EFFECTS O F NEUTRA.L SALTSox THE HAMMETT ACIDITYFUNCTION
1857
The Effects of Neutral Salts on the Hammett Acidity Function
by Mario Ojeda Universidad de Chhile, Facultad de Qulmkca
Farmacia, Santiago, Chile
and P. A. H. Wyatt Universidad de Chile, Centro de Quimica, Santiago, Chile
(Received February 6 , 1965)
The Hammett function H o is measured spectrophotometrically, using the indicator p-nitroaniline, for dilute solutions of HCl in the presence of 2 and 4 Jd NaBr, and 1, 2, 3, and 4 M CaClz and MgCln. Parallel solubility measurements permit the calculation of the activity coefficient of the basic form of the indicator in each case (and also for o-nitroaniline in KaC1 and KCl solutions) and log j~ is then applied as a correction to BO, thus eliminating the salting-out effect and leaving a rather larger effect, which is interpreted in terms of the dehydration of the hydrogen ion as the water activity is reduced. h simple model, developed previously for concentrated solutions of the strong acids alone, is found to give a good quantitative account of this part of the salt effect for 0- and p-iiitroaniline. Obstacles to the more general application of this treatment are, however, emphasized.
Although the effects of salts upon the ionization of acid-base indicators have been extensively studied in the region of concentration and acidity of interest in ordinary pH work, there have bPen relatively few a t higher concentrations where such effects can be related to theories of the Hammett acidity function, Ho. Our interest in this field has stemmed from two sources: one of us (31.0.)had the advantage of a period of experimental study on salt effects in Professor Kolthoff’s laboratory a t the TTniversity of A h nesota, while the other was interested in trying to extend to salt solutions the hydration treatment of concentrated acids developed independently by Bascombe and Bellg and Wyatt’O (cf. also Hogfeldtll). During the course of our work, Rosenthal and Dwyer have published papers treating the problem on rather similar They measured the color ratios of p-nitroaniline, o-nitroaniline, and 4-chloro-2-nitroaniline in solutions of LiCl up to 9 M containing small amounts of HC1 and interpreted the resultant H o values in terms of a Bascombe and Bell type of expressiong
Ho
=
-log C H +
+ n log cc - BC
(I)
where C H + is the concentration of the strong acid, C is the molar conceiitiation of the salt, a is the activity of water, and B is a constant which depends on the salt
and the indicator. This equation results when the proton is regarded as associated with n molecules of mater in a hydration complex, and the logarithm of the activity coefficient of the free indicator base varies linearly with the salt concentration, the other activity coefficient terms (after allowance for proton hydration) being assumed to cancel (or to contribute only linear terms) : cf. the form of the well-known expression’ for HO
HO
=
-log C H + - log
(.fBfI-I+/fBH4)
(2)
Rosenthal and Dwyere have also discussed in general terms the possibility of variations in the parameter n (1) M. A. Paul and F. A. Long, Chem. Rez.., 57, 1 (1957).
(2) G. Harbottle, J . Am. Chem. Soc., 73, 4024 (1961). (3) M. A. Paul, ibid., 76, 3236 (1954). (4) I. I. Moiseev and R . M. Flid, Zh. Prikl. K h i m . , 27, 1110, 1145 (1954); Chem. Abstr., 49, 7325, 7327 (1955).
(5) F. E. Critchfield and J. B. Johnson, A n a l . Chem., 31, 570 (1959). (6) D . Rosenthal and J. S. Dwyer, Can. J . Chem., 41, 80 (1963). (7) D . Rosenthal and J. S. Dwyer, J . Phys. Chem., 6 6 , 2687 (1962). (8) D. Rosenthal and J. S. Dwyer, Anal. Chem., 35, 161 (1963). (9) K. S . Bascombe and R. P. Bell, Discussions Faraday Soc., 24, 158 (1957). (10) P . A. H. W y a t t , ibid., 24, 162 (1957). (11) E. Hogfeldt, Acta Chem. Seand., 14, 1627 (1960).
Volume 68. S u m b e r 7
J u l y , 1964
1858
with concentration, on account of the existence of hydration equilibria. This is essentially the argument reported previously,1oand of course there are far too many variables a t present to be confident that any one particular choice of hydration model really represents a t all closely the actual state of affairs in these solutions. Nevertheless, in the original paper lo a particularly simple model for successive hydrations was found to lead to predicted values of Ho in excellent agreement with experiment over wide ranges of concentration for the strong acids, and it was of interest to see to what extent the same model was capable of explaining the results obtained in the presence of salts. Our approach has been to regard the log f B term of ( 2 ) as an experimentally determinable correction, and to compare the behavior of the corrected Ho* (ie., H o f log fB) with that to be expected from the simple hydration model. In terms of Paul and Long's discussion, we are attempting a quantitative explanation of the difference between the rates of change of H o and log fB with salt concentration (see their Table 17). Using a solubility method for determining the activity coefficient of the free base form of the indicator, we find that the theory describes very well our results for the effects of concentrated solutions of magnesium and calcium chlorides and sodium bromide on the protonation of p-nitroaniline in dilute acid solutions. After determining the solubility of o-nitroaniline in sodium and potassium chloride solutions, similar success is obtained with the interpretation of Paul's results3 with this indicator. Some limitations of this particular approach are, however, mentioned below.
Experimental The salts used were Rlerck guaranteed reagents : calcium chloride in the form CaClz.2Hz0,and magnesium chloride as R/lgCI2. The "pure" dry sodium bromide was recrystallized from water, and its purity was found to be at least 99.9% by potentiometric titration with silver nitrate. The indicators were recrystallized twice from water. p-Nitroaniline had m.p. 147.4-148'; o-nitroaniline, 71-72'. Saturated solutions of the salts were prepared as stock solutions, filtered a t the vacuum pump, and standardized with silver nitrate. Hydrochloric acid solutions were prepared by dilution of a formal solution, itself prepared by weight from the constant boiling point acid. The diluted solutions were standardized with sodium carbonate (previously heated a t 280-300' for 2 hr.). Stock solutions of p-nitroaniline were prepared by making an exact weight of the indicator up to a fixed volume with water. The indicator concentration ratios, CBH+/CB, were The Journal of Physical Chemistrg
MARIOOJEDAAND P. A. H. WYATT
determined on a Beckman DU spectrophotometer with 1.0-cm. or 0.5-cm. matched fused silica cells a t 25 k 0.5'. In determining the absorption characteristic of the basic form of the indicator, an aliquot of the stock solution was made up with sodium acetate so as to be 0.01 M in this salt. The absorption of the acid form was negligible for all salt concentrations at the maximum absorption wave length of the basic form. Absorptions were always measured against blanks of the same composition (apart from the absence of the indicator). Beer's law m7as obeyed in the concentration range of the indicator used, and it was also found that the molar absorptivity and the wave length of maximum absorption were hardly affected by the presence of the salts. The absorptivity which was 1.33 ( X lo4) in water, shifted to about 1.36 in the most concentrated NaBr solutions and to 1.31 in the CaC12 solutions, but was unaffected by MgClz, while the maximum shifted only from 382 mp to 384-387 (4M NaBr, 4m RIIgCI,), or 389-391 (4 M CaCI2). From the indicator ratios values of Ho were calculated from the formula
Ho = P K B H~ log CBH+/CB
(3)
using the ~ K B Hvalues + recommended by Paul and Long, viz., 0.99 for p-nitroaniline, -0.29 for o-nitroaniline. In this way Ho m7as measured for the following salt concentrations: NaBr, 2 and 4 ,!!; CaClz and R'IgC12, 1, 2, 3, and 4 M ; and for each salt solution four concentrations of hydrochloric acid were investigated, 0.0102, 0.0203, 0.0507, and 0.1005 M . The results are shown in Table I. For the solubility measurements, approximately 0.1 g. of p-nitroaniline or o-nitroaniline was added to 30 ml. of the appropriate salt solution (made 0.01 M in sodium acetate) in a 60 ml. flask, which was then shaken in a thermostat a t 25 f 0.1' for 48-120 hr. Samples of the saturated solutions were withdrawn through cotton-wool plugs into 1 or 2 ml. transfer pipets and diluted for the optical density measurements. The ratio of the activity coefficient of the base form of the indicator in the salt solution, ji,, to that in water alone, foB, was then calculated simply from the inverse ratio of the corresponding solubilities, So/X, and, since the usual exponential relation was found to hold, the results are sufficiently well summarized by the Setchenov (salting-out) parameters k , log
(j~/f'B)
=
log (So/S)= h C
(4)
where C is the molarity of the salt. For water a t 25', solubilities of 0.570 and 1.210 g. I.-' were obtained for p- and o-nitroaniline, respectively. The k , parameters (in mole-I 1.) for p-nitro-
EFFECTS OF YEUTRAL, SALTSON
THE
1859
HAMMETT ACIDITYFUXCTION
Table I” -HC1,
Ho Salt
NaBrc CaClzd
MgClzd
M
mb
2 4 1 2 3 4 1 2 3 4
2.11 4.49 1.03 2.11 3.29 4.56 1.03 2.10 3.26 4.50
LZH~O
0,9243 0.13155 0.9431 0.8507 0.7241 0.5527 0,9400 0.8368 0.6597 0.5082
exptl.
2.09 1.47 0.91 1.61 0.98 0.61 -0.09 1.45 0.87 0.35 -0.29
-HCI,
0.0102 M--
Bo*
HQ*
No
exptl.
calod.
exptl.
1.55 1.07 1.72 1.23 0.97 0.39 1.59 1.15 0.77 0.24
1.66 1.34 1.70 1.39 1.00 0.43 1.63 1.32 0.75 0.24
1.66 1.33 0.77 1.29 0.81 0.28 -0.37 1.23 0.77 0.24 -0.30
0.0203 M--Ho* Ho*
exptl.
1.31 0.93 1.40 1.06 0.63 0.12 1.37 1.05 0.66 0.23
oalod.
1.37 1.05 1.40 1.09 0.71 0.15 1.34 1.02 0.45 0.06
H ---C,I
0.0507 M--7
HO exptt.
1.28 0.84 0.37
0.88 0.43 -0.13 -0.65 0.84 0.43 -0.01 -0.76
Ho* exptl.
ho* oalcd.
0.92 0.98 0.53 0.65 1.00 1.01 0.68 0.70 0.31 0.23 -0.25 -0.17 0.95 0.98 0.71 0.63 0.08 0.41 -0.23 -0.45
v H C 1 , 0.1005 M--Ha Ho* Ho* exptl. exptl. oalcd.
0.98 0.60 0.68 0.21 0.37 0.57 0.69 0.11 0.36 -0.43 -0.07 -0.99 -0.51 0.53 0.68 0.01 0.29 -0.55 -0.13 -1.05 -0.52
0.68 0.36 0.73 0.41 0.02 -0.55 0.65 0.34 -0.06 -0.75
‘
Molalities were calculated from density data found in “International Critical Tables,” Vol. 111, McGrawValue of k used: 5.02. Hill Book Co., Inc., New York, N. Y., 1928, p. 80 (NaBr), p. 72 (CaC12),p. 71 (MgCl2). Corrected H,values for NaBr solutions Corrected Hovalues for CaC12 and MgCl, solutions were were calculated by using a value of 0.040 for the k, parameter (ref. 12). calculated by using our solubility data.
aniline were 0.120 in CaC12 and 0.139 in RIIgCl,; for o-nitroaniline, 0.093 in XaC1 and 0.046 in KCI. The corresponding parameter for p-nitroaniline in XaBr solutions, 0.04, was taken from Long and McIntyre.12 With these numbers log JB could be calculated for any salt concentration and added to the experimental Ho to obtain the corrected Ho* values shown in Table I. In this paper f~ is a molarity activity coefficient; conversion to the rational scale makes little difference to the work below, and in any case is probably not justified since no such conversion was made in the work on acids with which the present results are compared.
Discussion We may distinguish between two ways of talking about an activity coefficient ratio of the type fH+/ ,fBH +. When a simple stoichiometric equation such as (2) is under discussion, where C H + refers to the total concentration of acid in the solution, fa+ incorporates all the effects of changes in hydration of the proton as well as other effects due to ionic strength, etc. The large changes in N o with concentration are then said to be due principally to large changes in the term log (fH+/f~H+).l Values of this term with just this meaning have been calculated froin Ho, log f ~and , log CH+, in accordance with (2), but are not included here. They show the expected negative trend with increasing salt concentration, since any reduction of the water activity with added solutes will increase the acidity of the solution (and make log (fH+/fBH+) more negative), because of the tendency to shift hydration equilibria in the direletion of increasing the proportion of free H30+ (cf. the effects obseriied by Harbottle2).
On the other hand, in attempts a t the interpretation of the changes in the stoichiometric activity Coefficient ratio, it is often convenient to consider symmetrical equilibria between supposed unhydrated species, such as
B
+ H30+ = BH+ + HzO
(5)
It may then be reasonable to argue that the activity coefficient ratio for the unhydrated species, fx,o +! ~ B +,H should remain fairly constant over relatively large ranges of concentration, from the type of electrostatic argument familiar in dilute solution work. As Rosenthal and Dwyer emphasized,6 the possibility of the hydration of the indicator species must also be kept in mind, so that a treatment which only allowed for the hydration of the proton could still give rise to large apparent changes in such a ratio through not having quite reached the stage in the argument a t which (5) is actually applicable. But unbalanced hydration of the indicator species would also have much more serious and immediate effects upon the initial interpretation of the experimental indicator ratios in terms of protonation equilibria. To simplify the argument a t present we ignore this possibility, and assume that only the proton hydration need be considered in calculating the correct concentrations for the equilibrium (5). The experimental ratio CBH+/CBwill therefore (according to (5)) be proportional to XH,O?fB/a, where X H a O + represents the mole fraction of H 3 0 + in the solution, if the activity coefficients of the free species H 3 0 + and BH+ cancel. (12) F. A. Long and D. McIntyre, J . Am. Chem. Soc., 7 6 , 3243 (1954).
Volume 68, Number 7 J u l y , 1964
1860
MARIOOJEDAAXD P. A. H. WYATT
The concentration of H30+ has been written in mole fraction (ideal solution) terms because of the form of the water activity, which it has to offset in the equilibrium (5) since the units of B and BH+ are automatically cancelled by taking the ratio experimentally. Substitution of CBH+/CBin this way and adding log fB to both sides of (3) now gives
Ho*
=
No
+ log f~ = - k
-
log (XH,,+/U) (6)
The constant - k in this expression has actually arisen froin the difference between the PKBH+of ( 3 ) and the pK of the equilibrium (5) ; but a comparison with (3) immediately shows that it can also be regarded as the pK for the formation of the free species H30+ from water, as suggested by Den0 and Taft,I3 who argued that its value must be about -6.78 (corrected to the new Paul and Long scale'). Reasons were later givenlofor believing that it is actually nearer - 5 , and for concentrated acid solutions in the absence of salts the number -5.14 (on the new scale') gave the best fit for the simple hydration model adopted in this paper. In investigating whether or not the particular calculations of the right side of (6) used for the strong acid solutions are also applicable when salts are present, the assumption is now made that log f~ is small in the pure acid solutions.' Before embarking upon numerical calculations, however, a qualitative check that the data are going to be consistent with this type of explanation is provided by plotting Ho* for a fixed acid concentration against the water activity of the salt solutions. If the fraction of the fixed acid concentration in the forin of free H30+ depends only on the water activity,'O eq. 6 shows that the points for the different salts should lie on a common curve (apart from a slight difference to be expected in the correction for the conversion from CH+ to a mole fraction for the different valence types). Figure 1 shows the satisfactory outcome of this test for the highest and lowest concentrations of acid used: similar results were obtained a t the intermediate concentrations. Thus, we conclude that apart from the log f~ effect, which has already been allowed for, the principal reason why the neutral salts incrcase the acidity is that they simply reduce the water activity and thus tend to dehydrate the H3O -Iion. As was emphasized before,lo there are probably several different detailed theories which would account quantitatively for the trends found, but in this paper we confine ourselves to the simple model which was successful for the pure acids. The H30+ ion is regarded as being able to take on four water molecules in stages, the equilibrium constants for which are related by the statistical formula The Journal of Physical Chemist?'?/
Kn+l
=
Kn ( ( N
-
n(N - n) n l)(n
+
+ 1)
where N stands for the maximum number of ligand niolecules (4 here), and n t,akes values from 1 to 3. This implies that the energy of the successive stages does not alter, only the entropy, according to the availability and occupation of the sites around the central ion. It also means that only one number has 2.2
1.8
1.4
1 .o
$
0.8
0.4
0.00
-0.40
0.90
0.80
0.70
0.60
0.50
a.
Figure 1. Plot of 11, HCl 0.1005 M .
Ho*us. 0,
U H ~ O . Curves: I, HC1 0.0102 M ; NaBr; 0,CaClz; MgCL.
+,
now to be chosen to give all the equilibrium constants required. For acid solutions the number 20 was adopted for the first hydration constantlo so that the others became 7.5, 3.33, and 1.25, and the products K I K ~K, I K ~ KK1K2K3K4, ~, 150, 500, and 625. These products are required for expressing the concentration of each type of hydrated H30+species in terms of that of the free H30+ and the water activity alone. The (13) N. C. Den0 and R. W. Taft, J . Am. Chcm. Soc., 76,244 (1954).
EFFECTS OF NECTRSLSALTS ox THE HAMMETT ACIDITYFUNCTION
1861
sum of all these species concentrations must of course equal the total acid concentration
tables of Robinson and Stokes.I4 For the other solutions mentioned below a sinal1 correction was applied to the tabulated values when the acid concentration CH+ = CHS,+(l 21Da 150a2 500a3 was rather larger.l5 625a4) (8) Equation 12 was tested by inserting the experimental values of a and Ho*, finding a value of k for where a represents the water activity. A similar each solution, and examining the constancy of the k relation to (8) holds between the mole fractions XH+ values so obtained. For the 40 experimental soluand XHs0+,so it only remains to convert from CH+ tions, the mean k was 5.02, with a standard deviation to X H +to be able to calculate X H ~ O so+that (6) can of only 0.12; this compares well with the value be tested. The complete calculation of this conver5.14 obtained from solutions of acids alone, thus sion would require EL knowledge of the extent of hydraindicating that the same model also accommodates the tion of all the species in the solution, and this is not effects of at least some added salts. Values of Ifo* attempted here. Instead the water activity is used calculated from (12), adopting k = 5.02, are iiicluded again to get an idea of the total number of osmotically in Table I ; on the whole they agree excellently with active particles of all types associated with the same the observed values except, rather inexplicably, in quantity of solution as the salt molarity C (Le., 1 1.); the case of the higher acid concentrations in the this total number of particles is represented by q. presence of 3 and 4 M nlgClz (though not CaCI2). If the water activity is assumed to be equal to the mole Table I1 shows the results of applying (12) with the fraction of the free molecules in the solution (ie., those constant 5.02 to other experimental data in the literanot used in hydration), then the following relation ture, both for p-nitroaniline and o-nitroaniline. The must hold salting-out parameters for p-nitroaniline in LiCl a = (q - vc - 2C,+)/q (9) (0.082), NaCl (0.072), and KC1 (0.030),necessary for the log f~ correction, were all taken from Long and since all the particles in the solution which are not salt IllcIntyre,12 while those for o-nitroaniline in NaCl and or acid (hydrated or unhydrated) must be water, KCl were specially determined, as described in the which must therefore be measured by ( q - UC - 2cH +). experimental section. It is evident that the calculated ( U is the number of ions per molecule of the salt, and and experimental Ho* almost all agree to the required the number 2 represents the same quantity for HCI). 0.1 unit, and generally to very much better than this. On the same basis X H + is given by C H + / ~and , XH~O+ Thus, for these two indicators and the salts LiC1, NaC1, by CHSo+/q. But (!3) can be transformed to NaBr, KC1, CaCl,, and ?(/IgC12up to 4 M , eq. 12 seems q = 2cH+)/(1 - a) to be very satisfactory. (10) Anomalies occur, however, even with the indicator whence p-nitroaniline, especially among those salts which XH+ = - a)/(VC f 2cH+) “salt-in” the indicator. A glance at the final column of Paul and Long’s Table V’ shows that the residual and, from (8) effect on Ho, after allowing for salting-out and salting.)/(Vc 2cH+) x XHIO- = c H + ( l in, is much too large in the case of sodium p-toluene sulfonate to be explicable in our terms, but this may (1 20u . , . . . ) (11) be due to a special local distribution (complex?) effect Division by a and substitution into (6) gives the final on the indicator equilibrium with a large organic ion. equation A similar effect may operate in the case of tetraethylammonium bromide, but it is strange that the tetraHo* k = -log methyl salt behaves quite normally. A more perplexi C H + ( l - a) I ing, though smaller, anomaly occurs with sodium per\a(UC f 26H+)(1 $- 20a 1 5 0 ~ ’-tci00a3 f 625a4)J chlorate, which has a greater effect on H o than we predict from (12). This is the more difficult to ex(12) plain as perchloric acid fitted in well with the other For our experimental solutions CHAwas always small acids.1° compared with the salt molarity, and so the values of a for the various salt concentrations could easily be (14) R . A. Robinson and R. H. Stokes, “Electrolyte Solutions,” interpolated (after conversion to molarities) from the 2nd Ed., Butterworth and Co., Ltd., London, 1959. values for the salts alone in the osmotic coefficient (15) See ref. 14, p. 451.
+
+
(7jc
+
+
+
+
+
+
+
+
Volume 68, Number 7 Julg, 1964,
1862
MARIOOJEDAAND P. A. H. WYATT ~
Table TI Ref. exptl. data taken
M
Salt
LiCl
4
LiCl
8
NaCl
1 2 3 4 4 1 2 3
4 KCI
CaClz MgCh
1 2 3 4 4 1 2 3 2.66 2.66
CH+,M
4.98 x 10-3 7.97 x 10-3 1 x 10-2 2.99 X 1 x 10-1 4 . 9 8 x 10-3 7 . 9 8 x 10-3 1 x 10-2 1 . 0 x 10-1 1 . 0 x 10-1 1 . 0 x 10-1 1 . 0 x 10-1 1 . 0 x 10-1 1.o 1.0 1.0 1.o 1 . 0 x 10-1 1 . 0 x 10-1 1 . 0 x 10-1 1 . 0 x 10-1 1 . 0 x 10-1 1.0 1.0 1.0 1 x 10-1 1 x 10-1
No* calod.
1.53 1.33 1.23 0.76 0.24 0.38 0.17 0.08 0.84 0.72 0.60 0.45 0.45 -0.31 -0.40 -0.52 --0 ,63 0.85 0.75 0.65 0.53 0.53 -0.31 -0.39 -0.49 0.08 0.04
Ho* exptl.
1.57 1.37 1.30 0.83 0.36 0 .4 P b 0 . 27"5b 0.196'' 0.84 0.71 0.58 0.45 0.50 -0.31* -0.42' -0.52' -0.62b 0.86 0.75 0.63 0.52 0 54 -0.29' -0 39' -0.48' 0.19 0.06
from
7 7 7 7 5 7 7 7 3 3 3 3 5 3 3 3 3 3 3 3 3 5 3 3 3 5 5
a IC, = 0.084, calculated from solubility measurements made Experimental data obtained only for 8 M LiCl solutions. with o-nitroaniline. I n all the others p-nitroaniline was used.
Apart from these anomalies, which may yet be capable of some special explanation, the results examined seem to provide further confirmation of the usefulness of our equations as a method of allowing for the changes in the hydration of the H 3 0 +ion. Nevertheless, we emphasize that the model adopted may not
The Journal of Physical Chemistru
represent too closely the actual physical state of affairs in the solution. The same reservation also applies to other attempts of this kind, by comparison with which our model has certain advantages for handling concentrated solutions. The similar treatment of Bascombe and Bell,9 for example, owes some of its attraction to the simplicity introduced by adopting a single ion of structure16 H+(H20)+but much of the evidence supporting an ion of this structure would apply equally well to our model, which also requires an average hydration number of four for the proton over the range covered by Bascombe and Bell.$ But our model also incorporates a feature which has not yet been quantitatively attempted for the H+(HzO), form: it describes the buildup of the various hydration stages, and something of this kind is clearly required for more concentrated solutions. I n the light of our present results, the behavior of the indicators 2,4-dichloroaniline and diphenylamine is rather puzzling. Paul found3 that for these the whole of the salt effect could be accounted for by the magnitude of log f B , leaving nothing for the dehydration effect described above. This is contrary to the present trend of opinion about the importance of proton hydration in determining acidity function changes. Such results could still be accommodated by supposing that the two forms of the indicator may be hydrated to different extents, or that some new (structural) factor operates in such a way as to cause some cancellation of log f B by log f B H + , thus disposing of the observed correlation as accidental. But more work with other indicators will be required to see which class of behavior is really the more typical. Even the qualitative form of the hydration explanation can only be accepted with reserve until the Hammett functions for indicators of other charge types, and more particularly the J functions for secondary bases, can also be interpreted. The J functions especially are very puzzling a t present. ~~
(16) E. Wicke, M. Eigen, and T. Ackermann, Z . Physik. Chem. (Frankfurt), 1, 340 (1954).
