J. Phys. Chem. 1986, 90, 2757-2761 parameters were compared with the experimental results. The smooth curve included in Figure 5 shows the calculated time profile of the compound absorption, A(C2H5) A(C2H502),obtained with the values e(C2H5) = 670 M-' cm-' and t(C2HSO2) = 1360 M-l cm-' and the rate constant kloa = 3.2 X lo9 M-' s-I. The close agreement between the model and the experimental results supports the spectral assignments as well as the kinetic results. The calculated yield of C2H5O2 at low oxygen pressures was found to be very sensitive to the value of k12. With a value of k12= 6.0 X 1O'O M-' s-l we obtained a good fit to the experimental yields and formation half-lives as shown in Figure 6. The decay of C2Hs02followed simple second-order kinetics, and the calculated rate constant k13= 3.15 X lo7 M-' s-I in fair agreement with the results of previous studies, (6.0 f 0.6) X lo7 M-l s-' lo and (3.5 f 0.3) X lo7 M-I s-'." Using the same procedures, we studied the addition reaction (loa) in the range of 298-400 K, and the results are collated in Table 111. When presented in terms of an Arrhenius equation, the results show an apparent small negative activation energy, log k l o a / M 1s-' = (8.90 f 0.04) + (840 f 70 cal mol-')/2.303RT, where the error limits represent the standard deviations. Prior to this study, there were no direct measurements of the rate constant klh. Early, indirect estimates of the rate constant klh, derived from complex reaction systems, lie in the range from 6 X lo7 M-' s-' at 308 K and 47 mbar22to 1.2 X lo9 M-' 6' at 348-575 K and 6 mbar23and 4.2 X lo9 M-' s-* at 295 K and pressures of 6-137 mbar.24 Recently, very
+
(22) J. E. Jolly, J . Am. Chem. Soc., 79, 1537 (1957). (23) I. I. Avremenko and R. V. Kolesnikova, Izu. Akad. Nauk SSSR, Otd. Khim. Nauk, 806, 989 (1960).
2757
+
extensive studies of the overall rate constant klo = kloa klobfor the C2H5 O2reactions and their branching fractions have been performed by Slagle and Gutmanz5 and by Plumb and Ryan.26 In their studies the C2H5 radical decay was monitored directly by photoionization mass spectrometry, and special attention was given to obtain the pressure dependence in the falloff region, 0.5-10 mbar, of k1025and k10a.26If we take into account that at room temperature the fraction of reaction 10 producing C2H4and H 0 2 is small (ca. 5%)24*26and that this reaction is independent of the pressure,26 then our value of klOaat room temperature may be taken as the high-pressure limit of klo = klo,. The value of klOa = 3.2 X lo9 M-' s-l obtained in this study at 298 K and 1 atm of H2is in agreement with that of 2.7 X lo9 M-I s-l at 295 K and 1 atm of H e predicted by Plumb and Ryan26on the basis of their experimental data at low pressures and Troe's theory.27
+
Acknowledgment. We thank H. Egsgaard for carrying out the G C analysis and P. L. Genske for his technical assistance. We are also grateful to D. Gutman and I. R. Slagle for providing a preprint of their work. Finally, we acknowledge many valuable discussions with our highly esteemed colleague, the late Gosta Nilsson. Registry No. H, 12385-13-6; H2, 1333-74-0; C2H4, 74-85-1; 0 2 , 7782-44-7; C2H5, 2025-56-1; C2H5O2, 3170-61-4. (24) D. P. Dingledy and J. G. Calvert, J . Am. Chem. Soc., 85,856 (1963). (25) I. R. Slagle, Q. Feng, and D. Gutman, J . Phys. Chem., 88, 3648 (1984). (26) I. C. Plumb and K. A. Ryan, Int. J . Chem. Kinet., 13,1011 (1981). (27) J. Troe, J. Phys. Chem., 83, 114 (1979).
Intercalation Kinetlcs of Alkali-Metal Ions Into y-Zirconium Phosphate Using the Pressure-Jump Technique Naoki Mikami, Minoru Sasaki, Naofumi Kawamura, Kim F. Hayes,' and Tatsuya Yasunaga* Department of Chemistry, Faculty of Science, Hiroshima University, Hiroshima 730, Japan (Received: March 26, 1985)
The intercalation kinetics of a series of alkali-metal ions, Li', Na', K', Rb', and Cs', into y-zirconium phosphate (y-ZrP) were studied by using the pressure-jump technique with conductivity detection. In aqueous suspensions of y-ZrP containing alkali-metal ions, double relaxations of the order of milliseconds were observed under the experimental condition that the mol dm-I. Both amount of alkali-metal ion adsorbed is more than half of the ion-exchange capacity of y-ZrP, 3.12 X fast and slow relaxation times increase with the concentration of alkali-metal ion. From the kinetic and static results obtained, the fast and slow relaxations were attributed to the entering and interlayer diffusion processes of alkali-metal ions in y-ZrP, respectively. The forward rate constants for these processes are closely related to the interlayer distance of alkali-metal intercalation compounds of y-ZrP and are smaller than those reported in the a-ZrP-alkali-metal ions ,system. The difference between the rate constants for a-and y-ZrP could be interpreted by the difference between the acidities of the phosphate groups in a-and y-ZrP.
Introduction Zirconium phosphate, Zr(HP04)2.nH20, has several layered crystalline structures such as Zr(HPO,),.H,O (a-ZrP) and Zr(HP04)2.2H20 (y-ZrP) and forms various intercalation compounds as a result of intercalation of guest donors into its int e r l a y e r ~ . ~ In - ~ particular, y Z r P has interesting characteristics (1) Japanese Ministry of Education Research Scholar. Permanent address: Environmental Engineering and Science, Department of Civil Engineering, Stanford University, Stanford, CA 94305. (2) Clearfield, A.; Blessing, R. H.; Stynes, J. A. J . Inorg. Nucl. Chem. 1968,30, 2249. (3) Clearfield, A.; Smith, G. D. Inorg. Chem. 1969,8,431.
compared with a-ZrP; the y-form of the crystal not only has a relatively long interlayer distance of 12.3 8, but also a densely linked layered structure, and the phosphate groups in the compound exhibits strong a ~ i d i t y . ~The . ~ study of intercalation dy(4) Clearfield, A.; Duax, W. L.; Medina, A. S.; Smith, G. D.; Thomas, J. R. J. Phys. Chem. 1969,73, 3424. (5) Clearfield, A.; Garces, J. M. J . Inorg. Nucl. Chem. 1979,41, 879. (6) Alberti, G.Acc. Chem. Res. 1978,11, 163. Costantino, U. Intercalation Chemistry; Whittingham, M. (7) Alberty, G.; S., Jacobson, A. J., Eds.;Academic Press: New York, 1982; Chapter 5, p 147. (8) Yamanaka, S.; Tanaka, M. J. Inorg. Nucl. Chem. 1979,41, 45. (9) Sasaki, M.; Mikami, N.; Ikeda, T.; Hachiya, K.; Yasunaga, T. J. Phys. Chem. 1982,815,5230.
0022-3654/86/2090-2757$01 .50/0 0 1986 American Chemical Society
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The Journal of Physical Chemistry, Vol. 90, No. 12, 1986 I
I
Mikami et al. 1
I
I
I
1
100
-
v1
G
0
0.5
1 .o
50
time, s Figure 1. Typical double relaxation curve in the y-ZrP-M+ system observed by the pressure-jumptechnique with conductivity detection at Cp = 10 g dm-3 and 25 "C.
namics for a- and y-ZrP is necessary to attain a fundamental and structural understanding of the nature of the selectivity in the guest intercalation. Recently, the mechanisms for protonation-deprotonation of phosphate groups and proton intercalation-deintercalation in aand y-ZrP were proposed by the present author^.^ For various cation intercalations, however, it is necessary to perform further extensive kinetic studies and to clarify the relationship between the dynamic intercalation properties of the cation into ZrP and the layered structures. In a preceding paper,I0 the mechanism for the intercalation-deintercalation of alkali-metal ions, M+, into a-ZrP has been proposed. The purpose of the present study is to clarify the mechanism of intercalation-deintercalation of alkali-metal ions in y-ZrP by using the pressure-jump technique and to discuss the difference of the intercalation rate constants between a- and y-ZrP in light of their crystal structures.
0
5 6 7 CM'L ,lO'*mol dm-3 Figure 2. Dependences of the fast reciprocal relaxation time on the concentrationsof added alkali-metal ions in aqueous suspensions of y-ZrP 4
at Cp = 10 g dm-3 and 25 OC.
Experimental Section
The y-ZrP sample used was supplied by Daiichi Kigenso Chemical Industries Ltd. The X-ray powder diffraction pattern of this sample was the same as that reported,s showing that the sample has a layered crystalline structure. Scanning electron microscopic examination confirmed that the particle size of y-ZrP was 4.7 f 1.1 Km and approximately uniform. Reagent grade alkali-metal hydroxides were used without further purification. The supernatant solution of the suspension was separated by contrifugation for 1 h at 30000g, and then the bulk concentrations of alkali-metal ions were measured by an isotachophoresis apparatus (Shimazu Ind. Corp., IP-2A type).]' The pressure-jump apparatus with conductivity detection used was described previously in detail.I2 The kinetic measurements were carried out on samples with particle concentrations of C, = 10 g dm-3 at 25 O C under a nitrogen atmosphere. The {-potential of the y-ZrP particles was measured by the microelectrophoresis method. Results and Discussion
Figure 1 shows a typical double relaxation curve observed in an aqueous suspension of y-ZrP containing alkali-metal hydroxides, MOH, obtained by using the pressure-jump technique with conductivity detection at 25 OC,where the direction of the relaxation signal indicates a decrease in the conductivity of the suspension during relaxation. Relaxation curves were only observed at relatively high adsorption densities, where at least 3.12 X mol of M + were adsorbed per gram of y-ZrP. No relaxation was observed in the supernatant solutions of y-ZrP-M+ suspensions or in M O H solutions. On the other hand, it is well-known that M+ is intercalated into y-ZrP.2,5,7As mentioned (10) Mikami, N.; Sasaki, M.; Yasunaga, T.; Hayes, K. F. J. Phys. Chem. 1984, 88, 3229. ( 1 1) Sasaki, M.; Negishi, H.; Ohuchi, H.; Inoue, M.; Yasunaga, T. J . Phys. Chem. 1985, 89, 1970. (12) Hachiya, K.; Ashida, M.; Sasaki, M.; Kan, H.; Inoue, T.; Yasunaga, T.J. Phys. Chem. 1979. 83, 1866.
4
I
1
5
6
7
CM'L ,10-2mol dm-3 Figure 3. Dependences of the slow reciprocal relaxation on the concentrations of added alkali-metal ions in aqueous suspensions of y-ZrP at C, = 10 g dm-l and 25 'C.
in the previous paper,I0 the following two types of ion-exchange processes for y-ZrP occur: (i) first an ion-exchange stage at lower coverage Z I - ( H P O ~+ ) ~M+
Zr(HP04)(MP04) + H+
(ii) second an ion-exchange stage at higher coverage Zr(HP04)(MP04) + M+ === Zr(MPO&
+ H+
The above facts suggest that the relaxations may be attributed to the intercalation-deintercalation of M+ in the interlayer of y-ZrP during the second stage ion-exchange process. Figures 2 and 3 show the dependences of the fast and slow reciprocal relaxation times, T { I and T ; ~ , on the concentrations of added alkali-metal ions, [M'],, respectively. As can be seen from these figures, the values of both 7r1and 7c1decrease with increasing the concentration of added M+. These tendencies are similar to those reported in the a-ZrP-M+ system.l0 However, the magnitudes of both relaxation times, 7f1 and T [ I , are in the decreasing order of Cs+ > Rb+ > K+ > Li+ > Na+, in contrast to the order, Cs+ > Li+ > Rb+ > Na+ > K+, observed in the a-ZrP-M+ system.I0 This fact suggests that the selectivity of y-ZrP to alkali-metal ions in the intercalation-deintercalation process may be different from that of a-ZrP. The dependences of the pH of suspensions on the concentrations of added alkali-metal ions are shown in Figure 4. In this figure, each curve exhibits two plateaus, in agreement with those reported by Clearfield et aL2s5 The steep increase between two stages corresponds to the transition between the first and second ionexchange stages. The amounts of alkali-metal ions adsorbed, [M+),d,, were measured and are plotted as a function of the bulk
The Journal of Physical Chemistry, Vol. 90, No. 12, 1986 2759
Intercalation of Alkali-Metal Ions into y Z r P I
I
Ot 9
I
I
I
'
> -20 E w -40
-
-60 I
1
I
I
0
2
4
6
C M+lp ,
5
0
10
15
CM'I ,103mol6nW3 Figure 4. Dependences of the pH on the concentrations of added alkali-metal ions in aqueous suspensions of y-ZrP at C, = 10 g dm-3 and 25
TABLE I: Overall Equilibrium Constants of Ion Exchange of Alkali-Metal Ion for H+in y-ZrP at 25 O C
OC.
12
I
I
lo-*mol d m 3
Figure 6. Dependences of the {-potential on the concentrations of added alkali-metal ions in aqueous suspensions of y-ZrP.
alkali-metal ion Li+ Na+
I
K+ Rb+ cs+
Rb' CS'
102KT
10BKil 370 5.6 18 14 21
2.5 23 67 27 11
mechanism involving the entering of M', the diffusion of M+, and the releasing of H+:
8 I
SH
0.
-+ - i!2- (SHM+)* T (SHM+)
M+
4
SM
(I)
H+
(step 1. K , )
(step 2. K,)
(step 3. K,)
with
0
4
2 CM'Ia
,
6
8
mol dm-3
K2 = [(SHM')*]
/ [(SHM')]
Figure 5. Adsorption isotherms of the alkali-metal ions in aqueous suspensions of y - Z r P a t C, = 10 g dm-3 and 25 O C .
concentration of alkali-metal ions, [M'], as shown in Figure 5 . The plateau region of the curves in this figure corresponds to the transition shown in Figure 4. As described above, the value of mol g-' corresponds well with the onset [M+],& = 3.12 X of the second ion-exchange process in which the relaxation signals are observed. On the other hand, it has been reported that the hydrolysis reaction also occurs for large cations such as Rb+ and Cs' a t a higher pH range and then the ion-exchange capacities d e c r e a ~ e .As ~ can be seen from Figure 5 , however, such an effect for Rb' and Cs' was negligibly small in the present system; the hydrolysis reaction of the zirconium phosphate should be extremely slow ranging from 1 week to 1 month and thus agree with the results reported from aging of the samples. This fact suggests that hydrolysis does not occur in the present system. Meanwhile, the M+concentration dependences of {-potential were measured and the results are shown in Figure 6 . As can be seen from this figure, the value of {-potential for each alkali-metal ion decreases steeply in the first ion-exchange stage, and very slowly in the second ion-exchange stage. In addition, the effect of cations on the {-potential becomes smaller in the second ion-exchange stage. This fact also indicates that hydrolysis does not occur. The negative value of the {-potential may result from the negatively charged phosphate groups on the surface of y-ZrP. According to the mechanism for the intercalation-deintercalation of alkali-metal ion in a-ZrP reported previously,1° for each ion-exchange stage described above, let us consider the following
where the symbols SH, (SHM+), (SHM+)*, and S M denote the phosphate group, the adsorbed state of M' at the entrance of the interlayer, the deeply intercalated state of M+ in the interlayer, and the adsorbed state of M' on the phosphate group, respectively, \E, represents the electrostatic potential,'0,'',13,'4 the superscript int denotes intrinsic, the subscript s refers to the surface, and the other symbols have their customary meanings. The overall equilibrium constant, K, for mechanism I can be expressed aslo ([(SHM+)I + [(SHM+)*I + [SMI)[H+l PHI [M+l Kl([H+l + [H+IK2 + K2KA (1) The values of the overall equilibrium constants for the first and second ion-exchange stages, KI and KII, determined statically using eq 1 from the adsorption isotherms for each stage are shown in Figure 5 and are listed in Table I. The orders of magnitudes of KI and KI,are approximately equal for all of the alkali-metal ions except for Li+. If the kinetic experimental results for H + intercalation into ZrP is taken into account, the amplitude of the relaxation for step 3 K=
(13) Davis, J. A.; Leckie, J. 0. J . Colloid Interface Sci. 1978, 67, 90. (14) Negishi, H.; Sasaki, M.; Yasunaga, T.; Inoue, M. J . Phys. Chem. 1984,88, 1455.
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The Journal of Physical Chemistry, Vol. 90, No. 12, 1986
Mikami et al.
TABLE 11: Rate Constants and Eauilibrium Constants for the Ion Exchange of Alkali-Metal Ion for H+Obtained Kineticallv in wZrP at 25 OC alkali-metal ion
Li+ Na+
K+ Rb'
cs+ 0
10-lk in1 I ? mol-l dm3 s-l 11 2.8 12 43 100
lo-%''nt, 1.7 2.4 5.0 2.5 3.0
0.5
1 .o
I
I
k2, s-l 22 4.8 23 21 48
s-1
1.5 I
10k+ SKI 6.0 8.5 8.0 12 10
Klint,mol-' dm3 6.2 1.2 2.3 17 33
K2 37 5.6 29 23 48
1010K31n1, mol dm-' 230 82 36 6.0 5.5
60 -
2
10
'Jl
-.
"0
l &L
2
-
40 -
c
'cn
u)
-
-
F
5
aJ
F I 0
0.4 F1 ,mol dm-3
0.8
0
0
0.25
0.5
Fz Figure 8. Plots of
Figure 7. Plots of rrl exp(-eq8/(2kBr)) vs. F1in eq 2.
vs. F2in eq 3.
T ; ~ I
I
I
I
can be considered to be negligibly small under the present higher pH condition. Thus, from the kinetic analysis based on mechanism I for the second ion-exchange stage, the following two cases can be considered: (1- 1)
step 1: slow,
step 2: fast
(1-2)
step 1: fast,
step 2: slow
For 1-1, the theoretical value for T F I should be constant, which contradicts the experimental results in Figure 2. Consequently, case 1-1 is excluded. For 1-2, the expressions of T Na+ > K+),'O in the TiS2-M+ system (Li' > Na+ > K+ > Cs+ > Rb+),11914and in the zeolite 4A-M+ system (K+ u Rb+ u Cs+).15 This fact indicates that y-ZrP has a characteristic selectivity to alkali-metal ions in the entering process, which is not (15) Ikeda, T.; Nakahara, J.; Sasaki, M.; Yasunaga, T. J . Colloid Interface Sci. 1984, 97, 278.
J. Phys. Chem. 1986, 90, 2761-2764 found in a-ZrP, TISz, or zeolite 4A. The values of log klintand log k2 are plotted against the interlayer distances of the second ion-exchange stage reported by Clearfield et al.,5 where the interlayer distance for Cs+ (16.5A) was determined by the present authors, and the results are shown in Figure 9a together with those reported in the a-ZrP-M+ system.1° Figure 9a shows that the plots for y-ZrP give reasonable linear correlations, as well as those for a-ZrP. Consequently, the above results indicate that the forward rate constants for the entering and the interlayer diffusion processes are controlled predominantly by the interlayer distance of y-ZrP. As can be seen from Figure 9a, furthermore, the intercalation rates of alkali-metal ions into y-ZrP are slower than those into a-ZrP in spite of the larger interlayer distances of y-ZrP compared to those of a-ZrP. On the other hand, it is reported that the acidity of the phosphate group in y-ZrP differs from that in a-ZrP? Let us consider the contribution of the acidity of ZrP, K,i"t, to the forward rate constants klintand kz as follows: log ki* = log ki - p K P
( i = 1, 2 )
where the values of pK,int used have been previously determined to be 5.78 for a-ZrP and 3.60 for y-ZrP.' The values of log (klht)*
2761
and log k2* obtained are also plotted against the interlayer distances, and the results are shown in Figure 9b. This figure shows clearly that both plots for a-and y-ZrP give same linear correlations, suggesting that the differences in the intercalation rates between a-and y-ZrP in Figure 9a results from the differences of the acidities between the two compounds. In conclusion, the correlations in Figure 9 may be attributed to the acid properties of the phosphate groups in ZrP and the layered structure of ZrP which varies its interlayer distance with the size of the cations and water content, in contrast to the layered structure of T i s z whose interlayer distance is governed mainly by the number of hydration water layers surrounding the alkali-metal ions and to the concrete crystalline cage structure of zeolite.I6
Acknowledgment. We thank Daiichi Kigenso Chemical Industries Ltd. for the y-ZrP sample. Registry No. Zr(HP0.J2, 13772-29-7; Lit, 17341-24-1; Na', 17341-25-2; , ' K 24203-36-9; Rb', 22537-38-8; CS', 18459-37-5. (16) Whittingham, M. S.; Jacobson, A. J. Intercalation Chemistry; Academic Press: New York, 1982; Chapters 4 and 10.
Determining Kinetic Parameters from Pulse Voltammetric Data John O'Dea, Janet Osteryoung,* Department of Chemistry, State University of New York at Buffalo, Buffalo, New York 14214
and Thomas Lanet MIT Statistics Center, Cambridge, Massachusetts 02139 (Received: April 8, 1985; In Final Form: November 11, 1985)
A nonlinear least-squares method of analyzing pulse voltammetric data suitable for use with a small laboratory computer is presented. The method, which is applicable to all forms of pulse voltammetry, is based upon a nonlinear simplex optimization of the parameters of the problem in a normalized space derived from linear regression of the measured current on a calculated dimensionless current function. Hence, in contrast with procedures employing working curves, no extraneous scaling factors or biases are required to compare models directly with experimental data. The method also provides an efficient computation of the confidence ellipsoid surrounding the optimal values of the kinetic parameters.
Voltammetric techniques are widely used to investigate mechanisms of chemical and electrochemical reactions. Advances in theory and improved computing power have made it possible and practical to use detailed mathematical models for even quite complex mechanisms. Theories have been developed for different types of voltammetric techniques, especially linear scan (or cyclic) voltammetry,1-2ac voltammetry employing the fast Fourier t r a n ~ f o r mand , ~ pulse ~oltammetry.43~Today calculation for even multistep or second-order processes is commonplace; the above references are illustrative rather than comprehensive. Our special interest has been in pulse voltammetry to which mathematical modeling based on numerical integration as popularized by Nicholson is ideally suited.6 A common feature of mathematical models for voltammetric currents is that they are formulated in terms of dimensionless parameters and yield a dimensionless current. It is this dimensionless current, $, which can be calculated readily for many cases, given dimensionless values of relevant parameters. Of course the problem facing the experimenter is the more difficult converse, that is, given an experimental voltammogram, to obtain from it the values of the parameters which characterize the response. Present address: IBM Cambridge Scientific Center, 101 Main Street, Cambridge, MA 02142.
0022-3654/86/2090-2761$01.50/0
Confronted with this problem, we sought a procedure which would apply to many types of pulse voltammetric data, i.e. to different forms of $. Once implemented, this one procedure should then be readily usable for different types of pulse voltammetry (normal pulse, staircase, square wave, etc.) and for many reaction mechanisms (slow electron transfer, preceding reaction, and so on). Secondly, we wished the procedure to run in near-real time on the laboratory computer used to control the experiment. This has the advantage that estimates of the parameters characterizing the model are obtained while the system is still under test. Thirdly, we were particularly interested in devising a procedure which would give an unbiased measure of the quality of the results. This (1) Nicholson, R. S.; Shain, I. Anal. Chem. 1964, 36, 706-723. (2) Imbeaux, J. C.; Saveant, J. M. J . Electroanal. Chem. 1973, 44, 169-187. ( 3 ) Schwall, R. J.; Bond, A. M.; Smith, D. G. Anal. Chem. 1977, 49, 1805-1 8 12. (4) Hanafey, M. K.; Scott, R. L.; Ridgway, T. H.; Reilly, C. N. Anal. Chem. 1978, 50, 116-137. (5) O'Dea, J. J.; Osteryoung, Janet; Osteryoung, R. A. Anal. Chem. 1981, -7-3, ew-7ni -- - .- - . (6) Nicholson, R. S.;Olmstead, M. L. In 'Electrochemistry: Calculations, Simulation and Instrumentation, Vol. 2, Mattson, J. S., Mark, H. B., MacDonald, H. C., Eds.; Marcel Dekker: New York, 1972.
0 1986 American Chemical Society
