T H E CONDUCTIT’ITY O F ELECTROLYTES. 111. THE CORRELATIOIY OF STROIYG AND WEAK ELECTROLYTES BY C E C I L R. D A V I E S
I n considering the conductivities of solutions it is customary to classify electrolytes into two groups. The “weak electrolytes” give a constant for the expression A2.c 11, .(A, - A ) , so that for these electrolytes the hrrhenius theory of partial ionisation is satisfactory. The second group consists of the “strong electrolytes” for which the theory of Arrhenius appears t o fail. I t seems most unlikely that this classification corresponds t o any actual fundamental distinction, for the typically “strong” and “weak” electrolytes in water are connected by an extensive and unbroken series of electrolytes intermediate in strength, and in same non-aqueous solvents most of the electrolytes investigated would have to be classed as “transition” electrolytes. This being so, a satisfactory theory of solutions will be impossible until the arbitrary distinction has been removed and the behaviour of the two groups reconciled. Lnfortunately the tendency of much modern research has been to make such a reconciliation more remote, owing to the developnient of the “complete dissociation theory,” which is regarded by many chemists’ as the only satisfactoiy explanation yet advanced for the behaviour of strong electrolytes in dilute solution. According to this theory the decrease in conductivity of a strong electrolyte, at any rate in very dilute solutions, is to be attributed t o a diminution in the mobilities of the ions concerned rather than to a decrease in their number. Much recent work on strong electrolytes is interpreted with the aid of this theory, while conductivity changes in weak electrolytes on the other hand are explained by the unmodified ;\rrhenius theory, in which no account is taken of possible changes in the mobilities of the ions. The defects of this position are evident, especially mhen we find that for transition electrolytes neither theory affords a satisfactory basis?. On the other hand. reference should be made to the attempts t o reconcile the behaviour of strong and weak electrolytes 011 the basis of the original Arrhenius theory: many chemists hold the view that the strong as well as the ueak electrolytes “obey the mass action law,” that is, that they do give a constant in the expression A2.c 11, .(A,-A) =E;, a t sufficiently low concentrations. Evidence for this belief is found in the more recent conductivity See -4.A. Soyes [J. d m . Chem. So?., 46, 1080 (1924)] and the authors referred t o in reference j of that article. * T h e dissociation constants found hy Rendall IJ. Chem. Soc., 101, 1275, (1912) ] for several acids of ”transitional” strength were obtained by choosing an appropriate value for the mobility of the hydrogen ion. This value, AE = 347.2 has since been shown to be too low [Kraus and Parker: J. -4m. Chem. Soc., 44, 2446-8 (19221, and see Davies: J. Phys. Chem., 29, 973 (192j)I.
9 78
CECIL IT. DAVIES
work of Kendalll on many organic acids, of Keiland2 on KC1, of Kraus and Parker3 on HIOa and of Parkeri on HC1. This view is open to much criticism. I n the first place, the constancy of K in “the mass action expression” depends entirely on the value taken for A , , and this in turn on the method of extrapolation adopted. I n the cases of KC1, HI03 and HC1 a method rf extrapolation, different from those used by the authors mentioned but just as consistent with the experimental results, will lead to quite different values5 in each case for A, ; and in the case of the organic acids investigated by Kendall the mobility of the hydrogen ion was taken as 3 4 7 . 2 , a value which is quite incompatible with the results of Kraus and Parker on other acids. Direct evidence for the view that the mass action law is obeyed is therefore of a very uncertain character. Again, many consequences of the theory are difficult to explain. For instance, KC1 is stated to obey the law of mass action at low concentrations, the value of the ionisation constant being K = 0 . 0 2 ; for HI03, K = 0.07 which leads Kraus and Parker t o state that it is a much stronger electrolyte rhan KCI; yet, at a concentration of o . o ; S , the value6 of -I A, for HI03 is only 0.80, while the corresponding value’ for KCl is 0.89. Further, taking again the case of KC!, the figures of Keiland’lead one t o suppose that the mass action expression gives a constant K at concentrations under 0.0001 S , but that at 0.0005 S the value of K has already increased by 1005. The presence of very considerable disturbing influences must therefore be admitted even at such great dilutions as this, and there seems to lie no u przori reason for exrecting, just as there is little evidence for proving, that at 0.0001 S all such influences have entirely disappeared whereas dissociation into ions is still in complete. The opposite view seems more probable and more consistent with the facts. Thus from the standpoint of the complete dissociation theory these disturbing influences are attributed to the electrical charges on the ions and to the effect their presence has on ionic niohilities, and this will persist even at dilutions so great that dissociation can be regarded as complete. The behaviour of aqueous solutions of strong electrolytes seems to support this view*. The object of the present paper is to show that this factor must he recognised in discussing the conductivity of any electrolyte except the very weakest ones; and that the theory of Arrhenius must be modified so as t o take account of these changes in ionic mobilities. I n any electrolyte in any solvent the change in conductivity with increasing concentration is to be ’Kendall: J. Chem. Soc , 101,
127j,
(19121; Medd. k Vetenskapsakad. Sobelinst.
2, 38 (1911). -z
U - e ~ l a ~ dJ.: Am. Chem. Soc., 40, 131 (19181. Kraus and Parker: J. d m . Chem. Sac., 44, 2429 1922) Parker: J. Am. Chem. Soc., 45, 2017 (1923). Davies: J. Phys. Chem. 29, ooo ( 1 9 2 5 ) . Groschuff: Z. Anorg. Chem. 47, 331 ( 1 9 0 5 ) . Abegg’s Handbuch: 2, I, 349 ( 1 9 ~ 8 ) . Dnvies: J. Phys. Chem. 29, 473 (1925).
979
CONDUCTIVITY O F ELECTROLYTES
attributed t o two causes: in part, t o changes in the mobilities of the ions, and partly t o a decrease in their number on account of their combination t o form neutral molecules.' I n tracing the magnitudes of these two effects the strong electrolytes will be discussed first, for they comprise an extreme case in which the first of these tn.0 contributory causes alone is present in very dilute solutions. Strong Electrolytes
It has been shown2 that in aqueous solutims of uni-univalent strong electrolytes a t concentrations below 0.002 K, the conductance of any ion can be represented by the equation A, = A, - 2 . 1 2 . IO-'. T3. d F . d z (1) where A, is the conductance of the ion at infinite dilution and A, its conductance a t the concentration considered. This empirical equation is supported by conductivity data (the conductivity of a uni-univalent salt being given by the sum of two such expressions) and by the transport number data. The behaviour of such solutions can also be studied in other ways such as by the freezing point and E. M. F. methods, from the results of which the activities of the ions can be derived; the activity-concentration ratio in dilute solutions is then invariably found to show a regular decrease with increasing concentration. I t is not difficult t o reconcile this fact with a complete dissociation theory; the activity of an ion depends not only upon its concentration but upon such factors as the resistance offered t o the passage of the ion by the medium. Any factor which affects the mobility of an ion will also be expected to affect its activity. From the standpoint of the Kinetic Theory, the activities of two reactants depend on the number of impacts occurring, and this in turn depends both on the speed of the molecules or ions and on their concentrations. Now the decrease in the mobility of an ion with increasing concentration is governed by the simple relationship expressed in equation ( I ) . I t seems reasonable to suggest that the decrease in the activity coefficient of an ion will be due t o the same cause and may be calculated in the same way: that is, that the activity of each ion of a strong electrolyte will be given by an expression such as a
= c.
~1, -KdFdx I'
(2)
110:
(where k = 2 . 1 2 . I O - ~ . T = ~j.61 at z j O C . ) , and that the mean activity coefficient for a strong electrolyte will be expressed by the equation:
This suggestion is easily tested. I n Table I the activity coefficients for several ions are calculated by equation ( 2 ) , and in Table I1 the mean activity coeffiIn interpreting work at higher concentrations than 3re dealt with in this paper other influences may have t o he recognised, such as the complex ion formation postulated by Schneider and Bradley: [ J . -1m. Chem. SOC.45, I I Z I (1923) ] and others. ? Dal-ies: J. Phys. Chen:. 29, 473 (1925).
980
C E C I L W.DATIES
cient's calculated in the same way from equation (3) are compared with experiment'al results'.
TABLE I C=
Ton.
Am
.\/Ai.
.OOOI
0002
0005
,001
,002
c1
75.8 39.6 351.3 74.0
8.71 6.29 18.74 8.60 7.14
,9936 ,9910 '9970 ,9936 ,9922
,9908 .98j j '9956 ,9908 ,9888
,9856 ,9800 '9931 ,9854 ,9826
,9795 ,9716 '9906 ,9792 .97j2
,9712 ,9601 '9868 ,9707 ,9649
I03 H K Na
jI .O
TABLE I1 c
HC1 (calc.)
,9954
'
KC1 (calc.) NaCl (calc.)
,9936 ,9928
'
lXC1 (obs.) KaC1
'
__
HC1 (obs.)
\
993
OOOsj
00 I
002
9895 991
,9851 ,984
9788 971
9908 9900
'9855
9842
'9793 ,9772
'9709 ,9681
990
984
977
967
982
972
,961
946
CCO2
OCCI
993 1
'
__
'
Having regard t o the degree of accuracy of the experimental results a t these low concentrations (they are all based on some empirical extrapolation of experimental results to zero concentration) it is difficult t o criticize the agreement shown in Table 11. But it may be said that all experimental results, as far as they go at the present time, support the view that the whole behavior of these electrolytes can be explained by a regular diminution in the mobilities of the ions with increasing concentration. The table also shows that activity coefficients calculated by equation ( 2 ) can be employed in mass-action expressions in place of experimentally determined values without introducing serious errors. Weak and Transition Electrolytes Here it is necessary t o distinguish between two effects: changes in the degree of dissociation, and changes in the ion mobilities. For this purpose it seems reasonable t o suppose that the presence of neutral molecules of the electrolyte will not affect the mobilities of the ions except in so far as it modifies, at fairly high concentrations, the viscosity of the medium. If this is so, the mobility of oach ion will be given at any concentration by the equation: A' = A, - 5.61 & d K where c, is the ionic concentration Taken from Lewis and Randall: 'Thermodynamics," pp. 336, 344.
CONDUCT1 VITY O F ELE C'I'HOLYTES
-
5
-
(
c
.
m
.
.
.
.
g Y
N
982
CECIL W. DAVIES
Kow the equivalent conductivity of the solution is given by the equation c,/C. (A’ A’), where C is the total concentration and c, ’C represents the degree of ionisation. If K is the specific conductivity of the solution, A
+
=
this equation may be put in the form K
=
A.C
=
c,
{
11,
- j.61.J;
(Jh,+ .\/A’,)
]
(4)
and from this we can, for each value of C, the total concentration, calculate the corresponding ion-concentration c,, and C - c, = c,, the concentration of undissociated molecules. The calculation is applied t o acetic acid at 2j°C in Table 111. The first two columns show total concentration and equivalent conductivity taken from the data of Kendall’; column 3 shows the viscosity (taking the mean of the values of Reyher? and of Rivett and Sidgwick3), and the corrected conductivity values, Avc vo are giren in column 4. Column 5 gives the degree of ionisation calculated by the Arrhenius method, (Y = A ACc vc ‘qo , -1, being taken as 3 5 1 . 3 ~ 4 0 . ; ~ = 392.0, and columns 6 and 7 give the massaction constants IC1 and Iiz. calculated according to the Ostwald dilution law using in the first case the uncorrected and in the second case the corrected conductivity values. The remainder of Table III shows the results of applying the methods descrihed in this paper. Column 8 gives the ion-concentration obtained by means of equation ( 4 ) . where the constants have in this case the values: K = c, . (392.0--140.92/;): column 9 gives the degree of dissociation, and column I O gives values for c,? c, = K3, where K3 will be seen to increase steadily with increasing concentration. I n order to obtain the true mass-action constant for the dissociation of acetic acid, the activities and not the concentrations must be employed. For the neutral molecules at these low concentrations it may be assumed that the two are proportional, but in the cases of the ions this is no longer true, and the ionic concentration must be multiplied by appropriate activity coefficients as shown in equation ( 2 ) . These are given in columns 11 and 12, calculated by means of the formulae
+
f~ = 351 3 - 5 . 6 I d Z
351 3
6 K
f-\? = 40.7- j.61.\/; 40 7
Finally, column 13 shows the true mass-action constant K 4 =
d\/40.7 fH
. ci
’f.4~
c i a
cu I t will be seen that IC4 shows a very satisfactory constancy. I n finding the mean value for the series, the figure at the lowest concentration is excluded as the experimental data seem less trustworthy in this case, and the value at the greatest concentration is also neglected as the ionic concentration here Chem. SOC., 101, 12jj (1912). Z. physik. Chem., 2 , 744 (1888). Rivett and Sidgwick: J. Chem. POC.,97, 736 ir910:. Davies: ,J. Phys. Chem. 29, Bredig: Z. physik. Chem. 13. 2 1 8 (1894).
I,J.
* Reg’her:
COKDUCTIVITY O F ELECTROLYTES
983
is rather greater than those for which the relationships used in the calculation are found t o hold’. The mean of the remainder is K = o.1799.10-~. When the values of K4 are compared with the values that one obtains when no allowance is made for changes of ionic mobility, as shown in columns 6 and 7 of the table, it is found that K 4 is on the whole much more constant than K1 and KS. At the same time, however, K1 and Kz are both satisfactorily constant u p to concentrations of 0.01S , and a comparison of columns 6 and 7 shows that the variations of K in the stronger solutions might conceivably be attributed to inadequacies of the viscosity correction. I n the stronger acids shortly t o be discussed, however, there is no room for doubt that the true dissociation constant K 4 is satisfactory and t h a t the Ostwald K is not. Another point of interest in Table I11 is the fact that although the calculated degree of dissociation differs considerably at the higher concentrations from that given by the Arrhenius formula, yet the new value for the constant K differs by only 0.57; from the K z obtained by OstIyald’s dilution law. The reason for this is that in calculating Kz two factors are neglected, and the weaker the acid the more nearly do these counterbalance one another. I n the first place, by assuming constant mobilities for the ions one obtains values for the concentration of the ions that are too lorn; but by using these concentrations in the mass-action expression instead of the lower activities one arrives at a value for the mass-action constant which is not greatly in error. I n Table IT- are shown the results of applying the same method of calculation to the transition acids investigated by Kendall, and to Iodic acid. The table shows clearly that K1, calculated by the Ostmald dilution law, is not constant but increases steadily with increasing concentration; but that the “true dissocia,tion constant’’ K 4 ieniains satisfactorily constant except at the greatest dilutions, where it shows a falling off. This effect at low concentrations is only t o be expected since the conductivity values a t these extreme dilutions are known to be too low?. The effect on K 4 of a slight change in the accepted conductivity value for the lowest concentration is illustrated in Table IT- for the case of cyanoacetic acid, an arbitrary conductivity value being employed in calculating row I O . I n the case of iodic acid the data used are those of Kraus and Parker3. This is a much stronger acid than those previously discussed, and at the concentrations considered in Table IT less than half of the conductivity diminution, A, -.1, is t o be attributed to the effect of incomplete dissociation, t h e greater part of it being due t o the diminution in the ion-mobilities Yet it will be seen that when allowance is made as before for the second of these causes a satisfactory mass action constant k = 0.173 is obtained. Iodic acid is especially interesting since it seems to occupy a position midway between the typical “strong” electrolytes and transition electrolytes Davie-: J. Phys. Chem. 29, 473 ( 1 9 2 j ) . Kraus and Parker: J. Am. Chem. SOC.,44, 2446 (1922). J. Am. Chem. SOC.,44, z249’(1922).
CECIL W. DAVIES
w,
p!
rr)
W
0
m
2 11 c\
.-
i)
m u m ~
e a -
w
*
r r ) d
m * o m a w N W O r - w m a rr) i d m m i Q .t * m m m 3
k
0
010
0
0
0
o " ~ o o o o o m o. o. o . o . o . ' o . o o o o
+
II
x
wwom
r w r - m r r ) r-vrw * N
m w m12;rr)
o- :o
0
0
o ' o o o
0
+ o o o o o
rr:
8
14
d
P
3
i C
5
* u
-. *. a . a. 0.
N
.
i
.
N
.
W
.
O
rr)
CONDUCTIVITY O F ELECTROLYTES
w
0
m
d-
c:
m
w c a
Q
m a
*0
C - r r ) * c I
N Q W
m
i 0
u2 m w m e m 3 - a c : ~ o o o o 0
0
0
0
0
0
0
? ? ? ? ? ? ?
uiN\Dcc k u i m a e N
0 W i N
N Q W
0 c h a d - m i ~ 0 0 0 0 0 0 0 0 0 0
0. 0. 0. 0
0
0
N
-
r
0
0
0
0 0 9
986
C E C I L W. DAVIES
such as cyanoacetic acid. Thus its conductivity curve suggests‘ that a t concentrations below 0.0001N it can he treated--like the strong electrolytesas completely dissociated, and this is in keeping with its dissociation constant k = 0.173 which requires a degree of ionisation of 99.94% a t O.OOOI S . It seems natural t o infer that the “strong” electrolytes themselves differ from weak and transition electrolytes only in their degree of ionisation which must be much greater even than that of iodic acid. Thus, for example, an electrolyte with a dissociation constant k = 1.0would be 99.99Yc ionised at O.OOOI X, and 9 9 . 0 7 ~ionised at 0.01K.
Summary I n the following paragraphs a summaiy is given of the conclusions reached in this paper and the preceding papers of this series. I. I n considering dilute solutions of electrolytes whether qtrong or weak the change of conductivity with increasing concentration is t o be attributed partly to changes in the mobilities of the ions and partly t o a decrease in the degree of ionisation. 11. With strong e!ectrolytes at concentrations of 0 . 0 0 2 S or less the first of these causes alone is present. Evidence for this is found in the facts t h a t (a) the conductivities of salts a t these concentrations are additive, i. e., the sum of the ionic mobilities at the concentration considered, (h) the ionic mobilities so calculated vary with the concentration according t o a general relationship: A’, - A = 2.12 IO-^ T 3 4 F 4 - Z . ( e ) the activity coefficients of the ions when calculated thermodynamically show the same proportionate decrease as do the mobilities, and probably from the same cause. 111. It follows from the equation quoted above that the conductivity of an electrolyte at sufficiently low concentiations is directly proportional t o the square root of the concentration. This provides a satisfactory method of extrapolation. IT’. K i t h weak and transition electrolytes the changes in the ionic niobilities are again given by the equation quoted, c in this case being the ionic concentration. These changes are only negligible hen dealing with the very weakest electrolytes. T’. If allowance is made for these mobility changes the true degree of dissociation of a weak electrolyte can be calculated; and if the product of the ionic activities is then divided by the calculated concentration of unionised molecules the tiue dissociation constant is obtained. The E d w a r d Dai aes Chcmttal Lahorntcrze*, 7’nar rrszty Collegc of Jl’alea, Aheryaiir y t h . Davies: J. Phys. Chem. 29, 973 (1925)
