Kinetics of Acid D;ssorciation-lon Recombination theoretical curves for ail $w and Pe collapse into the single curve shown, as discussed earlier. Again, the agreement is very good. We would note that the apparent larger scatter of the data is misleading in the sense that it is due to the fact that the rejection coordinate has been stretched by ref,mrrcin.g it with respect t~ the maximum rejecticm Rmax.
A theory of salt rejection by hyperfiltration through rmcroporous membranes has been presented. The theory is based on a relatively simple capillary model, wherein the membrane is assuined to consist of a uniform distribution of cylindrical pores whose interior surfaces acquire a constant potential when in contact with the saline solution. Although the Bow phenomena in actual porous membranes is undoubtedly more complex, the simple model appears to provide a theory in excellent agreement with sait rejectkm data from experiments on bentonite clay a d cellophane. Furthermore, the theoretical characterization of the rejectioa in terms of a dimensionless wall potential, the r a t i o of the Debye length to an effective pore radius, and a Peelet number based on the filtration velocity and an effective membrane thickness seems appropriate for the materials and operating conditions tested so far. Our present macroscopic, continuum model requires that both the Debye length and the effective pore size be large in comparison with the moiecular dimensions. Although the Debye iicxigth always satisfied this requirement in the present tesrs, the same was not true of the effective pore radius, which was in the range 10-20 A. An explana-
Kinetics of
4023
tion is required as to why the theory holds so well down to geometrical dimensions of the order of molecular size. In addition, although the assumption of a constant wall potential independent of salt concentration was made in the present work this is known not always to be true, depen jng on the so!id-solution interface, so that the effect of re., laxing this assumption should be determined. Another" assumption in the present work is that the axial distribution of charge om the pore wall is such that the ion dlstrihuti.ons in the pore fo'ollow the functional ~ e ~ a t ~ o (eq n s 2b) h~~ suggested by Gross and Osterle.13 Alth.ougR the effect of this assumption is not expected to be iarge on such integrated characteristics as fractional salt rejection and streaming potemtiai, nevertheless a detailed study should be made of the rejection characteristics corresponding to various physically realistic wal.1 charge or potential specifications. I f the theory i s to be useful for c h a r ~ c t e r ~ z ~porous ng membranes for desalination or d e ~ ~ n e ~ ~ ~of i water ~ a t i o ~ by reverse osmosis it should be extended to the case where the fluid contains several ionic solutes of the same sign having different valences agld diffusion coefficients. It would also be of interest to extend the analysis to model geometries more complicated than a straight cylindrical pore. Finally, it is evident that a pnilch wider range of membrane materials and operating conditiions must be investigated experimentaliy before the full utility of the theory can be properly assessed.
Achnoltledgmnent This research wa6 sponsored by the ater, 1.; S.Department of the Interior, under Grant No. 14-30-2575.
issociation- Ion Recombination of andberg, Gary H. Henderson, Robert D. White, and an war^ Oepartment of Chemistry, Universityof Utah, Salt Lake City, Urah 841 12
(Received May 25, 7972)
Publication sosts essisted by the Air Force Office of Scienfific Research
In -&10-5 M acidic aqueous solutions of Methyl Orange the rate constant a t 25" for the recombination of a proton with the monoanion of the indicator is 2.9 =k 0.4 X 109 M - I sec-I. This specific rate, determined by a spectrophotometric electric field jump relaxation technique, is significantly smaller than that expected for a diffusion-controlled ion recombination. An explanation for this result foOiluws from a comparison with similar kinetic data for aqueous Methyl Red.
Methyl Orange (I) is one of the most frequently used acid-base indicators a t acidic pH values near its reported1 pKa = 3.47 in water a t 25". In general, when Methyl Qrange is used as the indicator in temperature jump or other re:.axation experiments exact numerical values for the dissociation and ion recombination specific rates are unnecessary if the relaxation time ( 7 ) of the coupled system is long campared to that of Methyl Orange. However, in
studying very fast reactions (e.g., metal hydrolysis) characterized by relaxation times of the order of microseconds, values for the recombinatiom and dissociation specific rates of the indicator should be known in order to reliably interpret the observed relaxation data. The dissociation kinetics of Methyl Red (IT) in dilute aqueous solution were reported2 in terms of a one-step (1) R. L. Reeves, J. Amer. Chem. Soc., 88, 2240 ('1986). ( 2 ) i. P. Holmes, A . Silzars, D. L. Cole, L, D. Rich, and E. M. Eyring, J. Phys. Chem., 13,737 (1989). The Journal of Physical Chemistry dol 76 No 26, 7972
R. G . Sandberg, G . H.
4024
mechanism (eq I ) with the values of k~ and kR given as 4.8 X 105 sec-I and 3.5 X loro M - I sec-I, respectively. Using these data and the normal relaxation expression (eq 2 and 3) for the one-step p r o c e ~ s one , ~ would predict relaxation times as s8hortas 20 nsec for Methyl Orange a t M in the p H range of 3.2 to 4.4 concentrations near where this indicator is customarily used. I t is known that the pK, of the P-azo nitrogen is quite different in Methyl Red and in Methyl Orange.4-7 Since the resonance contribution in both systems should be approximately the same, the pKa difference seems anomalous. I t was, therefore, our purpose to dettbrmine by means of the dissociation kinetics in what way the acid-base behavior is affected by variations in structure in azo dyes.
Henderson, R. D. White, and E. M. Eyring
Results Presented in Table 1 are the values of T, the pH, and associated statistical parameters obtained in the Methyl Orange experiments. Additionally, the -Methyl Red ionization kinetics were reevaluated using the modified E-jump and the results obtained were in good agreement with those reported by Holmes, et al.2 The calculated Methyl Red recombination and dissociation specific rates are 4.02 f 0.4 X l Q I O M - I sec-I and 5.0 f 0.4 X 105 sec-I, respectively, yielding a kinetic pKa = 4.91. This value is in good agreement with both the previously reported2 kinetic pKa of 4.88 and the literature thermodynamic value637 of 5.00. If the single-step ionization mechanism (eq 1) identical with that for Methyl ed i s followed, a plot of the reciprocal relaxation time (7-I) os. the sum of the concentrations of the Methyl Orange anion and hydrogen ion ( [ A - ] [H+]) (eq 2) should yield a straight line (see Figure 1). The slope of the line gives the recombination rate constant k~ = 2.9 k 0.4 x lo9 M-' sec-l and the intercept the dissociation rate constant k D = 7.9 +Z 1.0 X 105 see-1 a t 25". These constants yield a kinetic ~ K =A 3.57 in reasonable agreement with the spectrophotometrically determined*P-azo nitrogen pK, = 3.37 of Methyl Orange.
+
Experimental ~
~ ~ t ~ o ~ iscussion Methyl Orange (I:), p-[(p-dimethylamino)phenyl]azobenFrom the experimental work of Eigen and coworliers3,9J0 zenesulfonic acid, was obtained from J. T. Baker Go. we have come to expect that unless intramolecular hyThe indicator was twice recrystallized from hot water, and drogen bonding, intramolecular electronic rearrangement the precipitate wais washed with ethanol and ether and as in pseudo acids, or steric factors are present, the specifdried a t low tempierature for several hours. ic rate of an ion recombination should closely approach + the limiting value predicted by Debye's phenomenological (CH3),X-SO,equationll for a diffusion-controlled reaction. On this H basis, for a proton reacting with a monoanion as in the I case of M'ethyl Orange we would expect a h~ of about 5 x Al!1 solutions used in the study were freshly prepared 1Q1O M - I sec-l. Thus, the k~ = 2.4 x 109 M-1 sec-1 reusing doubly ciistill.ed deionized water (specific conductivported above is somehwat surprising since none of the above-mentioned factors appear &o be operative in the ity 0.9 x 10--6 ohm-I cm-I). The dilute sample solutions case of Methyl Orange we would expect a k R of about 5 x were prepared from a 2.14 X 1 0 - 4 M stock solution. AdMethyl Red anion does have a k R .02 X 1O1O M--? sec-I. justments of pH were made using either dilute hydrochloA possible explanation for the thyi Orange observed ric acid or dilute sodium hydroxide. (This correction ion recombination rate may be found in the coupled ionicaused less than 0.1% error in the final indicator concenzation mechanism proposed by Sawicki4 (eq 4 ) . The relaxtration.) The pH measurements were made using a Beckation times, T, for this coupled equilibrium system are man Model 1019 pH meter equipped with a combination given by expression 5 , where the specific rates are identiglass electrode (Beckman 40498). Sample solutions evified in eq 4. I t is obvious from eq 5 that a plot of 1," us. denced a pEI constant to within f0.03 pH units when ([A-] [H+])will not, in general, be linear. Agreement is tested prior to and immediately succeeding the E-jump obtained, however, between the expression 5 for 1 / T and experiment ( a time interval of up to 30 min). The electric field jump (E-jum p) relaxation method apparatus used the experimental data if the recomljination rate constant, K ~ R ,for the amine group Is near its diffusion-controlled was basically identical with that described by Olsen, et al.8 The major modification involved the use of a digital limit (e.g.> h~ = 1.5 X 101g M - I sec-l for imidazolel2) and the value of K,r, the ratio of the protonated P-azo to delay trigger generator having a time base referenced to a lO-Rill-l[zquartz crystal. This allows for more accurate con(3) M. Eigen and L. De Maeyer, "Techn;que of qrganic Chemistry," VOI. trol of the square, high-voltage pulse width. It also beV l l l , Part 1 1 , S. L. Friess. E. S. Lewis, and A . Weissberger, Ed., l n terscience, New York, N. Y., 1963, Chapter 18. comes possible by accurately varying pulse width to en(4) E. Sawicki, J. Org. Chem., 27, 605 (1956). hance the relative amplitude of a given relaxation time (5) G. E. Lewis, Tetrahedron, 10, 429 (1960). (6) i . M. Kolthoff, J. Phys. Chem., 34, 1466 (1930). compared to others in a relaxation spectrum. (7) S. W. Tobey, J. Chem. Educ., 35,514 (1958). All kinetic measurements were made spectrophotome(8) S. L. Olsen, R . L. Silver. L. P. HOlme5, J . J. Auborn, P. Warrick, Jr., trically in the absence of an applied electric field ( i . e . , and E. M. Eyring, Rev. Sci. Instrum., 42, 1247 (1971). (9) M. Eigen, Agnew. Chem.. 75, 489 (1963); Agnew. Chem.. lnt. Ed. immediately following the termination of the square, Engi., 3, 1 (1964). high-voltage pulse). 'The electrical resistance of the sam(IO) M. Eigen, W. Kruse, G. Maass, and L. De Maeyer, Progr. React. Kinet., 2, 287 (1964). ple cell in all cases, exceeded l o 4 ohms, so that the rise in (11) P. Debye, Trans. Eiectrochem. SOC.,82. 265 (j942). sample temperature incident to the E-jump was negligi(12) M. Eigen, G , G. Hammes, and K. Kustin,
