1058
J. Phys. Chem. 1980, 84, 1058-1059
COMMUNICATIONS TO THE EDITOR Comments on the Penetrant Time Lag In a Diffusion-Reaction System Publication costs asslsted by The Kroc Foundation
.15
Sir: In a recent communication,l Ludolph, Vieth, and Frisch derived an expression for the penetrant time lag in a system influenced by the effects of linear irreversible reaction and simple penetrant immobilization. They considered a thin membrane containing a bound catalyst that separates two bulk volumes. The diffusing penetrant reacts catalytically within the membrane according to a first-order kinetic expression with a rate constant k . In addition, reversible immobilization of the penetrant within the membrane can occur. Initially, the membrane is devoid of penetrant, and at time t = 0 one face of the membrane is exposed to a constant concentration C,, while the other face is maintained at zero concentration. Assuming that the immobilized penetrant within the membrane remains in local equilibrium with the free penetrant according to the equilibrium constant E, a conservation equation for this system was derived, and appropriate initial and boundary conditions described. Rewritten in nondimensional form, these expressions are
a c_--a2c _ -_
42c
az2
a7
C(z,O) = C(1,T) = 0
(1) (2)
c(0,r) = 1
(3) where we have introduced the following nondimensional parameter and variables: cb2 = k12/D = C/C, z =x/l
h
I
I
0
2.0
4.0
1
60
8.0
10 0
9 Figure 1. The nondimensional time lag as a function of to eq 11.
4 according
finite Fourier transforms2 and obtained the following solution: sinh 4 (1 - z ) C(Z,T) = sinh 4 nr exp[-(n2n2 + & T ] sin n m (8) 2 2 n = l n2*2
+ $2
from which an expression can be derived for the time lag -2 sinh 4 m n2$ -=D L (9) (1 + E)l2 4 n = i (n2.rr2 42)2
c
+
Further, it can be shown from complex variable theory3 that
c
7 = -
Dt
(4)
12(1 + 4) D is the diffusion coefficient and 1 is the membrane thickness. By manipulating eq 1 and its steady-state counterpart, they were able to obtain an expression for the time lag L without solving the full transient problem. In our notation, this expression is
This leads to the following useful expression for the time lag:
This expression can be checked by taking the limit of‘no reaction, which gives 12
L = -(1
c,,
where is the steady-state solution to eq 1-3, and u(z) satisfies the following boundary value problem: d2u - _ @*U = c,, dz2 u(0) = u(1) = 0
Cnnsistant with the physical independence of the reaction and adsorption processes, the parameter E is separable and independent of the terms on the right-hand side of eq 5 . From evaluation of the expression in eq 5 , Ludolph, Vieth, and Frisch concluded that the effects of both reversible immobilization and first-order irreversible reaction increased the time lag. In an attempt to verify this conclusion, we have solved the transient problem by using 0022-3654/80/2084- 1058$01.0010
+ E)
6D which is identical with eq 17 of Ludolph, Vieth, and Frisch.l The results of eq 11 above confirm the conclusion that reversible immobilization of the penetrant can increase the time lag. However, the effect of an irreversible first-order reaction is, on the contrary, to decrease the time lag. These results are presented graphically in Figure 1, where the nondimensional time lag given by eq 11 is plotted as a function of 4. We evaluated the time lag expression of Ludolph, Vieth, and Frisch as a function of 4 for different values of 4 (both as it appeared in eq 16l and elsewhere4). In contrast to the claim of the previous authors, we found that while the expression does predict an increase in the time lag with 4 for 4 greater than 1,for 4 between 0 and 1 it predicts an initial decrease in the time lag followed by a sharp increase. For 4 = 0 their expression becomes numerically identical 0 1980 Arnerlcan Chemical Society
J. Phys. Chem. 1980, 84, 1059-1060
with eq 11 above. The equations, however, are not identical for # 0. The discrepancy in the predictions of these two treatments cannot be attributed to error in the derivation of eq 5 but rather to evaluation of the expression therein. We have repeated the calculation in eq 5 and have found it to reduce to eq 11,which provides an independent confirmation of the time lag derivation. These findings suggest that experimental results which show an increase in the time lag4 cannot be satisfactorily explained by the presence of a first-order irreversible reaction alone and that other explanations should be sought.
References and Notes (1) Ludolph, R. A.; Vieth, W. R.; Frisch, H. L. J. Phys. Chem. 1979, 83, 2793. (2) Miles, J. "Integral Transforms in Applied Mathematics"; Cambridge University Press: London, 197 1. (3) Carrier, G. F.; Krook, M.; Pearson, C. E. "Functions of a Complex Varlable": McGraw-Hill: New York. 1966. (4) Ludolph, R. A.; Vieth, W. R.; Venkatasubramanian, K.; Constantinides, A. J . Mol. Catai. 1979, 5, 197.
The editors have received the following concurrence from Professor Vieth: '' We have checked the timelag equation of LeypoMt and Gough (the author's e9 11) and we find it to be correct. Owing to the number of approximations we were forced to make to evaluate the integrals in eq 14 of our paper,
the role of became somewhat obscured. Therefore, when we prepared a sensitivity profile (our Figure 7) we obtained an increasing trend of the time lag with changes in our reaction parameter, R. However, our approximate solution, valid when t is zero, is not valid for other values of t . (In our simulation, we employed a value of 5 equal to 1.05.)" Department of Applied Mechanics and Engineering Sciences Bioengineering Group University of California, San Diego La Jolla, California 92093
John K. Leypoldt David A. Gough"
Received December 26, 1979
1059
U and D may have obscured the significance of Zapas a polymer charge fraction. My purpose here is to question the entire concept underlying eq 1for the high-molecular-weightpolymer case. For, even if U and D are separately measured at the same salt and polymer concentrations, the friction coefficient in an electrophoresis experiment is not expected to be the same as the friction coefficient in a diffusion experiment. The reason, as discussed by Armstrong and Strauss: is the different nature of the transport of uncondensed counterions in the two cases. In electrophoresis these counterions migrate in a direction opposite to that of the polymer segment, thus nullifying hydrodynamic interaction between segments. The polymer becomes effectively free draining. In diffusion, however, the counterions move with the polymer, and the coil is nondraining. Inspection of the values of U and D listed by Meullenet et al. shows typical free-draining and nondraining behavior, respectively. This negative assessment of the significance of eq 1 for high-molecular-weight polymers may not apply to highmolecular-weight DNA, the intrinsic stiffness of which may render it free draining at low ionic strength under both sets of condition^.^ It would be interesting, therefore, to repeat the measurements with this polymer. References and Notes (1) H. Magdelenat, P. Turq, P. Tivant, R. Menez, M. Chemla, and M. Drifford, Biopolymers, 18, 187 (1979). (2) J. P. Meullenet, A. Schmltt, and M. Drifford, J. Phys. Chem., 83, 1924 (1979). (3) G. S. Manning, J . Phys. Chem,, 82, 2349 (1978). (4) R. W. Armstrong and U. P. Strauss, Encyci. Polym. Sci. Techno/., 10, 781 (1969). (5) R. E. Harrington, Biopolymers, 17, 919 (1978).
Department of Chemistry Rutgers, The State University New Brunswick, New Jersey 08903
Gerald S. Manning
Received August 9, 1979
Comments on "Electrophoretic Light Scattering of Linear Polyelectrolyte Aqueous Salt Solutions" by J. P. Meullenet, A. Schmitt, and M. Drifford Publication costs assisted by the National Science Foundation
Sir: The Nernst-Einstein relation Zap= kTU/D
(1)
has recently been used by two groups to obtain apparent charges Zap on a polyion from measured values of its electrophoretic mobility U and its self-diffusion coefficient D. Using a low-molecular-weight polyion (chondroitin sulfate), Magdelenat et a1.l obtained polyion charges that were in close agreement with values predicted from counterion condensation theory. With high-molecularweight polymers, however, Meullenet et ala2have obtained values of Zapthat are more than an order of magnitude less than predicted. The authors suggest that a possible explanation of this massive discrepancy could lie in deficiencies of the Debye-Huckel approximation used in the theory. I t is difficult to accept their argument, however, both because it cannot explain the successful application of the theory in the low-molecular-weightcase and because the Debye-Huckel approximation probably leads at most to an error in the description of the interactions between polyion and uncondensed small ions no greater than about 20% .3 An alternate explanation advanced by the authors, but then discounted as probably unable to explain the magnitude of the discrepancy, is that the different polymer concentrations used in the independent measurements of 0022-3654/80/2084- 1059$01.OO/O
Reply to Dr. Manning's Comment
Sir: The Nernst-Einstein equation we use' has been established on theoretical grounds, using linear irreversible thermodynamics.2 In such an approach, the linear dependence between fluxes and forces is taken into account, but phenomenological coefficients, and thus friction coefficients, do not depend on forces, Le., on the presence or absence of a macroscopic electrical field, as suggested in the foregoing comment. In addition, what our analysis shows is that the simple form of the Nernst-Einstein relation is only valid as long as the Debye-Huckel approximation holds. If this approximation is not valid, then one should have (ref 2, eq A 1 and A2) z3aP
= 4 + 6)
(1)
where 6 is a negative parameter, which can be numerically estimated. Consider a 0.1 M sodium chloride solution, for which friction coefficients have been ~ a l c u l a t e dand ,~ a linear polyelectrolyte near the condensation threshold, the dimensionless charge parameter 2: being E = 0.85. The Manning self-diffusiontheory predicts, in the excess-of-salt limit kl = kz = 0.14 (2) where k, is an expansion parameter characterizing the reduced self-diffusion coefficient of ion i in the polyelec0 1980 American
Chemical Society
