and 20 minutes) approximate % ’ OH could be calculated but is not reported since the true yo OH is unknown, nor could reproducible results be obtained with the pyridine-acetic anhydride reagen t . ACKNOWLEDGMENT
The authors thank Ethyl Corporation, Atlas Chemical Industries, E. I. du Pont de Nemours and Co., Union Carbide Corp., and Wyandotte Chemicals Corp. for data sheets and generous samples of several of the compounds, and Alan D. Wilks for help with some of the analyses.
LITERATURE CITED
(1) Berger, A., Magnuson, J. A., ANAL. CHEW36, 1156 (1964). (2) Burton, H., Praill, P. F. G., J. Chem. SOC.1950. 1203. (3) Ibid., 1951,-522. (4) Fritz, J. S., Schenk, G. H., ANAL. CHEM.31, 1808 (1959). (5) Ibid., 32, 987 (1960). (6) Gutnikov, G., Schenk, G. H., ANAL. CHEM.34, 1316 (1962). (7) Magnuson, J. A., ANAL. CHEM.35, 1487 (1963). (8) Pietrzyk, D. J., Belisle, J., Ibid., 38, 969 (1966). (9) Schenk, G. H., Ibid., 33, 299 (1961). (10) Schenk, G. H., Santiago, M., Microchem. J. 6, 77 (1962).
(11) Schenk, G. H., Wines, P., Mojzis, C., ANAL. CHEM.36, 914 (1964). (12) Siggia, S., “Quantitative Organic
Analysis via Functional Groups,” Third
Ed., Wiley, New York, 1963. (13) Stetsler, R. S., Smullin, C. F., ANAL.CHEM.34, 194 (1962). RECEIVEDfor review Ma Accepted July 20, 1966. & n % i J g z sistance came from a grant .(GM 123106-01) from the National Institutes of Health and from a Du Pont Fellowship (1965-1966) for one of the authors (JB). Presented in part at the First Midwest Regional American Chemical Society Meeb ing, Kansss City, November 4-5, 1965.
Models for the Analytical Applications of Dialysis Using Differential Kinetics R. F. BROMAN‘ and R. C. BOWERS2 Department of Chemistry, Northwestern University, Evansfon, 111. Mathematical expressions describing the dialysis of two-component mixtures of nonelectrolytes and of three-component mixtures of univalent ions are presented for both finite and infinite bath techniques. Experimental verification of these equations for single electrolyte dialysis has been carried out. Model calculations made using the expressions cast doubt on the general analytical utility of the method of dialysis based on differential kinetics for the determination of components in a mixture. Optimum values of the concentration ratio of components in the mixture and differences between diffusion coefficients of the components are discussed. The dialysis method of analysis appears to be most useful when the components are present in nearly equal concentrations and their diffusion coefficients differ by a factor greater than two. Brief experimental work with potassium chloride-hydrochloric acid mixtures supports these conclusions.
S
Hanna, and Serencha (2f,88) have applied the technique of dialysis using differential kinetics to the determination of components in binary and ternary mixtures. They reported good results for the analysis of such mixtures as nitric acid-acetic acid, potassium chloride-potassium bromide, and nbutylamine-tert-butylamine. The dialysis method was applied in a manner analogous to previous work with differential chemical reaction rates as analytical tools (16), and the differential physical reaction rate of dialysis was
6020 1
described using the mathematics of chemical reaction rates. The mathematical treatment of parallel first-order chemical reaction rates (9) is identical with that for the rate of dialysis of mixtures of nonelectrolytes, but it has long been recognized that ionic mixtures do not diffuse in the same manner ( f a ) . McBain and coworkers (18, 23) have studied in some detail the coupled diffusion of ions within a concentration gradient in a sintered glass disk. Earlier workers have noted the process of coupled diffusion of ions in solution (19, 20) and in membranes (4, 7 , 14, f7). If the dialysis technique is to be analytically useful, the theories must account for this coupled ionic diffusion. An appraisal of the theoretical models has been made here in an attempt to predict the conditions under which the technique may be satisfactorily utilized. A brief experimental verification of the theory and experiments of the type reported by Siggia, Hanna, and Serencha (21), designed to test the general analytical usefulness of their technique, is also presented.
IOGIA,
1512
ANALYTICAL CHEMISTRY
THEORETICAL
Solution of Fick’s second law for the appropriate boundary conditions leads to the general mathematical equation for diffusion through a membrane. Barnes (1) and Dvorkin (8) have solved the problem for dialysis of a single nonelectrolyte (which is also applicable to single electrolyte dialysis). These solutions include expressions for the initial establishment of a finite concentration gradient across the membrane. Steady-state
flux equations for coupled ionic diffusion have been derived by Gose (fl), who considered the diffusion of three ionic species. A steady-state solution to the dialysis problem is most convenient for our purposes. Initial establishment of the concentration gradient occurs rapidly, especially for thin, wide-pored membranes and for solutes with relatively large diffusion coefficients. The time required for the attainment of steady state is negligibly small compared to the time over which the entire dialysis experiment is conducted. The following assumptions have been made in the derivations: The membrane does not possess any ion exchange properties; activity coefficients are taken as being equal to unity; diffusion coefficients are independent of concentration; the thickness of the membrane is small enough so that the membrane volume is negligible compared to the solution volume; diffusion is restricted to the membrane; and for the electrolyte case, no electric current flows through the dialysis cell. It has also been assumed that specific solute-solute and solute-membrane interactions as well as osmotic effects would be accounted for in the magnitude of the membrane diffusion coefficient, an integral diffusion coefficient. Solutions have been obtained by solving the general flux equation 1 Present address, De artment of Chemistry, University of gebraska, Lincoln, Neb. 68508 2Present address, College of Liberal Arts and Sciences, Northern Illinois University, DeKalb, Ill. 60115
fj
Dj(dCj/dz)
+
ZjDjCj(F/RT)
(dFE/dz) (1)
for the appropriate boundary conditions. (Symbols are defined in the Nomenclature section.) The mathematical condition for steady state is written djj/dx = 0
> 0) 4, t > 0)
(z = 0, t (z =
j j
=
Dj(dCj/dz)
=
C j o exp(-ADjt/CVi)
(5)
which describes the behavior of either M X or N X during dialysis. If the total concentration of undialyzed M X and N X is denoted by C X ,we may write an equation for the dialysis of a mixture log c x = log [CMX"eXp( -ADMxt/dVi) CNX"exp(-AD.vxt/dVi)
+ 1
(6)
When the more rapidly diffusing species is essentially completely dialyzed out of the inner solution, Equation 6 may be written log c x
=
log CNX'
(z
cj,i< cjo cj,o< Ci"
(x = G, t > 0) (x = 0, t > 0)
(z =
c,
t = 0)
[x = 0, t > 0) (Z == 4, t
(14)
> 0)
A material balance equation can be written ViCj,i
+ VoCj,, = ViCi"
(9)
Solution of the flux equations yields for Cj.i
cj,i =
Cj"
+ V,
x
+
exp[-A(Vi V,)Djt/.! X ViVol)/(Vi Vo) (10)
+
This equation is the same as the steadystate solution given previously (1, 8). Writing C X , ~= C M X , ~ C N X , we ~ obtain
+
(4)
Solution of Equation 4 under the conditions of Equation 3 yields
Cj
0
(3)
Because z M X = Z N X = 0, the flux of each solute species, from Equation 1, is given by
Cj < Cj"
(x = 4, t = 0) 0, t = 0) (8)
CjO
{Vi
(x = t, t = 0)
Cj" -0
cj =
(2)
Dialysis of Two Nonelectrolytes into an Infinite Bath. A solution of volume Vi, initially containing the nonelectrolytes M X and N X at concentrations CMX" and CNX", respectively, is separated from an infinitely large volume of pure solvent by a membrane of thickness .!. The initial and boundary conditions are
cj < Cj"
tion of volume V,. The outer solution is initially free from M X and N X .
- AI)~xt/2.3CVi
(7)
Siggia, Hanna, and Serencha (21, 22) have applied this equation in their analysis of binary and ternary mixtures of both electrolytes and nonelectrolytes. They found that a plot' of log C X us. t would give a curve containing a straightline segment which, when extrapolated to t = 0, would yield a value for log
+
+ +
\-ADMX(V~ Vo)t/NiVoI)/(Vi v,) {vOCNxoexp [-AdNX(Vi Vo)t/c ViVol)/(Vi Vo) (11)
+
+
The first term on the right-hand side of Equation 11 represents the concentration of total solute at equilibrium. The second and third terms describe the rate of dialysis of M X and N X , respectively. T o obtain a linear portion in the log CX us. t plot with this equation, the second term must become very small while the third term is still larger than ViCx0/ (Vi Vo). Intuitively, i t can be seen that resolution of such a straight-line segmeqt will require a higher value of DMX/DNX than was necessary for the infinite bath case. Dialysis of Two Uni-Univalent Electrolytes into an Infinite Bath. An inner solution of volume Vi, contains two electrolytes, M X and N X , with the common anion, X - . The flux equations for the three ions are given by Equation 1, where Z M = Z N = 1 and zx = - 1. The electroneutrality conditions are
+
(19)
The boundary conditions are
cj = cj,i< c j o cj,,< cjo
(z =
e, t > 0) > 0)
(z = 0 , t
(20)
Following the derivation for the previous model, we write forfx
CNX'.
Dialysis of Two Nonelectrolytes into a Finite Bath. More commonly, the volume of the outer solution is not so large that it can be considered of infinite magnitude. I n the finite bath case, the increasing concentration of the diffusate in the outer solution must be accounted for. An inner solution of volume Vi containing the nonelectrolytes M X and N X is separated by a membrane from an outer solu-
(z = 0, t = 0) = e, t = 0)
C j = O ( oCj" (x
and the condition that no electric current flows in the system is written
f
- fX
@ = -
=0
(13) The initial and boundary conditions are .fM
fN
where
-+&+& ?(& Dx DN
and VOL 38, NO. 1 1 , OCTOBER 1966
e
1513
The existence of the exponent (@ fx)/@ in Equation 21 prevents explicit solution for fx. By comparing Equation 21 with Equation 16 for the infinite bath case, it can b_e seen that-the term in the numerator ( D M C M . ~DNCN..,)y is a correction term to account for the concentration buildup in the outer solution. At sh_ort times, the quantities (D.vC.V.~ DNCN,,)and CX.,/CX,~will be negligibly small because Cj,,
