J. Phys. Chem. 1985,89, 2965-2966 about 1 M after which what are presumably solvent effects begin to intrude. I have carried out the experiments summarized in Table I and Figure 1. These data can be expressed by a two-term rate law: rate = [S20!2-].(kl + k2[sulfoxide]). The second-order rate constant, k2,IS given by the slope of the l i i and is equal to 0.027 M-l min-’. The firstsrder rate constant, k l , is given by the intercept and is equal to 0.0019 m i d . Levitt was unaware of the k2 term because he worked under conditions where the sulfoxide concentration was very small and where the sulfoxide: persulfate ratio was never greater than one. Under these conditions, the kl term dominates. The k2 term probably represents nucleophilic displacement on peroxide ~ x y g e n . The ~ kl term is complex as it contains at least the following components: (1) the thermal decomposition of persulfate in water (O.OO0295 m i d ) , (2) a component involving the sulfone not affected by allyl acetate (ca. O.OOO4 min-’), and (3) a component sensitive to allyl acetate inhibition at low concentrationsof sulfoxide (see Table I and ref 10). The addition of sulfate ions has no detectable effect on the rate of the reaction under the conditions I have used (Table I, entry 2). I have not reinvestigated the reaction with diphenyl sulfoxide because this reaction is so slow in comparison with the water oxidation term and because of the low solubility of this substrate in water.” A few runs with dimethyl sulfoxide confirm Rahkonen and Tommila’s observations with the additional datum that sulfate ions again have no effect on the rate. Registry No. Na2S208,7775-27-1; diethyl sulfoxide, 70-29-1. (9) Curci, R.; Edwards, J. 0. In ‘Organic Peroxides”; Swern, D., Ed.; Wiley-Interscience: New York, 1970; Vol. 1, Chapter 4. (10) Davies, M. J.; Gilbert, B. C.; Norman, R. 0.C. J. Chem. Soc.,Perkin Trans. 2 1984, 503. (1 1) The oxidations of diphenyl sulfide and of alkyl aryl sulfides by persulfate proceed by very different mechanisms: Srinivasan, C.; Kuthalingam, P.; Arumugam, N. Int. J. Chem. Kind. 1982,14, 1139.
E. J. B e h a n
Department of Biochemistry The Ohio State University Columbus, Ohio 43210
Received: July 1 , 1984; In Final Form: March 15, 1985
2965
usually equimolar with the [S2OBZ-)r Oxidation of sulfides by H202 gives the same result.1° Therefore, rate equations of tlmfollowing form were deduced8v9
+ So)]
rate = -dP/dt = Pk,[So/(b
where P is the instantaneous [S20B2-],So is the initial substrate concentrations, and b is an empirical parameter that varies from substrate to substrate. Behrman’s rate equation consists of two terms: He correctly states that our investigations dealt with only the first (first order) term because we worked only up to equimolar concentrations: The second-order term in his equation begins to dominate at higher RzSO concentrations. However, he gives neither plots nor raw data to support this interpretation. From the results reported in ref 1, we cannot exclude an alternative interpretation: that the reaction is first order in [s2082-]and the first-order rate constant shows a complicated dependence on So as we and others have repeatedly found, not only for sulfoxides,2 but for thiols,’.” alcohols? and other types of organic substrates. We have several other concerns with Behrman’s experimental results. (1) Behrman finds for 0.02 M EtzSO k 0.140 h-’ compared to our 0.108 h-’, which has been repeated with an accuracy of *4%. (2) The lower three points in Behrman’s Figure 1 (for 0.005, 0.01, and 0.02 M Et2SO) indicate an approach to a limiting rate at around 0.02 M substrate concentration. The limiting rate we observed (see the last six points in our Figure 4)2 encompasses a 4-fold higher substrate concentration than this limiting value and, with thiodigylcol sulfoxide,11up to a 20-fold higher concentration. For thiols, Eager and Winkler’ increased the initial substrate concentration 10-fold higher than the concentration where the limit was reached and found no further increase in the first-order rate constant. When i-PrOH was oxidized: a 5-fold higher concentration produced no change in k. The same is true for a 50-fold increase in the oxidation of glycols.12 (3) Behrman’s rate equation gives for his psuedo-first-order rate constant & = kl k#, and the value for kl, 0.0019 min-],implies min-l. However, the that, when Sois zero, k, = kl = 1.9 X entry in his Table I for So = 0, k+ = 2.95 X lo-“ m i d , is clearly inconsistent with this value of kl and is simply his rate constant for the thermal decomposition of S20B2-in water. (4) Behrman’s runs with added sulfate, showing no effect on k, are in contradiction to other r e p ~ r t s ,though ~ . ~ we have never tested this effect. ( 5 ) The runs with added allyl acetate (a “radical trap”) arc surprising, since his mechanism involves nucleophilic displacement’ on persulfate and is not a free radical reaction. Behrman himself found no effect by allyl acetate or allyl alcohol on the rate of the persulfate oxidation of phenol^.'^ At any rate [sic], we have previously shown conclusively that allyl acetate13 or any other substrate oxidized more slowly by persulfate than the substrate in question reduces the rate because it is oxidized more slowly, 7
+
RqHy to “Comments on the Oxidation of Sulfoxides by Peraxydkultate Ions” Sir: In the foregoing Comment,’ Behrman makes two new observations on my study of the kinetics of the peroxydisulfate oxidation of diethyl and diphenyl sulfoxides.2 (1) Added sulfate ions do not depress the rate. (2) The reaction rate is first order in sulfoxide at higher [R2SO], as well as being first order in persulfate. In our previous studies, we did not test the effect of added sulfate on the rate but rather accepted findings of earlier investigators3*‘who reported a considerable retardation with this anion. Hundreds of papers have appeared in the past 60 years on the kinetics of peroxydisulfate oxidation^.^ When the reductant is an organic substrate, the kinetics of the uncatalyzed reaction are first order in S2OS2-with very few exceptions. In some 90% of the reactions studied, the order with respect. to organic substrate is In numerous cases,the observed first-order rate constants reach limiting values at higher initial substrate concentrations,Z@ (1) E. J. Behrman, J . Phys. Chem., preceding comment in this issue. (2) L. S. Levitt, J . P h y . Chem., 88, 1177 (1984). (3) K. Elbs and P. Neher, Chem.-Ztg.,45, 1113 (1921); A. Kailan and L. Olbrich, Monatsh. Chem., 47,449 (1926). (4) S. P. Srivastava and A. K. Mittal, private communication. (5) See,for example, D. A. House, Chem. Rev.,62, 185 (1962).
0022-3654/85 I2089-2965SOl .SO10 --, 1 I
I
-
-
(6) E. Larsson, Trans. Chalmers Uniu. Technol. Gothenburg, 87, 23 (1949). (7) R. L. Eager and C. A. Winkler, Can. J. Res., 26, Sect. E, 527 (1948). (8) L. S. Levitt, Can. J. Chem., 31,915 (1953). (9) L. S. Levitt and E. R. Malinowski, J. Am. Chem. SOC.,77, 45!7 (1955); 80, 5334 (1958). (10) J. Boeseken and E. Arrias, Red. Trau. Chim. Pays-Bas, 5 4 7 1 1 (1935). (1 1) L. S. Levitt and J. R. Pytcher, submitted for publication. (12) W. C. Vasudeva and S. Wasif. J. Chem. SOC.B. 960 (19701. (13j L. S. Levitt, B. W. Levitt, and E. R. Malinowski,’J. Org. Chem., ? I , 2917 __ - . 11962). ,----I-
(14) L. S. Levitt and B. W. Levitt, Can. J. Chem., 41, 209 (1963). (15) E. J. B e h n and P. P. Walker, J. Am. Chem.Soc., 84,3454 (1962); E. J. Behrman, ibid., 85, 3478 (1963).
0 1985 American Chemical Society
J . Phys. Chem. 1985, 89, 2966-2967
2966
and in any "double substrate" run with persulfate, the rate is always intermediate between the separate runs (instead of being their sum). This is discussed a t the bottom of p 1180 of ref 2 in terms of the simultaneous oxidation of Et,SO and i-PrOH. This effect is another unusual and intriguing aspect of oxidations by peroxydisulfate. (6) If we assume that Berhman's data are correct, the rate equations should be rate = klP + k2PS0,where Sois the initial sulfoxide concentration in a given run. This does not mean that the second term is second order but merely that the overall first-order rate constant, k , is a linear function of So at higher concentrations. In a recent study of the persulfate oxidation of sulfonamides: however, the reaction was truly second order at all Sovalues, since the integrated second-order plots were linear. Registry No. S2OS2-,15092-81-6.
The Institute for Theoretical Studies POB 13690 El Paso, Texas 79912
L. S. Levitt
Received: November 2, 1984; In Final Form: April 5, 1985
TABLE I: Reduced Potential at the Axis of a Cylindrical Microcapillary, &, for a Reduced Surface Potential 6, = 2 dn Ka
0.01 0.05 0.1 0.2 0.5 1 2 3
5 7
numerical 1,9999 1.9977 1.9909 1.9647 1.8050 1.4278 0.7652 0.3637 0.0667 0.0390
Oldham
Olivares
et al. 1.9999 1.9977 1.9910 1.9647 1.8054 1.4278 0.766 0.367 0.071 0.012
al. 1.9999 1.9977 1.9910 1.9647 1.804 1.415 0.723 0.316 0.048 0.007
Sir: The linearized Poisson-Boltzmann equation for the inner region of a cylindrical charged microcapillary immersed in an aqueous electrolyte solution, valid for small reduced surface potentials 4a< l , was solved by Rice and Whitehead.' For larger potentials Levine et aL2 gave an approximate analytical solution which has several regions of validity. In a previous work3 we extended the range of application of the simple linear solution of Rice and Whitehead for reduced potentials 0 C 4, C 6 and reduced micropore radius 1 < KU < 1000, using a variational approach for solving the nonlinear e q ~ a t i o n .The ~ variational trial function was a simple parametrization of the linear solution
where Io(x) is the zeroth modified Bessel function at the reduced distance x = Kr and p is the only variational parameter, obtained by minimization of the free energy functional
Is( z)+ cosh 4 - 1
J [ d ] = - 2 ? r ~ l i ( xdx 1 d 4
Levine
Martinov
1.9999 1.9977 1.9908 1.9640 1.801 1.409
1.9999 1.9977 1.9910 1.9647
$a2
12(A
+ B4:)
(4)
where A =1
+
(KU)2/3
(5)
and B = O.OI(1 - 5(KU)3/2)
(5)
for 0 C KU C 1 and & C 6. The correct values for the potential at the axis in the variational approximation are given in Table I and give very good agreement with the numerical data and other existing approximations for small values of KU and 4,. We also calculated the values corresponding to Oldham's approximation using our own iteration routine and obtained values slightly different from those reported by Sigal and Ginsburg. These values are in better agreement with the numerical solution of Sigal as shown in Table I. For application in systems such as zeolites, where a i= 10 A, we fall in the region KU < 1 for a 0.1 M 1-1 electrolyte at 25 O C . We thus would like to compare the existing approximations with special attention to this range of KU values. In the region of values of KU < 1, the potential satisfies the condition +(x) > 1 for all values of x inside the cylinder, particularly for surface potentials > 2. This falls in the so-called region I11 of Levine's solution, which actually has a simple expression for the potential
The parameter p is a function of both the reduced potential and the radius, which to within a few percent is given by the expression p2 = ex~(a4;) + P(4a/Ka)2
0.753 0.362 0.067 0.011
satisfactorily fitted the numerical data, our approximation was unsatisfactory. Unfortunately, these authors used eq 3 with the wrong value of and also applied it outside its range of validity, since for KU < 1, the parameter p has a completely different behavior that can be calculated instead by the relation p=l+
Comments on the Calculatlon of the Potential Inside a Charged Microcapillary
et
(7)
(3)
for 1 < KU C 100 and 0 < C 6. The fitting constants cy and /3 have the values 0.0412 and 0.0698, respectively. This last constant was erroneously transcripted as 0.698 in our original paper.3 In an attempt to compare our approximation with an exact numerical solution of the nonlinear Poisson-Boltzmann equation, Sigal and Ginsburg5 used eq 1 and 3 to calculate the potential at the axis of the microcapillary, do. The results obtained were also compared to the power series suggested by Oldham et a1.6 Sigal and Ginsburg concluded that, while Oldham's solution (1)C.L. Rice and R. Whitehead, J . Phys. Chem., 69,4017 (1965). (2) S . Levine, J. R. Marriott, G. Neale, and N . Epstein, J . Colfoid Interface Sci., 52, 136 (1975). (3) W. Olivares, T. Croxton, and D. A. McQuarrie, J . Phys. Chem., 84, 867 (1980). (4)W. Olivares and D. A. McQuarrie, Biophys. J., 15, 143 (1975). (5)V. L.Sigal and Yu. Ye. Ginsburg, J. Phys. Chem., 85, 3730 (1981). (6) I. E. Oldham, F. J. Young, and J. F. Osterle, J . Colloid Sci., 18, 328
(1963).
0022-3654/85/2089-2966$01 S O /O
where
40
In (16Bo/(~a)')
(8)
(4 + e q K a ) 2 ) ' / 2 - 2 (4 + e"(Ka)2)'/2 + 2
(9)
and Bo =
The values obtained for 4o by this analytical expression are essentially identical with those obtained by Oldham's iteration and compare very well to our approximation and to the numerical data, as can be seen in Table I. More recently, Martynovl gave analytical expressions valid for KU C 0.3 and $a C 5 and KU > 2 and 4, C 5 . Both of these expressions fare very well to the numerical solution, as shown in Table I. The region between 0.3 C K RC 1 is not covered by either of Martynov's expressions. (7)G.A. Martynov and S. M. Avdeev, Kolliodn. Zh., 44,702 (1982).
0 1985 American Chemical Society
