SUPERSATURATION AND LIESEGANG RING FORMATION. I1 ANDREW VAN HOOK1 Department of Chemistry, Lafayette College, Easton, Pennsylvania Received July 80, 1958
It has been reiterated (12) that a supersaturation theory of Liesegang ring formation is compatible with the fact that rings are formed in the presence of crystal seeds of the material to be precipitated] if it is assumed that the rate of diffusion is rapid compared to the rate of crystal grom’th. There should then exist a critical concentration of seeds above which rhythmic precipitation will not occur. This has been observed (7, 12), and for particular conditions it is approximately independent of the medium. For concentrations below the critical the clarity and number of bands should increase as the seed concentration diminishes. This has been confirmed qualitatively, and it is with the quantitative aspects of this variation that this paper deals. THEORETICAL
The classical case of strong silver nitrate diffusing into dilute potassium chromate may be taken as a model. In the usual experimental arrangement the source of silver nitrate is considered inexhaustible at the concentration of the saturated solution. Counterdiffusion of the weak potassium chromate solution will be neglected in the following simplified treatment. By Fick’s law the amount of silver nitrate diffusing regularly into a volume dx at a distance x from the origin is D-dc , where D is the diffusion dx constant. The amount leaving is
or the amount accumulating is D
D(2)
dx
dx. This is potentially -__
2
moles of silver chromate, provided that the chromate-ion concentration is sufficient.
* Present address: Department of Chemical Engineering, University of Idaho, Moscow, Idaho. 1201
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AKDREW VAN HOOK
If crystal seeds of surface s are dispersed uniformly throughout the medium, each one ivill grow a t the rate of
where (e-CCC) is the supersaturation of silver chromate. The total growth will bc nks[j(c-co)]dx, where n is the seed density. The concentration of silver chromate, c, refers t o a state of lion-stable equilibrium, and therefore it is reasonable t o interpret it in terms of a n equivalent concentration of silver nitrate, Le., potential silver chromate, generated by the incoming eilver nitrate. This presumption amounts t o neglecting the effect of the inner ion concentration on the rate of growth of the seeds, and seems appropriate in view of the incompletely known form of the rate of crystallization function, and also in view of the approximate experimental results obtained. Thp limit of band formation will correspond to the condition when the rate of crystal growth equals the rate a t which silver nitrate diffuses into thc section: namely, Znks d2e __ [f(c - col = D dx2 That the Koyes-Khitney (lO)-Nernst (9) theory of a unimolecular crystal growth is inadequate is evident from the work of Marc (6), who found that the order varies from unimolecularity to bimolecularity with decreasing temperature. Other workers (1, 2, 4, 5) also find a decided approximate nature for the first-order expression. Therefore, taking f(c-co) = (c-cCo)a empirically, the above equation may be solved by setting (c-cg) equal to y, and multiplying through by 2dl//dx:
Integrating :
At y
=
0, c =
Q,
and dy/dz = 0, whence K1 = 0, and
Integrating again and evaluating the constant by means of the condition that c = a constant, B , at x = 0 gives the solution in the final form:
SUPERSATURATION AND LIESEGANG RING FORMATION
1203
If the criterion of Liesegang ring formation is taken as
where H is the supersolubility product (3), then when everything but n is held constant, the extent of band formation will be given by nx2 = constant. The foregoing analysis is admittedly crude, mainly because of considering thc chromate-ion concentration constant and ignoring its counterdiffusion. This complication has been considered2 and the correction is deemed unwarranted in view of the necessity of either transforming silverand chromate-ion concentrations into un-ionized silver chromate concentration, or else expressing the crystal growth equation in terms of these two ion concentrations. Accurate information concerning these operations is not yet available. EXPERIMEKTAL
The relation, nx2 = constant, was tested with silver chromate in gelatin and in bentonite, lead iodide in agar, and magnesium hydroxide in agar systems. The usual capillary technic, with 1.5-mm. tubing, was utilized. Measurements of the extent of band formation were made t o the nearest 0.1 mm. with an ordinary scale and magnifying glass, or to 0.01 mm. by means of a Gaertner micrometer microscope. I n all rhythmic precipitation experiments there is a distance, immediately surrounding the point of inoculation, in which the precipitation is general. This “flushing-in” area, preceding the commencement of band formation, increases with the ratio of the concentrations of the reactants. The extent of band formation, 5, was usually measured from the head of the series of bands, but since it was not always easy to identify this position (especially as the seed concentration increases) , distances in such cases were measured to the point of inoculation and the results plotted in accordance with the modified equation: %($-a)* = constant No appreciable difference in behavior was observed with seeds prepared and added in three different ways. For preiiminary work seeds were precipitated “internally” (12). Externally precipitated seeds were prepared by bringing together the necessary reactants under conditions duplicating the experiment to be performed, and, after thorough mixing, the gel was allowed to set without agitation in a tall cylindrical vessel. Aliquot portions of a narrow horizontal slab were taken for seeding pur-
* Thanks are due Dr. R. Bailey of the Department of Mathematics of Lafayette College for assistance in this task.
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ANDREW VAN HOOK
poses. The third method consisted in precipitating the respective materials, washing, drying in an oven, and finally powdering. Graded portions were used, usually between 200 and 250 mesh in size. Each batch of experiments was inoculated simultaneously with an excess of concentrated reactant. Observations on the extent of ring formation were made at various times, and for each set of conditions it was found that there was an optimum period of observation for best results. Beyond this time the rings became indefinite, distorted, and even totally obliterated, owing to secondary effects, while earlier the distance of ring formation was not sufficient for comparative purposes. The best time usually corresponded with the formation of 10 to 20 mm. of rings in a blank experiment. RESULTS AND DISCUSSION
Representative results are presented in the following log-log plots (figures 1, 2, and 3). The points represent observed values, while the line is placed with a slope of -2 as expected from the equation: log n
+ 2 log
5
= constant
In general the relation is approximately valid over a considerable portion of the entire range, although serious discrepancies exist. The more serious deviations at large values of n may be associated with the arbitrary assumption of ignoring the chromate-ion diffusion, while at the other extremity the generation and initial rapid growth of new nuclei may be the cause of the deviation. That the presence of seeds does not seriously influence the fundamental diffusion nature of the Liesegang phenomenon is evident upon application of the Morse and Pierce (8) and Schleusner (11) relations: namely, h n / d n = constant, K ; and ~ , , / / L =~constant, A p, respectively. h and t are the distance and time of appearance of a particular band. For 0.01 M potassium chromate in 5 per cent gelatin with (a) no seeds, (b) “internally” precipitated silver chromate seeds equivalent to 0.001 M potassium chromate, and (c) 0.0025 M seeds, the results shown in table 1 were obtained. The rapid decrease in the values of k , as banding proceeds, for cases b and c is as expected. If the distance of the diffusion wave front (which is difficult to discern in case c ) is employed for h , the values found for k are better: Case
Values of k
Average
b e
4.3; 4.4; 4.2; 4.0; 3.9 5.0; 4.3; 4.5; 4.1; 3.8
4.2 4.3
The slight decrease in these values over a distance well beyond that of usual observation indicates that the total diffusion during the time of
SUPERSATURATION AND LIESEGANG RING FORXATIOK
LOG X
1205
MM.
FIG.1. 0.01 il4 potassium chromate in 5 per cent gelatin. Saturated silver nitrate. Seeds prepared as follows: 0 , ‘Linternally”; 0, “externally”; X, dry; 0 , “externally” in test tube experiment.
LOG X
LOGX
MM
FIG.2
CONSTANTS
P
‘I a
b C
K
a
b C
MM.
FIG.3
VALUES
AVERAQE
1.11; 1.08; 1.06; 1.11; 1.11; 1.10 1.06; 1.12; 1.05; 1.07; 1.09; 1.08 1.01; 1.05;1.03; 1.06;1.05;1.04
1.10
4.49;4.45;4.55;4.50 4.2 ; 4.40;4.2 ; 3.28 4.9 ; 4.51;4.2 ; 2.4
1.08 1.04
4.50
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ANDREW VAN HOOK
interest is little disturbed by the presence of seeds. That is, the number added is negligible compared to the number generated in the formation of a definite and distinct Liesegang band. Simultaneously with the above observations (extending over a distance of 30 to 50 mm. and up t o 60 hr. j , it was noted that the chromate-ion concentration remained undisturbed at a distance of only 2 to 5 mm. ahead of the diffusion wave front. In the capillaries used it would have been possible to detect visibly a variation of 10 to 15 per cent in concentration. This makes the early assumption of ignoring counterdiffusion of potassium chromate more agreeable. According l o the original complete expression, nsx2 should Ee constant. Since s is proportional to the average size squared, then 71L2z2= constant; or if n, is held Constant, 1, and x should vary in an invcrse manner. 'This was fourd to he only qualitatively true, R' being less dqxndent on L than indicated. SUMMARY AXD CONCLUSIOh'S
The extent of Ljesegang ring forination ( 2 ) in the presence of crystal seeds of the material precipitahg is found to decrease rapidly as the number of nuclei (n) increases. Over a limited range there is approximate confirmation of the expression nx2 = constant. More exact, information on the effect of ion concentrations on the rat,e of growth of slightly soluble substances is necessary for a more complete form of t,hc above expression. The initial presence of crystal seeds does not alter the fundament,al diffusion nature of thc Liesegang phenomenon. REFERENCES (1) CAmBmL, A. N., AXD CAMPBELL, J. R.: Trans. Faraday SOC.33, 299 (1937). (2) Gapox, E. N.:J . Russ. Phys. Chem. Soc. 61, 1729, 2319 (1929). (3) HUGHES,E. E . : Kolloid-Z. 71, 100 (1935); 72, 216 (1935). (4) JENKINS, J. D.: 3 . Am. Chem. Soc. 47, 902 (1925). (5) KURBATOV: 2. Krist. 77, 164 (1931). (6) MARC:Z. physik. Chem. 73, 688 (1910). (7) MORSE, 11. W.: J. Phys. Chem. 34, 1554 (1930). (8) MORSE,H. W., ASD PIERCE, G. JV,: Z. physik. Chem. 45, 589 (1903). (9) NERNST,W.: 2. physik. Chem. 4?,52 (1904). (10) NOYES-WHITNEY: Z physik. Chem. 23, 688 (1897). (11) SCHLEIJSNER: Kolioid-Z. 34, 338 (1924). (12) VAK H o m , A , : ,J. Phys. Chem. 42, 1191 (1938).
