T H E FREQTTESCT D E P E S D E S C E O F T H E ELECTRICAL CONDUCT-ISCE OF SOLVTIOSS OF GTROSG ELECTROLYTES* BY H A N ~FALKESHAGES'
ASD JOHS TVARKES
ii-m,uiis2
I. Introduction The interionic attraction theory of the behavior of ions in dilute solutions, based on the fact that the thickness! K of t,he ionic atmosphere of mean chargc surrounding each ion has to be proportional to the square root' of the concentration of the solution if the principal forces between ions are the ordinary C'oulomb forcesi, has been utilized by Debye and his students3 to show that the square-root law, found experinimtally by KohlrauschIi is the correct one to explain the change in electrical conductance of dilute solutions of electrolytes with concentration. It, v a s shonm that the ionic atmosphere, built about each ion, is characterized by cert,ain properties; for example, it has (as already mentioned) a definite thickness which is of extreme importance not only for the simple conductance theory but also for the activity theory5 now so well known in phyeics and chemistry. The ionic atmosphere possesses, in addition, a finite time of relaxation. This atmosphere can neither be destroyed nor created in an infinitely small period of time. If an ion is suddenly separated froni the solution, the regularity in its neighborhood nil1 cease t o exist because it oives its presence only to the force field of the central ion. The transfer to the regular orientation, with reference to the point where the ion was, will take place gradually. The time necessary for this change is the time of relaxation, and is dependent on a number of factors. Its order of magnitude in seconds has been shown? to he
n h t w y is the concentlation of the solution in moles per liter. The values of for y = G . O C I in wvnter at 18°Cfor several electrolytes are given in Table I. Included in this table are corresponding wave lengths whose significance will he recognized later. * Contribution irom t h e Physikalisches Institut der 1-niversitiit, Leipzig. Ft,llon. of the International Education Board. S a t i o n a l Ilesenrch Council Fellow. Dehye and Hiickel: Physik. Z., 2 4 , 305 (1gz3"; Onsager: 2 7 , 388 (1926:; 2 8 , 277 (1927). ,See also Hiickel: Ergeb. esakt. Saturniss.. 3 . 199 1924 ; Baars: Handbuch der Phvsik von Geiger-Scheel, 13: 35; (1928 ; TVilliams a n d Falkenhagen: Chemical Reviews, F i r t h caoming publication. Fee Iiohlrausch u n d Holborn: "Das Leitvermogen der Elektrolyte" (1916!. j Debye a n d Hiickel: Physik. Z. 2 4 , 185 (1923:; Debye: 25, 9; (1924).
'
1122
HAXS FALXENHAGEN AXD JOHX ‘TARREF ’TILLI4hlS TABLE
I
Values of the Time of Relaxation, Electrolyte
e.
Time of Relaxation
Corresponding Wave
Lengths
KC1 HC1 RIgC12 CdS04 LaCl3
0.5j3
X 10-’sec.
0 . I89
0.314
x x x
”
0.208
X
”
0.323
1 6 . 6 meters j . 6 7 ’’ 9.69 ” 9 . 4 4 ,’ 6.24 ”
”
Recently it has been shown that as a necessary consequence of the existence of this finite time of relaxation, an electrical conductance of dilute solutions of strong electrolytes which is dependent on the frequency must result. This effect has been discussed previously by Debye and one of US,^ and by ourselve~.~ The present paper reviews in some detail this important development; in addition a considerable amount of original material has been added. Before proceeding with the discussion, it may be interesting to state that the effect predicted by the theory has been shown to esist by Sacks in preliminary measurements made in this laboratory. The experimental study of this effect is being continued both here and a t the University of Wisconsin.
11. Discussion of Frequency Effect Consider an ion, surrounded by its ionic atmosphere, to be suddenly removed from the solution. The regular distribution of the ions in the neighborhood of the ion can no longer exist, because the central force-field, which is the cause of the regular distribution, has been removed. The change of the distribution of the charges of the ionic atmosphere to a random distribution with respect to the point where the central ion was, will take place gradually. The disappearance of this equilibrium position has been given mathematically in the original articles,6 it is sufficient for our purpose to give the result by means of a graph (Fig. I ) . For the quantity of electricity, dQ, which lies between spheres having radii, r and r dr, the following relation holds
+
where
Debye and Falkenhagen: Physik. Z., 29, 121 (1928). Falkenhagen and Williams: Z. physik. Chem., A137, 399 (1928 8 Sack: Physik. Z., 29, 627 (1928) 6
7
I
.
FREQUENCY DEPESDESCE O F ELECTRICBL CONDUCTASCE
t
and
T = =
e
where
I 123
1 e- = pi+p2 2kT PIP2
The quantity y is a measure for the density of the ionic atmosphere a t different times and in a shell of thickness dr a t different distances r from the ion in consideration. In Fig. I is shown y for T = 0,T = 0.2 j and T = I as a function of KT. For each time y has a maximum value which decreases as Kr is increased. Or, it may be said that the maximum of the thickness y at time t = o lies at K r = I , a t time t = 0 . 2 5 8 a t Kr 1.5 and finally at time t = 6 a t Kr 2 . The maxima themselves for these times are given approximately
-
-
FIG.I
by the numbers 0.36, 0.31, and 0.11. Fig. I shows how the density in the ionic atmosphere at the time 4 8 is small compared to the original density a t time zero. After the four-fold time of relaxation, 4 8 , the density has already disappeared. The time of relaxation 6 is, therefore, a measure for the disappearance of the equilibrium condition. I t has already been shown6 10-10
that it is of the order of magnitude - sec. In the treatment to follow it Y
is useful to introduce the time of relaxation 8 which is defined by the relation -
8
e , where q is given by the equation
=-
9
(3)
where E, and Lz are the mobilities of ions I and 2 , and z1 and z2 are their valence? (See Reference 3 ) . It is of course evident that either time of relaxation could be used in any discussion of this sort. * The reason why a dispersion of the electrical conductance must exist may be shown in the following manner. If an ion is in motion due to the action of an external electrical field, there will be a density of charge which is too small before and too large behind any ion upon which attention is fixed. Therefore, there will be a dissymmetry of the ionic atmosphere which becomes * For a detailed consideration of these two times of relaxation see Reference 7.
I124
HASS FALKESHSGEN A S D J O H S TT'ARRES KVILLIAMS
more and more important the greater the average velocity of the ion is. The result is a resisting force which has been called the electrical force of r e l a ~ a t i o n . This ~ electrical force of relaxation appears in the calculation ab a decrease in the mobility of the ion. It must be emphasized that the calculation of the dissymmetry can give an approximation only for small ionic velocities. In the usual cases an approximation is sufficient, but the case is different when one, as has been done by JViene* in his recent experiments, causes the existence of abnormally large ionic velocities by great field strengths. The order of magnitude of the velocities in the \Tien experiment was I nieter per second contrasted to the usual velocities of the order of magnitude, 0.01min sec. I t can be readily shown that ions which have these high velocities d l travel many times the thicknesses of their ionic atmospheres in the time of relaxation. Under these condit,ions, then, the ionic atmosphere can hardly be built. Therefore, in very strong fields the force which has been termed? the additional electrical force of relaxation is of little or no consequence, and the conductance will approach that value found at infinite dilution. This deviation from Ohm's law is exactly the effect discovered experimentally by Wien and explained in a more or less quantitative manner by Joos arid Blumentritt.'O The W e n Effect is illustrated by Fig. 2 in which the values of the field strength, 9, are
-i, -Ax plotted as abscissae against values of the function __-A%-Ax
as ordinates. = 0
If now, the experiments of Wien can be understood in t,he way which has been given, then the existence of the effect which is to engage our attention in this section is to be suspected. If it be assumed that an outer electrical field of oscillation frequency w acts on the ions in the solution, then apart from their Brownian movements there is imparted to the ions a back and forth (periodic) movement. If the frequency of the field is small, the ionic atmosphere will have in each moment a dissymmetry of distribution of charges which corresponds to the momentary velocity of the ion. I n other words the additional electrical force of relaxation n d l be the same for small frequencies of the outer field as it ITas for the stationary caye. In the other extreme case, where t,he frequency of the field is extraordinarily great conipared to I , e, the situation must be different. Fixing our attention on an ion in motion, under these conditions it may be shown thatfi,'the dissymmetry of the ionic atnioFphere cannot be built. This s h o m that if the frequency is great enough, the electrical force of relaxation disappears. Since the electrical force of relaxation is a force, which in the case of the frequencies ordiK i e n Ann. Physik, 83, 327 (1927); Physik. Z.. 28, SJA (1927,. See also Reference ;. * The discussion of the TVieri experiments docs not rightfully belong in a n article on this subject. It is introduced a t this point to assist the reasoning which leads to the explanation of t h e dispersion effect. Joos a n d Blumentritt: Physik. Z., 28, 836 { ~ g z i ) . * Professor X e n has very kindly sent us the results o i his most rerent experiments which are in good agreement Yvith th; requirements of the theory.
F R E Q U L S C Y D E P E S D E S C E O F ELECTRICAL C O S D C C T A S C E
1125
narily used for the ni,easurement of conductance, operates to resist the passage of an ion, it is evident that an increase in the conductance appear vvhen such frequencies are used that this force disappears. In other words, a dispersion of conductance must exi.st. In this article only the results of the calculations for the general case of an unsymmetrical electrolyte, taking into consideration the Brownian movement of the iom, will he presented. Further, a discussion of a number of factors which influence the phenomenon of the dispersion will be given. The molecular conductance, ,Ias a function of the frequrncy may now be nritten .-1 = A* - - 1 I W - -111 ( 4) where 3, = equivalent conductance a t infinite dilution.
__ A d K
A;A,=,
FIG. 2
electrical force of relaxation for frequency w,expressed as conductance.
ordinary force due to electrophoresis (See reference 3 ) > expressed as conductance.
D
=
k
=
T n,
=
=
77
=
el
=
y
=
dielectric constant of the solvent. Boltzmann's constant = 1 . 3 7 2 X 10-l6 erg 'degree. absolute temperature. Sumber of ions of the i t h kind in I cc. internal friction constant of the liquid. charge on ion of the ithkind. concentration in moles per liter of solution.
1126
H A S S F A L X E S H h G E S AND J O H S WARREN WILLIAMS
The dependence on frequency appears in the term X which is in turn dependent on w 0 and q. h knowledge of the time of relaxation is also necessary for the dispersion effect. I n the case of zero frequency the expression for the molecular conductance goes over to that of the original conductance t h e ~ r y . ~ It is thereby assumed that the electrophoretic part of the conductance is independent of the frequency. I n order to recognize the effect of frequency most clearly, it is desirable ~Iw/,&Q. The following method serves for the calculation of the dispersion effect for any simp!e electrolyte, provided solvent, temperature, electrolyte, and concentration are given. The valencies z1 and mobilities E, must also be known. According to the formulae
to examine the ratio
For water at 18'C K?
= 0.05332
K?
=
x
1oI6
y Z v,z,?
16a)
y
(6b)
at 25°C o.oj38j X
1 0 ~ ~v,z,?
K?, and 0 are determined. The quantity 1 1 11~ ~depends only on q and w e . I n Table I1 is given an interpolation table which is extremely useful;
q,
'1x1~
it contains values for the quantity L&w as a function of w 0 for five different values of q. The molecular lowering of the conductance due to the electrical force of relaxation for the stationary case AKro, and the lowering due to the electrophoretic effect, i%lare l calculated by means of the formulae
111. The Influence of Various Factors upon the Ratio ~ I w / ~ I , The influence of concentration, mobility of the ions, dielectric constant, temperature and the valence upon the function will now be discussed. The values of the valences, mobilities, and the temperature used in these sections are given in Table 111. The electrolytes are HC1, LiC1, KCl, MgC12, CdSOI, LaC13, K4Fe(CK);)6,and Ca2Fe(CKj6.
*x~~/~&~
According to this table values for the mobilities are used which agree with the best values given in the literature to within 1 % ; they are, therefore, sufficiently exact to be used to illustrate the effect of dispersion. The value
FREQUESCY DEPENDENCE O F ELECTRICAL CONDCCTAXCE
I127
TABLE 11* Interpolation Table-Values I .oo
I .oo
I .oo
0,999 0.997 0.990 0.981 0.959 0.934 0,909
0.999 0.997 0.990 0.981 0.962 0.938 0.913
0.882
0.888
0,999 0.997 0.990 0.981 0.965 0.942 0.918 0.894
0.833 0.791 0.753 0.691 0.603 0.540 0.496 0.41; 0.368 0,334 0.307 0.286 0.267
0.841 0,799 0,763
0.848 0.808
0.999 0.997 0.990 0.981 0.968 0.946 0.922 0.900 0.856 0.817 0,785
0.701
0.614 0.553
0.713 0.628 0.565
0.507
0.520
0.428 0,378 0.344 0.295
0.442 0,390 0,354 0.327 0.305
0.277
0.287
0.254
0.263
0.272
0.318 0.299 0.288
0.242 0.199 0.173
0.250
0.259
0.271
0.206 0.179
0.214
0.223
0.142
0,147
0.1232
0.12;s
0.19; 0.160 0 . I394
0.1011
0.1046 0.0812 0.0688 o.ojj6 0.02j8
0.186 0,153 0.1328 0 . I088 0.0844 0.0716 j 0,0599 0.0269 0.0190
0.0199
I .oo
I
0.1
40
0.999 0,997 0 ' 989 0,980 0.956 0.930 0.904 0.876 0.826 0,783 0.745 0.682 0,593 0 . 531 0.486 0,409 0.360 0.326 0,299 0.279 0.261
45
0,247
50
0,235
75
0.193 0.168 0.138 0.1195 0.0983 0.0759 0,0643 0.0538 0.0241 o . O I706
0.35 0.5 O.i.5
I I.2j
1.j 2
2.5
3 4 6 8 IO
15 20
25
30 35
IO0
150 200
300 j00
j 00 1000 j000
*
0.0783 0.0663 0.0555
0.0249
0.317
0.01824 T h e values given in this table are accurate to within I R.
10000
q = 0.30
.oo
0
0.2
h~~/d~,
q = 0.40
q
= 0.5
of
q = 0.4j
Le
0 . 0 1 j. i. ;
q
=
0.35
0.773
0.727
0.643 0.582 0,537 0.458
0,406 0.369 0,342
o.114j
0.0888 0.OjjI
0.0629 0.0282
of the molecular conductance a t infinite dilution for K4Fe(CN)6is taken from an article by Burrows.11 At temperature 2 j°C this investigator gives the value 680. If the mobility of the K+ ion a t 25°C is 75, then that for the Fe(CS)s=- is 380. In Table IT the characteristic quantities K?, q, and 87 are given. The values for the thickness of the ionic atmosphere I , ' K for different concentrations may also be taken from this table. Burrows: J. Chem. SOC., 123,
2026
(1923).
1128
HAKS FALKENHAGES . W D JOHS T A R R E S WILLIAMS
TABLE111 Xobilities of Certain Ions Ion
H K c1 Li
L,
t”C
I
315
I
65 65
18 18
I 2
SO4 Cd Ca La Fe(CX)
18
18
33 9’
I
wx
-
2,
18 18 18
136 6
2
2
92
2
I20
3 4
150
18
3 ‘30
25
2:
I ___f
Ltnm
FIG.3
TABLE I\’ Yalues of Electrolyte
HC1 KC1 LiCl MgClz CdSO, LaC13 K,Fe ((2%) CazFe(CS)8
q and 8
K?,
!ISF4;
q
K?
h
3 06
0 Io;
3 06
x
I0’6-f
0 . j
BY
0.189 X
IO-’O
107
”
0.5
,553
”
3 06
107
”
0.5
,732
”
77 53
321 428
”
0.421
”
”
0.5
,323 ,314
25
642
”
0.3j2 0,344 0.480
,208
”
.I02
”
,113
’’
1 1
0
96
o 88
I
08
I 29
‘I
‘’
”
FRCQCESCY DEPESDESCE O F ELECTRICAL COTDCCTlNCE
1129
Influence of the Concentration on the Dispersion of Electrical Conductance For the purpose of the study of the effect of concentration solutions of KC1 in water at 18OC viill be considered. (Dielect'ric Constant of the solvent = 81.3). According to Table 117for concentrations y = 0.01, y = O . O O I , and y = 0.0001 the corresponding times of relaxation are 8 = 0. j j 3 X IO-^ sec., 8 = o . j j 3 X 10-7 see., and 8 = 0.jj 3 x 10-6 sec. -According to Table IT the values of K? are, respectively, K? = 0.10; X IO-^^, K? = 0 . 1 0 ~X 1 0 - l ~ ~ and K ? = 0.10; X 1 0 - l ~ . Xs KC1 is a symmetrical electrolyt'e q = 0.j. Therefore, by means of the interpolation table values of ATIJATIs as a function of w e may be writt?n. Fig. 3, in IThich the vave lengths in meters ob-
&l o
i 95
FIG.4
tained from Table I1 are plotted as abscissae ivith the corresponding values as ordinate., s h o w the effect of concentration on the dispersion of effrct. For the value '-TI, = 0.; the wave lengths 1.16 m, 11.6 m, and 116 m correspond to the concentrations y = 0.01, y = 0.001 and y = 0.0001 respectively. For thrse wive lengths the dispersion effect has reduced the electrical force of relaxation to half its value in the stationary case. The effect of concentration on the dispersion effect is most simply expressed by saying that the t i n w of relaxation are inversely proportional to the concentrations. Influence of the Mobility The influence of different mobilities on the dispersion effect can be described by means of 0.001molar solutions of HC1 and IiCl in n.ater at I ~ T . From the tables the corresponding molecular conductances at infinite dilution are am^^^^ = 3 8 0 ; LKCI = 130
Ax~w,'Lx~,
*xri.
By means of Table IT- the times of relation are
OHCI = 0.189 x
10-7
sec.;
eKC1 = o.jj3 x
~o-~sec.
I
130
HANS FALKESHAGEK A S D JOHN WARREN WILLIAMS
The value of q is again 0.5. From the interpolation Table I1 and according to Table IV, the values of ~ I ~ /are~ given & ~ as functions of the wave length. The result of the calculation is given in Fig. 4. It is a t once evident that for the same wave length the values of l&w/I~, for HC1 are much smaller than for KC1.
Influence of the Dielectric Constant I n this case 0.001molar solutions of XCl in the solvents water and methyl alcohol a t 2 5 O C will be considered. The dielectric constants are, respectively,
DH-o= 78.8;
DairoH = 3 0
M
1 07
02
04
OS q8
7
2
FIG 5
The molecular conductances a t infinite dilution are i i a H 0 = 150; r i , ( \ f e O H ) = 108 The latter value was taken from the research of Fraser and Hartley.12 The times of relaxation are OHOo= 0.466 x 10-’sec. &OH = 0.245 x 10-’sec. Again q = 0.j . By means of the interpolatlon table A%Iw/-xIo as a function of the wave length is obtained. The result is shown in Fig. j . I n the case of the methyl alcohol there corresponds to the same XI^ value a smaller wave length
Lx~w
Influence of Temperature To illustrate the effect of temperature data for water solutions for LiCl a t 18OC and 100°C will be utilized. For the mobilities of the ions the following values are used Fraser and Hartley: Proc. Roy. SOC.,109.A, 3j1 (19zj).
FREQUENCY DEPENDESCE
OF ELECTRICAL COSDUCTASCE
Li+ a t 18' Li+ a t 100' c1- a t 18' C1- at 100' Equation 6 gives for the
K?
values
&" (4)o
I
4
10
FIG.6
The corresponding times of relaxation are
= 33 = 117 = 6j = 208
I 13 I
H B S S F h L K E S H h G E S A S D J O H N W A R R E S TT-ILLIAUS
1132
Lx~w
As in the previous cases '-TI, can now be expressed as a function of the wave length. The result for y = 0.0001is given in Fig. 6.At the' higher A&o temperatures appreciably smaller wave lengths go with the same 11, values. Influence of Valence To this point symmetrical electrolytes have been considered. Considering the case of unsymmetrical electrolytes, it will be seen that the valence also affects the dispersion of conductance. In this section are considered electrolytes of the valence types 1-2) 2 - 2 , 1-3,1-4,and 2-4for the concentration y = 0.0001.For unsymmetrical electrolytes q f o . j . According to Table IT the times of relaxation for the various types differ greatly from one another. The differences in q values, and especially the differences in times of relaxation govern the influence of the valence. In Fig. ;! are given the results of the calculations. In order not to unnecessarily complicate the figure values for KC1, CdSO?, LaCl3, and I&Fe (CS)6 only have been shown. The dispersion curve for MgCl? falls almost exactly along that for 3IgSO+ very slightly t o the left. The dispersion curve for CaiFe(CS)e would lie between the curves for K,Fe(CS), and LaC13. The following Table V gives the values of 31~ for different wave lengths for the different electrolytes which have been considered.
Lx~u,
TABLE
Talues of Kave Length in Meters
Kave Lengthq
3IgS0,
lIgC12
LaC13
0.0jj
O.O~O
0.0.jo
0.08
2
,073
. I02
. I2
5
,115
,098 'I54 ,216
,162
.I9
.22j
.28
I
IiCl
Y*
-31~ 31,for Different
.163
IO
Ca2Fe(CS)BI < J ' e f C S
0.116 ,167
0.138
. I90
,260 ,353
,29j
.40
20 50
IO0 200
500 1000 2000
5000 10000
I
I
* The values given In Table \
I
I
I
I
are accurate to a i t h i n z C c
The values of -IIo are known froni equation l a . In Table TI have been for the different types of electrolytes studied and collected values for ll~o From this table the quantities and Axll for each concentration may be obtained at ome.
Lx~o
I I33
FREQUEKCT D E P E S D E S C E O F ELECTRICAL COSDUCTANCE
TABLE VI
Values of Valence T? pe
XI -and xII(Different Valence Types) -
47 ”
2-2
I’
o 677 I79
1-3
743
0
”
I j z
1”’
1890 o 248j o
”
2 1 2
”
”
6 32
262
1-4 2-4
18 18
0 224xod/i
4 404 o
1-2
t”C
-11,
I l l
50 5
1-1
”
”
I8
$”)
18 25
”
’’ ’
25
If the molecular conductances at infinite dilution for the strong electrolytes considered in this section are introduced into the above table, Table VI1 results. There have been included in this table values of the quantity -XIo. TABLEVI1 Values of 311and A&o (Different Electrolytes) Electroh te -1m Ill 1I t”C KCl HCl LlCl lIgC12 NgS04 Lac13 Ii,Fe(CK)s CasFe(CK)O
130
50 5 4 f
29
380 98
5 50 5
8j
222
50
262
4
‘(
Ijo
404
0
743
0
335 680
1890 o
620
248; o
” 1L
L( LL
((
229
Id?
2 22 0
((
lL
310
524 1440
l(
3920
“
(1
I8 18 18 18 18 18
(1 ll
25
25
Finally it is of inteiest t o make certain remarks concerning the magnitude of the dependence of the electrical conductance on the wave length or frequency used The magnitude is recognized most simply when the difference betnern the molecular conductance for the frequency w and that for the frequency zero is compared to the molecular conductance at infinite dilution In Fig 8 are included data for solutions of CdSO, at 18°C and K4Fe(CN)e at 2 j”C, y being o 0001 in each case. For the CdqO, solution the moleculai conductance is 2 2 8 6 according t o Table 111 The .3lI value is 4 04 ( I g c 1 of and .xl0 1s 109 ( I 8% of *ye). The total lowering of the conductance is, thrrefore, 3.6cC An increase in the molecular conducterved using the wave ance at this concentration of about o length 1 = appioumatcly 60 m for wh The euampli. of X,Fe(CS I 6 is similar. The nioleculai conductance a t infinite dilution ( 2 j T ) is -Is = 680
(Axm)
Further The total lowering of the niolecular conductance
IC
1.8‘;.
An increase of
H A S S FALKENHAGEN A S D J O H S W A R R E S WILLIAl\lS
1134
about 170 in the molecular conductance should be observed using the wave length I = approximately 16.3 m, for which .%~w’ = 0.5. The maximum dispersion effect (extremely short wave lengths) to be observed would be of the order of magnitude of zTc. At higher concentrations, using suitable wave lengths, a greater increase in the molecular conductance should result. However, since the theory in its present form is applicable only to very dilute solutions, it is e\ldent that one cannot resort to measurements upon these more concentrated solutions to
P
I
,
, ! I , ,
1
4
IO
40
100
400
1000
-L
4000
10000
inm
FIG.8
verify the theory. The authors believe that when the terms of higher order have been considered the concentration region which may be treated will be increased, though one may not say how much a t this writing. It may be recalled that the activity theory has been treated in this manner by several investigators with the result-apparently a t least-that the concentration range to which it may be applied has been considerably broadened. The authors wish to thank Professor P. Debye of this Institute for the interest shown and counsel given by him during the course of the work. Leiprig, Germanu, Madason. Tzsconsan.
