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4436

Theoretical Analysis of Conduction in Acid and Base Solutions Aiexandre Laforgue,t Christine Brucena-Grlmbert,t Denlse Laforgue-Kantzer,t Laboratoire de &&anique Ondulatoire Appliqu6 and Laboratolre d'Electrochimie, Universit6 de Relms, F-5 1062 Reims Cedex. France

Giuseppe Del Re, and Vlncenzo Barone' Cattedra di Chimica Teorica, Instituto Chimico, I-80 134 Napoli, Italy (Received: May 2 1, 198 7; In Final Form: June 25, 1982)

The various effects playing a role in the conduction mechanism of acid and base solutions have been analyzed in detail in terms of a new general model. The principle of reciprocity between proton and proton vacancy is used to explain the similar conductivity of acid and base solutions. The two-oxygen molecule ions H502+ or H302-appear to be the essential structures responsible for both normal and anomalous conduction; this explains the strong coupling between the two kinds of conduction. On these grounds it is possible to perform a quantitative partitioning of the experimental ionic mobilities into normal and anomalous contributions and to stress the importance of both tunneling and concerted thermal jumps in the anomalous conduction. The reduction of tunneling frequency (otherwise largely predominant) is related to the fact that only symmetric bridges, a fairly rare situation, really contribute to tunneling. Quantum-mechanical computations of one- and two-dimensional potential energy surfaces for proton exchange and dynamic studies of proton motion are reported to substantiate quantitatively the model.

I. Introduction The activated-jump description plays an important role in different parts of the theory of electrochemistry: selfdiffusion,' which is a kinetic phenomenon associated with equilibrium; reaction processes, viz., evolution toward equilibrium; and conduction, which is a stationary nonequilibrium phenomenon. The analysis of a specific electrochemical problem in terms of activated jumps has therefore a wider scope than the concrete application studied. In the specific case of proton transport in water solutions the existence of barriers between equilibrium sites is supported by which can be interpreted as measurements of conductivity activation energies. Efforts have been made in recent times4+to take proton tunneling associated with those barriers into account. However, much remains to be done in order to better elucidate the mechanism of proton conduction. This demands a detailed analysis of two distinct problems: (i) What elementary steps make up the transfer mechanism? (ii) Which is the rate-determining step of proton transfer? In 1928 Huckellosuggested that in acid solutions proton transfer is followed by rotation of H30+. That rotation is responsible for the irreversibility of the transfer process and is the rate-determining step. Calculations by Bernal and Fowler" confirmed that tunneling must be controlled by some other phenomenon. In 1956, Conway et a1.I2 concluded that rotation is in fact the rate-determining step but suggested that its role consists in providing the vacant site to a proton and that it takes place before the proton crosses the hydrogen bridge. In 1979, that model was still considered substantially valid.138 Nowadays, it appears to be a general agreement that three phenomena are present in the conduction process-jumps of the carrier ions, proton transfer between ion and molecule, dipole rotation-but the order and interdependence of the three processes, the nature of displaced and rotated particles, and the mode of proton transfer are still a matter of discu~sion.'~~ Hofacker et al.14 investigated the role of proton-proton interactions and of collective motions in proton transfer. Laboratoire d e Mgcanique Ondulatoire Appliquge.

* Laboratoire d'Electrochimie.

0022-3654/82/2086-4436$01.25/0

Ladik et al.15J6pointed out that asymmetric double wells are little favorable to proton transfer. Busch and De La Vega" also concluded that proton transfer is easy only when the hydrogen-bridge double well is symmetric; no transfer takes place when sufficient difference exists between the depths of the initial and final potential energy wells. This implies that the rate-determining step of migration can actually be the crossover of the hydrogen bridge. The necessary frequency-reducing factor is provided by the probability that the channel is open (i.e., that the double well is symmetric). In a previous work'* we have suggested a conduction mechanism which takes the above considerations into account and gives a satisfactory picture of the whole mechanism. The present paper is an attempt of axiomatization of that model and its analysis from both electrochemical and quantum chemical standpoints. (1) P. Guerin and A. Laforgue, J. Phys. (Orsay,Fr.), 37, C7-779 (1976). (2) (a) D. Laforgue-Kantzer, A. Laforgue, and K. Tran-Cong, Electrochim. Acta, 17, 151 (1972); (b) K. Tran-Cong, Doctoral Thesis, Universitg de Lyon, Lyon, France, 1969. (3) A. Olivier, Doctoral Thesis, UniversitC de Reims, Reims, France, 1979. (4) P. 0. Lowdin, Adu. Quantum Chem., 2 , 254 (1965). (5) P. 0. Lowdin, J. Chem. Phys., 18, 365 (1950). (6) S. G. Christov, J. Res. Inst. Catal.,Hokkaido Uniu., 16, 169 (1968). (7) S. G. Christov, Ann. Phys., 7 , 87 (1965). (8) S. G. Christov, Croat. Chim. Acta, 44, 67 (1972). (9) S. G. Christov, J. Electroanal. Chem., 100,513 (1979). (IO) E. Huckel, Z . Electrochem., 34, 546 (1928). (11) J. D. Bernal and Rh. Fowler, J. Chem. Phys., 1, 515 (1933). (12) B. E. Conway, J. O'M. Bockris, and H. Linton, J . Chem. Phys., 24, 834 (1952). (13) (a) J. O'M. Bockris and S. V. M. Khan, -Quantum Electrochemistry", Plenum Press, New York, 1979; (b) F. H. Stillinger, Theor. Chem.: Adu. Perspect., 3, 177 (1978). (14) J. K. Fang, K. Godzik, and G. L. Hofacker, Bunsen. Berichte, 77, 920 (1973). (15) A. Karpfen, J. Ladik, P. Runegger, P. Schuster, and S. Suhai, Theor. Chim. Acta, 34, 115 (1974). (16) G. Biczo, J. Ladik, and J. Gergely, Phys. Lett., 13, 317 (1964); Hung. Phys. Acta, 20, 11 (1966). (17) J. H. Busch and J. R. De La Vega, J . Am. Chem. Soc., 99, 2397 (1977). (18) (a) D. Laforgue-Kantzer and A. Laforgue, lectures presented at the 3rd International Summer School on Quantum Chemical Aspects of Electrochemistry, Ohrid, 1974; (b) A. Laforgue, M. Ammar, and S. Roland, Ann. Uniu. ARERS, 12, 1 (1974).

0 1982 American Chemical Society

The Journal of Physical Chemisfry, Vol. 86, No. 22, 1982 4437

Conduction in Acid and Base Solutions

BASES

ACIDS

H

F

-

H

‘ /[Dl b

P

c

P@

,v” [AI A

Flgure 2.

The composition of mobilities is represented in Figure 1,where the only directly observable mobility is, of course, the resulting one p*, i.e.

Flgure 1. Composition of mobilities in acid and base solutions.

The paper is organized as follows. Section I1 is devoted to a general outline of the model and a tentative estimation of the geometrical parameters required for its quantitative application. Section I11 is concerned about the electrochemical implications of the model (hydration number of the carrier ions; decomposition of the ionic mobilities into normal and anomalous contributions; analysis of the relevance and physical meaning of concerted thermal jumps). Finally, section IV is devoted to a quantum chemical analysis of the model (evaluation of potential energy barriers; coupling of proton and oxygen vibrations; dynamics of proton movement).

11. Outline of the Model 11.2. Reciprocity between Proton and Proton Vacancy. Different models of conduction in acid or base solutions are associated with different structures of water and its ions. According to most authors, conduction in acid or base solutions results from the combined effect of a (small) normal conduction similar to that associated with heavy ions, and a (large) anomalous conduction.1° We have pointed out that the proton behavior is intermediate between electron and other nuclei.lsb Further, a close correspondence can be established between electron-hole and proton-proton vacancy reciprocities. In fact, in the water medium the 0-H bonds and the lone pairs possess two homogeneous and nearly equivalent electron distributions and they can be mutually exchanged by an easy transfer. Consequently each lone pair can be considered as the site of a proton vacancy in both the ionized and neutral molecules. On these grounds, our model assumes that the anomalous conduction results from the fast migration of a proton (acid solutions) or a proton vacancy (base solutions). By considering that ionic species in water may be written as H30+.mH20and OH-.mH20, we assume that normal conduction is associated with m I0, while anomalous conduction is associated with m = -1. In other terms the same ionic charge (a proton or a proton vacancy) can participate in either type of conduction depending on its degree of hydration.lg8 ~

~

~~

~

Schematic representation of concerted thermal jumps.

~

(19) (a) D. Laforgue-Kantzer,Electrochim. Acta, 9,151 (1964); (b) R. Parson, “Handbook of Electrochemical Constants”,Butterworth, 1959; (c) Landolt, Bernstein Tabellen.

+ pa+ = 35.0020 + 0.5968At p- = p(OH-) = -pn- + p[ = 19.7655 + 0.3472At p+

= p(H+) = pn+

(la) (lb)

at 22.5 “C in MKSA ~ n i t ~where ~ J the ~ ~subscripts , ~ n and a indicate normal and anomalous conduction, respectively. If eq l a is a trivial one, eq l b is not trivial and expresses precisely the correspondence between a proton and a proton vacancy, which accounts for the close similarity between anomalous conduction in acids and bases.20 11.2. Coupling between Normal and Anomalous Conduction. We shall assume the following in the anomalous conduction: (1) Crossing of the hydrogen bridge is a necessary step, possible by both thermal and tunnel effects. (2) Tunneling is possible only if the bridge is open, i.e., symmetric; this symmetry condition is realized for the bridge B-H-s-A only when B = A. (3) The energy AE, thermally transferred from A to B may be returned to A, proceed to another ion C,21or be dissipated in the motion of B. The second possibility is considered the most probable and gives rise to concerted transfer sequences of the kind represented in Figure 2, usually referred to as the Grotthus mechanism.22 By symmetry requirements, concerted jumps can be excluded in the case of tunneling. According to hypothesis 3, if the number of concerted thermal jumps is large, a not negligible contribution of the thermal effect to the anomalous conduction can be observed, even if its frequency is very small with respect to the tunneling frequency. We can therefore distinguish in the anomalous mobility pa” the contribution ps* due to “tunneling symmetricjumps” and the contribution pc* due to “thermal concerted jumps”. Hence, eq 1 can be rewritten as P+

= pn+

P- = -Pn-

+~

+~

s +

c +

+ ~ s +- PLC-

(24 (2b)

(20) (a) A. Laforgue and D. Laforgue-Kantzer, “Proceedings of the International Summer School of Quantum Mechanical Aspects of Electrochemistry, Skopje, 1972, Vol. 1, p 15; (b) E. G. Weidemann and G. Zihdel, Z.Phys., 118,288 (1967); (c) M.Ammar, Doctoral Thesis, Lyon, France, 1970. (21) R. R. Dagonadze, Y. I. Kharkatz, and J. Ulstrup, J. Chem. Soc., Faraday Trans. 2, 70, 64 (1974). (22) (a) T. Grotthus, ’Memoire sur la Decomposition de 1’Eau et des Corps que elk tient en Dissolution”, Rome, Italy, 1805; (b) M. De Paz, S. Ehrenson, and L. Friedman, J . Chem. Phys., 52, 3362 (1970).

4438

Laforgue et al.

The Journal of Physical Chemistry, Voi. 86,No. 22, 1982

a

r

a

7

\ -

r

b

I /i

b

C

C

\ -

Figure 3.

/

Model of conduction in acid solutions.

and, according to previous s t ~ d i e seach , ~ ~partial ~ ~ mobility given by

F ~ *is

pi*

= Avi*li

(3)

In eq 3 li is the distance covered by the charge in the direction of the external electric field and Aui the corresponding change of the frequency (for an unitary field starting from zero field) as a result of the jumps in the two opposite directions. We point out that anomalous conduction requires a movement of the carrier ion to obtain either the symmetry condition in the case of tunneling (hypothesis 2) or, for concerted thermal jumps, the appropriate orientation of the water molecule chain (hypothesis 3). This explains the coupling between anomalous and normal conduction and (23) (a) M. J. Polissar, J. Chem. Phys., 6, 833 (1938); (b) 0. Y. Samoilov, "Structure of Aqueous Electrolyte Solutions and the Hydration of the Ions", Academy of Sciences URSS, Moscow, 1957.

Figure 4.

Model of conduction in base solutions.

the impossibilityof the former without the latter. In terms of section 11. 1 this fundamental result can be written Av,* + Av,* = fAv,* (34 11.3. Analysis of the Motion of the Carrier Ions. As a hydrated ion has a nonvanishing dipole moment, its motion will be classically a rotation-translation one, whereby the ion tends to dispose itself parallel to the electric field F. The ion may be thought to "jump'! from one site to another across a potential energy barrier associated with both translational and rotational degrees of freedom. For the anomalous conduction, our model assumes that proton transfer takes place within a big surrounding ion, that is, e.g., between a carrier ion ( m = 0, see next section)

The Journal of Physical Chemistry, Vol. 86, No. 22, 1982 4439

Conduction in Acid and Base Solutions

and a water molecule. If just one bridge is opened to proton tunneling (as the result of a jump of the carrier ion), the mechanism is that shown in Figure 3 for acids and in Figure 4 for bases. The two mechanisms being interchangeable (reciprocity principle), from now onward explicit reference will be made to base solutions, which are easier to visualize. Starting from a water molecule C in equilibrium with a hydroxide ion A, in step 1 a hydroxide ion C is created by proton loss across a hydrogen bridge, so that A becomes a neutral molecule. In step 2 the new hydroxide ion C immediately moves (by rotation and translation) to a new equilibrium site C' heaving away from A and forming a molecular association with D. Eventually, in step 3, the ion C' will in turn be transformed into a water molecule through a new proton jump. We point out that proton tunneling (steps 1 and 3) occurs only within the molecule ions H30z- or H502+,which are rigorously symmetric with respect to it. 11.4. Self-Dissolution Principle. The electrostatic forces, which contribute to deform the molecule ions under vacuum, are strongly reduced in the water medium due to the effect of induced p ~ l a r i z a t i o n . As ~ ~ a consequence the neutral (for instance, H40J and ionic (for instance, H,Oz- or H502+)aggregates share very similar geometries, chemicophysical properties, and lifetimes. This general principle (self-dissolution principle) is confirmed by direct m e a s ~ r e m e n t s ~of~ *the ~ ~distance between 0 and OH-, which point out the practical invariance of the 0-0 distance with respect to the ionization. Several values of the interoxygen distance lo in neutral aggregates (at room temperature) have been reported. The two most precise m e a s u r e m e n t ~ give ~ ~ ~2.86 s ~ and 2.77 A; we shall adopt the mean between these two values and postulate a negligible influence of the temperature. Thus

lo = 2.81 (5) A

dlo/dT

N

0

On the same grounds the other geometrical parameters are as follows: d(0-H) = 0.958 A; LHOH(H,O and H-OH-) = 0 = 104.75'; LHOH(H,O+) = 0' = 109.5'. As a last point, we postulate a lifetime of the ionized dimer near lo-', s on the basis of the lifetime obtained for the neutral dimer by simulation method^.^^^^' 11.5. Progression of the Charge in the Direction of the Field. In the present model tunnel transfer occurs in a well-defined association (the molecule ion H 3 0 2or H502+) rigorously symmetricwith respect to the tunneling; thermal transfer occurs in a cluster ion (section 11.2 and Figure 2), whose average number of elements N* will be determined later. In each case we must be precise with regard to the orientation of the active 0-H-0 bonds. We assume that (i) the ionic dimer (H302-or H502+)within which the proton jump occurs orients its dipolcmoment in the average direction of the external field F , (ii) in the motion of the carrier ions the neutral half of the ionic dimer (a water molecule) does not perform any translation but only does a rotation correspondingto the electric-dipole change of the supermolecule, and (iii) in the concerted thermal jumps each bond is oriented as in the tunneling. (24) 0.Tapia and 0. Goscinski, Mol. Phys., 28, 1653 (1975). (25) (a) J. R.Bell, L. Y. Tyroll, and D. L. Wertz, J. Am. Chem. SOC., 95,1456 (1973); (b)G. W. Brady, J. Chem. Phys., 28,464 (1958); (c) C. L. Van Pantaleon Van Eyck, H. Hendel, and J. Fahrenfort, R o c . R. SOC. London, 247, 472 (1958); (d) M. D. Danford and M. A. Levy, J . Am. Chem. SOC.,84, 3965 (1962). (26) J. L. Finey in 'Water", Vol. 6, F. Franks, Ed., Plenum Press, New York, 1979, p 47: (27) D. W. Wood in "Water",Vol. 6, F. Franks, Ed., Plenum Press, New York, 1979, p 279.

To evaluate the distances li covered by the unitary charge in each elementary step, we set the following: Is*, the distance covered in the complete process of tunnel symmetric jumps (translation of oxygen atoms being excluded); &*,the distance covered in step 2 (Figures 3 and 4); IC*, the distance covered in the N* concerted jumps. Figures 3 and 4 (where cp* is the angle between the AC jump direction and the dipole of the supermolecule ion) allow us to calculate 1; = 0.333 A, 1, = 1.292 A, ' 1, = 0.304 8,and 1: = 1.320 A. Figure 2 then shows immediately (following assumption iii) that

1,"

= N"1,"

111. Decomposition of the Ionic Mobilities 111.1. Normal Conduction. The motion of a neutral molecule in a liquid is related to the fluidity (inverse of the viscosity) of the liquid, whose temperature dependence is governed by an activation energy W,. The fluidity is better understood in terms of the migration of a fluid vacancyB and W, is then the energy required for the jump of the molecule from an occupied site to a vacant one. The normal conduction is very similar, although the migrating particle becomes a carrier ion; the two processes are linked by Stoke's law and, in our model, also by the self-dissolution principle. As in the case of hosk ionsB we can define a crystal radius R* of the carrier ions by the well-known expression kn* = (B/R*)e-@-n*

(4)

(with 0 = l / k T = 0.02535 eV at 22.5 'C) where AE,* is the energy required for the jump of the ion from an equilibrium site to another one., Since the B parameter of Stoke's law can be written

B = Boe-@Wz

(5)

(where Bo is a constant) we can conclude that the temperature dependence of the normal mobility is governed by

W," =

w,+ AE,*

(6)

which probably coincides with the "conductivity activation energy" measured for the carrier ion. On the basis of the self-dissolution principle the sum W,+ W; for a pair of ions H30+,OH- should be practically equal to the sum W, + W z for a water dimer. Hence

+

AE,+ + AE,-= 0

AE," = *€

(7)

On the same grounds, we assume also that

R* = (L3/2)(1 f 0 with

r small.

(8)

From these assumptions, we get ,un* =

P*

with p*

= 2B/10 = (6.3996 + 0.1513At)X

MKSA at 22.5 "C (10)

Equation 9 shows that the values of normal mobilities depend on two parameters: the conductivity activation energy of the carrier ion (*e) and the difference in size (in (28) H. Ryring and Mu Shik John, "Significant Liquid Structures", Wiley, London, 1969. (29) E. Darmois and G. Darmois, 'Electrochimie ThCorique",Alcan, Paris, 1960.

4440

The Journal of Physical Chemistry, Vol. 86, No. 22, 7982

lo units) between acid and base carrier ions ({I. Before computing these two parameters, let us briefly discuss the physical meaning of a,'. In a rigid medium (e.g., a glass) the energy barrier to the motion of an ion is related to the jump from a fixed site to another fixed site. In a liquid, the initial and final sites are determined by a fluctuating environment. Let us assume that the reference frame is fixed with respect to some average disposition of the surrounding molecules. Then the energy of the ion under thermal fluctuation is successively smaller or larger than W,". In the former case the ion remains linked to its surrounding shell, and in the latter case it jumps until it is stopped by a collision. Let us next consider the same process in the local reference frame of the ion, where its kinetic energy is, of course, zero. The solvent molecules can be divided into two classes, depending on whether they undergo hard collisions with the ion (whose statistical result is Stoke's law) or smooth collisions with low fluctuating forces and distances in a significant time. This picture is detailed by Wolynes in his theory of ion mobility,30 where the author suggests a nonactivated motion of the ions. We believe that Wolynes' analysis points in the opposite direction in the particular case of acids and bases due to the role of the energy frontier Ecrit between the two classes of molecules. In fact, in our case, carrier ions and solvent molecules have approximately the same dynamical properties and Ecritis very near Wn*, which, in the reference frame of the solvent molecules, is just an activation energy for the motion of the carrier ions. The above discussion permits us also to conclude that W," is in the range of one thermal quantum (to be compared to activation energies of about 0.7 eV for the diffusion of ions in the solids31 of 0.26 eV for anomalous conduction (see later) and of 0.1-5 eV for chemical reactions) so that 6 should be smaller than the thermal quaneV, which will tum (in agreement with the value 1.7 X be derived in the next section). 111.2. Size of the Carrier Ions. Equation 9 can be written (introducing eq 3) as

The change of frequency Au,' due to the presence of the external field can be related21ato the zero-field jump frequency vn* by AV* = cov,*ln* (12) where co is a proportionality constant. The jump frequencies depend, in turn, on the conductivity activation energies W,' through v,+/u,- = ,-B(w"+-w"-) = e-28f (13)

Introducing eq 12 and 13 into eq 11, we finally obtain

or

R+ = 1.52 8, R- = 1.30 A If one considers the crystal radii of 0- and 02-to be 1.46 (30) (a) P. G. Wolynes, J. Chem. Phys., 68, 473 (1978); (b) P. G. Wolynes, Annu. Reu. Phys. Chem., 31, 345 (1980). (31) P. Guerin, Doctoral Thesis, UniversiG de Reims, Reims, France, 1979.

Laforgue et al.

and 1.32 A, respecti~ely,~~ the computed values of R+ and R- show that the carrier ions in the normal conduction are OH- in the case of the bases and H30+in the case of acids. This is a first check of our model. 111.3. Final Estimation of the Mobilities. We now proceed to estimate the three contributions pn* (for the normal mobility) and psi + pc' = pa* (for the anomalous mobility) to the total experimental mobilities p*. We have a total of 12 equations: 2 equations of the type psi + ICif pn" = pi (eq 21, 2 equations of the type pn* = [p*/(l f (eq 9), 6 equations of the type pCLif = Av? f li (eq 31, and 2 equations of the type Au,' = fAv,* (eq 3a). However, we have 13 unknown values (the 6 p i t , the 6 Avif, and E ) . The last relationship can be obtained as follows. We introduce the jump frequencies in both the anomalous (vcf) and normal ( u t ) thermal effects assuming that the same proportionality constant co (eq 12) holds in both cases. Hence Av,' N*ls* uc* -=-(14) AV,* I,* v,* These frequencies are, once again, related to the corresponding activation energies f

vn* = ,-8(w.*-w,f)=

e-B(AE,*-AE"*) = e-B(AE.*'Tf)

(15)

Substituting eq 15 into eq 14 we obtain two equations of the type Au,'~ Av," = (N'l? f lni)e-~(;lE~"tpf) (16)

From Figure 2 it can be seen that, neglecting the extremes, the transfer chain for concerted jumps is very similar for acids and bases and hence we can confidently set

N+ = N(17) The two quantities AE," and AEn* of eq 15 must be considered as independent (and, in general, different) parameters, in accordance with Samoilov's theory,23bwhich assumes different barriers for the dry ion and the solvated ion. In addition, due to the comparatively small mass of the proton, its motion is almost completely separated from that of the oxygen atoms; this explains why the energies Ma* and AE,' lie in different ranges and, therefore, can be evaluated without ambiguities. If we give to AEa the mean value discussed in previous papers1Sb120a AEai = AE, = 0.26 eV

(18)

we have a system of 15 equations with 15 unknown quantities. Such a system can be written for each temperature and, hence, the derivative quantities can be evaluated by considering two different temperatures. In particular, 6 should be considered as a free energy, whose derivative should give the entropy term. However, we have preferred to set (as in the case of AE,)&/at = 0. This leaves only 14 derivatives to be evaluated. The solution (32) L. Pauling, "The Nature of the Chemical Bond", Cornel1 University, Press, Ithaca, NY, 1960. (33) M. D. Newton and s. Ehrenson, J . Am. Chem. Soc., 93, 4971 (1971). (34) G. H. F. Dierksen, Chem. Phys. Lett., 4, 373 (1969). (35) P. A. Kollman and L. C. Allen,J.Am. Chem. SOC.,92,6101 (1970). (36) M. Ammar, A. Laforgue, and D. Laforgue-Kantzer, C.R. Hebd. Seances, Acod. Sci., 274, 2140 (1972). (37) C. Grimbert and A. Laforgue, Folia Chim. Theor. Lot., 5, 90 (1977). (38) W. J. Hehre, R. F. Stewart, and J. A. Pople, J . Chem. Phys., 51, 2657 (1969). (39) R. Ditchfield, W. J. Hehre, and J. A. Pople, J . Chem. Phys., 54, 724 (1971).

The Journal of Physical Chemistry, Vol. 86, No. 22, 1982 4441

Conduction in Acid and Base Solutions

of the system is tedious, but without mathematical difficulties, and leads to

= -1.7 X

t

eV

+ 0.1500At pn- = 6.493 + 0.1535At pa+ = 28.655 + 0.4468At pa- = 26.256 + 0.5007At pn+

= 6.347

+

do + -p1 1-0

(25) This uncertainty in the kinetic energy corresponds to an uncertainty in the momentum given by

P 6W 6p = - - = 2 m a

(a) 112

6W =

kg m/s

(26)

(m being the proton mass). The uncertainty in the momentum is, in turn, associated with the proton delocalization along the bond, 6 x , through 6x h/6p = 2 A (27)

+ 0.649At ps- = 25.182 + 0.595At

which is in the range of intramolecular distances. On the other hand, the uncertainty in the kinetic energy 6 W is related to the lifetime 6t of the molecular association by

pc+

= 1.203 - 0.202At

pc-

1.074 - 0.094At

6t t q w 2.9 x 10-14 9 (28) From eq 28 and 19, we obtain finally N 6 t N 0.5 X 10-l2 s which is coherent with the assumption of contemporary thermal jumps.

p8+

= 27.452

MKSA units at 22.5 O C and

N' = 17

1 P 6W d6W=dt t

6W = -; log (2(1 - a)]

From the value of o (eq 22) it follows that 6 W = 0.02 - 0.001At eV

in MKSA units at 22.5 "C. I t appears that H30+ is slower than OH- (normal conduction) despite the negative value of E . On the other hand, anomalous conduction is slower for bases than for acids (it is easy to show this latter result as a direct consequence of the inequality p* lf2(p+ 113). Incidentally, the normal mobility pn+ is found not far from the mobility of Na+ (5.46 X lo4 at 22.5 "C)and this is in agreement with the early suggestion of Huckel.lo We have also found

in

From eq 23 we obtain

d N ' / d t = - 1.9

at 22.5 OC

(19)

111.4. Grotthus Mechanism. Equations 19 imply that the Grotthus mechanism vanishes near 30 "C and that the mean number of concerted jumps increases in cold water. The ratio of the tunnel component and the thermal component in the anomalous conduction is

8'

= bL,f/pC')N*

from which Q+ = 388 and Q- = 399. Tunneling is hence largely dominant, in agreement with the conclusions reached by Conway et al.12 and Busch and De La Vega.17 However, the Grotthus mechanism plays a not negligible role and has a clear physical meaning. This can be shown as follows. Let o be the probability of proton transfer from a molecule to the following one in the chain (Figure 2); then the mean number of concerted jumps in an infinite chain is given by m

N =

k=O

Itak

-do = - - (1 -

0/(1-

(20)

dN

dt 1 + a dt Using the previous values of N and d N / d t (eq 19), we obtain O = 0.78 - 0.OllAt (22) On the other hand, the energy transmitted by tunneling must be greater than Ma, e.g., ma+ U. The thermal noise does not disturb the transfer if the thermal energy is lower than U + 6 W , 6 W being the uncertainty of the energy barrier. The probability of this limitation is

1 - l/,e-Oaw (23)

IV. Quantum-MechanicalAnalysis of the Proton Transfer IV.1. Proton-Transfer Mode. The results of the preceding section have confirmed in a quantitative way the relevance of proton tunneling in anomalous conduction. However, several questions remain to be answered in order to better clarify the detailed mechanism of proton transfer. A first point is to establish whether the energetic barrier separates (i) two narrow wells or (ii) two almost free regions. Ab initio17i33-35 as well as ~ e m i e m p i r i c a lcom~~*~~ putations suggest the existence of a well-defined double well for interoxygen distances in the range of interest. These results, however, do not solve the problem because the energy barrier at hand is established before each proton transfer and destroyed after the transfer by the slow motion of the carrier ions (Figures 3 and 4). If proton vibration within the carrier ions can be neglected, the problem reduces to the consideration of an ingoing and outgoing channel corresponding to weakly bounded states (approach of the molecule A at time 0 and removal of the molecule D at time 3 (Figures 3 and 4)) separated by an isolated barrier. This hypothesis, which corresponds to case ii, has been analyzed extensively by Christov.6-8 On the other hand, if the vibrational energy of the proton within the carrier ion plays a relevant role, we fall into case i, and this demands a detailed analysis of the double well. Obviously, each description has a range of application: case i if the time for proton transfer is shorter than the mean life of the supermolecule H302-(or H502+)and case ii if the transfer time is longer than the supermolecule lifetime. In the present study we shall be concerned mainly with case i and shall only briefly comment on case ii at the end of this section. IV.2. One-Dimensional Proton Transfer. As a fiist step we have examined in detail the energy barrier for the proton transfer in the H302-complex computing the supermolecule energy vs. the position of the proton along the 0-0 line. The other geometrical parameters have been

4442

The Journal of Physical chemistry. Vol. 86, No. 22, 1982

Laforgue et al.

t

dE I Kcal mol -'I

K I L O (ECKART)

\ 4-310

Flgwe 5. Potential energy curve for proton transfer in the H,Op- supermolecule as a function of the 0,-H distance by KILO and ab initio (STO-3G and 4 4 1 G basis sets)methods. The oxygen-oxygen distance is fixed at I , = 2.815 A. The cusp of the KILO curve at d(0,-H) = d(0,-H) (dotted line) is eliminated by an Eckart function approximation.

discussed in the Introduction and the computations have been performed by nonempirical (standard STO-3GS and 4-31G39 basis sets of the GAUSSIAN/76 package40) and P C I L O ~methods. ~ It is well-known that the PCILO method breaks down when applied to symmetric many-center bonds such as the 0-H-O bond with the proton equidistant from 01,02.41 On the other hand, the results from nonsymmetric situations can be expected to be quite s a t i s f a c t ~ r y .In ~ ~order ~ ~ ~to eliminate the spurious cusp from the PCILO results, we have fitted the potential energy curve by an Eckart function. This kind of function has been largely used in quantum and fits very well the PCILO results up to about 0.1 A from the bridge center. The results obtained by the different methods are reported in Figure 5 and Table I. The STO-3G basis set seems unable to provide reliable energy barriers as is also the case with several other hydrogen bridges.46 On the other hand, the 431G basis set and the PCILO method agree in forecasting significant energy barriers, even if the PCILO barrier is quite higher (15 vs. 10 kcal/mol). In order to verify the assumption that ma+ N AE; (see section 111.31, we have preformed some test computations also for the H502+ molecule at the 4-31G level. The results reported in Table I indicate that the difference between AEa+ and hE; is less than 10%. Furthermore, AE; is slightly higher (40)J. A. Pople et al., QCPE, 12,368 (1980). (41)J. P. Malrieu in "Modern Theoretical Chemistry",Vol. VII, G. A. Segal, Ed., Plenum Press, New York, 1977,p 69,and references therein. (42)(a) R. Lochmann and T. Weller, Znt. J.Quantum Chem., 10,909 (1976);(b) R. Lochmann, ibid., 12,841 (1977). (43)V.Barone, F.Lelj, and N. Russo, Mol. Pharmacol., 18,331 (1980). (44)S.G.Christov, Ann. Phys., 12,20 (1963). (45)S.G.Christov, Bulg. Acad. Sci. Comm. Dept. Chim., 4,19(1971). (46)Y.C. Tse, M. D. Newton, and L. C. Allen, Chem. Phys. Lett., 75, 350 (1980). (47)T. H. Dunning, Jr., and P. J. Hay in "Modern Theoretical Chemistry", Vol. 111, H. F. Schaefer, 111, Ed., Plenum Press, New York, 1977,p 1.

TABLE I : Ab Initio Computations for Proton Transfer in H,O; or H,O,+Supermolecules" ( A ) H,O;

d-

STO-3Gb

4-31GC

PCILO~

(01-

H)e

energyf

0.907 1.007 1.107 1.207 1.307 1.407

-149.06234 -149.08335 -149.08807 -149.08641 -149.08377 -149.08266

A E ~ 16.29 3.11 0.15 1.19 2.84 3.54

energyf

-151.18405 -151.19329 -151.19065 -151.18446 -151.17925 -151.17729

A E ~ 6 09 0.29 1.94 5.83 9.09 10.32

8.06 0.24 4.30 11.61 17.97

( B ) H,O,+ STO-3Gh

d(O,-H)e

energyf

0.907 1.007 1.107 1.207 1.307 1.407

-150.31376 -150.33329 -150.33543 -150.33096 -150.32615 - 1 5 0 . 3 2 4 23

4-31G' AE'

13.66

1.41 0.06 2.87 5.89 7.09

--

energyf

a E'

-152.13067 -152.14089 -152.13890 -152.13307 -152.12801 - 1 5 2 . 1 2 6 10

6.63 0.22 1.47 5.13 8.30 9.50

The oxygen-oxygen distance is always 2 . 8 1 5 A . = -149.08830. d(0,Heq) = 1 . 0 3 5 A ; E , , = - 1 5 1 . 1 9 3 7 5 . cl(0,-He,) = In a t o m i c units. g In kcal/ 1 . 0 9 2 f~ e In angstroms. mol. d ( 0 , - H e , ) = 1 . 0 8 9 A ;E,, = - 1 5 0 . 3 3 5 5 4 . d ( O l - H e q ) = 1 . 0 2 4 .A;Eeq= - 1 5 2 . 1 4 1 2 4 . a

' d(O,-Heq) = 1 . 1 3 3 A ; E,,

than AE,+and this result is in agreement with the finding of section 111.3 that anomalous conduction is faster for acids than for bases. Another interesting point is the modification of the barrier height as the cluster of water molecules increases in size by lengthening and branching. This study has been performed only by the PCILO method in view of its rapidity and substantial agreement with ab initio computations. For the complexes HZnMlOn-, our re-

The Journal of Physical Chemistry, Vol. 86, No. 22, 7982 4443

Conduction in Acid and Base Solutions

TABLE 11: Energies of the Successive Quantum Levels for the One-Dimensional Proton Transfer energy, e V

Es1

%:

PCILO

4-31G

H,O,+4 - 3 1 G

0.23 1 0.237 0.589 0.709 0.358

0.165 0.166 0.408 0.448 0.243

0.153 0.155 0.386 0.434 0.233

sults indicate that, when the hydrogen bridge is a terminal one, addition of a chain element to the proton-donor molecule increases the barrier height of about 0.2 kcal/mol for nonbranched chains and of about 0.5 kcal/mol for branched chains. The shapes of the different energy c w e s remain the same. The results of Table I allow one to determine the energy levels resulting from the one-dimensional motion of the proton in the double well. We have followed the method reported in a recent commun i c a t i ~ nfitting ~ ~ the potential energy curves by a polynomial Lagrange approximation of fourth degree. The energy of the successive quantum levels obtained by ab initio and PCILO methods are reported in Table 11. Es, correspondsto the minimum vibrational energy and is very close to the value estimated by Conway.12 The first excited symmetric level being very close to the top of the barrier, we assume that the thermal excitation So S1 corresponds to the thermal crossover, i.e.

(A )

a

-

m

a

1

2

I 1

4

o,H(A)

Es,- Es,

I t is gratifying that the 4-31G results for both acids and bases are very near to the experimental value AE, = 0.26 eV, notwithstanding the approximated theoretical treatment. IV.3. Potential Energy Surface for Proton Transfer. The value lo of the transfer distance is, of course, a sort of equilibrium value. Actually oxygens are mobile and their vibrations are necessarily coupled to the vibrations of the proton. In order to discuss this coupling and its role in the overall behavior of the hydrogen bridge, one must consider potential energy surfaces. We have built an energy map for the H302-supermolecule as a function of the two distances d(Ol-H) and d(02-H) of the proton from the two oxygens by both PCILO and ab initio (4-31G basis set) methods. The maps presented in Figure 6 have been obtained from sections defined by d(Ol-H) + d(02-H) = d(01-02) = constant. It can be seen that PCILO and ab initio methods give very similar results. In particular both methods agree in suggesting that the potential energy barrier increases with the distanct d(01-02). The absolute minimum J corresponds in both cases to a very short interoxygen distance (d(01-02) = 2.4 A). However, it is well-known that in polar liquids separation of charges is much easier than when considering the isolated supermolecule (as in our computations) and the self-dissolution principle suggests that the true minimum of potential energy in water should take place near I. Both surfaces reported in Figure 6 are consequently much distorted and will be used only for a qualitative discussion. If the masses of the oxygens are considered very large with respect to the mass of the proton, the point representing our system remains on the line IBF (d(Ol-0,) = constant). In the other limiting case (i.e., when the system undergoes a very slow transformation) the representative point would pass through a saddle point A; that process involves just a very weak potential energy barrier. The (48) P. Guerin and A. Laforgue, DECIS, Montpellier, J. Chim. Phys.

Phys.-Chim. B i d , 78,144 (1980).

b

Figure 6. Potential energy surface for the H302-supermolecule as a function of H,-H and 02-H distances: (a) KILO; (b) 4-31G.

actual situation will be intermediate between the limiting models just considered, and probably much closer to the first one, due to both self-dissolution principle and the large mass of the oxygen atom. Let us consider a classical jump. In the internal coordinates of the supermolecule, the proton H covers the distance between its initial site HI to the vacancy V d(H1-V) = 10 - 2d(01-HI) = 0.815 A

The Journal of Physical Chemistry, Vol. 86, No. 22, 1982

4444

However, we have made the assumption that during the proton jump the carrier ion is free and its center of mass moves only very slowly in the water medium. Accordingly, proton and oxygen atoms move in opposite directions so that the momentum of the oxygen jumps is equal to that of the proton jump (conservation of the center of mass of the carrier ion), i.e. 2MoARo = MH[d(HlV) - ARo] where ARo is the displacement of oxygen atoms and Mo and MH are the masses of oxygen and hydrogen. It follows that the “true” transfer distance for proton jump becomes l*o = 1, - A R O = 2.79 A = 11 More realistically the trajectory of the system can be assumed to stay in the region defined by the broken line ICDEF of Figure 6. The O2 jump is represented by IC, the H jump by CDE and the O1jump by EF. In fact the three jumps occur almost simultaneously and the broken line should be replaced by the curve IDF; anyway the point D is well determined by the crossing of CE and AB. In conclusion the thermal excitation from I to D is represented by a nonvertical line corresponding to the motion of the oxygen coupled with the proton transfer; it is realized by successive jumps of the three atoms of the carrier ion, which contribute to the propagation of the Grotthus mechanism. IV.4. Propagation of the Proton Wave. If the lifetime of the supermolecule H302-(or H502+)is long enough, the hopping of the proton back and forth between the two wells causes the separation of the energy levels 0, 1, ... of each isolated well into doublets So,&;Sl,Al; ..., which we have computed in section IV.2. The corresponding time delay for the crossover of the barrier by tunneling is 70

= ‘/,h/(Es,- E&) = 0.37 X lo-’’ s

and increases sharply with the transfer distance d(01-02) (for instance, at d(O1-o2) = 2.90 A, 7 0 = 0.45 X 10-12s). On the other hand, according to computer simulations, the lifetime of a hydrogen bond in liquid water is also in the neighborhood of s26127and this strongly limits the number of possible proton jumps within the carrier ion. In order to gain further insight into this problem, it is necessary to consider the dynamical behavior of the carrier ion. The system to be studied can be reduced to a linear chain consisting of two oxygen atoms 01,02 (with coordinates ~ 1 , and ~ 3 masses Mo) separated by a hydrogen atom H (with coordinate x 2 and mass MH). The potential energy of this system (internal degrees of freedom) can be determined as a function of the 01-H (Q1coordinate) and 01-O2 (Q2 coordinate) distances. This choice of the coordinates is not the most elegant but is physically satisfactory, because the first coordinate will then describe the motion of hydrogen in the field of the oxygen atoms kept at a fixed distance, and the second coordinate will describe the motion of the two oxygens one with respect to the other. As the potential energy is assumed to depend only on the internal coordinates, the center-of-mass motion is a free one and can be ignored because it separates out. Therefore, the total Hamiltonian for the internal motions of the system is5, eint

=

(49) H. S. Johnston, ‘GasPhase Reaction Rate Theory” Ronald Press, New York, 1966, Appendix C. (50) V. Barone and G. Del Re, to be submitted for publication.

Laforgue et al.

with With this expression for the Hamiltonian it is possible to solve numerically the time-dependent Schroedinger equation and to study the evolution of the system at hand.51*52Moreover, this new point of view can be used to explain the shortening of the 0-0 bond, for the interaction of the tunneling proton with its environment is in a sense a “negative dielectric f r i ~ t i o n ” . ’ ~Of~ course, ? ~ ~ the resulting shortening of the 0-0 distance is limited by “normal friction’! of the oxygen atoms with the surrounding medium, so that the equilibrium is reached for the observed value ARo = O.Oz5 A. As outlined in section IV.3., our potential energy surface V(Ql,Qz)is very distorted because it does not include the solvent effect; 54 hence, in the present context we consider the limiting case in which the Hamiltonian for the internal motion reduces to that of a hydrogen atom moving in the field of two oxygen atoms fixed at their “equilibrium” distance lo, i.e.

The corresponding time-dependent Schroedinger equation has been solved numerically by the method of ref 51. The method demands the specification of the wave function at time t = 0; we have chosen the first eigenfunction of the perturbed harmonic oscillator55corresponding to one well of the H302-molecule at very long interoxygen separation. At time t the probability of finding the proton between x i and xj (e.g., in the second well) is given by

1;

Q*(Ql,t) W Q d ) dQ1

and the integral is evaluated numerically by a Simpson discrete integration. The evolution of the wave function is shown in Figure 7. At time t = 0 the wave is completely localized in the first well. The quantum penetration begins at t > 10 and is accompanied by a partial reflexion, which leads to a deformation of the wave packet (t = 30). This is followed by the progress of the wave under the barrier and the flowing of the wave in the second well. Finally, the wave is reflected in the second well and flows back toward the first well; this gives rise to the interference effects which are apparent in Figure 7. A t time t = 150 (i.e., N 3 X s) the probability of finding the proton in the second well is 1176, in good agreement with the estimate of the tunneling time delay computed at the beginning of the paragraph. It must be pointed out that, if enough time is given to the hydrogen bridge, its properties will correspond to a situation where the proton is equally divided between the two oxygen atoms. This is an almost trivial conclusion; what really matters is that the time needed for a significant proton tunneling ( 10-13-10-14s) is such that an initial state where the proton is localized near an oxygen atom cannot survive for a time comparable to the lifetime of the supermolecule H302-or H502+( N IO-’’ s). Proton transfer is characterized by both spatial symmetry and time-reversal symmetry. Thus, Figure 7 can (51) E. A. McCullough, Jr., and R. E. Wyatt, J. Chem. Phys., 54,3578 (1971). (52) G. Del Re and A. Lami, Bull. SOC.Chim. Belg., 85, 995 (1976). (53) R. W. Zwanzig, J . Chem. Phys., 52, 67 (1970). (54) A. Warshel, J . Phys. Chem., 83, 1640 (1979). (55) C. Cohen Tanoudji, B. Diu, and F. Laluc, “Mechanique Quantique”, Val. 2, Hermann, Paris, 1973, p 1103.

The Journal of Physical Chemisrry, Vol. 86, No. 22, 1982 4445

Conduction in Acid and Base Solutions

*T

a

0

b Figure 8. Two limiting models for the transfer process: (a) The supermolecule lifetime is greater than the transfer time; (b) the supermolecule lifetime is shorter than the transfer time.

Figure 7. Time evolution of a proton wave which at t = 0 is localized in the first well. Each curve represents \k'(Q ,) \k(Q ,) as a function of Q , after the time marked near the curve. The oxygen-oxy en distance is fixed at I , = 2.815 A and the time unit is 2.15 X 10- s.

B

equally well represent the propagation of the wave associated with a proton vacancy (in the reverse direction) or the final steps of the barrier crossing by a proton initially localized in the second well (in the reverse time order). Finally, it should be noted that the quantum chemical calculations reported here neglect the proton kinetic energy, which can stabilize the 0-0 bond in length, direction, and symmetry. IV.5. Coupling between Tunnel and Thermal Effect and Temperature-Dependent Anomalies in Water. As a last point let us briefly comment on the implications of hypothesis ii of section IV.l. Figure 8 shows the two limiting representations, following the conduction path, of the transfer process. In case A the lifetime of the ionized dimer is longer than the transfer time; there are undoubtedly quantized levels and two distinct processes with or without thermal excitation. We have treated only this case, which is consistent with a mean lifetime of the supermolecule larger than the jump time. However, a nonnegligible number of carrier ions could have lifetimes shorter than the transfer time so that tunneling and thermal jumps occur simultaneously (case B of Figure 8). In this case the negative energy of tunneling and the thermal activation energy are of given sum (the height of the potential barrier) and the two phenomena are coupled. This process does not imply symmetry but excludes concerted thermal jumps, unless there are many supermoleculesof very short lifetime. We have neglected this process (estimated of low probability) in the partitioning of the mobilities; now we go on to discuss it in detail, by treating the barrier of Figure 5 according to Christov's theory.- We find a characteristic temperature TK (Le., the temperature for which thermal and tunneling transfers are equiprobable) of 2970.1 K. The

high value of TK implies that in water tunneling is largely dominant with respect to thermal transfer and this is in agreement with the computed value of ' Q (section 111.3). From the value of TK Christov's theory allows an approximate evaluation of the "tunneling correction" as a function of the absolute temperature T. At our standard temperature (22.5 "C) T K / T N 10 and we are beyond question in the range of very large tunneling, although the precise value of T K is quite sensitive to the introduced approximations. Some expressions suggest that there are peaks in physical observablesX ( T ) for ~ e r t a i n ~narrow v~~?~ (few tenths of a degree) temperature ranges around T,, = TK/2n,where n is an integer (e.g., for n = 5, T5 = 297 K and for n = 4, T4 = 371.3 K). As a matter of fact, anomalies of the properties of liquid water in very narrow temperature ranges have been described; for instance, HastedM reports an anomaly of the imaginary part of the dielectric constant in a very sharp interval around 296 K.

Conclusion In the present study we have been concerned with the mechanism of conduction in acid and base solutions. We have proposed a general model based on previous wellestablished evidence, which seems apt to provide a coherent explication of the large amount of experimental data collected on normal and anomalous conduction. The model has been further analyzed by performing quantum-mechanical computations based on the experimental geometry of water and well-known chemicophysical properties (PH,POH,ma, 4,). The principle of reciprocity between proton and proton vacancy allows one to explain the similarity in conductivity of acid and base solutions. The two-oxygen molecule ion (H302-or H502+)appears to be the essential structure responsible for both normal and anomalous conduction. In fact, this molecule ion probably has a lifetime allowing several proton exchanges (anomalous conduction) and is destroyed at each step by the formation of another identical ion (normal conduction). Our model also stresses the importance of both tunneling and thermal concerted jumps in the anomalous conduction and provides a sounder physical interpretation of the Grotthus mechanism. Possible refinements of the model involve the replacement of the self-dissolution principle by explicit consideration of large supermolecules and inclusion of solvent (56)J. B. Hasted, "The Physics and Physical Chemistry of Water",F. Franks, Ed., Plenum Press, New York, 1972, p 289.

4448

J. Phys. Chem. 1982, 86,4446-4450

e f f e ~ t ~ which , ~would ~ ~ allow ~ , the ~ computation ~ ~ ~ ~ ~of~a ~ more reliable potential energy surface for proton transfer. Such improvements are certainly desirable but are in no way essential as far as the scope of the present model study goes, for the results reported here (e.g., AE,and the tunneling time) already provide a sufficient support for the (57) 0. Tapia, F. Sussman, and E. Poulain, J. Theor. Biol., 71, 49 (1978). (58) 0. Tapia, Theor. Chim. Acta, 47, 157 (1978).

proposed mechanism. On the other hand, further extension of the numerical analysis is mandatory for a more detailed understanding of the dynamics of proton transfer.

Acknowledgment. We acknowledge Dr. M. Ammar for many helpful discussions, Dr. Langlet for the use of her PCILO program, Professor S. G. Christov and Professor G. Barone for their useful comments, and Miss C. Minichino for help in the computations. Financial support of CNR (Rome) is gratefully acknowledged by the Italian authors.

Application of Positron Annihilation to the Characterization of Zeolites Hlroshl Nakanishi and Yusuke UJlhlra’ Faculty of Engineerlng, The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113, Japan (Received: November 17, 1981; In Flnal Form: May 21, 1982)

Positron annihilation lifetime and Doppler-broadening measurements were carried out for synthetic zeolite 13X, SK-40, NH,-X, and NH4-Y by varying the evacuation temperature in order to study the character of the zeolite cages. Four components of the positron annihilation, derived from the lifetime spectra, were interpreted from the results of the authors’ measurements and other studies on zeolites. The 0-Ps lifetimes in the cages became longer as the desorption of water molecules proceeded. It was found that some active groups in zeolites interacted with 0-Ps and reduced the 0-Ps lifetime after all the water molecules had detached. Br~nstedacid in the zeolite acted not only as an oxidizer but also as an inhibitor of Ps formation. An attempt was made to estimate the amount of Br~nstedacids by the positron lifetime technique. The longest lifetime of 50 ns indicates 0-Ps annihilation in a pore with 60 A free diameter, which seems to exist irregularly in the faujasite zeolites. It was also found that 0-Ps was oxidized in this large cavity.

Introduction When positrons, antiparticles of electrons, are emitted from neutron-deficient nuclei such as 22Nainto a condensed medium, they slow down to thermal energy within the order of picoseconds. A portion of them form the bound state of a positron and an electron pair called positronium (Ps). A thermal positron or Ps interacts with the surrounding material and annihilates, emitting two or three y quanta in less than a microsecond. The lifetimes of a positron and Ps reflect the character and states of the surrounding materials. In solids, they are very sensitive to structural irregularities such as lattice defects or micro void. Extensive research has been successfully carried out on metals and crystalline solids. The potentia1 capabiIities of positron measurements have not yet been fully applied to the studies of porous or amorphous materials. This is partly due to the complexity of the positron annihilation process in those substances and the difficulties in the analyses of the data obtained and partly due to the lack of systematic investigations in the field. Hitherto several reports have been published on the application of positron techniques to the studies of zeolites or fine Unfortunately there are certain inconsistencies among the data or their interpretations. In this work, several kinds of synthetic zeolites, whose structures were well-known, were selected as samples, and (1) Perkal, M. B.; Walters, W. B. J. Chem. Phys. 1970, 53, 190-8. (2) Levin, M. V.; Shantarovich, V. P.; Agievskii, D. A.; Landau, M. V.; Chukin, G. D. Kinet. Katal. 1977, 18, 1542-7. (3) Gol’danskii, V. I.; Mokrushin, A. D.; Tatur, A. 0.;Shantarovich, V. P. Appl. Phys. 1975, 5, 379-82. (4) Gol’danskii, V. I.; Mokrushin, A. D.; Tatur, A. 0.; Shantarovich, V. P.Kinet. Katal. 1972,13,961-8. (5) Paulin, R.; Ambrosino, G. J. Phys. 1968, 29, 263-70. 0022-365418212086-4446$0 1.2510

the states of the cations in the cages and the oxidation processes of 0-Ps were investigated by measuring both the lifetime and the Doppler-broadening of annihilation y rays. Sufficient experimental data have been provided to resolve the discrepancies found in the interpretation of the preceding experimental results. The usefulness of positron measurement in porous materials was also demonstrated.

Experimental Section Sample Preparations. The powders of Linde-type synthetic zeolite 13X and SK-40, whose unit-cell compositions are Naes[(A102)86(Si02)10,1.276H20 and Na56[(Al02)56(Si02)13s]*264H20, respectively, and the powders of NH4-X- and NH,-Y-type zeolites, which were obtained by exchanging Na+ of 13X or SK-40 to NH4+following the manual of “Ion Exchanging Procedures” specified by Union Carbide Co Ltd., were used in the experiments. The details of ion-exchange procedures are described in procedures 1-3. Procedure 1 . The hydrated zeolite powder (100 g on an anhydrous basis) was slurried in 1 L of 2.23 M NH4Cl aqueous solution and heated to reflux temperature while stirring for 2 h. Procedure 2. The exchanged powder was filtered while hot, washed with an equal weight of deionized water, and contacted again with fresh NH4C1solution. Procedure 3. After the final exchange, the molecular sieve was washed free of all soluble salts and dried a t 50 “C to a free-flowing powder. Procedures 1and 2 were repeated several times to obtain a high exchange ratio. The exchange ratio of zeolite 13X after each exchange process was estimated by measuring the amount of residual Na by thermal neutron activation-yray spectrometry, a 23Na(n,y)24 Na nuclear reaction 0 1982 American Chemical Society