April, 1962
DIELECTRIC CONSTANT AND CONDUCTANCE OF ION PAIRS
An additional second order effect was the appearance of screw markings on the cylindrical surfaces, A slight additional amount of corrosion occurred along a helix. Free Convection.-Totally immersed samples and those with the liquid surface intersecting the sample showed SI, negligible difference in corrosion rate. Measurements of corrosion rate vs. distance from leading edge shown by lower points of Fig. 4 are proportional to the reciprocal fourth root of distance from the leading edge as predicted by eq. 9. The data for solution rate 0.1 in. beneath the leading edge in free convection over the temperature range 0 to 50" are summarized in the lower half of Fig. 3, along with the results predicted by Ellenbaas's expression for free convection, eq. 11, with and without the correction factors. The experimental results show very nearly the predicted temperature dependence and are about midway between the predictions using bulk and interface property values. The reasonable agreement of experimen tal results with those predicted and a lack of noticeable difference between partially and totally immersed samples indicateei that buoyancy differences and not surface forces are the driving force for free convection in thiis system. The free convection corrosion rates aire an order of magnitude lower than the rate of corrosion from a disc rotating a t 1200 r.p.m. over the entire range of measurements.
Conclusions The geometry, time dependence, temperature dependence, and liquid velocity dependence of corrosion, and the agreement of experimental results with quantitative predictions all indicate that the kinetics of sodium chloride dissolving in glycerol are controlled by diffusion in the liquid. Relations derived from boundary layer theory for both free and forced convection adequately de-
669
scribe results over a 40" temperature range in t,his high viscosity liquid system. The presence of velocity, boundary layers thick in comparison with experimental dimensions does not have an important effect on results. Apparently the interface reaction occurs more slowly on the cube faces (010) and the face diagonal planes (011) than on the other available surfaces. With a rotating cylinder there is rapid convection transport in the tangential direction; this permits the selective solution process to occur even though radial diffusion limits the over-all rate of solution from the originally cylindrical surfaces. Free convection samples do not have this tangential convective transport and hence do not show any indication of selective solution. The development of (010) planes is consistent with the observation that the normal habit of rock salt is cubic, indicating that cube faces also are slowest growing during crystallization. Typical experimental variations such as depth of immersion and eccentricity of rotation were shown to have only a slight effect on rates of solution. Correction factors can be calculated to take into account variations in diffusivity and viscosity in the boundary layer. Better information regarding the concentration dependence of properties is required to further test these calculations, but it appears that agreement of calculated and experimental results is improved by using weighted average values of the properties rather than bulk or interface values alone. Acknowledgments.-Appreciation is expressed to Michael Cable, who critically reviewed the manuscript, to Janis Kalnajs for the X-ray examination, and to the U. S. Atomic Energy Conimission for sponsoring the research, which was conducted at M.I.T. under contract No, AT(30-1)1852.
THE DIELECTRIC CONSTANT AND CONDUCTANCE OF ION PAIRS; AN EXTENSION OF ONSAGER'S FIELD EFFECT TO RELAXATION OF ION PAIRS 1.N AN ALTERNATING FIELD1 BY W. R. GILKERSON Department of Chemistry, University of South Carolina, Columbia, South Carolina R6ceived Aupuat I f , lg6l
Onsager's expression for the distribution function of ion pairs in an external field, in the form of a definite integral, has been solved and expressed as an infinite series, which reduces to the zero-field distribution and a term linear in the field for low field strengths. This result then is used to calculate the displacement current in the field direction in an alternating field, and equations for the resulting increase in dielectric constant and conductance as a function of frequency are derived. The resulting equations are compared to Pearson's, t o the Debye-Falkenhagen equations, and to experimental results in the literature.
The distance of closest approach, a, is one of the principal parameters used to describe the behavior of ion pairs. One measure of this is the dipole moment of the ion The first thought
concerning the relation between the observed moment and the distance of closest approach is that it is ea, where e is the charge on either ion of a symmetrical electrolyte. It has been pointed
(1) This work has been supported in part by a grant from the U. 9. Army Research Office, Durham, North Carolina. (2) J. A. Geddes and C. A. Kraus, Trans. Faraday Xoo., 82, 585 (1936).
(3) E. A. Richardson and K. H. Stern, J . Am. Chem. SOC.,82, 1296 (1960). (4) W. R. Gilkerson and K. K. Srivastava, J . Phys. Chem., 65, 272 (1961).
0111,~that
niorc realistic results arc nhtaiiied if the cffects of polarization are included iii the equation for the moment. I'carson has suggestedj that the correct approach l o the problem would be first to obtain an explicit solution for Onsagcr's distribution fuhction of the ion pair system in an extmnal electrical field,6 and then to calculate the average momtbnt of the pair using this distribution function. This is a report of results obtaincd as a consequence of Pearson's init ial suggestion. Thc work actually went beyond the original iiitcnt ions, hwliiig naturally, it serms to the author, to the final rrsults rcportcd here. Oiisagcr gavc 1ht. distribution funvtion in the form of eq. 1 f(r,79) =
n1722r-1
+ pl' cos 9 - al)r
cxp(2y/1.
(I)
where I =
lOZ'l
J " [ (- 86s)'/p cos $/a] csp( -
S/? ) (18
n1 and n2 arc the number of typc 1 and 2 ions per cc., T is the radial distance between the two ions, d is the angle between the line of rcnttrrs and the direction of the external field, E,, and , l o is thc Ucssel function of order zero. For the only c+:iscl wc consider here, that of symmetrical clcctrolytcs, wc also have 2q = z2e2/dcTl and 2/3 = zeh'/kT. S o t e that the factor 2q in eq. 1 is to be regarded as arising, not from Bjerrum's characteristic distance beyond which ions are no longer counted as pairs, but rather from Onsager's notation for expressing the reduced energy of interaction between two ions a t a distance r as 2p/r. The first part of the problem is to obtaiii an explicit solution for the intcgral I. Let 2
= [( -8ps)'/2 cos
9/21 = ( a s ) ' / z
Then
Now, making &e of the relation7 d [ z n J n ( x ) ] = z l ~ J ~ - ~dt, ( x arid ) integrating by parts, we find m
I
( 2 / a r ) n - l z n ~ ~ ( z )e x p ( - z 2 / a r ) l ~ ~ $
= 2a-1 tl
(2)
=1
That cq. 2 is r t d l y tho solution for the integral
I may be checked by differentiating cq.2 with respect t>o x . Xow, evaluating I between the limits arid substituting in cq. 1, we obtain f(r,79) =
cxp(f3r(cos 79
- 1))
presence of an t:xt,crnul field according to Onsagcr, one obtains his result, which is t o bc expected. The first term on the left in eq. 4 corresponds to a local Boltzmann distribution a t small values of r and results in orientation of the ion pair dipole in the field. The second term is due to the increased flow of ions in the direction of the field. Following Onsager, the mean relative velocitly is given by
(fia
= Icy'
+
(WI
u2) ( 1 '
VU -
(5)
V~I)
where fiis the distribution function givw by equation 4, w, is the ion mobility, and U is the rcduwd potciitial energy, given by 2q/y 2pr cos 29. Onsager evaluated the average rate of dissociation by taking the surface integral of the radial coiiiponent of cq. 5 . This yielded a value which of necessity had to be independent of the direction of the field. As we shall show below, however, the flow is in the direction of the field and thus leads to polarization. We now wish to calculate the 2, y, and z components (where the field E is in the z-+ direction) of the mean relative velocity, (flu) = fI(& ;tu zu), where;, and 5 are unit vectors in the .c, y, and x directions. We then shall show that the average over a spherical surface of thc first two vanishes, while the latter has a non-zero value. For small T
+
+ +
+ w2)(2qcos 8 [I + 2pq(cos 9 + l)l/r9 + 2Py(cos2 + 1)/vl (flu)= nlnzk2' ( w l+ w2)(2q sin 8 cos 9 [1 + 2&(cos 8 + - 264 sin 8 cos cos O / T ] (fw) = nln&!i" + 4 ( 2 y sin 79 sin 9 [1 + 2pq(cos + 1)],W - 2pq sin tp cos d sin 9 / r ) (6) (flu) = nlnzkil'
(a1
9
1)]/~2
(w1
$
Integration of say (fiw) over a spherical surface of radius r gives the net mean relative component of velocity in the y-direction. It is seen that J ( f i w ) i 2 sin 79 d 8 d 9 = J (f1u)r'sin 79 d79 d 9 = 0 This means that the components in the pohitive E a i d y dircc'tioiis balancae out those in thc opposite directions. This is not so in the case of the 2component. Using the distribution fuiiction given by Onsager,6 fi = n1n2,for free ions, where n1 and nt are the frec ion concentrations, the x-component of the mean relative velocity is given by (fzu) =
7 ~ l r i 2 k T(w1
+ ~~)[-2rj7cos 8 / r 2 + 2131
(7)
Sow we introduce the valuc for ?L1n2 a t small values of r in equation 6 as rL1n2
m
=
(YO
-~
) K o
(2(1/1.)"
where yo is the zero field ion pair concentration, AV is the increase in free ion conceritratioii due t o [28Q(COS s + l)]"/rn'(rn + n)1 (3) the field effect, and K Ois the zero field ion pair dism-0 sociation constant. The value of n1n2 in eq. 7 Xow, we really desire f(r,d) for small p. Then then is (no A v ) ~where no is the zero field concentration of the frce ions. We consider only the f ( r , 8 ) = fo exp(2prcos 8) - n&?pq(cos 79 + 1) (4) symmetrical electrolyt e case. The net component where fo = nln2 [exp(2q/~) - I]. This latter of the flow of pairs in the z-direction is then ( f l u ) result is correct t o the first p o m r of p. If eq. 4 (fit)). Kow, we will drop terms of the order of is used t o calculate the rate of dissociation in the PAY, and ( A v ) ~since , thrsc are second order. Also, ( 5 ) R. G. Pearson, p i ~ v a t ecommunication following Onsager, WC' will drop the I / r tcrm in (6) L. Onsagrr, J Chem Phys., 2, 509 (1934). comparison to the 1/ r 2 term, sinw most of thc eff (7) E. T. U hittukor and G. N. Watson, "hIodern i\nnl, w," Cainooriirs at small wlueq of r . \VP thni obtain bridnis Prrss, 4th ~d , 1952, p. 380. fllfl2
c
n=l
+
+
DIELECTRIC COXSTANT AND CONDUCTANCE OF ION PAIRS
April, 1962 ( f ~ )= k~ (wl
+ +
W : ’ ) { V O K O ~ &(cosz ~ 6
+
4co6 a)/rz -
+
(KO 2nd Av2q cos 6/rz 2p(no A V ) ’ ] (8) The last term on the right will be seen to correspond t o the ionic conductance of the free ions in the field. At equilibrium, the first two terms must go to zero, so that the equilibrium value of Av is given by
+
+
671
tion 10 is in suitable form to obtain A P in an alternating field. Now in a field of circular frequency u( = 2nf,fin cycles per sec.) E, = EOexp(iwt) Now d AP/dt = KlE0 exp(iwt) - AP/r‘
Av, = voi’co 2& (008 6 1 ) / ( K o 2no) The value of AV to be substituted into eq. 8 is not Av, but is a function of the time and direction.
The solution of this equation for A P , aside from a transient which rapidly disappears is
When the field is first applied, there is a charging current which disappears at equilibrium. We will suppose that Av = Av(t) h(6,q). Comparison with Av, shows that h = (cos 9 1). We are principally interested in the z-component of the flow of ions, not pairs. Now the average number of ions per cc. having the component of velocity in the z-direction of v is
The relaxation time for A P is 7’)the Langevin relaxation time. Separating O P into its real and imaginary parts, and using the relation to the increase in the complex permittivity,s Ae = 4 n A P I E = A d - i Ad’, we obtain
+
= f(fu)
dV
The volume V (== 4ns3/3) over which the integration is to be carried out is just the reciprocal of the free ion density, i.e., l/(no Av). Separating (fv) into an ionic term and a charging term we get (nu), = 1cT ( w i WZ) [ 1 6 ~ v o K o & ~ ~-/ 3(KOf 2no)
+
+
A v( t ) 8?rgs/3]
for the charging part. It is interesting to note that the value of the ionic part, at equilibrium, is just that given by Onsager.6
(Z)i= kT
(wi
+ w2)2@(no+ 8 3 , )
where
+
A;, = J A v , dV/ f d V = 2pqvoK0/(K0 2nd We thus see that the ratio of the equivalent conductance a t high field strengths to that a t low field strengths is given by
h(E)/A(O) = 1
+ ASe/n0 = 1 + ( 1 .-
-
c~)2Bq/(2 a )
for fields not too large. Returning now t o the charging portion, we may write = vok$[fQS/3 - (hi+ 2 ? ~ r ~ k A , ) v( ~ t ) / 3 (9) where IC, = 8nplcT(wl w2),the specific rate constant for ion recombination, in cc./ion sec., and 1Cd is the specific rate constant for pair dissociation. Now (G), is the x-component of the mean ion flow due to the charging process. The specific relation between this quantity and AVmust be found before a solution of eq. 9 can be written down. Onsager’s method of calculating the increased rate of dissociation due to the field is to integrate the normal component of the flow, f1;, over a spherical surface. If in this manner we calculate the rate of dissociation due t o the x-component of the flow, we obtain
(z).
+
+
f ( f 1 v ) cos 6 rz sin 8 d 6 dq = dcd(l 2pg)/3
This result is one-third of the total sate of dissociation. Thus
(&XI
=
( s / 3 ) d A v/ d t
Now, inserting this result in eq. 9, and letting A P = xes Av/3, then we obtain d A P / d t = v0k~2fiq.zes/3- (kd 4- an&,) AP ( 1 0 ) where A P is the increased polarization density of the solution due to the development of Av. Equa-
AP = K1r’EO e x p ( i w t ) / ( l
Ae‘ = 4 k K 1 r 1 / ( 1 $-
+iw~’)
w’T’’)
Ae” = 4?rK10~’’/(1 $- d r ” )
(11)
where AE‘ is the increase in dielectric constant and A€‘’ is the increase in loss factor due t o the ion pair relaxation process.
Discussion We already have seen that the increase in ionic conductance due to the field effect calculated by the foregoing method is the same as that found by Oiisager, Le., A(E)/A(o)
=
1
+ ( 1 - a)2pg/(2 - a )
The present treatment goes further and yields the effect of frequency on low field dielectric constant and loss factor due to relaxation of the ion pair equilibrium. These effects will be discussed in relation t o the Debye-Falkenhageng effect and the power loss due to ion pair relaxation as calculated by Pearson.10 Taking the latter first, the power loss, 4, in ergs/l. sec., as obtained by Pearson, is related to the loss factor, Ad’, by
+ = 1000wEo2~er~/8T We then see that Pearson’s loss factor is A d ‘ = 64z2e2qzN,cao ( 2 w r t ) / 9 0 0 0 ~ k T( 1
while eq. 11gives
+ 4w2rt2)
(12)
+
Ae” = ~ P ~ ~ ~ ~ ~ s ~ ~ . c ~1 ~ ww’rt2) T ’ / ~( 1O3 )O O ~ T (
where c is the stoichiometric salt concentration, a0 the degree of dissociation, and the other factors have their usual significance. The differences between eq. 12 and 13 arise principally from Pearson’s assumption that the only energy absorbed from the field was that in increasing AV during the charging process. He neglected the energy dissipated in actually transporting the ions from the origin t o the completely dissociated state. This is most apparent when one considers his statement regarding the corresponding calculation of 46 from the current-voltage integral. The voltage drop for dissociation is not ‘‘clearly Ep.” It is true that eq. 11 suffers from some uncertainty regarding the integration of eq. 8 over the volume. The distance parameter, s, thus introduced, should (8) C. J. F. Bottcher, “Theory of Dielectric Polarization,” Elsevier, New York, N. Y., 1952,p. 232. (9) P. Debye and H. Falkenhagen, Phyaik. Z., 29, 121, 401 (1928). See also “Collected Papers of P. J. W. Debye,” Interscience Poblishers, Ino., New York, N. Y., 1954, p . 374, ff. (10) R. G. Pearson, Discussions Faraday Sac., 17, 187 (1954).
672
W. R. GILKERSON
Vol. 66
dielectric increments fall to one-half their low frequency value a t about the same frequency, and with the assumption that SK = 1, have about the same magnitude. The justification for assuming that KS = 1, rather than s being the distance calculated from a cube-root law is the same as that involved in the original Debye-Huckel derivation’l of the ion atmosphere potential. The conductance functions, on the other hand exhibit different behavior. We will discuss these in the forms (assuming SK = 1)
1.0
0.5
0 -1
0
1
2
Log ( W T ) . Fig. 1.-Frequency dependence of dissociation relaxation effect and Debye-Falkenhagen effect. Descending solid line has !(UT’) as ordinate. ascending solid line is ~ Z d 2 f ( ~ 7 ’ ) ; descending dashed fine is g ( W T ) ; ascending dashed lme is (1 h); all plotted aa a function of frequency.
-
be some sort of average distance the dissociating ions attain during the charging process. There is a point which the author feels is strong justification for setting 4ns8/3 = l/(no Av). This choice results in a value for the free ion current which is in agreement with Onsager’s. In connection with this, it might be noted that if one confines the volume of integration to this value, then Bjerrum’s distribution function, nln2 exp( -2q/r), is normalized in a satisfactory manner. The frequency dependent terms in eq. 12 and 13 show that Pearson’s relaxation time is twice that in eq. 13. This is a result of using the absolute value of the field in his calculation. It is true that the q u a n t i t y x must be independent of the direction of the field, from the manner of its definition, but the charging current is dependent on field direction, and it is this latter quantity which determines the power loss in the field. Comparison with the results of Debye and Falkenhagene for the dispersion of the conductance and dielectric constant of a strong electrolyte shows some interesting parallels and differences. Let subscript 1 denote the Debye-Falkenhagen result, while 2 denotes that in eq. 11. Then, for the dielectric increment we have
+
Ae’l = Z % * ~ g ( w ~ ) / 2 4 k T
As‘z
22~2KzSf(WT’)/24kT
(15)
where ~ ( w T ’ ) = 1/(1 3. ~ ~ 7 . ’ Recalling ~ ) . that s is the result of integration over the volume containing a free ion, if we set s to be 1 / ~ which , is the average radius of the ion atmosphere, then AE’z/AE’I = f ( W T ‘ ) / g ( W T )
The two functions, ~ ( o T ’ ) and g ( w T ) , are plotted vs. log (07) in Fig. 1, as the quantities decreasing as log ( W T ) increases. For slightly dissociated electrolytes, 7 = 27’. It is seen that even though the relaxation times, 7 and TI, are different, the
31
(AG/G&
=z
&?K(
- h)/10.24q,kT
1
2’e2KWPT’2f(WT’)/12(;okT
{ 16)
(17)
where AG/Go is the relative increase in the conductance of the system, Go being the low frequency value, h is the quantity . A I ~ / A = I ~ 0 tabulated in Harned and Owen,’2 and eo is the solvent dielectric constant. The remaining are the same as in eq. 14 and 15. The magnitude of the two conductance increments are seen to be the same, except for the frequency dependency. The latter may be compared in Fig, 1 as the ascending curves with log (UT). It is seen that the ion pair relaxation effect reaches its maximum value faster than the DebyeFalkenhagen effect. An important question occurs a t this point: do the two foregoing effects occur under the same experimental conditions? I n the case of a weak electrolyte, the average lifetime of a free ion is n = l/k,n. Now if k, is given by Onsager’s value, then ~i is just the characteristic time of formation (or relaxation) of the ion atmosphere. It might appear that under these conditions, for a given ion, the only atmosphere it sees is that due to the oppositely charged ion from which it has just dissociated. But other free ions are being formed in the neighborhood also, so that on a time average basis, one would expect the Debye-Falkenhagen effect to be additive to the ion pair relaxation effect. One still might expect strange behavior in systems where the DebyeHuckel limiting law cannot apply at accessible concentrations such as solvents of low dielectric constant. In systems where the degree of dissociation is large, then eq. 13, 15, and 17 must be modified to remove the assumption that a is much less than one. Then 7’
(14)
where K is characteristic of the ion atmosphere and g ( w 7 ) is a complicated function of w and the ion atmosphere relaxation time T . Now when the degree of dissociation is small, we find
( AG/G&
(1
-
CY)T/(2
-
CY)
(1 -
a)T/2
and i s much smaller than T . As a approaches unity, the ion pair relaxation effect disappears, so that only the Debye-Falkenhagen effect remains. As an example of the expected effect of dielectric constant on the magnitude of (AG/G&, the results of calculations for 0.001 M tetra-n-butylammonium picrate in benzene-0-dichlorobenzene solvent mixturesls for W T ‘ much greater than unity are shown in Fig. 2 as a function of solvent dielectric constant. The ion pair dissociation constants range from 1.9 X 10-5 in o-dichlorobenzene to (11) P. Debye and E. Hackel, Physilc. Z.,24, 185 (1923). (12) H. S. Httrned end B. B. Owen, “Physical Chemistry of Electrolytio Solutions,” Reinhold Publ. Corp., New York, N. Y., 1950, 2nd Ed,, p. 132. (13) W. R. Gilkerson and R. E. Stamm, J. Am. Chem. SOC.,82, 6295 (1900).
April, 1962
DIELECTRIC CONSTANT AND CONDUCTANCE OF ION PAIRS
1.8 X in benzene. Note the maximum. This is due t o the fact that (AG/Go)z is proportional to the half-power of the ion concentration (which increases with increasing EO) and is inversely proportional to the three-halves power of the dielectric constant. Comparison with Experiment -Since recent data from this Laboratory will be the subject of another paper we confine ourselves here to a comparison of the calculations based on eq. 11 with those of Pearsonlo and the meager data available in the literature. It first must be noted that observations reported by this author14 on boric acid in water must be regarded as in serious error. They cannot be reproduced. Efforts to find the source of the original absorption have proved fruitless. The earliest available data are those of Whitmore,16 as cited by Pearson,lo on the conductance of propionic and tartaric acids in .water relative to potassium iodide a t 400 Mc. Assuming for the propionic acid an ionization constant of 1.3 X 10-5 and a recombination rate constant of loll M-l see.-' (as Pearson did) then for a 0.1425 M solution, using eq. 17, we calculate a relative increase in conductance of 0.72y0. This is to be compared to Pearson's result of 0.1% and an observed increase of 1.1% for this concentration. There is enough scatter in the observed results to render any hope for a more detailed comparison fruitless. Cole and Strobel16 reported that tetra-n-butylammonium picrate (9.55 X M ) in anisole a t 25" exhibited a minimum in A d ' around 10 Mc. I t has been suggested1' that the increase in A€" toward lower frequencies possibly is due t o ion pair relaxation. We subscribe to this and further will include the Debye--Falkenhagen effect. From low frequency conductance datal8 and assuming that all dissociating species behave identically in an external field, we calculate, at 1 Mc., A d ' = 0.037 due to Debye-Falkenhagen effect, and de" = 0.078 due to ion pair relaxation. At 1.8 Mc., the lowest frequency observed, the A d ' was about 0.05. Since we calculate that the maximum should be around (3.8 Mc., the foregoing calculated values seem reasonable. Using Pearson's equation, one would calculate a value of A d ' = 0.024 due to ion pair relaxation. Comparison in this case, where the daka are good, is difficult because of the complexity of concentrated electrolyte solutions in solvents having dielectric constants as low as anisole (4.30). Other maxima in the loss curves observed by Cole and associates16 are a t such high (14) W. R. Gilkerson, .I. Chem. Phys., 27,914 (1957). (15) B G.Whitmore, Rhysik. Z., 34, 649 (1933). (16) R. H. Cole and H. A. Strobel, Ann. N. Y. Aead. Sci., 64, 807 (1949).
(17) M. Davies and G. Williams, Trans. Faraday Soc., 66, 1619 (1960). (18) G. Rien, C. A. Kraus, and R. M. Fuoss, J . Ana. Chem. Soo., 56, 1860 (1934).
673
5 10 Dielectric constant. Fig. 2.-Relative maximum increase in conductance, calculated from eq. 17, of tetra-n-butylammonium picrate in benzene-o-dichlorobenzene as a function of solvent dielectric constant. 0
frequencies that they could not be due to the simple process trsated here. The same may be said for recent observations" on 0.01 M solutions of tetran-butylammonium bromide in benzene where a maximum was found around 7 Mc. Conclusion The definite integral form of the distribution function for ion pairs in an external field, as given by Onsager,s has been integrated to give a series form. This result has been used to derive equations for the increase in dielectric constant and conductance of a weak electrolyte in an alternating field due to relaxation of the ion pair equilibrium. These results are felt to be an improvement over Pearson's equationlo for the conductance increase. There are three principal differences in the two approaches: the dielectric increment appears in the present work as a natural consequence of calculating the displacement current (it is absent in Pearson's treatment); the relaxation time for the present result is just the Langevin time T', not 27' as in Pearson's; the magnitude of the effect for small degrees of dissociation has prospect of being much larger than Pearson indicated since the ratio of the maximum value of the conductance increase as calculated here to that calculated by Pearson is 1 / ~ q . For instance, in water at 25", for an ion concentration of 10-5 M , 1/Kq is approximately 300. The theory appears to be in agreement with the two experimental results known to the author. Much experimental work needs to be done to indicate the approaches that must be taken to improve the formulation of the problem. The uses to which dispersion measurements could be put in the kinetics of fast ionic reactions already has been discussed by Pearson. lo
