J. Phys. Chem. 1889, 87,4471-4478
4471
pH Jump: A Relaxationai Approach E. Plnes and D. Huppert’ Depattment of ChemMty, TeCAviv University, 89978 Ramet Avlv, Israel (Received November 15, 1982)
-
Back-protonation of the 8-hydroxy-l,3,6-pyrenetrisulfonate anion (RO- + H+ ROH ( k b ) ) initiated by a laser pulse photolysis of the neutral molecule (the pH jump method, pHj) is used in order to establish the chemical equivalence between the generated protons and water equilibrium protons. The diffusion-controlled nature of the back-reaction is established by checking the temperature and isotope (H/D) dependence of the reaction. A relaxation kinetics analysis is carried out in order to follow the time evolution of [H+]. The same analysis is proved to be applicable for the description of the bleaching kinetics of an acid-base indicator (BCG) present in the solution at the time of the.pH jump.
Introduction 1. The pH Jump Method. Some aromatic alcohols, such as 8-hydroxy-l,3,6-pyrenetrisulfonate (HPTS), become more acidic when excited to their first singlet state.’ The HPTS molecule which in water solution has a pK* value of 0.4 f 0.1,2 compared to a pKo = 7.8 f 0.1 of its ground state,3 is used in the present study in order to create a laser-induced excited-state photolysis of the molecular hydroxy group. This measurable generation of an excited-state anion (RO-) and a proton is known as the “pH jump” (pHj),4 or, alternatively, the concentration “jump” (I$of the anion. Both terms are due to the sudden rise in the concentration of these species, observed after a laser pulse excitation of the neutral molecule solution. The induced concentration jumps are only temporal since the generated excited HPTS anion relaxes back, within Tf = 6 ns,S to its ground state, where it resumes its basic nature. Thus, a back-reaction of neutralization takes place. Consequently, the jump concentrations are decaying to their equilibrium values. The overall process is in principle a Forster cycle one.’@J The cycle is ideally summarized in Scheme I, where ROH symbolize the neutral molecule and RO- the anion. kb and kf are the ground-state protonation and deprotonation constants, kb* and kf* are the corresponding excited-state constants, and Tf and ffare the apparent radiative lifetimes of the excited ROH and RO-, respectively. Molecules going through a Forster cycle were previously used as the doorway step for the initiation of such processes as ultrafast diffusion-controlled reactions,38 excited-state proton induced proton transfer on and in micelles,’OJ1and Coulomb cage effecta,6J2and for the creation of the (1)J. F. Ireland and P. A. H. Wyatt, Adu. Phys. Org. Chem., 12, 131-219 (1976). (2)K. K. Smith,K. J. Kaufmann, D. Huppert, and M. Gutman, Chem. Phys. Lett., 64,522 (1979). (3)Th. Fbrster and S. Vblker, Chem. Phys. Lett., 34, 1 (1975). (4)K. Breitechwerdt, Th.Fbrster, and A. Weller, 2.Naturwbsen., 43 (1956). A. J. Campillo, J. M. Clark, S.L. Shapiro, and K. R. Winn in ‘Picosecond Phenomena”,C. V. Shank, E. P. Ippen, and S. L. Shapiro, Ed.,Springer, Berlin, 1978,pp 319-26. J. H. Clark, S.L. Shapiro, A. J. Campillo, and K. R. Winn, J. Am. Chem. SOC.,101,746 (1979). (5)M. Hauaer, H. P. Haar, and U. K. A. Klein, Ber. Bumenges. Phys. Chem., 81,27 (1977). (6)K.Lundy-Douglas and Z. A. Schelly, Adu. Mol. Relaxation Processes, 8, 79 (1976). (7)Th. Farater, 2.Electrochem., 54, 531 (1950). (8)K. Breitschwerdt and A. Weller, Ber. Bumenges. Phys. Chem., 64, 395 (1960). (9)D. Huppert and E. Kolodney, Chem. Phys., 63,501 (1981). (10)M. Gutman, D. Huppert, E. Pines, and E. Nachliel, Biochim. Biophys. Acta, 642,15 (1981). (11)U. k l e , U.K. A. Klein, and M. Hauser, Chem. Phys. Lett., 68, 291 (1979). (12)H. P. Haar,U. K. A. Klein, and M. Hauser, Chem. Phys. Lett., 58,525 (1978). 0022-3654/83/ 2087-4471$015010
Scheme I
RO ?H- ,
ti
RO-
t H+
&b
In this study we describe a ultilization of the HPTS system for a kinetic study of the ground-state backprotonation reaction at intermediate solution pHs, where pH. is of the same order as the equilibrium pH (pH, ). h e then proceed with a temperature and isotopic (H$) dependence study of the same reaction, referred to as the “back-reaction”. Finally, we describe a kinetic study of the perturbation caused by the pHj to proton-sensitive materials such as acid-base indicators. This is done basically with a relaxation kinetics approach. This analysis was originally developed by Eigen15-17for the kinetic study of such perturbation methods as the temperature jump (Tj),electric field jump (E.),and the pressure jump (Pi). 2. Kinetic beatment of the Back-Reaction. Following Debye’s treatment of diffusion-controlled r e a c t i o n ~ , ~ J ~ J ~ the diffusion rate constant of the back-reaction at zero ionic strength is given by
k ~ ’= 4?rN’D&~[exp(R~/Ro) - I]-’ or when lRDl
>> Ro and RD C 0 k ~ ’= ~TN’DABIRD~
(1)
where kD is the diffusion-controlled rate constant, N’ is Avogadro number per mM, DABthe relative diffusion coefficient of the reactants, Ro the distance of closest approach, and RD, finally, is the Debye radius. The absolute value of RD is usually pictured as the practical reaction radius for diffusion-controlled reaction between ionic species. RD is given by (13)M.Gutman and D. Huppert, J. Biochem. Biophys. Methods, 1, 9 (1979). (14)M. Gutman, D.Huppert, and E. Pines, J. Am. Chem. SOC.,103, 3709 (1981). (15)M. Egen and L. de Maeyer in “Technique of Organic Chemistry”, Vol. VIIIb, A. Weiesberger, Ed., Wiley, New York, 1963,p 895. (16)H. Strehlow and W. Knoche in ‘Fundamentals of Chemical Relaration” (Monographs in Modem Chemistry, Vol. IO), Verlag Chemie, Weinheim, 1977. (17)M.Eigen, Angew. Chem. 3, 1 (1964). (18)P. Debye, Trans. Electrochem. Soc., 82,265 (1942). (19)M. Eigen, W.Kruse, G. Maas, and L. de Maeyer, Prog. React. Kinet., 2,287 (1964).
0 1983 American Chemical Society
4472
The Journal of Physical Chemistry, Vol. 87, No. 22, 1983
Pines and Huppert
TABLE I: Activation Energy of the Back-Reaction
where 2, and ZBare the charge numbers of the reactants and ro is the Onsager lengthmwhich represents the distance between two unit charges at which the mean thermal energy and the electrostatic energy are of the same magnitude. ro is given by
ro =
e2 tkBT
(3)
where e is the electron charge, e the dielectric constant of the medium, T the absolute temperature, and kB the Boltzmann constant. ro has the value of 7.1 X 10" cm in water (t = 78.3) a t 298 K. Equation 1 is a steady-state solution of Debye's original expression for diffusion-controlled rate c0nstants.2~fl It repreaenta a physical situation in which a steady flux of one reactant is entering the "reaction sink" of ita counterion where they both disappear immediately to form the reaction product. Numeric evolution of kD of the HPTS back-reaction is possible since all the needed parameters are known. The calculation yields a value of 2.0 X 10" M-'s-l at T = 298 K, in a good agreement with the experimental value measured by us and ~ t h e r s . ~ J ~ In finite concentrations ionic screening must be taken into a c ~ o u n t For . ~ ~dilute ~ ~ ~solutions, this effect is adequately treated according to Deybe and HuckeP~" kD
= kDo eXp(RDK)
(4)
where K is the ionic atmosphere and has the units of cm-l. In water solutions K is given by K
= 5.03 X
lo@($,)''2
cm-'
(5)
where I is the ionic strength.
Experimental Section Reagent grade &hydroxy-1,3,6-pyrenetrisulfonate(HPTS) was purchased from Eastman. Sample pH was adjusted by addition of perchloric acid or NaOH and was checked with a Radiometer pH meter. Two experimental setups were used in this work. Both were described elaewhere.13J4 In the first one, a ruby laser was paasively Q switched by Cryptocyanine dye, generating a giant pulse at 694.3 nm with a time duration of 30 ns full-width at half-maximum and energy of approximately 1J. The laser pulse passed through a KDP crystal, generating the second harmonic frequency at 347.2 nm with approximately 10% efficiency. The visible light was removed by a filter. Samples were irradiated in a quartz 5 X 10 X 40 mm cuvette. The sample interrogation source was either a continuous 150-W Xe arc lamp or, for the specific wavelengths of 632.8 and 441.6 nm, a 2-mW Spectra Physics He-Ne laser and 5-mW Liconix He-Cd laser, respectively. The interrogatingbeam passed through the sample either collinear with or perpendicular to the excitation pulse and then was focused onto the entrance slit of a 250-mm Jarrell-Ash monochromator. The inter(20)L. Onsager, J. Chem. Phys., 2, 599 (1934);Phys. Rev., 54, 554 (1938). (21)R. M.Noyes, Prog. React. Kinet.,1, 129 (1961). (22)S. A. Rice, P. Robin Butler, M. J. Pilling, and J. K. Baird, J. Chem.Phys., 70,4001 (1979). (23)(a) R.E.Weston, Jr., and H. A. Schwarz in "ChemicalKinetics", PrenticeHaU, Englewood Cliffs, NJ, 1972,p 165. (b) R.A. Robinson and R. H. Stokes in 'Electrolyte Solutions",2nd ed,Butterworths,New York, 1959,Chapter 4,p 73. (24)P. Debye and E. Hiickel, Phys. Z., 24, 185 (1923).
HPTS concn, excitation M source
pH,,
4.30 4.50 6.10 6.10 4.70 6.10
lo-' lo-' 10''
N, laser N, laser ruby laser ruby laser N,laser ruby laser
temp range, O
20 t o 20 t o 23 t o 23 t o 19 t o -5 t o
meas
calcd
3.3 f 0.3 3.5 f 0.2 3.6 f 0.2 3.6 f 0.2 4.0 i: 0.2 3.8 f 0.2
3.5 3.6 3.6
c
2 6 -5 -12 -13 -12
TABLE 11: Diffusion Coefficients, Mobility, and Conductivity Values of Protons in Supercooled Water "7
cm
T, OC
25 -10 -12 -15
s2
DH*,c m 2 s-' 9.31 X (4.1 f 0.1) X (3.8 f 0.1) X (3.6 i: 0 . 1 ) X
uH*,c m 2 V
3.63 x (1.81 f (1.73 f (1.60 i
I
s-'
10-3 0.05) X lo-' 0.05) x 0.05) X
-I
equiv-I
349.8 175 i 5 167 i: 5 155 i 7
rogating light intensity was measured by means of a photomultiplier (Philips TVP-56 or RCA 1P-28)coupled to a Tektronix 454 or 7623A oscilloscope. In the second setup, a pulsed nitrogen laser (337.1 nm, 50-kW peak power, l-ns full-width a t half-maximum, 30 Hz) was used for the excitation. The interrogating setup was identical with the one described above. The analog signal derived from the photomultiplier was digitized with a Biomation 8100 transient recorder (time resolution, 10 ns per channel), and the digitized waveform was stored in a Nicolet 1170 signal averager. Between 128 and 4098 pulses were averaged, yielding plots of transmitted interrogating lights vs. time as shown in Figure la. Kinetic analysis of these plots yielded rate constants for the investigated systems. The overall error associated with the measuring process is estimated to be &5%.
Results Preliminary measurements of the back-reaction in strong acidic media (HCl, HC104)yielded the pseudo-first-order kinetics predicted by Debye's treatment for diffusioncontrolled reactionsl8 and matched the results previously reported by other^.^ This behavior was checked by us up to pH 3.8. In adition to the observation of pure second-order kinetics of the back-reaction in water,3 we observed a fmt-order tail in all the decay curves at intermediate pHs (5.0 < pH < 6.0), even in cases where up to 40% of the HPTS molecules were photolized. Figure l a shows such a decay pattern of (R0-). One can notice that the reaction order is gradually changing with the progress of the reaction, from second to first order. The results of the kinetic analysis of Figure l a is shown in Figure lb, where agreement between the calculated reaction order and the observed one is achieved with the use of relaxation kinetics. Temperature and isotopic (H/D) dependence of the back-reaction in acidic media (4.0 < pH, < 6.1) was found to be in accordance with a diffusion-controlled mechanism. The results for the back-reaction temperature dependence between 23 and -13 "C are shown in Figure 2 and listed in Table I. These results enable us to calculate DH+ in supercooled water up to -13 "C, and to estimate DH+ at -15 "C where the solution started to freeze. These values are listed in Table 11,with the corresponding proton mobility and conductivity values. The kinetics of DPTS anion's back-reaction in D20 were monitored between 15 and -10 "C (supercooled D20).
The Journal of Physical Chemistry, Vol. 87, No. 22, 1983 4473
pH Jump
a
--
ROH H P T .
N
I O - 3 ~
pH 5.44
%
1 ' 18'C
a I
1
[XI.
8.85*10-6 M
I
I
I
WATER T = O0C [H*]f 3.1 s 10.8 M [W] j u n p L ( a l O - (
M
f I/ T'K
1
1
2200
I
1
1
4416
1
1100
I
I
1114
t ( nsec ) Flgure 1. (a) Translent transmltted llght of HPTS solution at 441 nm where the HPTS anion absorbs. The plot Is an average of 1024 consecutive N, laser excitation experiments. The curve represents the decay of the HPTS a n h to Its equlllbrlum concentratkn. The order of the decay changes with the progress of the reaction from second order to first order. This behavlor is typical of the back-reactbn when pH, N pH, (b) Relaxation klnetlcs analysis of la. The order of the reaction changes with time (2to 1). The HPTS concentratlonIs 10" M, pH, = 5.44.
Using these measurements we calculated the ratio kD(H')/kD(D+) and the DD+ and UDC values (Tables I11 and IV) Finally, the HPTS system was coupled with an acidbase indicator (Bromocresol Green, BCG) which was perturbed by the induced change in the solution pH (Figure 3). The relaxation kinetics analysis of the resulting
.
io5
Figure 2. Actlvation energy plots of the back-reaction. The full circles represent 10-mJ,304s fwhm ruby laser pulse excitation of lo4 M HPTS at pH, 6.10. All other plots represent 5 0 4 , l-nsfwhm N, laser pulse excitation of lo-' M HPTS at different solutlons pHs: open 4.70;full diamonds, pH, 4.50;full squares, pH, 4.30. dlamondsl E, values whl p% are calculated between extreme pdnts represent only averaged values. The differences in the absolute value of k , measured in different experlments Is explalned partly by the differences in Ionic strength and mainly represents the overall experimental error.
/ i
'I
transient bleaching of the indicator, a process which is described in detail elsewhere,13J4provided the indicator steady-state protonation rate constant. The observed and calculated indicator rate constants in different solution pH's and indicator concentrations are listed in Table V. While the observed rate constants differ significantly at different initial reaction conditions, the calculated ones show no such tendency. It is clear therefore that the latter indeed represents the true steady-state rate constants of the indicator. For additional proof, the same kinetic analysis was used in order to calculate kl which is the known back-reaction rate constant. kl was found to be (1.2 f 0.2) X loll M-' s-l. This value is well in accord with the expected kD values for the given systems (-1 mM total HPTS concentration and -0.1 mM total indicator (BCG) concentration). Discussion 1. Time Dependence of the Back-Reaction. The use of steady-state kinetics for the description of the backreaction must raise some questions about the validity of the approach, since the reaction "opens* when the reactants are only a few water molecules apart. This situation is in marked contrast to the assumed steady-state density distributions of the reactants. However, the reaction re-
4474
Pines and Huppert
The Journal of Physical Chemlstty, Vol. 87, No. 22, 1983
TABLE 111: Isotope Dependence of the Back-Reaction (9.5 f 0 . 2 ) x 1O'O 4.58 15 (7.3 f 0.2) x 1 O ' O 4.95 3 (5.25 f 0 . 2 ) X 10" 4.95 -10 a Values extracted from ref 35.
(6.6 f 0.2) x 10" ( 4 . 8 0.2) X 1Olo (3.4 f 0.2) x 10'O
TABLE IV : Diffusion Coefficients, Mobility, and Conductivity Values of D' in Supercooled D,O AD>
cm
T, "C
DD+, c m 3 s-'
25 0 -5 -10
6.41 X (3.5 f 0 . 2 ) x 10-5 (3.0 f 0.2) X (2.6 f 0.2) x 10-5
a-l
.'
.-*.
equiv-'
uD+, cma S-'
2.50 x 1 0 4 (1.5 f 0.1) x 10-3 (1.3 f 0.1) X (1.1 0.1) x 10-3
240.9 145 5 127 f 5 114 5
BCG BLEACHING KINETICS
w
a
.
-
94 nscc IS
*.
*.
2
J
n A W
I-
k
s
VI
z 4 a
c
Flgure 3. BCG bleaching kinetics at 632.8 nm represented by the transient transmltted light of HPTS-BCG solution after a N, laser excitation. [HPTS] = 800 pM, [BCG] = 95.2 pM, p& 5.60, [RO-], = 6.4 pM [In-], = 83.3 pM, x , = 2.5 pM. (a) 1.25 ps full scale. (b) 20 ps full scale.
laxes within 100 ns to ita steady-state rate. In very short times, the major contribution to the reaction is due to a geminal recombination of the generated ion pair. The ions react within their mutual "Coulomb cage".12 This period is estimated to last, for the HPTS system, for about 2 ns. The escape probability of the ions is given by25 exp(RD/rd - exp(RD/Ro) Q(m) = (6) 1 - exP(RD/Ro) O ( m ) is the escape probability at t after the creation of the ion pair. In our case lRDl >> Ro and RD < 0, and eq 6 is reduced to Q(m) = exP(RD/rAB) (7) where rABis the mutual distance a t which the ions are generated at t = 0 of the reaction. The measured yield of photodissociation of HPTS after saturation excitation of the HPTS molecule was found to
-
(25) K. M. Hong and J. Noolandi, J. Chem. Phys., 68,5163 (1978).
1.44 t 0.07 1.52 f 0.07 1.54 i 0.07
1.49 1.54 ( 5 "C)
be around 50% in relatively concentrated HPTS solutions (HPTS 1 100 pM). Other investigators measured even higher yields of separation (-70%) in lower solution con~entrations.~ The very high yield implies either a reduced reactivity of the excited anion, which has a lifetime of about 6 ns, or a water-assisted mechanism which causes an immediate long-range separation of the charges. Since, according to eq 7, the rm value needed to satisfy a 50% separation is of the order of 2RD,which is more than 50 A, the possibility of reduced reactivity in the excited-state seems more reasonable. This conclusion is partly backed by the fourfold reduced rate of reaction measured in the excited state at a steady-state condition: 5 X 1O'O M-l s-l compared to 2.0 X 10" M-' s-l for the ground state.26 When a two-stage reaction mechanism is sed,^^,^' this fourfold overall rate reduction can be further shown to result from a two order of magnitude reduction in the molecular reactivity.28 At longer times, the geminal recombination process decays and the bimolecular reaction provides the major contribution to the back-reaction. This period is characterized by a time-dependent rate of reaction between the newly created ions and the bulk concentration of the opposite ionic species. The time dependence is due to the decay of the density distribution of the ions to their steady-state values, as imposed by the reaction. The rate of approach to steady-state concentrations will be affected by the initial prereaction density distribution of the reactants around each other.21v22The general form of the time-dependent rate constant is given byz2
where p is the density distribution of one ion around its "central" counterion. For a diffusion-controlled reaction the boundary conditions are =0 p(r-.m,t) = 1 In the steady state p ( r ) is given by
(9)
(10)
which yields the steady-state expression for kD (eq 1). An extreme deviation from the steady-state conditions is the case where we assume that the reaction opens with positive Boltzmann distribution or p(r,t=O) = exp(-RD/r) (RD < 0, r 2 Ro) (12) and that lRDl is the effective reaction radius. In this case kD(t)' is given by22
(26)K. K. Smith, K. J. Kaufmann, D. Huppert, and M. Gutman, Chem. Phys. Lett., 64,622 (1979). (27)P. Warrick, Jr., J. J. Auborn, and E. M. Eyring, J. Phys. Chem., 76, 1184 (1972). (28)E.Pines, M.Sc. Thesis, Tel-Aviv University, 1981.
The Journal of Physical Chemistry, Vol. 87, No. 22, 1983 4475
pH Jump
TABLE V: PH
4.70 5.00 5.20 5.60 6.14 6.10 6.10
Kinetic Analysis of In-ROH System Following HPTS Photolysis (pHj)
W'l,, PM
[RO-l,,a PM
0.8
20 10 6.3 2.5 0.7 0.8 0.8
E1n-lO,b PM
71,
47.6 61.0 72.3 83.3 91.9 46.0' 23.0d
1.6 2.0 6.4 21.5 20.0 20.0
Total HPTS concentration = 800 PM. Total BCG concentration = 23.8 AIM. a
meas
4.0 X 4.0 X 4.7 x 5.0 X 7.6 X 5.4 x 6.1 X
lo6 lo6
106
lo6 10'
lo5 10'
+
It follows that for t = 2 ns k&) = 1.36kD(-) and that even in this case kD(t)after 100 ns is only 5% greater than its steady-state value. Thus, one can conclude that on the microsecond time scale and up the back-reaction is essentially a steady-state reaction. Since all our experiments were done on the microsecond time scale we can indeed treat the back-reaction as a steady-state one. 2. The Back-Reaction a t Intermediate pH's. The Kinetic Equivalence of the Generated Protons to Water Equilibrium Protons on the Microsecond Time Domain. The peculiar behavior of the back-reaction at solution pH's where pHj = pH is readily explained when we take into account the equifikrium concentration of the reactants. In other words, we have a relaxing system which is characterized by an equilibrium constant, K(ROH), equilibrium concentrations (ROHIo and (H+I0, and the amount of change induced on the equilibrium concentration ( x ) . The steady-state equilibrium is given by kf
kb
+ H+
Assuming that the excitation and decaying processes are (1)fast enough compared to the meburement time scale and (2) completely reversible so that the preexcitation and postexcitation equilibria are identical, one can proceed to write the relaxation rate equation. In this case we must keep the second-order terms and get (15)
or -i = kl**x
2.03 X 10' 1.15 X l o 6 9.6 x 105 3.5 X 10' 4.2 X 10' 5.4 x 106 3.6 X 10'
1.02 x 10" 1.14 X 10" 1.49 X 10" 1.02 x 10" 1.39 X 10" 1.22 x 10" 1.05 X 10" (1.2 i 0.2) X 10"
+ k2**x2
(16)
where kl** = kb([RO-]o + [H+]o) + kf
5-l
4.46 X 10" 4.21 X 10" 4.48 X 10" 4.20 X 10" 4.38 X 10" 5.10 X 10" 4.94 x 1O'O (4.5 i 0.3) x 10"
Total BCG concentration = 47.6 pM.
Clearly, eq 17 holds only if complete identity exists between the pHj protons and the protons already present in the solution from other sources such as strong acids. The success of this simple analysis is shown in Figure l b and with it three very important conclusions follow: (1) the pHi is indeed completely reversible and indifferent to its origins, (2) the pHj is capable of introducing available protons to the solutions; and (3) on the microsecond time scale, the pHj is equivalent to the introduction of a strong acid to the solution. 3. Temperature Dependence of the Back-Reaction. 3-1. Reactions in Water above the Normal Freezing Point. A convenient way to simplify eq 1is to assume that Dm = DH+ + DRo-= DH+,ors kDo =
~N'DH+RD
(18)
In order to compare our results with proton conductivity measurements, one can use the Nernst equation29
where ui is the mobility of the ion and Zie its charge. Substituting for DH+ in eq 18, one gets
The activation energy associated with kD is of the form
kf[ROH]o - ki-,[RO-]o[H+]o = 0
+ [H+]o) + k f ] +~ k g 2
k , , M-I
(14)
and
-1 = [k,([RO-]o
k , , M-I s-l
Total BCG concentration = 95.2 pM.
where kp(-) is given by eq 1. Substituting for Dm = 10-4cm2s-l and RD = 28.4 X lo4 cm one gets the relation kD(t) E kD(-)[l 1.6 x 10-6t-'/2] (t 2 2 nS)
ROH r RO-
72, meas
(16a)
k2** = kb (16b) The immediate integration of (16) yields for the case when [H+l0> [RO-1, + k b and with some algebraic rearrangements
where r0 is t he initial change in the concentrations and xt is the change at any time t. k,' is the pseudo-first-order rate constant kb[H+Io.
Taking into account the finite ionic strength (eq 4), one gets
Equation 22 is of the forml6 = &(mob) - E,(elec)
+ E,(ion)
(23) E,(mob) is calculated from mobility or conductivity measurements, E,(elec) is calculated from the known temperature dependence of the dielectric constant of water, and E,(ion) is approximated by28 E,(ion) = -3.0(1)1/2
(24)
where I is the ionic strength. The calculation was carried out by taking q,= 78.3, T = 298 OC,and dao/dT = -0.356 O W . &(mob) is the major contributor to the total activation energy with almost an 80% share. The agreement between the calculated E, and the directly measured one for the back-reaction, confirms the assumption of a diffusion-controlled mechanism and enables us to use our (29) Walter J. Moore in *Physical Chemistry",3rd ed,Longmans,New York, 1961, pp 446-8. Nernst, 2.Phys. Chem., 2, 613 (1888).
4476
The Journal of Physical Chem/sfv, Vol. 87, No. 22, 1983
results in order to calculate &(mob) and the H+ mobility values in cases where these data were not available. 3-2. Supercooled Water. Data concerning the conductivity of protons in supercooled water were scarce. To our knowledge, reliable measurements had been done only up to -6 0C,30Thus, we can use our own measurements in order to estimate the proton mobility and diffusion coefficients down to -15 "C. Using eq 23, one gets E,(mob)
-
E,(u+) = E, (meas) + [EJelec)
- E,(ion)]
(25)
where E,(meas) is the value of E,(kD) measured for the back-reaction. &(ion) was calculated in the usual way, and E,(elec) was calculated from the empirical relation31 €0 =
532/T'
+ 233.76 - 0.9297T + 0.1417 X 10-'TZ
-
0.8292 X 1O4P (26)
and adjusted to yield to = 87.74 at T = 273 K. The slight temperature dependence of E,(u+) is a characteristic of transport processes in water, as well as of water's physical properties. Supercooled conditions do not inflict sudden changes in E,(u+) and the behavior of kD is a simple extension of the above zero temperature behavior. Since the viscosity of water rises very fast in that region (from 1.8 cP at 0 "C to 2.6 cP at -10 "C), the observation must be a manifestation of the nonclassical nature of proton motion in water. The classical mechanism is inversely proportional to the viscosity. It is described by the Stokes-Einstein e q u a t i ~ n ~ ' , ~ ~ (27)
where 9 is the solvent viscosity and ri is the radius of the ion i. At 25 "C some 85% of the total proton mobility is due to the nonclassical mechanism. Since the totalE&+) value is much less than E,(& the classical motion of the proton should be more dependent on temperature than the nonclassical one. In other words, when cooling the water the classical mobility decreases much faster than the nonclassical one. As a result the ratio DHt(nonclassical/ DHt(classical) increases. At -15 "C we estimate the ratio to be roughly 1O:l compared to 5:l at 25 "C. Following the same line of reasoning, one can conclude that the mechanism of proton diffusion (or mobility) in ice should be purely nonclassical. With favorable structure conditions the mechanism for ice should yield much higher rates of diffusion in ice compared to liquid water, indeed, UHt values up to 100 times greater than those of water were extracted by Eigen and coworkers from conductivity measurements done on pure ice.32 However, those measurements are now believed to be erronous.33 More recent conductivity measurementsM indicate reduced mobility of protons in ice. This is explained by the influence of structure and ionic defects present in the ice lattice.33 Although the overall picture is not yet clear some preliminary measurements of the back-reaction in ice indicate a reduced rate of diffusion (30) Heise, 2.Naturforsch. A , 13, 547 (1958). (31) D. Eisenberg and W. Kauvnann in "The Structure and Properties of Water", Oxford University Press, London, 1969, p 189. (32) M. Eigen, L. De Maeyer, and H. Spatz, 2. Electrochem., 68, 19 (1964). (33) Lars Onsager in 'Physics and Chemistryof Ice", E. Walley, S. J. Jones, and L. W. Gold, Ed.,Royal Society of Canada, Ottawa, 1973, p 7. (34) A. Von Hippel, A. H. Runck, and W. B. Westphal in 'Physics and Chemistry of Ice", E. Walley, S. J. Jones, and L. W. Gold, Ed., Royal Society of Canada, Ottawa, 1973, p 236.
Pines and Huppert
of protons in slightly doped ice compared to supercooled water at the same temperature. These results confirm the more recent conductivity results. This investigation will be reported in detail in a separate paper. 4. Isotopic (HID) Dependence of the Back-Reaction. The ratio kD(H+)/KD(D+) = 1.44 f 0.07 found at 15 O C is in a good agreement with the isotopic (hH+/hDt) effect found in conductivity measurements and yields 1.49 at 15 oc.35
Polarographic measurements yielded a similar ratio, = 1.52 f 0.03 at 25 0c.36 Temperature dependence of kD(D+)is slightly greater than the corresponding kD(H+)dependence. This is in accord with the greater DD+dependence on temperature. The reason for this is that the fraction of nonclassical diffusion in DDt is smaller than in DHt (roughly 75% compared to 85% for DH+). It follows that viscosity changes effect DDt more than DH+. This is supported by the fact that in all the cases E,(q) is greater than Ea(DDt). The different dependences of DHt and DD+cause the isotopic effect to be temperature dependent. On going from 15 to -10 "C (supercooling condition), the ratio kD(H+)/ kD(D+)changes between 1.44 f 0.07 and 1.54 f 0.07. This slight change points again to the fact that supercooling conditions are much the same as the conditions above the normal freezing point (+4 "C for D20). The nonclassical contribution to the isotopic effect is usually estimated to be3' (AH+ - hNat)/(hDt - AN,+) = 1.57 at 25 "C.One should notice that this value differs slightly from the square root of the isotope masses or (mD+/mH+)'I2 = 1.41. Extrapolating Table I11 results to lower temperatures should yield a similar value as a limit for DHt/DD+. From the previous discussion it follows that this limit indeed represents the pure nonclassical isotope effect of DHt and DDt. 5. Relaxation Kinetics Treatment of the HPTS-Indicator System. The availability of the protons produced by the pHj's is demonstrated by the bleaching of acid-base indicators present in the solution at the time of the pHj. The overall kinetic system of the HPTS-indicator (symbolizedby ROH-In) solutions in water is represented by the following six possible reactions.28 ROH equilibrium k
(1)
ROH & ROkl
+ H+
indicator equilibrium In-
(2)
k4
direct proton transfer (3)
k
+ H+ & InH
ROH
+ In-
-
RO-
kai
+ HIn
reactions involving water equilibrium (4)
(5) (6)
ROH
+ OH--& RO- + HzO k41
InH
+ OH-
-In- + HzO 5 k51
H+ + OH-
&
HzO
k6l
(35) hHt Values from R. A. Robinson and R. H. Stokes in 'Electrolyte Solutions",Butterworths, New York, 1959, p 465. AD+ values extracted from Lewis and Doody, J . Am. Chem. SOC.,55, 3504 (1933). (36) N.K.Roberta and H. L. Northey,J. Chem. SOC.,Faraday Trans., 68,1528 (1972); 70, 253 (1974). (37) T.Erdey-Gruz and S. Lengyel, Mod. Aspects Electrochem., 12, l(1977).
The Journal of Physical Chemktry, Vol. 87, No. 22, 1983 4477
pH Jump
The simultaneous solution of all six coupled reactions is poissible,17but not needed since reactions 3 are ionically unfavorable with Z1Z2 = 9+ for k13 and Z1Z2= 8+ for k31. At pH, < 6.5, one can further neglect reactions 4 and 5 because of the small OH- concentration, the ionic repulsion between negative charges in k14 (Z1Z2= 3+) and k15 (Z1Z2 = 2+), and the very slow nature (hydrolysis) of k41 and k51. The omission is also justified by the super fast nature of reactions 1and 2 which involve protonation of negatively charge species in diffusion-controlled mechanism. Reaction 16 involves water production with k16 = 1.4 X loll M-ls-l. Yet at pH's I6, the hydroxyl concentration is low enough to make reactions 6 negligible altogether. So that only first-order (linear) terms are considered in the rate equations the equilibrium concentration of all species in reaction 1 and 2 should be greater than the perturbation (Cj x , with pK(HPTS) = 7.8 and pK(BCG) = 4.7). The desired (pH), range is 4.7 IpH, < 6.1. In this range all concentrations are at least of the order of lo* M. It is also desirable to keep the pHj ( x ) below lo4 M in order to achieve minimum perturbation of the system. With all these conditions fulfilled one has to consider only the two coupled reactions 1and 2 which involve the equilibrium reactions of the two species. From the stoichiometry it follows that x(RO-) = -x(ROH)
(28)
y(1nH) = -y(In-)
(284
From eq 28 and 28a it follows that x(H+) = x(RO-) - y(1nH)
(29)
(From here onward, the sign of the charges is omitted from the equations; x represents the induced change in the equilibrium concentration of the HPTS molecule and y is the induced change in the indicator (BCG) concentration.) Substituting into the rate equations of reactions 1and 2 one gets d([ROH] - x)/dt = ki([ROl + x ) W I d([InH] + y]/dt = -k4([InHl + Y)
+ x - Y) - k,([ROHI - x ) (30)
+ k3([Inl - y)([HI + x - Y) (31)
cancelling the equilibrium rates of reactions one gets the two coupled differential equations 2 = -[ki([RO]
+ [HI) + k2lx + (k2[RO])y
P = (k3[Inl)x- [k3([Inl + [HI) + k41y
(32) (324
Solving for y in the usual way and with the initial conditions x ( 0 ) = xo
(33)
Y(0) = 0
(334
one gets (34) Quation 34 describes the induced transient change in the concentration of the unprotonated indicator form (In-),at any time after the creation of the pHp yl,zare the observed protonation (yl) and deprotonation (yz)rate constants of
the indicator. Solving in the usual way the two coupled differential equations of (32) one gets for yl,2
where
= -ki([RO]
+ [HI) - kz
(354
= kl[RO]
(35b)
u3 = k3[In]
(354
~2
+ [HI) - k4
u4 = +([In]
(35d) from eq 34 it follows that the time delay between the creation of the pHj and the maximum bleaching of the indicator is given by t, =
In
(Yl/Y2)
71-
Yz
At t, the rates of protonation and deprotonation exactly cancell one another. Substituting t, into eq 34 and after some algebraic rearrangements one gets the maximum bleaching of the indicator achieved by the pHj.
Solving eq 32 for kl and k3, which are the protonation constants of the HPTS anion and indicator, respectively, one gets klk3 = YlY2
+ [ROI + [HMKI,, + [In] + [HI) - [InROl) (38) where KRoH
E
k3
=
k2/k1
Kh
E
k4/k3
d f (d2- 4~bc)'/' 2c
(39)
where
u = k1k3 d = Y1 + Y2 c = Kh + [In] [HI
+
d = 71 + Y2 Of the two possible solutions of (39) the (+) solution yielded the better results. In Table V all k3 values are calculated accordingly. In much the same way it is possible to calculate directly any pair of constants among kl, k2, k3, and k4. But the calculation of k4 or k2 is not desirable because they are comparatively very small and thus very sensitive to the experimental error in yl,z. However, since Kh and KRoH can be measured independently from steady-state titration, all rate constants can be evaluated.
Conclusion The full reversibility and relaxational nature of the back-reaction are demonstrated. The success of the kinetic analysis of an acid-base indicator present in the solution at the time of the pHj can lead to a straightforward application of the pHj as a perturbation method. To initiate the process, high-repetition low-intensity light sources can
J. Phys. Chem. 1983, 87, 4478-4484
4478
be used. This makes the pHj method available for biochemical research on proton-sensitive systems. Temperature and isotope dependence of the back-reaction confirm its diffusion-controlled mechanism. This makes the back-reaction a possible research probe for protic reaction medium properties.
Acknowledgment. This research was supported by the Israeli Commission for Basic Research and the United States-Israel Binational Science Foundation. Registry No. DP,7782-39-0;BCG, 76-60-8;&hydroxy-1,3,6pyrenetrisulfonate anion, 86527-89-1.
Absorption Cross Section and Kinetics of I O In the Photolysis of CH31 in the Presence of Ozone R. A. Cox' and 0. 6. Coker Environmental and Medical Sciences Divlslon, A.E.R.E. liarwell, Oxon OX1 1 ORA, England (Received: December 28, 1982)
The photolysis of CH31 in the presence of 03 was used as a source of IO radicals in N2 + 02 diluent at l-atm pressure and 303 K. IO was detected in absorption by using the molecular modulation technique. The absorption spectrum in the region 415-470 nm, arising from the A211 X211 transition of IO, was recorded and the absolute X cm2 absorption cross section at the band head of the (4-0) band at 426.9 nm determined to be ;:13:. molecule-'. IO decayed by a rapid reaction which yielded an aerosol of probable formula 1409 as final product. The observed rate coefficient for IO decay was near the gas kinetic collision rate which probably reflects an efficient attachment of IO radicals to the growing aerosol. The significance of the photochemical and kinetic parameters for atmospheric iodine chemistry is briefly discussed.
-
Introduction The gas-phase chemistry and photochemistry of the halogen monoxide radicals C10 and BrO has been the subject of intense study in recent years, because of their important role in the catalytic cycles resulting in ozone destruction in the stratosphere. These XO species are rather unreactive toward stable closed-shell molecules but they undergo a variety of interesting rapid bimolecular and termolecular reactions with other odd-electron species.l These reactions determine the atmospheric behavior of active C10 and BrO species which are released as a result of the breakdown of C1- or Br-containing molecules by photodissociation or free-radical a t t a ~ k . ~ - ~ The chemistry of IO radicals and related I-containing species is much less well-defined. In a recent paper Chameides and Davis5have suggested a possible influential role of iodine photochemistry in the budget of ozone in the troposphere, particularly in the marine boundary layer where there is a significant flux of gaseous CH31from the ocean to the atmosphere." In their paper Chameides and Davis5 have formulated the expected chemical behavior of gaseous I-containing species in the atmosphere, based largely on the analogy with C10 and BrO. Their review of the available kinetic and photochemical data revealed large gaps in knowledge but it was clear that photochemically driven cyclical reactions involving I, IO, Os, NO,, and HO, could be important in the atmosphere. This arises principally because (a) the electronic absorption spectra (1)M.A. A. Clyne and J. A. Coxon, R o c . SOC.London, Ser. A, 308, 207 (1968). (2)R.S.Stolarski and R. J. Cicerone, Can. J.Chem., 62,1610(1974). (3)Y.L. Yung, J. Pinto, R. T. Watson, and S. P. Sander, J. Atmos. Sci., 37, 339 (19t%). (4)D. L. Baulch, R. A. Cox, P. J. Crutzen, R. F. Hampaon, J. A. Kerr, J. Troe, and R. T. Watson, J . Phys. Chem. Ref. Data, 11, 327 (1982). (5) W. L. Chameides and D. D. Davis, J. Geophys. Res., 85, 7383 (1980). (6)J. E. Lovelock, R. J. Maggs, and R. J. Wade, Nature (London),241, 194 (1973).
of most I-containing compounds extend into the intensity-abundant near-UV and visible regions of the solar spectrum and (b) the weakness of the H-I bond makes formation of relatively stable hydrogen iodide, in H-atom abstraction reactions of I atoms, very slow at prevailing atmospheric temperatures and trace gas composition. The key iodine reactions leading to ozone destruction are as follows: I O3 IO O2 (1) IO + IO 21 + 02 (2) Kinetic data for these reactions have been reported by Clyne and Cruse,' who used absorption in the A21T X211 electronic transition near 427 nm to monitor IO in a lowpressure flow system. They found a relatively rapid recm3 molecule-'s-') to action of I with O3 (k, N 8 X produce IO in both ut' = 0 and u" = 1 vibrational levels of the ground state. The results for the decay of IO were consistent with a second-order process with a rate coefficient of kz = 5.2 X cm3molecule-' s-l. However, the extent to which reaction 2 was occurring was not clear since O3was not apparently removed in the presence of IO, and wall removal of I atoms was reported to be very fast. The A X transition of IO was first reported by Gaydon and co-workers,*as the so-called methyl iodide flame bands; the transition has subsequently been investigated in absorption by Ramsay and co-workersgJO using flash photolysis techniques. The intense absorption bands due to the vibrational ground state lie in the region 470-400 nm. Absorption in this region leads to predissociationg and therefore the atmospheric photolysis of IO
+
-- +
+-
-
(7)M.A. A. Clyne and H. W. Cruse, Trans. Faraday SOC.,66,2227 (~1 -~ -7_,_ .n ) .
(8) E. H. Coleman, A. G. Gaydon, and W. M. Vaida, Nature (London), 162,108 (1948). (9)R. A. Durie and D. A. Ramsay, Can. J.Phys., 36,(1958). (10)R. A. Durie, F. Legay, and D. A. Ramsay, Can. J.Phys., 38,444 (1960).
QQ22-3654/83/2Q87-4478801.5QfQ 0 1983 American Chemical Society
