800
The Journal of Physical Chemistry, Vol. 82, No. 5, 1978
J. L. Anderson, F. Rauh, and A. Morales
It was found that, for the systems examined here, the concentration range could be extended up to -8 X M (PK 0.8) before any systematic deviations between the calculated and experimental values appeared; these deviations would suggest that we were “forcing a fit” to the data (i.e., absorbing deviations arising from inadequacies in the theoretical treatment into one or more of the parameters, a, KA,etc.), so that the equations are valid within the concentration ranges we used. In conclusion, use of the Quint-Viallard equation makes possible a more rigorous analysis of data for many systems, for which at present no satisfactory treatment is available. For the alkaline earth halides this has enabled the association constants for the equilibrium M2+ X- = MX+ and the limiting conductivity of the charged ion pair MX+ to be determined.
information is available on any current masthead page.
References and Notes (1) T. L. Broadwater, T. J. Murphy, and D. F. Evans, J . Phys. Chem., 80, 753 (1976). (2) R. J. Lemire and M. W. Lister, J. Solution Chem., 5, 171 (1976). (3) F. Ferranti and A. Indelli, J . Solution Chem., 3, 619 (1974). (4) T. L. Broadwater and D. F. Evans, J. Solution Chem., 3, 757 (1974). (5) E. C. Righellato and C. W. Davies, Trans. Faraday Soc., 26, 592
(1930). (6) (7) (8) (9) (10) (1 1) (12) (13) (14)
+
(15) (16) (17) (18) (19)
Acknowledgment. One of us (R.J.W.) thanks the Science Research Council for their financial support. Supplementary Material Available: (i) A table listing the molar concentration and corresponding equivalent conductances for BaC12 and SrClz in MeOH at 25 “C; (ii) Appendix I, a corrected version of the Quint-Viallard equation; and (iii) Appendix 11, an outline of a “leastsquares” curve fitting computer program for the analysis of data for associated 2:l and 3:l salts (7pages). Ordering
(20)
(21) (22) (23) (24)
R. M. Fuoss and L. Onsager, J. Phys. Chem., 61, 668 (1957). L. Onsager and R. M. Fuoss, J. Phys. Chem., 38, 2689 (1932). K. A. Krieger and M. Kilpatrlck, J. Am. Chem. Soc., 59, 1878 (1937). J. Quint and A. Viallard, J. Chim. Phys., 72, 335 (1975). T. L. Murphy and E. G. Cohen, J . Chem. Phys., 53, 2173 (1970). J. Quint and A. Viallard, J . Chim. Phys., 69, 1095 (1972). J. Quint and A. Viallard, J . Chim. Phys., 69, 1100 (1972). J. Quint, Thesis, University Clermont-Ferrand, 1976. R. M. Fuoss and F. Accascina, “Electrolytic Conductance“, Interscience, New York, N.Y., 1959, p 195. W. H. Lee and R. J. Wheaton, J . Chim. Phys., 74, 689 (1977). R. M. Fuoss, Proc. Natl. Acad. Sci. U.S.A., 71, 4491 (1974). C. F. Mattina and R. M. Fuoss, J. Phys. Chem., 79, 1604 (1975). M. A. Copler and R. M. Fuoss, J. Phys. Chem., 68, 1177 (1964). E. M. Hanna, A. D. Pethybridge, and J. Prue, Electrochlm. Acta, 16, 677 (1971). W. C. Duer, R. A. Robinson, and R. Bates, J. Chem. Soc., Faraday Trans. 1, 68, 716 (1972). E. M. Hanna, A. D. Pethybridge, J. Prue, and D. J. Spiers, J. Solutlon Chem., 3, 563 (1974). H. Hartley and H. R. Raikes, Trans. Faraday Soc., 23,393 (1927). R. M. Fuoss, J. Am. Chem. SOC.,80, 5059 (1958). J. E. Lind and R. M. Fuoss, J . Phys. Chem., 65, 1414 (1961).
Particle Diffusion as a Function of Concentration and Ionic Strength John L. Anderson,* Department of Chemical Engineering, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213
Francis Rauh, FMC Corporation, P.O. Box 8, Prlnceton, New Jersey 08540
and Antonio Morales Eli Lilly Industries, Incorporated, G.P.O. Box 71325, San Juan, Puerto Rico 00936 (Received September 19, 1977)
The mutual diffusioncoefficient for bovine serum albumin and polystyrene particles was measured as a function of concentration in aqueous solutions of potassium chloride. A simple capillary method based on a principle of hydrodynamic stability was used to obtain the data which were interpreted assuming a linear concentration dependence: D = Do[l + k C ] ,where C is the albumin or latex concentration. We found for both species that (1)Dowas independent of ionic strength, (2) the coefficient h was positive (at pH ~ 6 ) and , (3) the value of k increased as ionic strength decreased. For example, with serum albumin k increased from 0.6 to 130 cm3/g when the ionic strength was reduced from 0.1 to 0.001 M. Our results were compared with calculations from theoretical models of interacting Brownian particles and the agreement was good. This work demonstrates that the concentration dependence of the diffusion coefficient is not unique, even for a given species, but rather is sensitive to solution properties such as pH or ionic strength which affect the interaction energy between two particles. It may be that practical application can be made of this sensitivity.
Introduction The manner in which an electrolyte affects the diffusion rate of a compact, charged macromolecule is of central interest in this work. Charged macromolecules interact directly through electrostatic forces which are mediated through Debye-Huckel screening by small ions in the solution. Because the macromolecule diffusion rate depends on such interactions, one might expect that changes in ionic strength can effect changes in the mutual diffusion 0022-3654/78/2082-0608$01 .OO/O
coefficient. Such behavior is what we have observed experimentally and is discussed in this paper. At “infinite dilution”, i.e., at concentrations low enough such that particle interactions are negligible on the average, the diffusion coefficient of neutral, rigid Brownian particles each of radius a is given by the Stokes-Einstein equation:
Do = h g T / 6 i ~ ~ o a (1) where v0 is the solvent viscosity and the subscript zero @ 1978 American Chemical Society
Electrolyte Effect on Particle Diffusion
denotes the limit of zero particle concentration. However, charged particles are surrounded by small ions whose own Brownian motion may influence the motion of the particle. Stephen1 attempted to account for this coupling by assuming the existence of a local self-consistent electrical field and applying the time-dependent Nernst-Planck equations to the motion of both the particle and the small ions, with the following result for the perturbed particle diffusion coefficient (DO’):
Do’ = Do[l
+ 1411
where q is the net charge on the macromolecule in terms of the fundamental electronic charge. Ermak2 investigated the same system by computer simulation of the Brownian motion of all species. His conclusion was that small ion effects are typically much smaller in magnitude than predicted by Stephen. For example, interpolation and extrapolation of his numerical results predict that for a charge q = -6, a particle radius equal to 36 8, and low solution ionic strength (50 cm) such that the particles never reached the bottom (z = L ) even at the stability condition. If eq 8 is used to substitute for dz in the integrand of eq 9 then
Thus, the measurement of C‘ provides information about an integral average of D over the concentration range of the experiment. Suppose for a moment that Co is sufficiently small such that D and q are well-approximated by the infinite dilution values Do and qo. Equation 10 can then be integrated directly to give
Do= Co2/2cKLqo
(11)
and hence the diffusion coefficient for noninteracting particles can be directly calculated from the measured value C‘. To obtain values of D ( C ) from the data vs. Co), the following mathematical expressions for D and 7 were assumed:
(e
D =D o [ l + h C ]
(12b) The parameters hl and h2were determined by independent experiments as described later. The terms of O(C2)for 7 and O(C) for D reflect two-particle interactions affecting the solution viscosity and the particle diffusion rate, respectively; therefore, the use of a quadratic in eq 12a and a linear expression in eq 12b is consistent on physical grounds. By combining eq 1 2 with 10 we obtain n
2
(13) From data of C vs. Co, the parameters Do and k were fit by a least-squares criterion applied to eq 13. Materials. The water was purified by distillation and then passed through a Milli-Q2 ion-exchange system (Millipore Corp.); the initial resistivity of this purified water was greater than 5 X lo6 ohm cm. Potassium chloride was added to achieve the desired ionic strength. The protein solution was made by first adding the required amount of crystalline bovine serum albumin (BSA) to a given volume of salt solution and then dialyzing for 24-48 h against the same salt solution at 4 “C. The BSA (crystallized and lyophilized) was purchased from the Sigma Chemical Co. Although no buffer was ever added, the pH of all BSA solutions was 6.5 f 0.2 and quite steady. The mean BSA concentration in a capillary was determined by measuring the absorbance at 278 nm of the diluted sample taken from that capillary. There is some concern that commercially available BSA contains appreciable amounts ( 15% 17) of dimers and higher order associations, presumably connected by disulfide bonds. To check for aggregation we passed several BSA solutions through a 45 cm long column packed with
(e)
N
Sephadex G-150 beads; this column was capable of separating catalase (MW = 250000) from BSA. We observed only one peak for the BSA each time, implying that the BSA we used did not have a large degree of association. Furthermore, we tested 1%solutions at 0.1 M KC1 made from BSA (obtained from Miles Laboratory) which had its exposed sulfide groups blocked, and the measured diffusion coefficient of this “guaranteed” monomer agreed with the determinations made with the Sigma BSA. From this evidence we may conclude that association of the BSA was not a dominant factor in our experiments, allthough admittedly we cannot precisely determine the fraction of dimers that were present in our protein solutions. Monodisperse polystyrene (latex) particles 910 8, in diameter were purchased as 10% aqueous stock solutions from the Dow Chemical Co. (Diagnostics Division, Indianapolis). The “solvent” was made from purified water and potassium chloride as before except this time a surfactant, sodium dodecyl sulfate (SDS, Eastman), was added to a level of 0.22 g/L to prevent coagulation of the particles. A small amount of the stock latex solution was added to a certain amount of solvent to reach the desired particle concentration, and the solution was then dialyzed at least 48 h against the latex-free salt/surfactant solution. After dialysis the solution was passed through a Millipore filter of pore size 1.2 or 3.0 pm to remove small amounts of aggregated particles; however, turbidity checks showed that only a small mass (
