Physical Significance of Transfer Activity Coefficients for Single Ions Orest Popovych Department of Chemistry, Brooklyn College of the City University of New York, Brooklyn, N. Y. 172 10
would therefore be 285.1, 262.1, and 250.1 kcal/mole, reH by spectively. If we chose to correlate the above ~ U values referring all three of them to the aqueous standard state (G'H' = -258.0), then according to Equation 1, the PUH of the ammonia solution would become 19.6 and that of the formic acid solution, -5.6 on the common aqueous scale. Thus, solutions of the same nominal pH in different solvents may represent vastly different acidities. A similar correlation could be achieved for standard potentials in the above solvents by choosing the aqueous SHE as the only reference point. Then, the SHE in ammonia would become negative by about one volt when referred to the aqueous S H E as the zero point. However, the correlations of emf series and activity scales described above require knowledge of the differences between the standard free energy of an ion of interest in , in nonaqueous solvents, b G ~ oThis . difwater, u G ~ oand ference can be viewed loosely as the free energy required to transfer the ion from the aqueous (or some other reference) standard state to the given nonaqueous standard state, hence the term transfer free energy, (i):
A summary is presented of the most striking examples where the transfer activity coefficients (medium effects) of single ions provide reasonable and useful interpretation of major changes in the solvation properties of ions upon their transfer between pairs of solvents. Key evidence in support of the validity of the tetraphenylborate assumption for the estimation of the transfer activity coefficients of single ions, ,,^/, is reviewed. From the parallel behavior of the AplCs of primary ammonium acids and of the medium effect for the proton, ,,,yH, in ethanol-water and methanol-water solvents, it is concluded that values of ,^/ for single ions estlmated by the tetraphenylborate assumption may be capable of reflecting even minor changes in solvation energy. A correlation is shown between the rate constant for the hydrogen-ion catalyzed mutarotation of a-glucose and ,,,yH in ethanol-water mixtures. Preliminary values of ,^/ for several single ions in methanol-water solvents estimated by the tetraphenylborate assumption are presented in graphical form.
Acto
Traditionally, the physicochemical properties of electrolytes were studied almost exclusively in aqueous solutions, so that there was little need for concern about the nature of the standard states for the solutes or the implications of the arbitrary reference point for the emf series represented by the aqueous standard hydrogen electrode (SHE). More recently, however, the widespread use of nonaqueous solvents in acid-base and electrochemistry has also given impetus to attempts of correlating emf and activity data determined in different solvents. I t then became obvious that the convention of referring electrode potentials in each solvent to the arbitrary zero of the S H E in the same solvent generates as many thermodynamically unrelated emf series as there are solvents and that referring activity scales to infinite dilution in each solvent produces an analogous multiplicity of unrelated scales. To illustrate the problem, let us compare solutions having conventional ~ U values H of 3.0 in water, liquid ammonia, and formic acid. From the definition of activity in terms of the Gibbs partial molal free energy of the proton in the given solution, &, (Le., a t pH 3.0) and in the standard state in the same solvent, @H", the ~ U isH given by: -
c," - G ,
(1)
= 2 . 3 RT Thus, the fact that in each of the above solutions the ~ u value is 3.0 means that the difference (GH" - G'H) is equal for all three solvents, its value a t 25 "C being 4.1 kcal/mole. However, it does not follow that the proton activities in these three solutions are equal, because the values of @ H o are likely to be very different for solvents of such diverse acid-base properties. In fact, Izmaylov ( I ) has estimated the values of (-G'H") in kcal/mole to be 281.0, 258.0, and 246.0 in ammonia, water, and formic acid, in that order, The corresponding values of CH for solutions of ~ U = H 3.0 (1) N A Izmaylov, Dokl Akad. Nauk SSSR. 126, 1033 (1959)
H
Accordingly, the corresponding activity coefficient, m ~ i has , been recently renamed "transfer activity coefficient," but it should be noted that it is identical with the "medium effect" of the earlier literature. The analytical uses of transfer activity coefficients of ions have been discussed by this author before (2-6). Of course, only transfer activity coefficients for uncharged molecules and electroneutral combinations of ions are accessible by rigorous thermodynamic methods. For single ions, they can only be estimated via extrathermodynamic assumptions. This author's critical evaluation of the existing methods for the estimation of single-ion transfer activity coefficients has been expressed first in a brief survey ( 3 ) , followed by two comprehensive reviews ( 4 , 6). Although estimates of the transfer free energy for a given ion may differ by several kilocalories/mole even among plausible assumptions, such discrepancies generally do not obscure the physical significance of very large values of ACto for individual ions. Izmaylov's ( I ) estimate (by extrapolation methods) that the free energy of the proton is lower (more negative) in ammonia than in water by 23 kcal/mole, which indicates that the proton is much more tightly solvated by ammonia than by water, is hardly surprising, but it is a datum that could not have been obtained from thermodynamics alone. Strehlow (7, 8) has repeatedly pointed out that whereas the transfer of alkali halides from water to ( 2 ) 0. Popovych, Anal. Chem., 38, 558 (1966). (3) 0. Popovych and A. J. Dill, Anal. Chem., 41, 456 (1969). (4) 0. Popovych, Crit. Rev. Anal. Chem., 1 , 73 (1970). (5) 0. Popovych, A. Gibofsky, and D. H. Berne, Anal. Chem., 44, 811 (1972). ( 6 ) 0. Popovych in "Treatise on Analytical Chemistry," I. M. Koithoff and P. J. Elving, Ed.. Wiley-lnterscience, New York, N.Y., Part I, Vol. 1, Chapter 16, Second edition, in press. (7) H. Strehlow in "The Chemistry of Non-Aqueous Solvents," J. J. Lagowski, Ed., Academic Press, New York and London, 1966, Chapter 4 . (8) H. Strehlow and M . Schneider, Pure Appl. Chem., 25, 327 (1971).
ANALYTICAL CHEMISTRY, VOL. 46, NO. 13, NOVEMBER 1974
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ammonia involves energy changes of the order of a few kcal/mole, the corresponding transfer of acids is accompanied by AGto values in the vicinity of -20 kcal/mole. In view of Izmaylov's estimate that AGt0(H+) = -23 kcal/ mole, it is clear that the proton dominates the energetics of the above transfer process, regardless of which anion is transferred along with it. Equally convincing evidence for the physical significance of the transfer activity coefficients for single ions comes from Parker's (9) studies of transfer processes between trifluoroethanol (TFE) and hexamethylphosphoramide (HMPA). The silver ion is preferentially solvated by HMPA to such an extent that the Acto(Ag+) from T F E (reference) to HMPA amounts to -22 kcal/mole (log m ~ = -16). The reverse is true for the chloride ion, which is tightly solvated by the powerful H-bond donor TFE, but has no affinity for the aprotic HMPA, so that AGt0(C1-) = + I 5 5 kcal/mole (log m ~ = ~ 11.4). l The above results are obtained by several extrathermodynamic assumptions. As a consequence of these extreme opposite preferences on the part of the Ag+ and the C1- ions for TFE and HMPA, the corresponding transfer activity coefficient of AgCl is comparatively small (log mYAgC] = 4.9). Thus, a misleading view of relative ionic solvation in T F E and HMPA would be obtained if one relied solely on the thermodynamic transfer activity coefficients for the complete electrolyte. In general, small anions are preferentially solvated by protic solvents via H-bonding and their transfer to dipolar aprotic solvents is energetically disfavored, as shown by large positive values of the corresponding transfer activity coefficients. This is why equilibria involving the chloride ion are reversed upon transfer from protic to aprotic solvents (9). Transfer activity coefficients of single ions can also account for the drastic reversal of the copper-silver equilibrium when the solvent is changed from water to acetonitrile. In water the oxidation of silver by cupric ions is disfavored by 0.64 V, but transfer to acetonitrile causes a positive shift in the above E o by 1.3 V. This fact is explained by the preferential solvation of the Ag+ and the Cuf ions by acetonitrile as compared to water (log m ~ = ~-3.9R ( I O ) and log m ~ ~ = u -8.7 + (111) and by the preferential solvation of the Cu2+ ions by water relative to acetonitrile (log m ~ ~ u z =+ +9.2 (11)).It should be noted that the strong interaction between the Ag+ ion and acetonitrile as indicated by the ~ been g confirmed from meanegative value of log m ~ has surements of Hittorf transference numbers (12), as well as by polarography (13,1 4 ) , NMR ( 8 ) ,and Raman spectroscopy (15). Subsequent discussion in this paper makes use of both literature and new data. The new data include the values for the solubilities of KBPh4, KPi and Ph4PPi in methanol-water mixtures as well as the rate constants for the hydrogen-ion catalyzed mutarotation of a-glucose in some ethanol-water mixtures. EXPERIMENTAL M a t e r i a l s . The preparation and purification of KPi, KBPh4 (161, and Ph4PPi (17) as well as the purification of ethanol-water
solvents (18) and of methanol (16) were described. a-Glucose was prepared from anhydrous dextrose (Baker Analyzed Reagent) by the procedure of Hudson and Dale (19). S p e c t r o p h o t o m e t r i c D e t e r m i n a t i o n of S o l u b i l i t i e s . Saturated solutions of KBPh4, KPi, and Ph4PPi in methanol-water solvents were prepared by ultrasonic generation followed by shaking of the suspensions at 25.00 "C using apparatus and procedures already described (17, 18).The concentrations of the saturated solutions were determined by UV spectrophotometry on a Cary Model 17 spectrophotometer. K i n e t i c s of Mutarotation of a - G l u c o s e . Measurements of optical rotation were made on a Bendix Automatic NPL polarimeter Model 1159 with digital readout. The 0.5-dm cell was jacketed, with water from a 25.00 "C bath circulating through it. Readings of the angle of rotation were taken on solutions of a-glucose (-0.4 g/ ~ 100 g ml) in ethanol-water solvents with and without added HC1 a t 3-minute intervals for a period of about one hour, with the infinity reading taken after about 24 hours. The first-order rate constants for the mutarotation were evaluated in the usual manner from linear plots of the function log (Rt - R,) us. time by the method of least squares. Rt and R , are the observed rotations at time t and a t equilibrium time, respectively.
RESULTS A N D DISCUSSION Evidence f o r the Validity of t h e Tetraphenylborate Assumption. T o date, the majority of transfer activity coefficients for single ions have been estimated on the basis of the assumption that the transfer activity coefficient of the tetraphenylborate anion can be equated to that of the tetraphenylarsonium cation ( m Y B p h 4 = m Y p h 4 A s ) . In this most popular form, the tetraphenylborate assumption has been employed by Parker and his associates (20-23), Kolthoff and Chantooni (10, 24, 25) and Popovych and his coworkers ( 5 ) . It was introduced originally by Grunwald et al. (26) in the form m Y B P h 4 = m Y p h r p , where Ph4P+ is the tetraphenylphosphonium ion. According to a comparative study reported earlier from this laboratory ( 5 ) ,the results calculated via the Ph4As-BPh4 and the Ph4P-BPh4 assumption differ by less than 0.1 log unit, which is roughly in the range of error for determining differences between solubility products in two solvents. The observed equality of m Y P h 4 A s and m Y p h 4 p , which is thermodynamically rigorous, represents strong support for the plausibility of the assumption that either m Y p h 4 A s or m Y p h 4 p is also equal to m ~ B p h 4 . Additional support for the above assumption comes also from the observed equality of the transfer activity coefficients for tetraphenylgermane, tetraphenylmethane, and tetraphenylsilane (27). These observations point to the conclusion that the transfer free energies of tetraphenyl solutes are not very sensitive to small differences in their size and to the nature of the central atom. Therefore, the reported NMR evidence to the contrary (28) may project an exaggerated view of the differentiating solute-solvent interactions. Our findings are in sharp contrast to the reported discrepancies among the enthalpies of transfer for tetraphenyl cations (29, 30). In the latter study, however, the single-ion enthalpies were derived on the assumption that AH,' (as well as IC,') are equal to zero for the trans(18) (19) (20) (21) (22)
(9) A. J. Parker, Pure Appl. Chem., 25, 345 (1971). (10) I. M. Kolthoff and M. K. Chantooni, Jr., J. Phys. Chem., 76, 2024 (1972). (1 1) W. E. Waghorne, Ph.D. Thesis, Australian National University, Canberra, 1973. (12) H. Strehlow and H. M. Koepp, Z.Nektrochem., 62, 373 (1958). (13) I. M. Kolthoff and J. F. Coetzee, J. Amer. Chem. SOC.,79, 1852 (1957). (14) S. E. Manahan and R. T. Iwamoto, J. Nectroanal. Chem., 14, 213 (1967). (15) E. G. Oliver and G. J. Janz, J. Phys. Chem., 74, 3819 (1970). (16) 0. Popovych and R. M. Friedman, J. Phys. Chem., 70, 1671 (1966). (17) D. H. Berne and 0. Popovych, J. Chem. f n g . Data, 17, 178 (1972).
2010
(23) (24) (25) (26) (27) (28) (29) (30)
ANALYTICAL CHEMISTRY, VOL. 46, NO. 13, NOVEMBER 1974
A. J. Dill and 0. Popovych. J. Chem. Eng. Data, 14, 156 (1969). C. S. Hudson and J. K. Dale, J. Amer. Chem. SOC.,39, 320 (1917). R. Alexander and A. J. Parker, J. Amer. Chem. SOC., 89, 5549 (1967). R. Alexander, A. J. Parker, J. H. Sharp, and W. E. Waghorne, J. Amer. Chem. SOC., 94, 1148 (1972). E. G. Cox, G. R. Hedwig, A. J. Parker, and D. W. Watts, Aust. J. Chem., 27, 477 (1974). A. J. Parker and R. Alexander, J. Amer. Chem. Soc., 90. 3313 (1968). i. M. Kolthoff and M. K, Chantooni, Jr., Anal. Chem., 44, 194 (1972). I. M. Kolthoff and M. K. Chantooni, Jr., J. Amer. Chem. SOC., 93, 7104 (1971). E. Grunwald, G. Baughman, and G. Kohnstam, J. Amer. Chem. SOC., 82, 5801 (1960). D. H. Berne and 0. Popovych, Anal. Chem., 44, 817 (1972). J. F. Coetzee and W. R. Sharpe, J. Phys. Chem., 75, 3141 (1971). C. V. Krishnan and H. L. Friedman, J. Phys. Chem., 73, 3934 (1969). M. H. Abraham, J. Chem. SOC., Faraday Trans. 7, 1973, 1375.
Table I. Comparison o f Transfer F r e e Energies f o r Ph4C a n d f o r P ~ ~ A S - B P ~ ~ ~ Solvent:
DMSO
DMA
DMF
iiMePy
HlWA
PhdC PhdAs-BPh,
-1.1
-1.1
-1.0
-2.2
-2.4
-2.6
-1.6 -3.4
-1.2 -2.9
-
AGLo,kcal/mole, (acetonitrile solvent) (molar scale), calculated from the data in Ref. (21). DMSO = dimethyl sulfoxide; DMA = N,S-dimethylacetamide; DMF = ,V,N-dimethylformamide; NMePy = ?i-methyl-2-pyrrolidone; HMPA = hexamethylphosphoramide. I
Table 11. ApK Values o f Primary Ammonium Ions (Molal Scale, 25 "C)(l L$ t $4 alcohol
10 20 30 50 70 80 90 100 ( 3 2 )
Ethanol-water solvents
... -0.24 rt 0.03 -0.35 0.03 -0.73 0.04 -0.91 x 0.10 -0.90 x 0.04
*
*
... +1.0
&
0.3
Methanol-water solvents
-0.11
* 0.01
...
-0.40 i 0.03 -0.75 0.07 -0.95 i 0.09
*
... -0.77 +1.2
* 0.11 * 0.2 20
a ApK = P ( ~ K) p ( , K ) . Unless otherwise stated, the ApK's were calculated from the data compiled in Ref. ( 3 2 ) .
fer of the Me4N+ ion from water to several solvents. This author finds it difficult to accept the above assumption as physically valid. Of course, significantly different results are obtained when the tetraphenylborate assumption is applied to the estimation of single-ion transfer enthalpies (22). If the enthalpies for the transfer of tetraphenyl cations are indeed different, we may be observing a compensation between the enthalpies and the entropies of transfer, leading to approximately equal transfer free energies. A relevant datum for this discussion is the magnitude and sign of AG to for the transfer of PhcAsBPh4 from water to acetonitrile. The Born equation predicts for it a value of +1 kcal/mole, while the observed value is about -16 kcal/ mole (21).It is significant that the corresponding value of Act"for the transfer of two moles of Ph4C is about -13 kcal ( 2 2 ) . Thus, it appears that transfer free energies of tetraphenyl ions are determined primarily by interactions of the four phenyl groups with the solvents, i.e., by the nonelectrostatic components of their solvation energies. This is corroborated further by Parker's (23) data for transfers between media of similar dielectric constants and dipole moments, which reveal that AGtO values of tetraphenyl molecules and ions are approximately equal for these media (Table I). All of the above evidence reduces but does not altogether eliminate the possibility that certain pairs of solvents may differentiate between the tetraphenylborate anion and the tetraphenyl cations. One of the strongest indications that the tetraphenylborate assumption might be capable of reflecting minor changes in solvation energy is the parallel behavior of the 1ph"s of primary ammonium acids and of the log m~~ function in ethanol-water and methanol-water solvents. Table I1 lists average values of ApK calculated from literature data for 13 primary ammonium acids in methanolwater solvents and in ethanol as well as for six of the acids in ethanol-water mixtures. Although the acids range in size from ammonium to octylammonium and substituted anil(31) R . P. Bell, "The Proton in Chemistry," Cornell University Press, Ithaca, N.Y., 1959,p 44. (32) R. G. Bates, J. Nectroanal. Chern.,29, l(1971).
40
60
80
roo
W T -% ETHANOL
Figure 1. Transfer activity coefficients for the proton, log Y , H, and ApK's of primary ammonium acids in ethanol-water solvents (molal scale, 25 "C)
I
0
I
20
I
40
60
I
80
1
1 co
W 1 . X METHANOL
Figure 2. Transfer activity coefficients for the proton, log m-(H, and ApK's of primary ammonium acids in methanol-water solvents (molal scale, 25 "C)
iniums and in strength from a p(,K) of about 2 to 10, the ApK values are remarkably constant for each medium, their average deviations being generally no worse than the repeatability of pK determinations between laboratories. A comparison of the ApK and log ,,,YH functions in ethanolwater (Figure 1) and in methanol-water solvents (Figure 2)
ANALYTICAL CHEMISTRY, VOL. 46, NO. 13, NOVEMBER 1974
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4 1
I
o
1
20
40
I
1
60
80
i
100
W T . % METHANOL
Transfer activity coefficients, log for single ions in methanol-water solvents (molal scale, 25 OC) from Equation 6 why ApK's are always slightly negative as compared to the corresponding values of log m ~ ~ . The values of log m Y H plotted in Figure 1 have been reported from this laboratory earlier ( 5 ) .Those for methanolwater solvents depicted in Figure 2 have been calculated from the literature data in the case of HC1 (33) and KCl (34, 35) and from preliminary data for the transfer activity coefficients of KPi, KBPh4, and Ph4PPi determined in this laboratory (36). The latter are based on solubility products for which activity coefficients were calculated from the Debye-Huckel equation with ion-size parameters. Precise values of transfer activity coefficients in methanol-water solvents will be reported as soon as our experimental determination of the concentration activity coefficients is completed. At this time, the tentative values of log in methanol-water water media are displayed in Figure 3 and the : corresponding values for single ions apportioned on the basis of the tetraphenylborate assumption are shown in in methanol-water Figure 4. The general pattern of log solvents is very similar to that observed earlier in ethanolwater solvents ( 5 ) .For the small inorganic ions, such as K+ and C1-, values of log m~ rise steadily as the dielectric constant of the solvent decreases, in qualitative agreement with the Born equation, as shown in Figure 4 for a hypothetical ion of r = 2 .& The Ph4P+ ion, on the other hand, displays the behavior of a typical organic compound, being more favorably solvated in media of lower dielectric constant, as compared to water. The picrate ion assumes an intermediate position. The Medium Effect on the Mutarotation of a-Glucose. Since the transfer activity coefficient of the proton, m ~ is ~ an index , of hydrogen-ion activity on a single solvent-independent scale, there should be a correlation between the solvent dependence of m~~ and that of the rate Figure 4.
WT.% METHANOI
Transfer activity coefficients, log m methanol-water solvents (molal scale, 25 OC)
Figure 3.
for electrolytes in
~ .
shows that they both describe the characteristic pattern with the minimum and that ApK's consistently exhibit a small negative deviation from log m ~ The ~ above . parallelism can be explained by formulating the transfer activity coefficient of an ammonium ion, m ~ ~ ~ as~ a 3composite + , of electrostatic and nonelectrostatic contributions:
and then, as a n approximation, equating the above nonelectrostatic contribution to the transfer activity coefficient of the corresponding conjugate base, m ~ ~ ~ ~
log , ~ ~ ~ ~ ? , ( n o n e log l)
RNHZ
(4 )
Since ApK defined as (p(,K) - p(,K)) can be expressed as:
ApK = 10g,rH - log
mYRNHS+ ~'RNH
(5 )
2
it follows. that:
'pK
l0gmVH
- log m Y R N H 3 * (el)
(6)
The approximate cancellation of mYR"3+(nonel) with m ~ ~ which ~ ~ leads 2 from , Equation 5 to 6, is reasonable, as the energy required to make a cavity in the solvent, the hydrophobic structure-making, and the dispersion interactions should be very nearly equal for a given acid and its conjugate base. The electrostatic component, mYR"3+(el), is approximately constant regardless of what R group is attached to the ammonium function apparently because the electrostatic interactions are localized a t the functional group, where the charge is. Electrostatic components of transfer activity coefficients are positive for transfers from water to media of lower dielectric constants, and in the case of transfer of ammonium ions to alcohol-water solvents, they were estimated (7) to be small. It is therefore clear 2012
2
(33)M. Paabo, R. G. Bates, and R. A. Robinson, Anal. Chem., 37, 462 (1965). (34)A. L. Andrews, H. P. Bennetto. D. Feakins, K. G. Lawrence, and R. P. T. Tomkins, J. Chem. SOC.,A,, 1486 (1968). (35) M. Alfenaar and C. L. DeLigny, Red. Trav. Chim. Pays-Bas, 86, 929 (1967). (36)P. LaBrocca, S.Goldberg. and 0. Popovych, unpublished research.
ANALYTICAL CHEMISTRY, VOL. 46, NO. 13, NOVEMBER 1974
constants kH for reactions catalyzed by hydrogen ions. To investigate this hypothesis, a preliminary study was carried out on the kinetics of mutarotation of a-glucose in ethanolwater mixtures ranging from 0 to 80 wt % ethanol, with and without catalytic concentrations of HC1 (37). Using standard polarimetric procedure, the first-order rate constants for the mutarotation were obtained in each of the pure solvents as well as in the same solvents with added HCl in the concentration range of to 1O-2M. Our rate constants for the solvent catalysis, ko, agreed well with the literature values (38).In the presence of catalytic amounts of added acid, the overall rate constant k is usually equated to (ko k H U H ) , where a~ is the activity of hydrogen ions. Values of k~ in each of the ethanol-water solvents were calculated from the above equation by substituting literature data for a HC1 (39) as an approximation of a H. The rate constants for the hydrogen-ion catalysis obtained in this manner are shown as a function of ethanol-water composition in Figure 5. One drawback in choosing the mutarotation of a-glucose for study of medium effects on the proton is the large contribution of k~ to the overall rate constants (25-75% a t the acid concentrations employed). The high solvent "corrections'' in determining k H are undoubtedly responsible for the considerable discrepancies (up to 10%) between values of k~ calculated from rate constants at different acid concentrations. Statistical studies a t several HC1 concentrations are now in progress with the objective of improving the reliability of the results. Nevertheless, even the preliminary data depicted in Figure 5 indicate that there is a fair correlation between the behavior of k~ and of ,,,YH (Figure 1) in ethanol-water mixtures. The curve in Figure 5, however, is much more shallow. This considerable dampening of the medium effect on the catalytic rate constant can be explained qualitatively by analyzing the relationship between the rate constants in water, w k H , and in nonaqueous solvents, a k H , in terms of the transfer activity coefficients of all the species involved in the rate-determining step. Our formulation is analogous to that which Parker (9) applies to one-step processes, but it is more complicated, as we are dealing with the reversible reaction:
"I
I
70-
+
a-glucose
kl a P-glucose k2
where k l and kp are the rate constants for the forward and the reverse process, respectively, while the experimentally observed constant is their sum. If we assume that the ratedetermining step is the protonation of glucose (40), and that the transition-state complex is some protonated form of glucose, GH+*, it can be shown that
(37) (38) (39) (40)
A. Sasala. D. Scherr. and 0 . Popovych, unpublished research. H. H. Rowley, J. Amer. Chem. Soc., 62, 2563 (1940). I. T. Oiwa, Sci. Rept. Tohoku Univ., First Ser., 41, 129 (1957). S. W. Benson. "The Foundations of Chemical Kinetics," McGraw-Hill, New York, N.Y., 1960, p 561.
'1 d
io
2'0
