cyclopropanes, but the number and location of the methyl substituents do not present a clear pattern of increasing basicity.
Conclusions
The matrix isolation technique has provided the first opportunity to isolate and characterize the 1:1 hydrogen-bonded complexes of the hydrogen halides with a number of methyl-substituted cyclopropanes. In all cases, the hydrogen stretching frequency of the HX subunit shifted substantially to lower energy, while certain vibrational modes of the base subunit were perturbed to a measurable degree. These perturbations allowed a determination of the site of interaction for the more highly symmetric (C2J substituted cyclopropanes, namely, between the most and least highly substituted carbons in the three-membered ring. For the remaining complexes, the most likely structure was suggested, but no definite conclusions could be reached. The agreement of the conclusions reached here with previous solution studies of the acid-catalyzed ring opening reactions of substituted cyclopropanes further suggests that the complexes identified here may either represent or closely resemble the initial intermediates in these solution reactions. Acknowledgment. The authors gratefully acknowledge support of this research by the National Science Foundation under Grant C H E 8400450. B.S.A. also acknowledges the Dreyfus Foundation for a Teacher-Scholar Grant, while C.E.T. acknowledges the University of Cincinnati University Research Council for a summer fellowship. Registry No. HC1, 7647-01-0; MCP, 594-1 1-6; 1,l-DMCP, 163094-0; tr~ns-1,2-DMCP,2402-06-4; cis- 1 ,ZDMCP, 930-1 8-7; TMCP, 4127-47-3; HBr, 10035-10-6; HF, 7664-39-3.
pK Values from Acidity Function Type Methods: Inherent Uncertainties John F. Wojcik Department of Chemistry, Villanova University, Villanova, Pennsylvania 19085 (Received: January 24, 1984; In Final Form: December 1 1 , 1984)
The uncertainties inherent in the measurements of thermodynamic pK values by means of excess acidity functions are examined by using a statistical approach. The key to the problem is the mathematical form of the equation relating the activity coefficients of two weak bases. The analysis presented here, using data for 10 weak bases, shows that, although an overlap method for determining pK values is possible from a mathematical viewpoint, experimental exigencies lead to serious problems since the data, derived from limited overlap acidities, give no hint as to the form of the activity coefficient relationship. The magnitude of the uncertainty in the determined pKvalues due to the uncertainty in the form of this relationship is given and is shown to have a high probability of being several pK units for very weak bases.
The relative basicities of organic bases has long been of interest both as a source of information about electronic structures and reactivities and as a means of estimating the amounts of various protonated forms of a base in solutions of different acid concentration. The most common measure of basicity has traditionally been the pK of the conjugate acid of the base. In order to make meaningful comparisons among bases, the px"s must be referred of a common standard state, the usual choice being infinite dilution (hypothetical 1 m). Numerous methods have been devised for the measurement of p 6 s . One requirement for all methods is that measurements are made under conditions where significant amounts of the unprotonated and protonated forms of the base are present simultaneously in solution. A second requirement for obtaining a pK value referred to the infinite dilution standard state is that activity coefficients be taken into account in some fashion since measurements are usually made in terms of concentrations rather than activities. Accounting for activity coefficients can amount to using a form of the Debye-Huckel equation or to extrapolating concentration constants to zero ionic strength, at least in those cases where the solutions are essentially dilute, that is, with 1 < pH 0022-3654/85/2089-1748$01.50/0
< 13. Many interesting bases, however, are very weak, and, in order to achieve the first requirement, they must be studied in concentrated acid. Fulfilling the first requirement complicates achievement of the second one. A promising solution to what might be called the extrapolation problem was presented by Hammett' in terms of acidity functions. In the many years since, these methods have been extensively studied and modified, a recent product of this work being the excess acidity function. The history of these functions along with extensive references has been given in a recent review.2 In a recent a r t i ~ l e however, ,~ the utility of acidity function methods for determining pK values was questioned and it was demonstrated that traditional methods, including excess acidity methods, logically yield not pK values but rather a quantity which is the sum of the desired pK value and an indeterminate constant. In the latter work it was suggested that the value of this indeterminate constant could be substantial, but the method used to (1) Hammett, L. P.; Deyrup, A. J. J . Am. Chem. SOC.1932, 54, 2721. (2) Cox, R. A.; Yates, K. Can. J . Chem. 1983, 61, 2225. (3) Wojcik, J . F.J . Phys. Chem. 1982, 86, 145.
0 1985 American Chemical Society
The Journal of Physical Chemistry, Vol. 89, No. 9, 1985 1749
pK Values from Acidity Functions TABLE I ~~
~~
~
~
av dev ref dye“ A B D G G I
test dye
]inb
B C E H I H
0.021 0.030 0.014 0.03 1
0.071 0.102
~~
~~
PK
var
quad 0.021 0.029 0.01 1 0.028 0.070 0.023
quad 0.21
N
lin
6
0.12
0.75
0.14 0.34 0.26 0.22
0.51 6.47 10.40 2.39
6 7
-2.41 -3.71 -6.55 -9.88 -9.65 -10.36
lin
0.07
6 5 4
quad -2.44 -3.43
-7.28 -5.63 -7.12 -7.64
“Indicators: A, 2,5-dichloro-4-nitroaniline;B, 2-chloro-6-nitroaniline; C, 2,6-dichloro-4-nitroaniline; D, 2,6-dinitroaniline; E, 4-chloro-2,6-dinitroaniline; F, 2-bromo-4,6-dinitroaniline; G, 3-chloro-2,4,6-trinitroaniline;H, 2,4,6-trinitroaniline; I, 3-bromo-2,4,6-trinitroaniline;J, 3-methyl2,4,6-trinitroaniline. bAbbreviation: av dev, average deviation; var, variance; N , number of data points used, lin, linear; quad, quadratic. ascertain the possible magnitude of these constants was based on a somewhat indiscriminate examination of activity coefficient data and left much to be desired. In the above-mentioned review it is stated that the indeterminate constant “is thought to be small in most cases, however, as large discrepancies would cause noticeable effects in several of the above treatments”.2 Since there appeared not to exist any analysis that would substantiate this assertion, a more careful analysis of the “indeterminate constant” problem has been undertaken and, contrary to this assertion, it will be shown that significant values for the indeterminate constant can exist in seemingly “good” data, implying unacceptable uncertainties in estimated pK values. For the sake of brevity, the background given in the review paper2 will not be repeated here. As has already been acknowledged, “the acidity function problem then becomes, essentially, a question of how to deal with the activity coefficient term”.2 In the zeroth-order approximation as orginally introduced, a direct proportionality between activity coefficient terms was assumed. If Fx is defined by the zeroth-order approximation becomes FA = FB where A and B represent different bases. In order to generalize this, the first-order approximation was given; that is F A = m*X = nl‘FB (3) Here the proportionality constant was allowed to deviate from unity. The first-order approximation leads to the development of the excess acidity scales. Whatever the true form of the relationship between activity coefficients for a pair of bases, it is safe to assume that such a relationship can b2 expressed in a power series; that is FA
=
niFBi i= I
(4)
The assumption of infinitely dilute standard states requires that there be no constant term in eq 4 since F A = FB= 0 at infinite dilution. Using eq 4 as adequately describing the relationship between FA and FBat all acidities, one can show that an overlap method for determining pK values is possible in principle. From the definition of pK one has where and P&H+ is the analogous expression in terms of concentrations. Writing eq 5 out for a different base, A, and combining this equation with eq 4 and 5 , one finally obtains, at a given acidity ppAH+ = PKAH++ Cni(ppBH+ - PKBH+)j i= 1
(7)
If we assume that sufficient terms are taken in eq 4 to span all acidities, eq 7 is likewise valid at all acidities. Now, given experimentally determined values of the hydrogen ion concentration in concentrated acid and given indicator ratio
values for weak bases, PFXHt is experimentally measurable. With such p P values, eq 7 can serve as a basis for determining pK values, By expanding the summation term one can rewrite eq 7 as ppA,+ = A0
+ A ~ P ~ B+HA+~ ( P ~ B H++...) ~
(8)
where
(9)
+
+
A1 = nl - 2 n 2 P K ~ ~ t 3 n j ( p K ~ ~ t )...~
(10)
and so on for the other coefficients. The number of terms in eq 8 is one more than the number of terms taken in eq 4. If sufficient data giving &)KCAH+ as a function of p&H+ is available, then the coefficients Ai can be determined by a curve-fitting procedure. And if pKBH+is also available, then pKAH+and all of the ni are capable of being determined. Equation 8 can serve as a basis for a logically correct overlap method of determining pK values. Although the above analysis does logically yield true thermodynamic pK values, a serious problem arises in deciding the number of terms to be retained in eq 4 as adequately describing the function. In order to illustrate the nature of the problem, indicator ratio data for 10 bases studied4 in H2S04was treated by using the above equations, first taking eq 4 with one term and then with two terms. Hydrogen ion concentrations as a function of percent HzSO4 were taken from the l i t e r a t ~ r e . ~In both cases, the pKc values were calculated and fit to the linear and quadratic forms of eq 8 by using an iterative least-squares procedure6 which allowed calculation of the variances in the final parameters. Pairs of bases were chosen which were studied in overlapping acidity ranges. Rather than attempting to relate all of the bases to each other by successive overlap, each pair was treated independently of all of the other pairs, and the pK of one member of the pair (called the reference base) was assumed known and taken equal to the value reported in the literature. The pK of the other base in each pair was then calculated relative to its reference base. Since the subsequent discusion is centered on the uncertainties in the measured pK‘s rather than their absolute values, this simplified procedure is adequate. The results are given in Table I. Consider first the columns labeled av dev (average deviation). These numbers are defined as the square root of the average of the square of the deviation of the experimental values of pKE from that calculated by the least-squares fit. It should be noted that, in all cases except one, there is no significant improvement in the fit of the data on going from a first-order to a second-order equation, that is, eq 4 with one term as compared with two terms. The pK values calculated in the two cases differ significantly. The uncertainty in the calculated pK values (taken as the square root of the variance of these quantities) for the two fits are also given. These were calculated by using the covariance matrix and Wentworth’s equation (43).6 The average deviations listed in Table I were used in this calculation. It should be clear that, in going from the linear fit to the quadratic fit, the uncertainty in the pK value greatly (4) Jorgenson, M. J.; Hartter, D. R. J . Am. Chem. SOC.1963, 86, 878. (5) Cox, R. A.; Yates, K. J . Am. Chem. SOC.1978, 100, 3861. (6) Wentworth, W. A. J . Chem. SOC.1965, 42, 96.
1750 The Journal of Physical Chemistry. Vol. 89, No. 9, 1985 increases, in some cases being several pK units. Which case, linear or quadratic, do the results support? Since going to the quadratic fit does not significantly improve the fit (except in one case), there is little reason to choose the quadratic fit. (An application of the F test leads to the same conclusion.) However, great care must be exercised here. That the linear fit is an adequate fit is only valid in the range of acidities for which data were collected and may not be true if the range were greatly extended. Equation 4 can be expanded about some value of FB corresponding to the acidity in the center of the overlap range. If the series expansion polynomial is of a lower order than the original polynomial, then the new polynomial will contain a constant term, even though the original polynomial (eq 4) did not. For example, if a second-order equation underlies the data, then eq 4 can be expanded to a linear function, i.e. FA
=a
+ bFB
+ A1pKCBH+
(12)
+ a - bpKBH+
(13)
where A0 = pKAH+
Al=b
(14)
Here we see that a fit of eq 12 gives two fitting constants which are defined in terms of three parameters (assuming that pKBH+ is known). Thus, given pKBH+,pKAH+ is undeterminable due to the presence of the indeterminate constant, a. Whether eq 4 should be taken with one term or two (or perhaps even more) is simply undecided because of the limited amount of data, and, if the second-order equation (or perhaps even higher order) were known to be correct beforehand, unacceptable uncertainties exist in the calculated pK values.’ In attempting to improve upon the acidity function method for determining pK values, the excess acidity method has been prop o ~ e d . ~Resulting ,~ from this analysis is the equation log I - log CH+ = pK’+ m*X
FA
(15)
where I is the observed indicator ratio and X i s the excess acidity scale. The latter scale is derived from indicator ratio data for many bases. Inherent in the definition of X and in the establishment of eq 15 is the assumption that eq 4 contains only the first-order term and is valid as such at all acidities. As demonstrated3 this assumption and the assumption of a more general function which is linearized over a limited range of acidities both yield eq 15, the more general equation, however, giving rise to the indeterminate constant. As the preceding analysis of actual data illustrates, there is little hope of deciding from the data whether the actual function is first order at all acidities or whether it is higher order but nearly linear in the overlap acidity range. What can be done is to consider two possibilities, first order and second order, and determine the conditions under which they become indistinguishable due to random errors and then estimate in a general fashion the uncertainty in the measured pK value due to the uncertainty in the actual form of eq 4. In view of the recent success of the excess acidity method in fitting a large amount of data to a single acidity function, the analysis will be expressed in terms of excess acidities. Assume that a set of bases exists each of which is known to follow eq 3 exactly at all acidities and that a scale of excess acidities has been set up by the methods outlined in ref 5. Assume further that a second set of bases (second-order bases) exists which follows exactly a second-order equation, i.e. (7) A recent paper (Traverso, P. G. Can. J. Chern. 1984,62, 153) on the lack of the need for a nonlinear term in an excess acidity function equation does not affect the conclusions drawn here since the analysis given here focuses specifically on the question of what can be hidden under ‘good” (Le., linear) data. That should be clear from the fact that the bases chosen for examples in this work give linear indicator ratio plots.
= nlX + n2X‘
(16)
(where FBis taken to be equal to X). This leads to the secondorder form of eq 15 log I - log CH+= pK
+ n l X + n2X2
(17)
The problem can now be rephrased. Given a set of indicator ratio data generated by a second-order base (with random errors only), for what sets of their parameters, pK, n,, and n2, will the fit of this data by the first-order equation (eq 15) fail statistically to show the curvature expected due to the form of eq 17? In solving this problem, the F statistic8 was used in the following manner. The left-hand side of eq 15 and 17 is renamed Y. F is defined as the ratio of the reduced x2 values on the basis of these two equations; i.e.
(11)
this function, of course, only being valid over a limited range of acidities. Using eq 11 one can derive equations analogous to eq 8-10, that is pKcAH+ = A0
Wojcik
F=
x12/x22
where
where Y, is the observed value of Y and YC that calculated by a given fitting function. Here v is the number of degrees of freedom and is given by v=N-n,-1
(20)
where “p is the number of parameters and N is the number of data points. n2 is the variance of each of the Y values and is assumed the same for each Y value. Actually the latter assumption is not accurate in that, for the function studied here, variances at either end of the acidity range (extreme values of indicator ratios) are greater than those in the center of the range. Here 2 can be taken as an average variance. Now one has for F
Here Yl is the value of Y given by the first-order approximation and Y2 that given by the second-order approximation. Equation 21 can be recast in a more useful form in the following manner. Define AY by AY = Y2 - Ul (22) Substituting eq 15 and 17 into eq 22 yields AY = ApK
+ AnX + n2X‘
(23)
where ApK = pK - pK’
(24)
An = nl - m*
(25)
and Solving eq 22 for Yl and substituting in eq 21 gives F=
+ AV2
~2C(ci
VlCti2
where t, is the difference between the observed Y and the true Y, that is, that given by eq 17 and is clearly random error. Before continuing it is necessary to examine the significance of F. A use of the F statistic is to compare two variances. Thus, for two sample variances, if a certain critical value of F, Fc, is exceeded (for a given confidence level and given degrees of freedom), the hypothesis that the variances are the same can be rejected. If, for a given confidence level, the calculated F value is less than Fc, there is no reason to reject the hypothesis that the variances are different; that is, given the random error in the data, there is no statistical basis for stating that the first-order fit is worse than the second-order fit. In terms of the system considered (8) Bevington, P. R. “Data Reduction and Error Analysis for the Physical Sciences”; McGraw-Hill: New York, 1969; p 196.
The Journal of Physical Chemistry, Vol. 89, No. 9, 1985 1751
pK Values from Acidity Functions here, this amounts to saying that if F is less than Fc, then one cannot confidently decide whether the base is a first-order base or a second-order base. Alternatively, if F is greater than Fc,then the differences in the fits are significant. Now Fc depends upon the degrees of freedom for the two fits and also upon a,which is the confidence level. For instance, if F > Fc leading one to conclude that the second-order fit is better than the first-order fit, and if one chose cy = 0.50, then there is a 50% chance that the conclusion is wrong. Thus, the conclusion strongly depends upon the choice made of (Y and, therefore, the choice of Fc. Incorporating this critical value of F into eq 26 one has
TABLE I1 ApK range
fraction
0.0-1 .o 1.0-2.0 2.0-3.0 3.0-4.0 4.0-5.0
0.074 0.072 0.069 0.065 0.060
B C
D
Now since AY and ci are uncorrelated, the sum in the numerator of the first term on the right-hand side of eq 28 will tend toward zero as the number of data points increases. Neglecting that term, one solves for C A P and obtains
This is an upper limit on E A F , the lower limit being zero. Equation 29, in a sense, gives the maximum value for the sum of the deviation of the first-order fit from the true, second-order fit, Z A P , that is hidden by a given sum of random errors, Zt:. Calling the right-hand side of eq 29 R and using eq 23 for AY, one obtains C(ApK
+ AnX + n2X2)2IR
(30)
Taking the equal sign, one has the equation of a three-dimensional surface in a space, the axes of which are ApK, An, and n2. The surface is expected to be closed by the nature of the problem and, in fact, is shown to be so with reasonable data. The inequality is thus satisfied anywhere within this closed surface. The volume defined by eq 30 has the following interpretation. Suppose the indicator ratio data with random error for a second-order base defined by pK, n,, and n2 are analyzed by the first-order equation yielding pK‘and m*.Then the volume defines a set of second-order base parameters which have a significant probability of yielding the same value of pK’and m*. O r from another viewpoint, given pK’and m*,there is no significant way of deciding which of the sets of parameters in the defined volume the data belong to. (Note that one base within the volume corresponding to ApK, An, and n2 = 0 is the first-order base.) Given this indeterminacy, one adopts a type of a priori equal probabilities hypothesis. That is, the probability that the true parameters describing the base lie within a given subvolume of the total volume is proportional to that subvolume and is independent of the location of the subvolume within the total volume. Equation 30 can now be used to estimate these probabilities in the following manner. For a given ApK value, that is, in a plane perpendicular to the ApK axis, eq 30 defines a closed curve, suggesting an ellipse. The area of this closed curve is easily obtained numerically. If this area is then multiplied by a small range in ApK, Le., a AApK value, the result is an approximation to the volume enclosing those points within a specified range of An and n and whose ApK values lie within the range ApK (1/2)AApK to ApK (1/2)AApK. By examining all of the ApK values which fall within the entire closed surface in this way, one takes the sum of the individually calculated volumes as the total volume of enclosing surface. The ratio of any of the smaller volumes to the total volume is approximately the fraction of possible sets of parameters having ApK values within the above-mentioned range. By combining subvolumes, the probabilities for other ApK ranges can be obtained.
+
5.0-6.0 6.0-7.0 7.0-8.0 8.0-9.0 9.0-10.0
fraction 0.052 0.044 0.034 0.022 0.010
TABLE 111 A
Expanding and rearranging eq 21 leads to
ApK range
Ci
XO
Ax
0.017 0.029 0.033 0.048 0.03 1 0.068 0.080 0.115 0.154 0.046
0.548 0.865 1.450 3.215 3.618 4.338 8.130 8.028 7.651 6.169
1.347 1.659 1.531 2.003 2.070 2.203 1.636 3.059 2.977 2.126
N D,,, 10 11 9 9 9 11
5 13 9 9
0.10 0.22 0.58 1.81 1.27 3.19 91.74 7.29 1 1.94 4.55
PK‘ -1.77 -2.40 -3.57 -5.55 -5.91 -5.72 -10.64 -10.63 -9.59 -7.69
Now hypothesize that the bases discussed in the first part of this paper were, in fact, second-order bases. They can be examined in terms of the eq 30. First, the indicator ratio data for each was fit to eq 15. The values of pK’and m* were then used to obtain calculated values of Y and from these an estimate of ti was made, where
In all cases Fc was taken corresponding to cy = 0.1. In this case, if F > Fc leading one to conclude that the second-order fit is distinguishable from the first-order fit, there is still a 10%chance that the conclusion is wrong. Detailed results for dye J are given in Table 11. ApK corresponds to the deviation of the pK measured by the first-order method from the true value for a given second-order base, and volume fraction is the fraction of the total volume of the ApK, An, n2 space corresponding to the given ApK range. (Note that, due to symmetry, the same values of the fractional volume are obtained for the corresponding negative ApK range values.) By the equal probabilities hypothesis, the fractional volume is also the probability that the true parameters for the base (if it is second order) fall in the given ApK range. Using the table, one can deduce that, for example, there is about a 32% chance that the true pK is more than 5 units away from the measured value. In Table I11 results for all of the indicators studied in ref 4 are given. Rather than show fractions for different ApK ranges, a root mean square deviation is given which is defined by
wherefi is the fraction of the total volume corresponding to the volume centered at ApKi. Since the data in Table I1 suggest a symmetrical, bell-shaped curve, D R M s can be taken as a measure of the width of the curve. In fact, analysis of results like those in Table I1 shows that about 62% of the entire volume is included in the ApK range from - D R M S to + D R M S . The results in Table I11 demonstrate that uncertainties in pK, defined as being D R M S , can be substantial, amounting to several pK units. Since the data in Table I11 refer to bases studied under vgrying conditions of precision over different X ranges and with different numbers of data points, quantitative comparison is difficult. Examination of the equations leading to D R Mshow ~ that they depend only on the values of X spanned, the number of data points, and finally on the individual uncertainties in Y and ci. If one considers measurements on any base starting at X , and spanning a range hx with N equally spaced (in terms of X ) measurements, then, by considering a large number of such cases with different values for X,, hx, N, and ti, one arrives, by trial
1752
J. Phys. Chem. 1985, 89, 1752-1755
and error, at the following expression for DRMs
DRMS= ( X o / a x ) ' ~ s10 ( + 46/(N - 4.5))ti
(33)
This is purely empirical and only approximate, the DRMsvalues calculated from the equation differing from those obtained above by amounts up to 30%, expecially at large N . It is valid for X o greater than 1 and the dependence on ei is exact. Equation 33 shows that the larger ei, the larger DRMs. Similarly, the smaller the range of acidities over which measurements are made, that is, the smaller AX, the larger DRMsbecomes. These conclusions simply reflect the fact that it is more difficult to detect curvature in a plot, the shorter the segment of the curve taken and the greater the experimental uncertainty in the data. The dependence on X o is reasonable, greater uncertainties being expected for the greater extrapolations needed for weaker bases. Finally, the dependence on N is reasonable. At low N , curvature would be hard to detect. Increasing N makes curvature easier to detect (that is, reduces DRMs), but beyond N equal to about 20 there is little further improvement. Dms can be taken as an estimate of the indeterminate constant defined in the earlier work.3 Clearly it is only a probabilistic estimate in the sense that there is a significant probability that it can be as large as DRMS. In a given case it could be larger or smaller. However, in a given case, the actual value of the constant remains unknown, indeterminate. The analysis given here might seem appropriate as a basis for actually estimating the uncertainty in pK. However, a word of caution is in order. It is assumed throughout that the base under study is a second-order base. Third- and higher-order bases would be expected to generate similar conclusions but with different equations. There would be no way of knowing which order best describes a given base. There is also the well-recognized problem of solvent effects. These can add curvature to data but they may also cancel curvature. Reducing solvent effects by limiting the range of acidities used in counterproductive as eq 33 illustrates. That the indeterminate constant exists and that it is substantial now seems well proven. How important is the results? The pK of acetic acid has been determined to fO.OO1 pK unit^.^,'^ Instructions for measuring p l r s of weak acids yielding results with (9) Harned, H. S.; Ehlers, R. W. J . Am. Chem. SOC.1932, 54, 1350. (10) MacInnes, D. A.; Shedlovsky, T. J . Am. Chem. SOC.1932,54, 1429.
an uncertainty of f0.02 units can be found in physical chemistry laboratory manuals. By comparison, an uncertainty of, let us say, f2.0 units seems outlandish, especially when one realizes that this represents an uncertainty in the K itself of a factor of 100. Excess acidity methods clearly fail to give a reasonable estimate of pK values. The results presented here can be characterized as negative, casting a doubt as they do on an existing method for determining pK values. However, since the uncertainties are, and always have been, inherent in the method, presenting the limitation of the method is certainly useful. Since the analysis presented here did not consider the case of third- or higher-order bases and since it did not account for solvent effects or systematic uncertainties in the concentration of the hydrogen ion in strong acid, it is only indicative of possible magnitudes of uncertainties. For even if real curvatuve were detected in the data, there is no reliable way of deciding the source of the curvature. For these reasons, I am somewhat pessimistic about the possibility of ever improving the method, short of determining the function for eq 4 for all acidities for each base of interest. And although much work has been done on determining activity coefficients;" these results seem to suggest that the form of eq 4 will remain elusive except, perhaps, in special cases (and excepting model systems where a form of one base is taken as being a good model for a different base). The utility of acidity function methods for defining basicities of very weak bases is problematic. A common measure of basicity is the pK value. Any comparison of weak bases on the basis of their pK values, those values being determined by acidity function methods, should be considered meaningless, except perhaps in the case where the p k s differ by at least several units. Finally, the foregoing analysis focuses attention solely on the uncertainty in the pK value taken as a thermodynamic property. It does not address itself to the possible utility of an excess acidity scale as an empirical scale nor to the possible utility of the constants derived from such a scale as empirical constants which can figure in empirical correlations. To be more in tune with the demands of rigor, it is suggested that the quantity determined by these methods and called pK be renamed pX and treated as an empirical parameter. pX is suggested to reflect the fact that the parameter is derived from the X scale and to reflect its unknown character. (11) Yates, K.;McClelland, R. A. Prog. Phys. Org. Chem. 1974, 1 1 , 323.
An FTIR Study of the Kinetics and Mechanism for the Ci- and Br-Atom-Initiated OxMatlon of SiH4 H. Niki,* P. D. Maker, C. M. Savage, and L. P. Breitenbach Research Staff; Ford Motor Company, Dearborn, Michigan 481 21 (Received: May 10, 1984; In Final Form: October 22, 1984)
On the basis of FTIR product analysis of the UV-visible (A E 300 nm) photolysis of Cl,-SiH, and Br2-SiH, mixtures in 700 torr of N2-02, both C1 and Br atoms were shown to exclusively undergo an H-atom abstraction reaction rather than a displacement reaction with SiH4. The corresponding rate constants k{C1+SiH4)and k(Br+SiH4) were determined at 298 K to be 4.4 X and 1.0 X lo-" cm3 molecule-' s-l, respectively, by the competitive kinetic method with reference to the reactions C1+ C3H8and Br + CH20. IR absorption bands attributable to an aerosol product were detected in the subsequent oxidation of the primary radical SiH3. This product was tentatively identified as a polymeric form containingthe [-HSi(0H)O-] group. A possible mechanism for its formation is discussed.
Introduction While the C1-atom reaction of alkane hydrocarbons has been the subject of extensive studies over the past several decades,' there is not much information on the corresponding reaction of their
closest structural analogues, i.e., silanes ( S ~ , H Z ~ +Namely, ~). the only previous kinetic study of the C1 + silane reaction appears to be that of Schlyer et al.2 for the C1 SiH4 reaction at room temperature using the discharge flow-resonance fluorescence
(1) Lewis, R.S.;Sander,S.P.;Wagner, S.; Watson, R. T. J . Phys. Chem. 1980, 84, 2009, and references therein.
2633.
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(2) Schlyer, D. J.; Wolf, A. P.;Gasper, P.P.J . Phys. Chem. 1978, 82,
0022-3654/85/2089- 1752%01.50/0 0 1985 American Chemical Society
