evaluated and w l ( t ) is also given by Equation 68. Thus these successive procedures give the terms u nas
w, =
2%
{-wn-t
+ (a,? - ~
ACKNOWLEDGMENT The authors thank M. Hattori of Tokyo Institute of Technology for his helpful discussions.
~ ~ ) ~ f ~ - ~ ~ , ~(77) , ~ - Received ~ ~ w " -for, dreview ~ } June 25, 1973. Accepted September
a0
24, 1973.
Precipitation in Gels under Conditions of Double Diffusion: Critical Concentrations of the Precipitating Components Zvonirnir PuEar, Biserka Pokric, and Ante G r a o v a c lnstitute "Ruder BoSkovic," Zagreb. Yugoslavia
A new method is proposed for determining critical concentrations at which the precipitation of two components begins in gels under conditions of double diffusion in Ushaped tubes. The method can be applied to the evaluation of both inorganic and organic precipitating systems, as well as of antigen-antibody immunoprecipitates. The exact solution of Fick's second-order differential equation was applied to one-dimensional diffusion in U-tubes. To solve the double diffusion problem for practical purposes, the algebra of the Gauss error function was processed in a computer and the data were summarized in a table. In this way, it was possible to perform calculations of "precipitating titres" of solutions and critical concentrations of precipitating components in gels and to obtain exact results. The precipitating system, Pb( NO3)*K2Cr04 in agar gel, at different concentrations at 25 "C was treated as an example. The critical molar concentrations in agar gel of both of the components were 5.6 X and were independent of their initial molar concentration ratios. The concentration of and/or the aging of agar gel had no influence on the critical concentration of the components. It has been proved both theoretically and experimentally that precipitations under conditions of double diffusion obey the "equivalency rule" CA = CB = ccr,t rather than the ionic solubility law CACB = Ks.
Early investigations of the formation of precipitates in gels are related to immunology. H. Berchhold ( 1 ) has shown that immunoprecipitates are formed if two reactants, antigen and antibody, instead of being mixed in a test tube, are allowed to meet each other by diffusion across a gel. The description of Liesegang's phenomenon (2-9) has initiated considerable interest in precipitation processes and diffusion in gels. The method became known through immunological studies (10-26), although (1) H.Berchhold, Z.Phys. Chem. (Leipzig), 52,185 (1905). (2) R. E. Ltesegang, Z. Phys. Chem. (Leipzig), 88,1 (1914). (3) S . C. Bradford, Biochem. J., 10,169 (1916). (4) D . N. Ghosh, J. Indian Chem. Soc.. 7,509 (1930). (5) R. Fricke. Koliofd-Z.. 64,62 (1933). (6) R . Fricke, Z. Elektrochem., 39,629 (1933). (7) H. Erbing. K o l l o d Z . . 56,194 (1931). (8) B. Tetak, Koiioid-Z., 68, 60 (1934). (9) C . J. van Ossand P. Hirsch-Ayalon, Science, 129,1365 (1959). (10) C. G.Pope and M . Healey, Brit. J. E x p . Pathol., 19,397 (1938). ( 1 1 ) J. Oudin, C. R . Acad. Sci.. 222,115 (1946); 228,1890 (1949). (12)J. Oudin. Ann. inst. Pasfeur, Paris. 75,30 and 109 (1948). (13) S.D. Eiek, J . Ciin. Pathol., 2,259 (1949). (14)0.Ouchterlony, Ark. K e m i , 1 , 4 3 a n d 55 (1950). (15) C. L . Oakiey and J. Fulthorpe, J. Pathoi. Bacterio:., 65,49 (1953)
considerable attempts to study the nucleation and growth of crystals in gel matrices were also made (27-34). Various processes concerning precipitations in gels have also been treated theoretically (16-18, 20-23, 25, 26). However, the problem of precipitation under conditions of double diffusion has not yet been treated in the way that would make it possible to determine parameters of analytical interest by applying exact solutions of the diffusion law. Our studies on tissue calcification have shown that the double diffusion method in gels may give valuable results concerning critical concentrations of soluble mineral components a t the place of gel mineralization. In this paper, we propose a simple and rapid method for determining critical concentrations a t which the precipitation of two components starts in gels under conditions of one-dimensional double diffusion in U-shaped tubes. This method will also meet requirements of various studies of precipitation processes in gels.
THEORETICAL General Experimental Behavior. One-dimensional double diffusion experiments in a certain gel may be performed in U-shaped tubes, which are schematically shown in Figure 1A. The two vertical branches of the U-tube contain solutions of the precipitating components A and B, their concentrations being coA and c o ~ respectively. , The gel is situated in the horizontal part of the tube (16)J. R. Preer. Jr., J. lmmunol., 77,52 (1956) (17) A. Polson. Biochem. Biophys. Acta, 29,426 (1958). (18) A. Poison, sci. roois, 5, 18 (1958). (19)0. Ouchterlony, "Progress in Allergy," P. Kallos, Ed., Karger. New York, N.Y., 1958,Vol. 5, p 1. (20) J. A. Spiers and R. Augustin, Trans. Faraday Soc., 54,287 (1958) (21) M. H. V. van Regenmortel, Biochem. Biophys. Acta, 34, 553 (1959). (22) J. Engelberg, J. lmmunol., 82,467 (1959) (23) J. van Oss and Y. S. L. Heck, Z. lmmunitaetsforsch.-Allergieforsch., 122,44,(1961). (24) G. Mancini, J. P. Vearman, A. 0. Carbonara, and J. F. Heremans, "Protides of Biological Fluids, Proceedings of the 1 l t h Colloquium, Bruges, 1963,"H. Peeters. Ed., Eisevier, Amsterdam, 1964,p 370. (25) F. Aladjem, J. lmmunol., 93,682 (1964). (26) R. Trautman. Biophys. J., 12, 1474 (1972);13, 409 (1973). (27) H. K. Henisch, J. Dennis, and J. I. Hanoka, J. Phys. Chem. Solids,
26,493 (1965). (28)J. Dennis and H. K. Henisch, J. Electrochem. Soc.. 114, 263 (1967). (29) S. E. Halberstadt, H. K. Henisch, J . Nickl, and E. White, J . Coiloid lnterface Sci., 29,469 (1969). (30) B. Rubin and A. Saffir, Nature, ( L o n d o n ) , 225,78 (1970). (31) H. M. Liaw and W. Faust. Jr., J. Cryst. Growth, 13/14,471 (1972). (32) 2 . LeGeros and J. P. LeGeros, J. Cryst. Growth, 13/14, 476 (1 972). (33) E. Banks, R. Chianelli, and F. Pintchovsky, J. Cryst. Growth, 18, 185 (1973). (34) A. R. Patel and S. K. Arora, J . Cryst. Growth, 18,199 (1973) A N A L Y T I C A L C H E M I S T R Y , VOL. 46, N O . 3, M A R C H 1974
403
B
A
A
I
'B
p'. x;
1
4
C Figure 1. Schematic representation of the experimental arrange-
ment designed for precipitation under conditions of double diffusion in U-shaped tubes X A and XB are the distances of the precipitation plane P.P. from the gelsolution interfaces at x = 0. n is the dilution and concentration factor, respectively. Other notation given in text (shaded area). The two precipitating components are allowed to diffuse through the gel against each other until the first visible precipitate appears in the gel in the form of a narrow film perpendicular to the direction of diffusion. This film is situated in the precipitation plane a t the distance X A and XB from the vertical parts of the tube, respectively. At the onset of precipitation, i.e., during the period of nucleation, the concentrations of both components in the precipitation plane are CA and cB, respectively. These concentrations may be designated as critical concentrations at which the precipitation in the gel starts. As long as both fluxes A and B are equilibrated in the precipitation plane, the developed, visible, precipitate becomes stationary, although the free diffusion of the components is disturbed along the gel by the precipitation process. However, as soon as the fluxes are no longer equilibrated, the precipitating component with the smaller flux will be exhausted in the precipitation plane, thus allowing the component with the larger flux to pass it, which ultimately results in the periodicity of precipitation (Liesegang's phenomenon). On the other hand, if the component with the larger flux dissolves the precipitate by complexing, the precipitation plane will not be stationary but will move in the direction of diffusion of the complexing component. The parameters X A and XB are therefore defined as the place in the gel column where the first visible precipi404
tation occurs, while the critical concentrations C A and CB represent the concentrations of the precipitating components in the precipitation plane just at the moment of appearance of the solid phase. If the experiment shown in Figure 1A is repeated so that the solution A is diluted by a dilution factor n, the first precipitate will appear closer to the side A of the Utube than in the first experiment (Figure 1B). Moreover, if the solution B is at the same time concentrated by the same concentration factor n, the precipitate will also appear closer to the side A (Figure 1C). This is general experimental behavior, independent of the type and mechanism of the formation of the precipitate. The same is also true of antigen-antibody systems in immunoprecipitations as well as of diverse inorganic precipitating systems. This observation is of fundamental importance in further mathematical treatment of the problem. U-shaped tubes of the type presented schematically in Figure 1 have the advantage that the concentrations of the precipitating components in the vertical branches of the U-tube may be regarded as constant during experiment in every part of the vertical branch, except the thin diffusion layer a t the gel-solution interfaces a t X A = 0 and XB = 0. The reduced concentrations and densities of the solutions a t the bottom of the vertical branches are spontaneously balanced by convection during the experiment. Because of homogeneity of solutions outside the gel, the diffusion in the U-tube is practically insensitive to vibration. The geometry of the vertical branches should be suitably chosen with regard to the geometry of the horizontal cylindrical gel column, and the initial concentrations of the precipitating solutions should be at least several times higher than their critical concentrations of precipitation. Application of Fick's Second Law of Diffusion to the Proposed Experimental Arrangement. In a U-tube the precipitating substances A and B will diffuse against each other into the gel through the interfaces at x A = 0 and XB = 0 according to Fick's second law of diffusion in the x direction. This can be written in the form ( 3 5 ) :
be/&
=
Db'c/bx2
(1)
D is the coefficient of diffusion of the substance under consideration in the corresponding gel, and is assumed to be constant, i.e., independent of the concentration; t is the time of diffusion (sec); x is a linear distance in the direction of diffusion (cm); and c is the concentration. Hence the dimensions of D are expressed in units cm2 sec-l. Considering an infinite cylinder on both sides of the gelsolution interface. and the initial distribution of concentration given by c = co for x < 0 and c = 0 for x > 0 a t t = 0, the solution of Equation 1 is c = 1 / 2 co erfc(x/2v'Dt) (36),where erf(u) is the Gauss error function, and erfc(u) = 1 - erf(u). For our experimental arrangement (Figure 1) where no concentration gradient is expected in free solutions in the vertical branches except in the thin diffusion layer at x = 0 (see preceding section), the following solution of Equation 1 would apply: c = c.
erfc(xi2)'E)
(2)
However, the law of precipitation in the precipitation plane under conditions of double diffusion should be taken into account, We may in general assume two possibilities: (35) A. F i c k , Ann. Phys Chem. ( L e i p z i g ) ,94. 59 (1855). (36) W. Jost, "Diffusion in Sollds. Liquids, Gases" Academic Press, New York, N . Y . , 1960. p 20
A N A L Y T I C A L C H E M I S T R Y , V O L . 46, NO. 3, M A R C H 1974
(i) The two precipitating components are present a t the onset of precipitation in the precipitation plane a t critical concentrations cA and cB, respectively. If the concentrations are expressed in equivalent concentrations, the relation CA = CB = ccrit should be valid. This is generally assumed for antigen-antibody interactions. (ii) In the case of precipitation of inorganic salts, the solubility law CACB = Kso, where Kso is the solubility constant, should be taken into account. It is evident that if the solubility law is applied to situations illustrated in Figure 1;A and C using Equation 2, no shift of precipitation plane can be expected:
1quadrant (no dilution of solutions):
D, inverfc(y,) 2 quadrant (solution A is diluted by the factor nA):
inverfc ( y s )
3 quadrant (solutions A and B are diluted by the factors nA and nB, respectively):
4 quadrant (solution B is diluted by the factor n B ) :
In other words, in this case the erfc products should be constant, and a t the onset of precipitation these products amount to K S o / ~ o A ~ oThe B . minimum time necessary for the erfc product maximum to reach this value is the same for both erfc products, hence t = t“ and therefore XA = X”A. Obviously, this is in sharp contrast to the experimental evidence. In all other cases, i.e., in situation B (Figure l ) , as compared to situation A, an insignificant shift of the precipitation plane can be expected, depending mainly on the ratio DA/DB. Therefore, the solubility law should be neglected in the case of precipitation under conditions of double diffusion, and the “equivalency rule” should be accepted as the only possible interpretation. Algebraic Treatment. Equation 2, which describes the distribution of concentration during a single one-dimensional or linear diffusion process, may be rearranged as follows: c/co = e r f c ( x / 2 d E ) , x / 2 0 = inverfc(c/c,), x = 2 d E i n v e r f c (c/co). Here inverfc is the inverse function of erfc. Substituting c/co by y , where the ratio co/c may be called the “precipitation titre,” and, y may therefore be assigned as the reciprocal of the titre, we obtain: x = 2 m inverfc(y)
xB =
-
inverfc(yA)
l/oB inverfc(yB)
(8)
From Equations 5 and 6 we obtain:
\ XB)? From Equations 8 and 7 :
From Equations 8 and 5:
From Equations 7 and 6:
(3)
The double diffusion of two precipitating components A and B which form a precipitate in the precipitation plane, characterized by distances X A and XB and critical concentrations CA and cB in the precipitation plane, may be written in the form: xA -
D, inverfc(n,y,)
(4 1
To obtain the relationship between the position of the precipitation plane in the gel column and the “titre” of one of the precipitating solutions from Equation 4, a t least two U-tubes are necessary in the experiment to be performed, provided the diffusion coefficients are not known. If, for instance, the “titre” of the solution A is to be found, the solution should be diluted by the dilution factor n A in the second U-tube. This is schematically shown in Figure 1, A and B. If the “titre” of the solution B is to be determined simultaneously, a third U-tube should be added containing solution B diluted by the factor nB. Sometimes addition of a fourth U-tube with both solutions diluted may be advantageous in checking the results obtained in the preceding three U-tubes. The characteristic experimental arrangement in each of the four U-tubes may be designated in further text as a “quadrant.” Thus, Equation 4 may be rewritten for all four “quadrants: ”
\xB/,!
The values z are ratios corresponding to two quadrants, marked by the indices outside of the parantheses, each of the quadrants consisting of the ratios of distances of the precipitation plane in the gel column from the interfaces gel-solution A and gel-solution B. Obviously, there is a certain symmetry, because Z A I ~ = ~ zA4:3 and 2 8 4 1 1 = zB3/2*
The resulting Equations 9-12 have the same general form z = inverfc (y)/inverfc ( n y ) . z is an experimental parameter as given by Equations 9-12. The solution of these equations should have the form .y = f ( z , n ) . It is not possible to obtain an explicit solution of this relation in analytical form, and y has to be computed numerically. In order to obtain a simple evaluation of y from the experimental data, a table of results of these computations is given in the Appendix. This table allows immediate readings of the titres, Le., l / y = c,/c, as a function of z for dilutions 2, 5, 10, and 20. The values for z were selected as follows: the gel column was assumed to be 30 mm long, whereas the precipitation plane was assumed to lie between the 5th and 25th mm of the column, and inside this region the X A and XB values varied in steps of 0.5 mm. All possible z values were selected ( z > 1) under this criterion. The A N A L Y T I C A L C H E M I S T R Y , VOL. 46, NO. 3, M A R C H 1974
405
verfc(y) and un = inverfc(ny) were calculated, and the u values were obtained as roots of the equations y - erfc (01) = 0 and ny - erfc(u,) = 0. This was done by the Regula falsi method. The function erfc(u) was approximated by (37):erfc(u) = (alt azt2 a d 3 ) exp(-u2), 0 5 u < m, t = 1/(1 + pu), p = 0.47047, a1 = 0.3480242, a2 = -0.0958798, a3 = 0.7418556. The absolute error of this approximation is I t ( v ) I 5 2.5 x The smallest step of iteration in y is yapprox was obtained as a final result of iteration. zapproxwas calculated from Equation 13 and compared with the given z value (at least four significant figures were reproduced). The error of l / y changes with z and n, and it was estimated for the z values of interest. These errors are taken into account in the results given in the table (see Appendix). All computations were performed on the C.I.I. computer, Model C 90.40, at the “Ruder BoSkoviC” Institute.
+
+
EXPERIMENTAL Apparatus. U-tubes were machined in blocks of Lucite (Figure 4) and consisted of two halfs, A and A‘, cut along the longitudinal I
1
2
3
z
Figure 2. Dependence of log l / y or log co/c on the experimental parameter z for several dilution factors n. This diagram is an illustration of the computed table given in the Appendix
plane, t o enable the removal of t h e gel column and cleaning a t the end of experiment. T h e two parts were mounted tightly together by means of metallic screws. The dimensions of the horizontal gel column were as follows: the cylinder was 6 m m in diameter, and 30 m m long; the vertical branches were 7 X 7 m m in cross section and 35 m m long. T h e horizontal part of the U-tube was filled with warm agar solution, and the hollow pistons B (Figure 4) were pushed into t h e vertical branches. After solidification, t h e excess of agar gel was removed by suction from the vertical holes of the pistons, a n d then the pistons were removed. In this way, the interfacial planes a t x;\ = 0 and x B = 0 were formed. The vertical branches were then filled with double-distilled water to equilibrate and t o age the gel. The water was emptied by suction immediately before the beginning of the experiment. At the end of the experiment, the x parameters were measured in multiples of 0.5 m m by the naked eye. Materials. Solutions of lead nitrate and potassium chromate were prepared by dissolving Merck p.a. salts in double-distilled water. The concentration of lead nitrate standard solution was determined by complexometric titration with EDTA (38). T h e agar (“Difco” Special Agar-Noble) was washed with distilled and double-distilled water a t room temperature for several days under continuous stirring and frequent change of water, and then it was dissolved in double-distilled water under heating in a water bath.
RESULTS AND DISCUSSION Figure 3. Maximum total error of the “titre” in per cent for a gel column 30 m m long, as a function of the “titre.” The error was obtained using the values from the table (Appendix) and measuring the x parameters in multiples of 0.5 m m maximum overall error of evaluation would thus correspond t o an uncertainty of 0.25 mm of the X A and XB value, respectively. Figure 2 illustrates roughly the results of computation in the form log c o / c us. z for dilution factors 2, 5 , 10, and 20. Figure 3 shows 100A(co/c)/(co/c)US. co/c--i.e., the maximum total error of the titre determinations (in per cent) in a gel column 30 mm long as a function of the titre, provided the calculated values from the table (Appendix) are used and the x parameters are measured in multiples of 0.5 mm. Algorithm of Computation of the y = f ( z , n ) Function. The reciprocal of the titre was calculated from z =
inverfc(y) inverfc(ny)
u1
3 -
un
(13)
The dilution factor n limits the interval of feasible y values to [0, l l n ] . The calculation of y was performed by iteration. For each value of y, two functions u1 = in406
The precipitating system Pb(N03)2-K2Cr04 in agar gel a t 25 “C (thermostat) was treated as an example (Figure 5). The precipitating component A is represented by P b ( N 0 3 ) and ~ component B by K2Cr04. First Set of Experiments. General Test of Theory and Experimental Accuracy. The coA/coB ratios of the initial molar concentrations of the precipitating components were 111, 112, 113, 114, 211, 311, and 411, and the initial molar concentrations coA and coB varied from 0.1 to 0.4. The dilution factors n were 5 and 10. The duration of experiments (the period of time until the first visible precipitate appeared) varied between 6 and 12 hours. The critical concentrations C A and C B a t which the precipitation occurs in the gel under conditions of double diffusion with standard errors of the means were found to be: CA = (7.7 f 0.3) x M , and C B = (7.6 f 0.3) x 10-4M. The precipitation starts a t equivalent concentrations of the precipitating components. Two-component precipitating systems in homogenous free solutions were character(37) M. Abramowitz and I . A . Stegun, Eds., “Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables,” National Bureau of Standards, Washington, D.C., 1968, p 299. (38) “Methods d’analyses complexometriques par les T l T R l PLEX.” 3rd ed., E. Merck AG. Darmstadt.
ANALYTICAL C H E M I S T R Y , VOL. 46, NO. 3, M A R C H 1974
.
1
Figure 4. Two halves of the U-tube A and A‘ for precipitation under conditions of double diffusion. The hollow pistons E are used to produce gel-solution interfaces ized by B. Teiak (39,40)through four main types of precipitation bodies (PB) representing neutralization, ionic solubility, formation of ionic pairs and associates, and unsymmetrical ionic reactivity. In comparison with this, our results indicate that the precipitating process behaves as simple neutralization. The critical concentrations a t which the precipitation of the two components starts in gels under conditions of double diffusion are independent of concentration ratios of the diffusing components. The error of the mean in this set of experiments was about +3.9% and +4.2%, respectively. Second Set of Experiments. Dependence of Critical Concentrations on the Concentration of Agar Gel. The results of the preceding experiments seemed not to depend on agar concentration. Therefore, in order to perform closer examination, experiments were performed in 0.3, 0.5, 0.75, 1, 2, 3, 4, 5, 7.5, and 10% agar gel. The initial concentrations of both precipitating components were 5.0 x 10-3M, and the dilution factor was 2 for both components. The duration of experiments for agar gels between 0.3 and 1% was about 7 hours, while for gels between 2 and 10%. it was up to 43 hours. dependinn- on the concentration of agar gel. Results: cA = 5.64 x i o - 4 ~ C~ . = 5.61 x 1 0 - 4 ~ . The critical concentrations do not depend on the concentration of agar gel, although the positions of the precipitation planes in the gel were found to he dependent on the concentration of the gel. The reason for this behavior lies in considerable differences in diffusion coefficients of the precipitating components. The error of the mean value was much smaller than in the preceding set of experiments, owing to constant initial concentrations of precipitating components. The critical concentrations were found to be about 26% smaller than in the first set of experiments, because the initial concentrations of the components were considerably smaller and therefore the flux of the components onto the precipitation plane was weaker (kinetics of precipitation). Third Set of Experiments. Dependence of Critical Concentrations on the Aging of Agar Gel. The 1% agar gel was aged 6, 24, 48, 72, 96, 168, and 264 hours. The initial concentrations of both components were 5.0 x lO-aM, and the dilution factorn was 2. Results: cA = 5.64 x 1 0 - 4 ~ cB , = 5.62 x 1 0 - 4 ~ . No dependence on the aging of the gel was found in the case of aging and equilibrating the gel for at least 6 hours. Fourth Set of Experiments. Check of the Validity of the “Equivalence Rule” for Precipitations of Inorganic (39) 0. Teiak. Dacuss. FaradaySoc.. b2.175 (1966). (40) B Tefak. Crost Chem Acta. 40.63 (1968).
L
.
4
FWre 5. Precipitarim disks in U-shaped tubes photographed after 13hoursof diffusion The photographs iepresent the first (1). Second ( Z ) , third (3). and lourth (4) quadrant of the precipitating System O.ZM Pb(NO&-O.lM K&rO,. The dilution lactors Were n A = ne = 10. Lead sol~tion~ dilfUSed lrom the left to the right. whereas chromate solutions dinused from the right to the Iell. The pOSitionS where disks 01 precipitates began to form in the precipitation Planes are marked with arrows. The large black points In the pictures come lrom metallic screws. Note the formation of Llesegang’s rings In the gel COiumnS corresponding lo the Second and fourth quadrant
Salts, a s Well as of the Critical Concentrations Determined Previously. Experiments were performed in 1% agar gel. The ionic product of critical concentrations was calculated from the second and the third set of experiments (5.6 X 10-4)2 = 3.1 X lo-”. Initial concentrations of precipitating components in molar concentrations [MI and ionic products of initial concentrations in [MI2 of the five systems used in experiments are: o*
4
5 . 5 x io-’ 5.5 x 10-4 5 . 5 X lo-* 5 . 5 X 10-4
5
5.5
i 2 3
x io-*
cos
5 . 5 x io-‘ 5 . 5 x io-* 5 . 5 X lo-‘ 5 . 5 X, 5.5 x 10-4
*AW*
’.
3.0 3.0
x 10-7 x 10-
3 . 0 X 103.0 X loa ~
3.0
x io?
. .
Results: No formation of the precipitation,plane in the gel column was observed in any of the systems even after several days of diffusion, although in systems 2 and 3 the ionic product exceeded the ionic product of critical concentrations about 10 times, and in systems 4,and 5 about 100 times. In systems 2 and 4, the .precipitate was formed after several days in the free solution-ie., in the vertical branch filled with Pb(NOa)z-whereas in systems 3 and 5, it was formed in the vertical branch filled with the KzCr04 solution, APPENDIX Table of the “titrea” c./c as a function of the experimental parameter z and the dilution factor n for precipitations under conditions of double diffusion in a gel column, provided the experimental arrangement allows the assumption that in the bulk of solution outside of the gel, the initial concentrations e, of the diffusing substances may be regarded as constant during experiment. The z values were selected for a gel column 30 mm long under the assumption that the precipitation plane lies between the 5th and 25th mm of the column and that the distances x of the precipitation plane from both ends of the column were measured in multiples of0.5 mm. For other dimensions of the gel column and/or more accurate measurements of the x parameters, the values c./e may be found by interpolation.
ANALYTICAL CHEMISTRY. .VOL. 46. NO. 3. MARCH 1974 -.
407
Table of Titres z
n = 2
1.0690 1.0691 1.0694 1.0699 1.0706 1.0714 1.0725 1.0737 1.0752 1.0769 1.0789 1.0813 1.0840 1.0872 1.0909 1.0952 1.1003 1.1064 1.1136 1.1225 1.1427 1.1429 1.1434 1.1442 1.1455 1.1471 1.1491 1.1515 1.1544 1.1579 1.1619 1.1667 1.1722 1.1786 1.1860 1.1948 1.2051 1.2174 1.2217 1.2222 1,2233 1.2250 1.2273 1.2302 1.2321 1,2338 1.2381 1.2432 1,2493 1.2500 1.2564 1.2647 1.2744 1.2857 1.2990 1.3061 1.3065 1.3077 1,3097 1.3125 1.3147 1.3162 1.3209 1.3265 1.3333 1.3414 1.3509 1.3557 1 3620 1.3750 1.3830 1.3902 1.3968 1.3979 1,4000 1.4032 1.4076 4oa
714 709 694 676 641 595 543 500 455 405 375 311.5 269.5 231.0 190.1 157.5 128.7 103.4 81,77 64.06 40.95 40.78 40.39 39.78 38.82 37.69 36.38 34.89 33.25 31.437 29,560 27,248 25.316 23.392 21.317 19.414 17.440 15.547 14.981 14.916 14.780 14.575 14.364 14.039 13.833 13.656 13.226 12.753 12.232 12.176 11.682 11.078 10.500 9.844 9.245 8.950 8.934 8.887 8.809 8.7032 8.6222 8.5675 8.4027 8.2163 8.0026 7.7652 7.5086 7.3866 7.2343 6.9444 6.7806 6.6414 6.5206 6.5011 6.4641 6.4090 6.3351
n - 5
n
=
10
n
-
2900 2800 2700 2630 2560 2380 2220
1.4082 1.4132 1,4202 1.4286 1.4293 1.4386 1.4505 1.4545 1.4644 1.4808 1.4848 1,4938 1.4945 1.4966 1.5000 1.5049 1.5114 1.5195 1.5217 1.5226 1 5294 1,5414 1,5495 1.5556 1.5723 1.5814 1.5921 1.5983 1.6000 1.6034 1.6087 1.6154 1.6158 1.6198 1.6250 1.6364 1.6429 1.6502 1.6667 1.6754 1.6863 1.7095 1,7101 1.7111 1.7143 1.7196 1.7273 1.7368 1.7373 1.7500 1.7611 1.7656 1.7692 1.7843 1.8067 1.8077 1.8182 1.8308 1.8333 1.8385 1.8464 1.8537 1.8571 1.8649 1,8710 1.8882 1.9023 1.9091 1.9343 1.9467 1.9600 1.9615 1.9643 1,9662 1.9740 1.9767 1.9852 I
2000
1820 1590 1390 1190 1020 833 694 562 448 351 326 323 316 306.8 295.0 281.7 273.2 266.0 248.8 229.4 208.8 207.0 189.8 170.4 151.5 133.51 115.74 108.23 107.87 106.61 104.60 101.94 100.00 98.62 94.70 90.25 85.40 80.13 74.68 71.84 68.73 63.01 59.92 57.37 55.19 54.83 54.17 53.22 51.92
z
20
25000 20000 17000 13000 9100 6700 5000 3600 3100 3000 3000 2900 2700 2500 2380 2330 2130 1920 1670 1639 1449 1250 1064 885 730 667 662 654 633 610 595 581 549 515 474 433 394 376 352.1 313.5 292.4 275.5 261.1 259.1 254.5 248.1 239.8
33000 29000 28000 26300 25000 23800 21300 20400 19600 17500 15200 12700 12500 10530 8850 7190 5680 4440 3940 3910 3850 3720 3550 3410 3330 3100 2820 2564 2304 2020 1908 1758 1504 1379 1282 1193 1180 1156 1119 1072
ANALYTICAL CHEMISTRY, VOL. 46, NO. 3, MARCH 1974
n = 2
6.3255 6.2445 6.1365 6.0132 6,0035 5.8754 5.7130 5.6651 5.5568 5.3792 5.3387 5.2507 5,2441 5.2244 5.1929 5.1488 5.0919 5.0294 5.0075 5,0005 4.9478 4.8589 4,8015 4.7533 4.6432 4.5865 4.5230 4.4877 4,4781 4.4593 4.4287 4.3943 4.3919 4.3708 4,3440 4,2872 4,2560 4.2223 4.1494 4.1130 4.0692 3,9818 3.9796 3.9761 3.9646 3,9462 3.9197 3.8883 3,8867 3,8463 3.8123 3.7990 3.7885 3.7455 3.6854 3,6828 3,6562 3.6253 3.6193 3.6070 3.5886 3.5721 3.5644 3.5472 3.5341 3.4981 3.4699 3.4538 3.4061 3.3838 3.3606 3.3581 3.3532 3.3501 3.3370 3.3326 3.3187
n - 5
n = 10
n = 20
51.76 50.38 48.71 46.77 46.60 44.50 42.09 41.32 39.56 36.96 36.36 35.13 35.03 34,75 34,32 33.70 32.93 32.01 31.77 31.67 30.96 29.78 29,04 28.50 27.10 26.44 25.70 25,28 25.13 24,94 24,59 24.18 24.15 23.93 23.70 23.05 22.70 22.33 21.55 21.16 20.70 19.79 19.77 19.74 19.62 19.43 19.16 18.84 18.83 18.42 18.08 17.95 17.85 17.43 16.86 16.83 16.58 16.29 16.23 16.12 15.95 15.80 15.73 15.57 15.45 15.13 1 4 87 14.76 14.36 14.150 13.957 13.935 13.895 13.868 13,759 13.723 13,607
238,l 229,9 217.9 205 8 204.9 192.3 178,6 174.2 164.2 149.7 146.6 139.3 138,7 137.4 135.0 131.6 127.6 122.7 121.4 120.9 117.1
1066 1019 948.8 881.1 875,7 807.8 735.3 712.8 664.5 588.2 572.1 539.4 536.8 530.8 518.4 501.8 481.2 457.7 451.9 449.4 432.0 403.7 386.9 375,l 343.8 328.52 313.28 304.60 302.30 297.80 290.95 281,85 281,37 276.70 270.71 258.53 251.95 244.98 230.41 222,82 214.50 198.93 198.57 197.90 195.89 192.60 188.01 182.65 182.35 175.62 170.13 167.98 166.31 159,62 150.63 150.24 146.39 142.01 141.16 139.45 136.93 134.70 133.67 131.39 129.65 125.02 121.48 119.83 114.16 111.43 108.85 108.57 108.04 107.69 106.26 105.78 104,26
I
I
111.1
107,3 104.6 98.0 94.6 90.9 88.9 88.3 87.3 85.7 83.8 83.61 82.51 81.10 78.13 76.57 74.85 71.23 69,49 67.43 63.41 63.29 63.13 62.62 61.81 60.64 59.24 59.17 57.47 55.96 55.40 54,98 53.28 50.89 50.81 49.73 48.52 48.29 47.87 47,26 46.64 46.36 45.73 45.25 43.97 42.96 42.50 40.88 40.15 39.39 39.31 39.15 39.05 38.63 38.49 38.04
2
2.0000 2.0187 2.0417 2.0425 2.0643 2.0695 2.0931 2,1000 2.1029 2.1037 2.1111 2.1224 2.1379 2,1429 2.1538 2.1579 2.1828 2.1905 2.2132 2.2273 2.2474 2.2500 2.2522 2.2587 2.2698 2.2857 2.2941 2.3067 2.3158 2.3171 2.3333 2.3469 2.3662 2.3986 2.4063 2.4103 2.4130 2.4182 2.4286 2.4444
n = 2
3.2953 3.2669 3.2335 3.2325 3.2025 3.1956 3.1651 3.1565 3,1529 3.1519 3,1429 3.1293 3,1113 3.1056 3.0934 3.0888 3.0621 3.0541 3,0312 3.0175 2.9985 2.9962 2,9941 2,9882 2.9782 2.9643 2.9571 2,9465 2,9390 2.9380 2.9249 2.9143 2.8995 2.8758 2.8703 2.8675 2.8657 2.8630 2.8558 2.8452
n - 5
13 13 12 12 12 12 12 12 12 12 12 12
416 182 913 905 665 609 367 300 271 264 194 088 11 947 11 903 11 809 11 774 11 570 11 509 11 335 11 232 11 085 11 071 11 057 11 013 10 937 10 833 10 781 10 702 10 646 10 638 10 543 10 466 10 359 10 190 10 150 10 131 10 117 10 092 10,042 9.9621
n = 10
37.30 36.42 35.41 35.37 34.47 34.270 33.378 33.135 33.025 33.003 32.744 32.352 31.847 31.686 31.348 31.221 30.488 30.266 29.656 29,283 28.785 28.727 28,637 28.490 28.217 27.863 27.678 27.412 27,226 27.203 26.882 26.617 26,254 25.681 25,549 25.484 25.439 25,355 25.183 24.931
n = 20
101.75 98.65 95.29 95.18 92.17 91.48 88.50 87.67 87.33 87.24 86.38 85.10 83.42 82.91 81.79 81.374 78.983 78.278 76.289 75.115 73.513 73.314 73.142 72.648 71.829 70.686 70.107 69.252 68.653 68.573 67.540 66.707 65.569 63.771 63.363 63.151 63.012 62.743 62.220 61.440
ACKNOWLEDGMENT The authors wish to thank T. Zivkovii: and H. FurediMilhofer for valuable discussions in the course of preparing the manuscript.
z
2.4545 2.4662 2.4865 2.4943 2.5000 2.5126 2.5294 2.5726 2.5789 2.5824 2.5879 2.5909 2.6000 2.6154 2,6250 2.6374 2.6667 2.6883 2.6923 2.7039 2.7500 2.7647 2.7690 2.7787 2.7857 2.8000 2.8065 2.8219 2.8519 2.8571 2.8750 2.8908 2.8947 2.9397 2.9580 2.9697 2.9835 2.9876 3.0000
n - 2
2.8386 2.8309 2.8182 2.8133 2.8099 2.8022 2.7923 2.7679 2.7659 2.7640 2.7609 2.7592 2.7543 2.7460 2.7410 2.7346 2.7198 2.7092 2.7075 2.7026 2.6813 2.6749 2.6732 2.6691 2.6661 2,6602 2.6576 2.6514 2.6397 2.6378 2.6310 2.6252 2.6238 2.6078 2.6015 2.5976 2.5930 2.5917 2.5877
-
n - 5
n = 10
n
9.9167 9.8639 9.775 9.742 9.718 9,666 9.598 9.433 9.410 9.397 9.377 9.366 9.335 9.281 9.247 9,206 9,111 9.043 9.031 8.996 8.862 8.822 8.810 8.784 8.764 8.727 8.710 8.670 8.590 8.577 8.534 8.496 8.488 8.386 8.346 8.322 8.293 8.284 8.258
24.777 24.600 24.301 24.190 24,114 23.935 23.719 23.164 23.084 23.041 22.973 22.936 22.831 22.650 22.543 22,401 22.099 21.877 21.839 21.706 21.263 21.128 21,093 21.004 20.947 20.820 20.768 20.640 20.396 20.358 20.218 20.101 20.072 19.751 19,627 19.550 19.459 19.433 19,354
60.961 60.412 59.495 59.154 58.910 58.374 57.683 55.963 55.732 55,605 55.408 55.301 54.978 54.448 54.124 53.714 52.784 52.127 52,007 51,666 50.380 49.990 49.878 49.628 49.449 49.089 48.931 48.558 47.881 47.762 47.362 47.017 46.933 46.000 45,637 45.411 45.149 45.071 44.841
20
Received for review July 16, 1973. Accepted September 12, 1973. This work was supported in part by the Public Health Research Grant No. 02-002-1, from the National Institutes of Health, Bethesda, Md.
ANALYTICAL CHEMISTRY, VOL. 46, NO. 3, MARCH 1974
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