SUPERSATURATIOX AND LIESEGASG RING FORMATION.
V
ANDREW VAW HOOK Department of Chemistry and Chemical Engineering, University of Idaho, Moscow, Idaho Received October $0, 1940
The chief objection which has been raised to a simple supersaturation theory of the formation of Liesegang rings is the fact that the rings do form in the presence of seeds of the precipitating material. It has been demonstrated in a qualitative way (17) that this behavior is not incompatible with a supersaturation theory, if a kinetic interpretation (3, 8, 12) is used in place of the static one ordinarily employed (4, 13, 14, 15). For a quantitative application of this interpretation it is necessary to know the rate of crystallization of the precipitate forming the band. Since silver chromate is the substance most commonly employed in Liesegang ring experimentation, and since its precipitation is typical of the behavior of difficultly soluble substances (3), it will be adopted as the model in the following analysis. The rate of precipitation of silver chromate has been determined in both aqueous and gelatin media (18, 19), and the results may be represented byethe common expression:
S is the supersaturation of silver chromate when its concentration is expressed in the sense of the mass action law, S = c - CO, where c is the ion product (u&+ ac,oi-), and co is the solubility product. A is the density of added seeds of the insoluble material, and t is the time. Various velocity constants are grouped in the constant coefficients. It has been observed (10, 20) that gelatin reduces the effective activity of silver ion in an exponential manner, and this introduces an awkward complication into the already cumbersome equation if ordinary concentrations are to be employed. However, the difficulty may be circumvented by adopting water as the medium in our model. This substitution is agreeable, for it has been amply demonstrated (11) that the Liesegang ring phenomenon is independent of the medium. It is also true that the decrease in activity of silver ion with dilution, in gelatin medium, is favorable to the mechanism developed below. In water medium of constant ionic strength, the conversion of ion activities to concentrations may be included in the constant coefficients of the above expression. 879
880
ANDREW VAN HOOK
For mathematical analysis of the Liesegang phenomenon the simplest model is the infinitely extended tube filled with one reactant, coupled to a supply of the other reactant of constant concentration. This is the model proposed by Morse and Pierce (13) in their analysis of the problem, and it is realized very closely by the capillary-tube technic indicated in figure 1. Distances and time are reckoned from the point of contact and placing of the inoculating solutions. The diffusion of silver nitrate, and counter diffusion of potassium chromate, are not undisturbed, for with the inauguration of precipitation there is introduced into the system a small, periodic occurrence which exerts its influence on the course of diffusion. However, if the initial difference in concentration of the two reactants is great (as it must be for the best Liesegang rings), this deviation becomes less significant than the more practical limitation of an inexhaustible supply of reactants.
J--x-c dX
FIG.1
Watanabe and Fujisaki (21) have examined the effect of precipitation on the velocity of diffusion, and find that the usual condition, x / d = constant, becomes modified to x / d / t = constant - constant’ t. The value of constant’ is small, and becomes smaller the greater the difference in concentration of the two reactants and the larger the amounts taken. The correction is insignificant under the ordinary conditions of conducting the Liesegang experiment; hence it may be assumed that the precipitation does not seriously influence the otherwise free diffusion. The concentrations of reactants a t any point x, and a t a time t , are then:
u and v are the concentrations of silver nitrate and potassium chromate, respectively, while uo and vo are the corresponding strengths at the commencement of the experiment. a and b are the respective diffusion coefficients.
881
LIEBEGANG RING FORMATION
&)
is defined as
and fl is an integration variable. c, the concentration of silver chromate in the sense of the mass action law, then becomes:
I t has already been mentioned that expression 1,
-* dt
= kl(c - co)
+ BA(c -
CO)'
$-
kdc
- eo)'
represents the rate of precipitation in the silver ion-chromate ion system.
If the concentration c is developed by diffusion, the amount of precipitation will accordingly vary in a periodic manner, because the crystallization tendency opposes the continuous increase in concentration of silver chromate in any particular section. The precipitation will thus reach a series of maxima with respect to both time and distance. These maxima will correspond to the head of the bands in a series of Liesegang rings, and the particular values should be calculable by means of either of the following operations: dc = d
[h(c
-
CO)
4- h A ( c - CO)'
+ h t ( c - a)']dt
+ hA(c -
+ k8t(c - a)*]dt = 0
=0
C'C0
2
d =
'
[kl(c - cg)
CO)'
0-co
The former quadrature has not yet been developed, while the latter evolves into a solution which is too cumbersome for practical use and interpretation. The following approximation1 is a satisfactory substitute: The number of crystals of silver chromate per unit volume may be considered &s a measure of the visible extent of precipitation in any locality. This seed density will then be a maximum at the head of each band, and 1 The complete solution of the exact expression may account for the very slight, almost imperceptible, drifts in the value of z/t/?in the very early and the later parts of a given experiment.
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ANDREW VAN HOOK
we may employ it in place of the total precipitation utilized in the above equations. The equation is: 71.
=
A
+ k(c - co)(t - t,,,)
(t - tCqO) is the time after precipitation starts at the saturation value of the salt. However, the high supersaturations obtainable in rhythmic precipitation indicate that ,t is small compared to the total time; hence it may be neglected in the above expression. In that case
and, upon substituting the values of c attained by diffusion,
I"
" 2 d i
In the case that co -+ 0 this equation reduces to
--
02
1
e
4dt
-_ 52
1
e
4821
4
The necessary and sufficient condition for the validity of both these equations is x/.\/t = constant, a condition already deduced and sufficiently substantiated in an experimental way by Morse and Pierce (13) and by many other workers (5). Their derivation is based upon the questionable existence of a definite supersolubility product, and this is avoided in the above development. The foregoing equations lead to a further relation which is partially verified by experiment. It has been observed that the formation of Liesegang rings can be prevented if sufficient crystal seeds are previously planted in the medium. At this critical seed density the condition for ring formation, xc/&o = constant, must still apply, and the growth equation l mill then reduce to the second term, since the added seeds predominate. Compatible with this approximation one may assume co to be negligibly small compared to working values of c . Hence, dc -= kzAc2 dt
IJESEGANG RING FORMATION
883
In this situation the seeds will grow as fast as material is delivered to their respective surfaces (16); which is to say, the concentrations in the above expression may be calculated by the diffusion equations. Doing this, and grouping all constants into K’s, the result is
%- = K~A(u:vo)
K~(uEvo)
t,
uivo. A x: = constant This is the same expression derived in a previous paper (18) in a more which is experielaborate manner. Here it has the added factor of Ho, mentally valid only in so far as the extent of banding is less the greater is uo (the other factors remaining constant). Of the four most prominent theories of the Liesegang phenomenon ( 5 ) , the two principal contending ones have been the supersaturation theory of W-i. Ostwald (14) and the adsorption theory of Bradford (2). Hedges (6) has reported some experiments on the rhythmic precipitation of sodium chloride from aqueous solution by hydrochloric acid and also others, which are seriously at variance with Bradford’s adsorption theory. Hedges and Henley (18, 19), Bechhold (I), and others find that periodic structures result from the diffusion of electrolyte into colloidally dispersed precipitate, thus suggesting that the formation of the precipitate is a secondary part of the Liesegang ring formation. Hedges pictures the process as consisting of the formation of the precipitate in a highly critical (colloidal) condition (apparently dispersed in a continuous or regular manner), followed by the more primary mobilization of this material, probably as a coagulation or salting-out process. Under the conditions used to disperse the precipitate (gels), it is true, for silver chromate a t least, that the solubility is tremendously enhanced, and the subsequent periodic structure may be the result of the interaction of this dissolved material and the incoming electrolyte, rather than that between a dispersed material and an inoculant. As Hedges himself (6) points out: “It is unwise to attempt to distinguish between highly supersaturated molecular solutions, and highly disperse colloidal sols.” Hence the supersaturation theory seems to offer the advantages of directness and simplicity. It does not require a mechanism to account for the marshalling of the ultimate precipitate in the manner found in banded precipitates. It applies correctly to the situation in all essential qualitative and quantitative aspects, and the few objections which have been raised may be readily explained. SUMMARY A N D CONCLUSIONS
1. The Morse and Pierce relation for rhythmic precipitation, z / d / t = constant, has been derived independently of the usual assumption of the existence of a definite supersolubility product.
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ANDREW VAN HOOK
2. In the presence of an inhibitory seed density, the extent of band formation decreases as the initial concentration of inner reactant is increased. The experimental results in the working region are indefinite and show only this trend. 3. There are no particularly valid objections to a supersaturation theory of Liesegang ring formation; moreover, it offers the advantages of directness and simplicity. REFERENCES (1) BECHHOLD, H.: Z. physik. Chem. 48, 418 (1904);67,47 (1906).
(2) BRADFORD, S.C.: Chemistry & Industry 4478 (1929). (3) FISCHER, W. M.:Z.anorg. Chem. 146, 311 (1925);163,62 (1926). (4) FRICKE, R.: 2. physik. Chem. 107,41 (1923);124, 359 (1926). (5) HEDGES,F. S.: Liesegung Rings. Chapman-Hall, London (1932). (6) HEDGES, F. S.: J. Chem. Sac. 1%9,7779; Chemistry & Industry 48,78,233 (1929). (7) HEDGES,F. 5.: J. Chem. Soc. 1938, 2718. (8)HEOELMANN, E.:in Berl's Chemisehe Ingenieur, Val. 111, p. 193 (1935). (9) JOST,W.: Difluaion und Chmrische Reaktion. Steinkopff, Dresden (1937). (10) KRUYT,H. R.,AND BOLEMAN, A. B.: Kolloid-Beihefte 36, 183 (1932). (11) LLOYD,F. E.,ANDMORAVEK, V.: J. Phys. Chem. 36,1512 (1931). (12) MORSE,H. W.:J. Phys. Chem. 34, 1554 (1930). (13) MORSE,H. W., AND PIERCE, G. W. : 2. physik. Chem. 46,589 (1903). (14) OSTWALD, WI.: Lehrbuch der allgemeinen Chernie. Steinkopff, Leipaig (1891). (15) OSTWALD, Wo.: Kolloid-2. 36, 200,380 (1925). (16) ROLLER,P. S.:J. Phys. Chem. 39, 221 (1935). (17) VANHOOK,A.: J. Phys. Chem. 42,1191,1201 (1938). (18)VANHOOK,A,: J. Phys. Chem. 44,751 (1940). (19) VANHOOK,A.: J. Phys. Chem., in prese. (20) VANHOOK,A.: J. Phys. Chem. 46,422 (1941). (21) WATANABE, M.,AND FIJJISAKI, T.: Science Repts. TBhoku Imp. Univ. 2, 105, 2-39 (1925); Chem. .4bstracts 21, 2588 (1927). (22) WILLIAMS, J. W., AND CADY,L. C.: Chem. Rev. 14, 171 (1934).
